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• Mediator vs. Moderator Variables | Differences & Examples

Mediator vs. Moderator Variables | Differences & Examples

Published on March 1, 2021 by Pritha Bhandari . Revised on June 22, 2023.

A mediating variable (or mediator ) explains the process through which two variables are related, while a moderating variable (or moderator ) affects the strength and direction of that relationship.

Including mediators and moderators in your research helps you go beyond studying a simple relationship between two variables for a fuller picture of the real world. These variables are important to consider when studying complex correlational or causal relationships between variables.

Including these variables can also help you avoid or mitigate several research biases , like observer bias , survivorship bias , undercoverage bias , or omitted variable bias

Table of contents

What’s the difference, mediating variables, moderating variables, other interesting articles, frequently asked questions about mediators and moderators.

You can think of a mediator as a go-between for two variables. For example, sleep quality (an independent variable ) can affect academic achievement (a dependent variable) through the mediator of alertness. In a mediation relationship, you can draw an arrow from an independent variable to a mediator and then from the mediator to the dependent variable.

In contrast, a moderator is something that acts upon the relationship between two variables and changes its direction or strength. For example, mental health status may moderate the relationship between sleep quality and academic achievement: the relationship might be stronger for people without diagnosed mental health conditions than for people with them.

In a moderation relationship, you can draw an arrow from the moderator to the relationship between an independent and dependent variable.

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A mediator is a way in which an independent variable impacts a dependent variable. It’s part of the causal pathway of an effect, and it tells you how or why an effect takes place.

If something is a mediator:

• It’s caused by the independent variable.
• It influences the dependent variable
• When it’s taken into account, the statistical correlation between the independent and dependent variables is higher than when it isn’t considered.

Mediation analysis is a way of statistically testing whether a variable is a mediator using linear regression analyses or ANOVAs .

In full mediation , a mediator fully explains the relationship between the independent and dependent variable: without the mediator in the model, there is no relationship.

In partial mediation , there is still a statistical relationship between the independent and dependent variable even when the mediator is taken out of a model: the mediator only partially explains the relationship.

You use a descriptive research design for this study. After collecting data on each of these variables, you perform statistical analysis to check whether:

• Socioeconomic status predicts parental education levels,
• Parental education levels predicts child reading ability,
• The correlation between socioeconomic status and child reading ability is greater when parental education levels are taken into account in your model.

A moderator influences the level, direction, or presence of a relationship between variables. It shows you for whom, when, or under what circumstances a relationship will hold.

Moderators usually help you judge the external validity of your study by identifying the limitations of when the relationship between variables holds. For example, while social media use can predict levels of loneliness, this relationship may be stronger for adolescents than for older adults. Age is a moderator here.

Moderators can be:

• Categorical variables such as ethnicity, race, religion, favorite colors, health status, or stimulus type,
• Quantitative variables such as age, weight, height, income, or visual stimulus size.
• years of work experience predicts salary, when controlling for relevant variables,
• gender identity moderates the relationship between work experience and salary.

This means that the relationship between years of experience and salary would differ between men, women, and those who do not identify as men or women.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Degrees of freedom
• Null hypothesis
• Discourse analysis
• Control groups
• Mixed methods research
• Non-probability sampling
• Quantitative research
• Ecological validity

Research bias

• Rosenthal effect
• Implicit bias
• Cognitive bias
• Selection bias
• Negativity bias
• Status quo bias

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A mediator variable explains the process through which two variables are related, while a moderator variable affects the strength and direction of that relationship.

A confounder is a third variable that affects variables of interest and makes them seem related when they are not. In contrast, a mediator is the mechanism of a relationship between two variables: it explains the process by which they are related.

Including mediators and moderators in your research helps you go beyond studying a simple relationship between two variables for a fuller picture of the real world. They are important to consider when studying complex correlational or causal relationships.

Mediators are part of the causal pathway of an effect, and they tell you how or why an effect takes place. Moderators usually help you judge the external validity of your study by identifying the limitations of when the relationship between variables holds.

If something is a mediating variable :

• It’s caused by the independent variable .

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Article contents

Moderator variables.

• Matthew S. Fritz Matthew S. Fritz Department of Educational Psychology, University of Nebraska - Lincoln
•  and  Ann M. Arthur Ann M. Arthur Department of Educational Psychology, University of Nebraska - Lincoln
• https://doi.org/10.1093/acrefore/9780190236557.013.86
• Published online: 25 January 2017

Moderation occurs when the magnitude and/or direction of the relation between two variables depend on the value of a third variable called a moderator variable. Moderator variables are distinct from mediator variables, which are intermediate variables in a causal chain between two other variables, and confounder variables, which can cause two otherwise unrelated variables to be related. Determining whether a variable is a moderator of the relation between two other variables requires statistically testing an interaction term. When the interaction term contains two categorical variables, analysis of variance (ANOVA) or multiple regression may be used, though ANOVA is usually preferred. When the interaction term contains one or more continuous variables, multiple regression is used. Multiple moderators may be operating simultaneously, in which case higher-order interaction terms can be added to the model, though these higher-order terms may be challenging to probe and interpret. In addition, interaction effects are often small in size, meaning most studies may have inadequate statistical power to detect these effects.

When multilevel models are used to account for the nesting of individuals within clusters, moderation can be examined at the individual level, the cluster level, or across levels in what is termed a cross-level interaction. Within the structural equation modeling (SEM) framework, multiple group analyses are often used to test for moderation. Moderation in the context of mediation can be examined using a conditional process model, while moderation of the measurement of a latent variable can be examined by testing for factorial invariance. Challenges faced when testing for moderation include the need to test for treatment by demographic or context interactions, the need to account for excessive multicollinearity, and the need for care when testing models with multiple higher-order interactions terms.

• interaction
• multilevel moderation
• latent variable interactions
• conditional process

Overview of Current Status

When the strength of the association between two variables is conditional on the value of a third variable, this third variable is called a moderator variable . That is, the magnitude and even the direction of the relation between one variable, usually referred to as a predictor or independent variable , and a second variable, often called an outcome or dependent variable , depends on the value of the moderator variable. Consider baking bread in an oven. In general, the higher the temperature of the oven (independent variable), the faster the bread will finish baking (dependent variable). But consider a baker making two different types of bread dough, one with regular white flour and the other with whole-wheat flour. Keeping the temperature constant, if the bread made with whole-wheat flour took longer to finish baking than the bread made with white flour, then the type of flour would be a moderator variable, because the relation between temperature and cooking time differs depending on the type of flour that was used. Note that moderating variables are not necessarily assumed to directly cause the outcome to change, only to be associated with change in the strength and/or the direction of the association between the predictor and the outcome.

Moderator variables are extremely important to psychologists because they provide a more detailed explanation of the specific circumstances under which an observed association between two variables holds and whether this association is the same for different contexts or groups of people. This is one reason why contextual variables and demographic variables, such as age, gender, ethnicity, socioeconomic status, and education, are some of the mostly commonly examined moderator variables in psychology. Moderator variables are particularly useful in experimental psychology to explore whether a specific treatment always has the same effect or if differential effects appear when another condition, context, or type of participant is introduced. That is, moderator variables advance our understanding of the effect. For example, Avolio, Mhatre, Norman, and Lester ( 2009 ) conducted a meta-analysis of leadership intervention studies and found that the effect of leadership interventions on a variety of outcome variables differed depending on whether the participants were all- or majority-male compared to when the participants were all- or majority-female.

The most important issue to consider when deciding whether a variable is a moderator of the relation between two other variables is the word different , because if the relation between two variables does not differ when the value of the third variable changes, the third variable is not a moderator variable and therefore must be playing some other role, if any. As illustrated in Figure 1 , a third variable is a confounder variable when it explains all or part of the relation between an independent variable and an outcome, but unlike a moderating variable, the magnitude of the relation between the independent and dependent variable does not change as the value of the confounder variable changes. A classic example of a confounding effect is the significant positive relation between ice cream consumption and violent crime. Ice cream consumption does not cause an increase in violent crime or vice versa; rather, the rise in both can be explained in part by a third variable—warmer temperatures (Le Roy, 2009 ). Moderator variables are also often confused with mediator variables , which are intermediate variables in a causal chain, such that changes in the independent variable (or antecedent ) cause changes in the mediator variable, which then cause changes in the outcome variable (or consequent ). For example, receiving cognitive-behavioral therapy (CBT; independent variable) has been found to cause reductions in negative thinking (mediating variable), and the reduction in negative thinking in turn reduces depressive symptoms (outcome variable; Kaufman, Rohde, Seeley, Clarke, & Stice, 2005 ). Moderator variables are not assumed to be part of a causal chain.

Figure 1. Path model diagrams for mediator, confounding, and moderator variables.

Interaction Models

When a moderator variable is present, such that the strength of the relation between an independent and dependent variable differs depending on the value of the moderator variable, the moderator variable is said to moderate the relation between the other two variables. The combined effect of the moderator variable with the independent variable is also called an interaction to reflect the interplay between the two variables, which differs from the individual effects of the independent and moderator variables on the dependent variable. This means that although the moderator variable changes the relation between the independent variable and outcome, the strength of the relation between the moderator variable and the outcome in turn differs depending on the values of the independent variable. Hence, the independent and moderator variables simultaneously moderate the relation between the other variable and the outcome. When an interaction term is statistically significant, it is not possible to interpret the effect of the independent variable alone because the effect depends on the level of the moderator variable.

Categorical by Categorical (2x2)

To illustrate the idea of an interaction, consider the finding by Revelle, Humphreys, Simon, and Gilliland ( 1980 ) that the relation between caffeine consumption and performance on a cognitive ability task is moderated by personality type. Specifically, Revelle et al. ( 1980 ) used a 2x2 between-subjects analysis of variance (ANOVA) design to examine the impact of consuming caffeine (independent variable; 0 mg or 200 mg) and personality type (moderator variable; introvert vs. extrovert) on cognitive performance (outcome; score on a practice GRE test). 1 Examination of the mean performance for the main effect of caffeine, which is the effect of caffeine collapsing across the personality type factor and shown in Figure 2a , demonstrates that the participants who received caffeine performed better than those who did not receive caffeine. Hence, one might categorically conclude that caffeine improves performance for everyone. In turn, the mean performance for the main effect of personality, which is the effect of personality type collapsing across the caffeine factor (Figure 2b ), shows that extroverts performed better than introverts. When the means are plotted for the four cross-factor groups in the study (Figure 2c ), however, it is apparent that although caffeine increased the performance of the extroverts, it actually decreased the performance of the introverts. Therefore, personality moderates the relation between caffeine and performance. In turn, caffeine moderates the relation between personality and performance because although introverts performed better than the extroverts regardless of caffeine consumption, the difference in performance between introverts and extroverts is larger for those who did not receive caffeine than those who did. Note that the vertical axis only shows a limited range of observed outcome values, so the response scale may have limited the real differences.

Figure 2. 2x2 Interaction: (a) Main effect of caffeine; (b) Main effect of personality type (black = introvert, white = extrovert); (c) Interaction between caffeine and personality type on Day 1 (black/solid = introvert, white/dotted = extrovert); and (d) Interaction between caffeine and personality on Day 2.

Finding a statistically significant interaction term in an ANOVA model tells us that moderation is occurring, but provides no further information about the specific form of the interaction (unless one looks at the coefficient for the interaction, which is usually ignored in ANOVA, but will be considered when moderator variables are discussed in the multiple regression context). Full understanding of the relation between the independent and moderator variables requires examination of the interaction in more detail, a process called probing (Aiken & West, 1991 ). Probing an interaction in ANOVA typically involves testing each of the simple main effects , which are the effects of the independent variable at each level of the moderator. In the caffeine example, there are two simple main effects of the independent variable at levels of the moderator variable: the simple main effect of caffeine for introverts, represented by the solid line in Figure 2c , and the simple main effect of caffeine for extroverts, represented by the dashed line. The plot makes it clear that caffeine had a larger effect on performance for the extroverts than the introverts (i.e., the ends of the dashed line are farther apart vertically than the ends of the solid line), but the plot alone cannot show whether there is a significant effect of caffeine in either of the personality groups; hence the need for statistical tests.

Another way to conceptualize moderation is to say that moderation occurs when the simple main effects of an independent variable on an outcome are not the same for all levels of the moderator variable. If the effect of caffeine on performance was the same for both introverts and extroverts, the two simple main effects would be the same and the two lines in Figure 2c would be parallel. Instead, the two simple main effect lines are not parallel, indicating different simple main effects (i.e., moderation). Despite the moderating effect of personality on the relation between caffeine and performance illustrated in Figure 2c , the introverts always performed better than the extroverts in this study. As a result, though the lines are not parallel and must cross at some point, the lines do not intersect in the figure. When the simple main effect lines do not intersect within the observed range of values, the interaction is said to be ordinal (Lubin, 1961 ) because the groups maintain their order (e.g., introverts always outperform extroverts). When the simple main effect lines cross within the observed range of values, the interaction is said to be disordinal because the groups do not have the same order for all values of the moderator. A disordinal interaction is illustrated in Figure 2d , which again shows the same simple main effects of caffeine on performance for the different personality types, but for individuals who completed the same protocol the following day (Revelle et al., 1980 ).

What is important to consider when probing an interaction is what effect the moderator has on the relation between the other two variables. For example, the relation between the independent and dependent variables may have the same sign and be statistically significant for all values of the moderator, in which case the moderator only changes the magnitude of the relation. Alternatively, the relation between the independent and dependent variables may not be statistically significant at all values of the moderator, indicating that the relation exists only for specific values of the moderator. A third possibility is that the relation between the independent and dependent variables is statistically significant, but opposite in sign for different values of the moderator. This would indicate that the direction of the relation between the variables depends on the moderator. These are very different interaction effects that the statistical significance of the interaction term alone will not differentiate between, which is why probing interactions is essential to describing the effect of a moderator variable.

There are two additional issues to consider. First, the labeling of one variable as the independent variable and the other variable as the moderator is guided by theory. Because a significant interaction means that caffeine is also moderating the effect of personality on performance, the simple main effects of personality at levels of caffeine may also be considered; in this case, the simple main effect of personality type on performance for the 0 mg caffeine group and the simple main effect of personality type on performance for the 200 mg caffeine group. Since the statistical model is the same regardless of whether personality is the independent variable and caffeine is the moderator or vice versa, the assignment of roles to these variables is left up to the researcher. Second, while the 2x2 ANOVA framework is a simple design that lends itself to probing interactions, splitting a continuous variable at its mean or median in order to force continuous variables to fit into the ANOVA framework is a very bad idea, as it not only results in a loss of information that decreases statistical power, but also increases the likelihood of finding spurious interaction effects (Maxwell & Delaney, 1993 ).

Categorical by Categorical (3x3)

Probing a significant interaction in a 2x2 ANOVA is relatively straightforward because there are only two levels of each factor. When a simple main effect is statistically significant, there is a difference in the average score on the dependent variable between the two levels of the independent variable for that specific value of the moderator. The significant overall interaction then tells us that the difference in the means for the two levels of the independent variable are not the same for both values of the moderator. When there are more than two levels, probing an interaction in ANOVA becomes more complicated. For example, imagine if Revelle et al. ( 1980 ) had employed a 3x3 ANOVA design, where participants were randomized to one of three levels of caffeine (e.g., 0, 100, and 200 mg) and personality type was also allowed to have three levels (e.g., introvert, neutral, extrovert). In this case, a significant main effect of caffeine would only tell us that the mean performance in at least one of the caffeine groups was different than the mean performance in the other two groups, collapsing across personality type, but not specifically which caffeine groups differed in mean performance. Determining which groups differed requires a main effect contrast , also called a main comparison , which specifically compared two or more of the groups. For example, a main effect contrast could be used to examine the mean difference in performance between just the 100 mg and 200 mg groups.

The same issue extends to probing the interaction because a significant interaction in the 3x3 ANOVA case only demonstrates that the simple main effects of caffeine are not the same for all levels personality type (and vice versa), but not specifically how the simple main effects of caffeine are different or for which of the three personality types. One way to probe a 3x3 (or larger) interaction is to first individually test all simple main effects for significance. Then for any simple main effects that are found to be significant (e.g., the effect of caffeine just for introverts), a comparison could be used to test for differences between specific levels of the independent variable for that simple main effect (e.g., 100 mg vs. 200 mg just for introverts), called a simple effect contrast or simple comparison . Alternatively, instead of starting with simple main effects, a significant interaction effect can be probed by beginning with a main comparison (e.g., 100 mg vs. 200 mg). If the main comparison is significant, then one can test whether the main comparison effect differed as a function of personality type (e.g., does the difference in performance between 100 mg and 200 mg differ between any of the personality types), which is called a main effect contrast by factor interaction . If the main effect contrast by factor interaction was significant, the effect can be further examined by testing whether the main effect contrast on the independent variable (e.g., 100 mg vs. 200 mg) differed at specific levels of the moderator (e.g., neutral vs. extrovert). That is, a contrast by contrast interaction specifies contrasts on both factors. For example, testing can show whether the difference in mean performance between the 100 mg and 200 mg caffeine groups differed for neutrals compared to extroverts, which essentially goes back to a 2x2 interaction.

Probing interactions in ANOVA when the factors have more than a few levels can lead to a large number of statistical tests. When a large number of these post hoc tests are examined, there is a danger that the probability of falsely finding a significant mean difference (i.e., making a Type I error) increases beyond a reasonable level (e.g., 0.05). When that happens, a Type I error correction needs to be applied to bring the probability of falsely finding a significant difference across all of the post hoc tests, called the experiment wise Type I error rate , back down to an appropriate level. The most well known of these corrections is the Bonferroni, but Maxwell and Delaney ( 2004 ) show that the Bonferroni overcorrects when the number of post hoc tests is more than about nine. Alternatives to the Bonferroni include the Dunnett correction for when one reference level is to be compared to each other level of the factor, the Tukey correction for all pairwise comparisons of levels, and the Scheffé correction for all possible post hoc tests.

Continuous by Categorical

Although not discussed in detail here, interactions between categorical variables can also be assessed using multiple regression rather than ANOVA. When one or both of the variables involved in the interaction is continuous, however, multiple regression must be used to test moderation hypotheses (Blalock, 1965 ; Cohen, 1968 ). The regression framework permits a moderation hypothesis to be specified with any combination of categorical and continuous variables. Consider the continuous by categorical variable interaction from Sommet, Darnon, and Butera ( 2015 ), who examined interpersonal conflict regulation strategies in social situations. 2 When faced with a disagreeing partner, people generally employ either a competitive strategy or conform to their partner’s point of view. Specifically, they found that the relation between the number of performance-approach goals (e.g., “did you try to show the partner was wrong”; continuous predictor) and competitive regulation scores (continuous outcome) differs depending on the person’s relative academic competence compared to their partner (same, superior, or unspecified; categorical moderator). The significant interaction indicates that performance-approach goals have a higher association with competitive regulation for both superior partners and partners with unspecified competence compared to partners with the same competence (Figure 3 ).

Figure 3. Categorical by continuous variable moderation.

Probing a significant interaction in multiple regression when the predictor is continuous and the moderator variable is categorical differs from probing interactions in ANOVA, but it can be straightforward depending on how the categorical moderator is incorporated into the regression model. There are many methods for representing nominal or ordinal variables in regression equations (e.g., Cohen, Cohen, West, & Aiken, 2003 ; Pedhauzer, 1997 ), though this article focuses only on dummy codes . Creating dummy codes for a categorical variable with k levels, requires k – 1 dummy variables ( D 1 , D 2 , … D k−1 ). Using the Sommet et al. ( 2015 ) example, where competence has three groups ( k = 3), two dummy variables are needed: D 1 and D 2 . Dummy variables are created by first selecting a reference group , which receives a zero on all of the dummy variables. Each of the non-reference groups receives a one for one dummy variable (though not the same dummy variable as any other non-reference group) and a zero for all other dummy variables. If same-competence is selected as the reference group, then one potential set of dummy codes is: D 1 = {same = 0, superior = 1, unspecified = 0} and D 2 = {same = 0, superior = 0, unspecified = 1}. Both dummy variables are then entered into the regression model as predictors. To create the interaction between the predictor and the dummy variables, each of the dummy variables must be multiplied by the continuous predictor and added into the regression model as well. For the interpersonal conflict example, the overall regression model for computing predicted competitive regulation scores (^ denotes a predicted score) from the number of performance approach goals, relative academic competency, and the interaction between goals and competency is equal to:

If regression coefficient b 4 , b 5 , or both are significant, then there is a significant interaction between goals and competence.

Interpreting and probing the interaction between a continuous predictor and a categorical moderator in regression is much easier when using the overall regression equation. Consider what happens when the values for the competency reference group (i.e., same competence) are substituted into the overall regression model.

Since the same-competence group has 0’s for D 1 and D 2 , the overall regression equation reduces to just include b 0 and b 1 . This reduced regression equation represents the relation between performance approach goals and competitive regulation scores for individuals who have the same academic competency as their partners. Equation 2 is called a simple regression equation because it is analogous to the simple main effect in ANOVA. The b 1 coefficient, which represents the relation between goals and competitive regulation for individuals with the same competency, is called the simple slope . But what do the other coefficients in the overall regression model represent?

If the dummy variable values for the superior-competency group are substituted into the equation and then some terms are rearranged, the result is:

Since b 0 and b 1 are the intercept and simple slope for the same competency group, b 2 is the difference in the intercept and b 4 is the difference in simple slope, respectively, between the same- and superior-competency groups. This means that if b 4 is significantly different than zero, the simple slopes for the same- and superior-competency groups are different from one another, and academic competency therefore moderates the relation between goals and competitive regulation. The simple regression equation can also be computed for the unspecified-competency group. These three simple regression lines are illustrated in Figure 3 and show that higher performance-approach goal scores are significantly associated with greater competitive regulation behaviors, although it is now known that this effect differs based on the relative level of competence of the partner.

The significance of b 4 and b 5 demonstrates whether or not the relation between the predictor and outcome variable is moderated by the categorical moderator variable, but a significant interaction does not explain whether the relation between the predictor and the outcome is statistically significant in any of the groups. Since b 1 is automatically tested for significance by most statistical software packages, there is no need to worry about testing the simple slope for the reference group. Aiken and West ( 1991 ) provide equations for computing the standard errors for testing the other two simple slopes, [ b 1 + b 4 ] and [ b 1 + b 5 ], for significance. Alternatively, the dummy coding could be revised to make another group the reference category (e.g., superior-competence), then the complete model could be re-estimated and the significance of the new b 1 value would test the simple slope for the new reference group.

Another characteristic of the simple regression equations that may be of interest is the intersection point of two simple regression lines, which is the value of the predictor variable at which the predicted value of the outcome variable is the same for two different values of the moderator variable. Looking at Figure 3 , the superior- and same-competence simple regression lines appear to intersect at around 5 on the performance-approach goals variable. The exact value of the intersection point can be calculated by setting the simple regression equations for these two groups equal to each other and then using algebra to solve for value of goals. While the intersection point is where the predicted scores for two simple regression equations are exactly the same, the points at which the predicted scores for two simple regression lines begin to be statistically different from each other can be computed. Called regions of significance (Potthoff, 1964 ), this is conceptually similar to a confidence interval that is centered-around the intersection point for two simple regression lines. For any value of the predictor closer to the intersection point than the boundaries of the regions of significance, the predicted outcome values for the two simple regression lines are not statistically significantly different from one another. For any value of the predictor farther away from the intersection point than the boundaries of the regions of significance, the predicted outcome values for the two simple regression lines are statistically significantly different from one another.

Continuous by Continuous

Interactions between a continuous predictor and continuous moderator variable can also be examined using the multiple regression framework. An example of a continuous by continuous variable interaction is that although injustice (continuous predictor) has positive relationships with retaliatory responses such as ruminative thoughts and negative emotions (continuous outcomes), mindfulness (continuous moderator) reduces these associations (Long & Christian, 2015 ). That is, high levels of mindfulness reduce rumination and negative emotions (e.g., anger) by decoupling the self from experiences and disrupting the automaticity of reactive processing. Long and Christian administered measures of mindfulness, perceived unfairness at work, ruminative thoughts, outward-focused anger, and retaliation behavior. They found that lower levels of mindfulness were associated with increased anger, whereas higher mindfulness was associated with lower anger (see Figure 4 ).

Figure 4. Continuous by continuous interaction.

Similar to continuous predictor by categorical moderator interactions in multiple regression, with continuous predictor by continuous moderator interactions each variable is entered into the regression model, then the product of the two variables is entered as a separate predictor variable representing the interaction between these variables. For the anger example, the overall regression model predicting anger from perceived injustice, mindfulness, and the interaction between injustice and mindfulness is equal to:

As with a continuous by categorical interaction, interactions between two continuous variables are probed by investigating the simple regression equations of the outcome variable on the predictor for different levels of the moderator. Unlike categorical moderator variables where one can show how the simple slopes differ between the groups, a continuous moderator variable may not necessarily have specific values of interest. If there are specific values of the continuous moderator that are of interest to the researcher, then the simple regression equation can be computed these values by substituting these values into the overall regression equation. In the absence of specific values of interest, Aiken and West ( 1991 ) recommend examining the mean of the moderator, one standard deviation above the mean, and one standard deviation below the mean. While it may seem that these values are somewhat arbitrary, these three values provide information about what is happening at the average score on the moderator, as well as providing a good range of moderator values without going too far into the tails, where there are likely to be very few observations.

A trick that makes interpreting a continuous by continuous variable interaction easier is to mean center the predictor and moderator variables, but not the outcome variable, prior to creating the interaction term. When injustice and mindfulness are mean centered before they are entered into the complete regression equation and the simple regression equation is calculated for the mean of the moderator, which is zero when the moderator is mean centered, the overall regression model reduces to:

Then b 0 and b 1 in the overall regression model are equal to the intercept and simple slope for participants with an average level of mindfulness, rather than for a person with zero mindfulness.

One issue not yet considered is the values of the regression coefficients themselves. There are two possibilities. When the regression coefficients for the predictor and the interaction are opposite in sign, buffering or dampening interactions occur, which results in larger moderator values decreasing the relationship between the predictor and the outcome. The distinction is based on whether a beneficial phenomenon is being decreased (dampening) or a harmful phenomenon is being decreased (buffering). The mindfulness effect in Figure 4 is a buffering moderator because it further reduces the effect of the independent variable. Alternatively, if the signs of the regression coefficients for the predictor and interaction term are the same, positive or negative, then increasing values of the moderator are related to a larger relationship between the predictor and the outcome variable. This is called a synergistic or exacerbating interaction depending on whether the phenomenon being examined is beneficial or harmful to the individual, respectively. Mathematically, buffering and dampening interactions (or synergistic and exacerbating interactions) are identical, so the distinction is based purely on theory.

Standardized Interaction Coefficients

Given many psychologists’ preference for reporting standardized regression coefficients, researchers should be aware that when regression models include higher-order terms (e.g., interaction terms or curvilinear terms), the standardized coefficients produced by most statistical software packages are incorrect. Consider the unstandardized regression equation for a dependent variable Y and two predictors X 1 and X 2 :

The standardized coefficients can be calculated by multiplying each unstandardized coefficient by the standard deviation of the corresponding predictor divided by the standard deviation of Y (Cohen et al., 2003 ) or equivalently by creating z -scores for Y , X 1 , and X 2 (i.e., standardizing the variables by mean centering each variable, then dividing by its standard deviation) and then estimating the model using the standardized variables ( Z Y , Z X 1 , and Z X 2 ) such that:

where a standardized regression coefficient is denoted with an asterisk.

As previously described, in order to test whether X 2 moderates the relation between Y and X 1 , a new variable must be created in the data set that is the product of the two predictors, X 1 X 2 , and enter it into the regression model as a separate predictor, resulting in the equation:

The software program is unaware that this new predictor X 1 X 2 is, in fact, an interaction term and not just another continuous predictor, however. This means that, when the software is calculating the standardized coefficients, it converts all of the variables in the model into z -scores such that the standardized coefficients come from the following regression equation:

Unfortunately, Z X 1 X 2 is not equal to the value of the product term created from standardized variables, Z X 1 Z X 2 . Hence, b 3 * is not the correct estimate of the standardized interaction coefficient. To obtain the correct estimate of the standardized interaction coefficient, a researcher must manually create Z Y , Z X 1 , Z X 2 , and Z X 1 Z X 2 , to fit the model:

and then use the unstandardized value b 3Z . While using the unstandardized solutions from a regression of standardized variables to get the correct standardized values of the regression coefficients seems counterintuitive, the discrepancy between the unstandardized coefficient b 3Z computed using the standardized variables and the standardized coefficient b 3 * using the unstandardized variables is quite evident in the output. And though the difference in the coefficients may be small, this difference can lead to large differences in inference and interpretation (Aiken & West, 1991 ; Cohen et al., 2003 ; Friedrich, 1982 ).

Curvilinear

Though not always included in discussions of moderator variables, curvilinear change that can be described with a polynomial regression model (i.e., quadratic, cubic, etc.) is a form of moderation, albeit one where a variable moderates itself. Consider the classic finding in psychology that the relation between physiological arousal and task performance is U-shaped (i.e., quadratic; Yerkes & Dodson, 1908 ), illustrated in Figure 5 . If the relation between arousal and performance for very low levels of arousal were described using a straight line, the result would be a regression line with a very steep positive slope. That is, when someone has low arousal, even small increases in arousal can lead to large increases in predicted performance. Describing the same relation for medium levels of arousal would result in a regression line with a very shallow slope, such that a slight increase in arousal would only be met with a slight increase in predicted performance. For very high levels of arousal, the regression line would again have a very steep slope, but now the slope is negative, such that small increases in arousal lead to large decreases in predicted performance. Therefore, the relation between arousal and performance is different depending on the level of arousal, so arousal is both the predictor and the moderator variable. This dual role is shown clearly in the regression equation for the quadratic relation between performance and arousal:

because the squared quadratic term that represents the U-shape is the product of arousal times arousal, the same form as the interaction terms between the predictor and the moderator variable in the two previous examples.

Figure 5. Quadratic relation between arousal and performance.

Three-Way Interactions

Up until this point in the discussion of moderators, the focus has been only on the interaction between two variables, an independent variable and a single moderator, which are known as two-way interactions . But there is no reason why two or more moderator variables cannot be considered simultaneously. Returning to the Revelle et al. ( 1980 ) example, the researchers believed that time of day also had an impact on the relation between caffeine and performance, so they collected data from participants in the morning the first day and in the afternoon on the second day. Figures 2c and 2d clearly show that the interaction between caffeine and personality type differs depending on whether the participants completed the study in the morning (Day 1) or in the afternoon (Day 2). That is, personality type moderates the relation between caffeine and performance, but time of day moderates the interaction between personality and caffeine. The moderation of a two-way interaction by another moderator variable is called a three-way interaction . As withtwo-way interactions in ANOVA, a significant three-way interaction is probed by testing a combination of post hoc effects including simple main effects, simple comparisons, contrast by factor interactions, and contrast by contrast interactions (Keppel & Wickens, 2004 ). In regression, probing a significant three-way interaction involves selecting values for both moderator variables and entering these values simultaneously into the overall regression equation to compute the simple regression equations (Aiken & West, 1991 ). Three-way interactions can also come into play with curvilinear relations. For example, the relation between two variables may be cubic, necessitating a X 3 term, or the quadratic relation between two variables may vary as a function of a third variable.

There are two very important considerations when examining three-way interactions. First, whenever a higher-order interaction is tested in a model, all lower order effects must be included in the model. For a three-way interaction, this means that all two-way interactions as well as all main effects must be included in the model (Cohen, 1978 ). This is more easily illustrated in regression. For example, consider if the two-way interaction between injustice and mindfulness in the Long and Christian ( 2015 ) example was found to differ depending on the person’s gender. 3 The correct regression equation would be:

which includes the three-way interaction between injustice, mindfulness, and gender, the three two-way interactions between these variables, as well as the three first-order effects. As described before, when the highest-order term is significant, no lower-order terms should be interpreted without consideration of the levels of the other variables.

Current Trends in Moderation

After defining moderator variables, providing an overview of the different types of interactions most likely to be encountered by psychologists, and discussing how to probe significant interactions between variables, the next section summarizes current trends in moderation analysis. Recent advances in moderation research have been focused in three areas: (1) moderator variables in the context of clustered data, (2) moderation with latent variables, and (3) models that have both moderator and mediator variables.

Multilevel and Cross-Level Moderation

Multilevel models (Raudenbush & Bryk, 2002 ; Snijders & Bosker, 2012 ), also called hierarchical linear models , mixed models , and random effects models , are a type of regression model that is used when participants are nested or clustered within organizational hierarchies, such as patients within hospitals, students within classrooms, or even repeated-measurements within individuals. Nesting is of interest because nested data violates the assumption of independence between participants, which causes the estimates of the standard errors for the regression coefficients to be too small. For example, two children in the same classroom might be expected to be more alike than two children who are in different classrooms. The degree of similarity of participants within a group or cluster is called the intraclass correlation coefficient , which is the proportion of the total variance that is shared between groups. Multilevel models work by dividing the total variability in scores on the outcome variable into different levels that reflect the nested structure of the data. Two-level models are most commonly used, although any number of levels are possible, such as students (Level 1) nested within teachers (Level 2) nested within schools (Level 3) nested within school districts (Level 4), and so on. Once the variability in the outcome has been attributed to the different levels of nesting, predictors, moderators, and interactions can then be added to the model to explain the variability at the different levels in the exact same manner as in single-level regression models.

Where multilevel models differ from single-level regression models regarding moderation, however, is that multilevel models can specify how variables occurring at one level influence relationships with variables at another level. Seaton, Marsh, and Craven ( 2010 ) use an example of the Big-Fish-Little-Pond effect to illustrate this concept, which states that although individual mathematics ability has a positive relationship with mathematics self-concept, higher school-average ability reduces this association. Here a two-level model is used because the students (Level 1) are nested within schools (Level 2). 4 In a simplified version of their model, Seaton et al. predicted individual mathematics self-concept (outcome variable) from individual mathematics ability (Level 1 predictor):

where i indexes individuals, j indexes schools, r ij is the Level 1 residual, and individual mathematics ability has been centered at the mean for each school.

The Level 1 model is at the student level and predicts self-concept for student i in school j . This model has an intercept ( β 0 j ) representing self-concept for mean mathematics ability across all schools and a slope ( β 1 j ) representing the effect of mathematics ability on self-concept across all schools. It is possible, however, that the effect of mathematics ability on mathematics self-concept is not the same for all schools. To explain the differences between self-concept and math achievement between students, β 0 j and β 1 j are allowed to vary across schools, hence the subscript j and why they are called random coefficients . In other words, each school is allowed to have its own intercept and slope. To model the variability in the intercept and slope of the Level 1 model between schools, two Level 2 models are created which are at the school level:

The Level 1 intercept ( β 0 j ) is partitioned into a mean intercept across schools ( γ 00 ) and a random effect ( u 0 j ), which represents the difference between the mean intercept across schools and the specific intercept for each school. In the same way, the Level 1 slope ( β 1 j ) is partitioned into the mean slope across schools ( γ 10 ) and a random effect ( u 1 j ), which represent the difference in the effect of individual mathematics ability averaged across schools and the effect of individual mathematics ability for a specific school.

Since β 0 j and β 1 j are allowed to vary by school, this variability in the random coefficients may be explained by adding school-level predictors to the Level 2 equations. For example, Seaton et al. ( 2010 ) added average school mathematics ability, centered at the grand mean, as a Level 2 predictor of both the Level 1 intercept and slope:

While a complete dissection of this model is beyond the scope of the current discussion, when the Level 2 equations are substituted into the Level 1 equation to get:

the interaction between student-level mathematics ability and school-level mathematics ability becomes obvious.

When a multilevel model contains a moderating variable from one level and an independent variable from another level, it is called a cross-level interaction (Raudenbush & Bryk, 2002 ). For the current example, students of all abilities had lower mathematics self-concepts if they attended high-ability schools compared to students of similar ability who attended average- or low-ability schools. The decrease in mathematics self-concept was more dramatic for higher-ability students. This phenomenon led Davis ( 1966 ) to warn parents against sending their children to “better” schools where the child would be in the bottom of the class. For multilevel models, it is not necessary to create a product term to estimate a cross-level moderation effect. Rather, if a Level 2 variable has a significant effect on the Level 1 slope, the moderation hypothesis is supported. Interactions between variables at the same level (e.g., a Level 1 predictor and Level 1 moderator) must still be entered manually.

Moderator variables in multilevel models share many of the challenges of moderators in single-level regression. For example, centering is recommended in multilevel models to facilitate interpretation, unless the predictors have a meaningful zero point. When adding Level 1 explanatory variables, centering becomes especially important. There are two ways to center Level 1 variables: grand mean centering (individuals centered around the overall mean) and group mean centering (individuals centered around group means). To avoid confusing a within-group relationship with a between-group relationship, it is recommended to group mean center Level 1 predictors, while grand mean centering Level 2 predictors. For more about centering in multilevel applications, see Enders and Tofighi ( 2007 ).

Moderation in Structural Equation Models

Structural equation modeling (SEM) is a collection of techniques that can be used to examine the relations between combinations of observed variables ( manifest ; e.g., height) and unobservable construct variables ( latent ; e.g., depression). As such, SEM can be used for examining many research questions, including: theory testing, prediction, estimating effect sizes, mediation, group differences, and longitudinal differences (Kline, 2011 ). SEMs can include both a measurement model , which describes the relation between each latent construct and the observed items used to measure individuals’ scores on that latent construct, and a structural model , which specifies the relations between latent constructs, as well as manifest variables.

Multiple-Group Analysis.

Testing for moderation in SEM can be conducted in multiple ways. If both the predictor and the moderator are manifest variables, then an interaction term can be computed by taking the product of the predictor and moderator, which is then added to the SEM as a new variable, just as in multiple regression. Provided the moderator is an observed categorical variable, moderation can also be tested in SEM using a multiple-group analysis. In a multiple-group analysis , the SEM model is fit with the path between the predictor and the outcome variable constrained to be the same in all moderator groups, and then a second time with the path unconstrained, such that the effect is allowed to be different for each group. The overall fit of the two models (i.e., constrained vs. unconstrained) is then compared. If the unconstrained model does not fit significantly better than the constrained model, then the effect is the same for all of the groups and moderation is not present. If the unconstrained model fits significantly better than the constrained model, however, it is concluded that the effect is different for at least one of the groups and moderation is present.

When variables are not perfectly reliable, as routinely occurs in psychology, it is often preferable to create latent variables to provide a mechanism for explicitly modeling measurement error. Latent moderator approaches are divided into partially latent variable approaches, where at least one variable is latent and at least one variable is observed, and fully latent variable approaches, where all variables are latent (Little, Bovaird, & Widaman, 2006 ; Marsh, Wen & Hau, 2006 ). A multiple-group analysis with a latent predictor variable is a partially latent variable approach since the moderator must be observed. Two other historical partially latent approaches include using factor scores in regression and a two-stage least-squares method (Bollen, 1995 ), although these methods are generally inferior to SEM approaches and therefore are not recommended. Fully latent approaches can also implemented within the context of an SEM (e.g., creating a third latent variable to represent the interaction of the two other latent variables), but some issues exist concerning the practicality and interpretation of a latent construct that represents the interaction between two other latent constructs. Several approaches have been proposed for modeling fully latent interactions (see Marsh et al., 2007 , for a review), but most approaches are based on the Kenny and Judd ( 1984 ) product indicator model.

One of the most common reasons for testing for moderation with latent variables in SEM is invariance testing (Mellenbergh, 1989 ; Meredith, 1993 ). Invariance testing is used to determine the degree to which a specific model fits the same in different groups or across time. Invariance is tested by imposing progressively stricter constraints across the groups and then comparing the model fit of the constrained model to a model with fewer constraints. Two types of invariance are discussed: factorial invariance and structural invariance.

Factorial invariance tests the factor structure or the measurement model across groups or time. Five levels of factorial invariance are commonly tested. The first level, dimensional invariance , is used to test whether the number of latent factors is the same across groups—this level of invariance is more commonly assumed than tested. The next level, configural invariance , tests whether the general pattern of item loadings on the latent constructs is the same across groups. If the factor loadings are found not just to have the same general pattern but to be exactly equal across groups, the model has loading or weak invariance across groups, which is the third level of factorial invariance. Loading invariance is the minimal level of invariance needed as evidence that a construct has the same interpretation across groups or time. The next level is intercept or strong invariance , which occurs when, in addition to the observed item loadings, the item intercepts are also equal across groups. The final level of factorial invariance is strict or error invariance , in which the observed item loadings, intercepts, and relations between the residual error terms are equal across groups. With strict factorial invariance, we have evidence that the measurement portion of the model is exactly the same across groups. In other words, this states that any group differences in scores are not due to how the constructs were measured, but rather are due to differences in mean ability levels or differences in the relationships between variables (Angoff, 1993 ; Millsap, 2011 ). We can also test for differences between groups in their average level and variability on a latent construct. Factor (co)variance invariance constrains the factor variances and covariances to be equal across groups, and if this is met, then the variance across groups is homogeneous. The highest level of factorial invariance is latent mean invariance , in which the latent means are constrained to be equal across groups. This is equivalent to a latent t -test or ANOVA, for which homogeneity of variance is an assumption.

To test for group differences that are due to differences in the relations between variables, structural invariance is used, which assumes full factorial invariance and imposes additional constraints on the regression coefficients in the structural model across groups. This is what is generally tested within the multiple-group SEM analysis described previously, which tests whether the path coefficients are the same across observed groups. It is not necessary for group membership to be observed, however. When subgroups are hypothesized, latent class analysis (McCutcheon, 1987 ) is a method used to identify individuals’ memberships in latent groups (i.e., classes), based on responses to a set of observed categorical variables. The latent group membership can be extracted and included in SEMs as a latent moderating variable. Additionally, changes in class membership over time can be examined using latent transition analysis (Collins & Lanza, 2010 ).

A different context in which latent variable models are useful is for modeling measurement error when the moderator variables or the corresponding independent variable have missing data. Enders, Baraldi, and Cham ( 2014 ) showed that re-specifying manifest independent and moderator variables as latent variables with one indicator each, factor loadings of one, and residual errors of zero preserves the intended interpretations but deals with the missing data using the multivariate normality assumptions in maximum likelihood estimation. Latent variables can easily be centered by constraining the latent means to zero, which provides meaningful and interpretable results without the need for transformations. Alternatively, multiple imputation has been shown to produce similar results as maximum likelihood, so the methods are interchangeable for this purpose.

Conditional Process Models

Given that the structural model is often used to reflect causal relations between variables, another topic that can be discussed in the context of SEM is moderation of mediated effects. Conditional process models combine moderator and mediator variables in the same model (Hayes, 2013 ) with process standing for the causal process that is mediation and conditional representing the differential effects of moderation. Consider the Theory of Planned Behavior (TPB; Ajzen, 1991 ), which is an example of a conditional process model. In the TPB, changes in attitudes and subjective norms (antecedent variables) change intentions (mediator variable), which in turn change observed behaviors (outcome variable), but the relation between intention and behavior differs depending on the level of an individual’s perceived behavioral control (moderator variable). The minimum requirements for a conditional process model are a single mediator variable and a single moderator variable, but conditional process models can be much more complex with multiple mediator and moderator variables operating simultaneously. This is the main reason the general term conditional process model has begun to replace the rather confusing historical terms moderated mediation (e.g., Little, Card, Bovaird, Preacher, & Crandall, 2007 ) and mediated moderation (Baron & Kenny, 1986 ). Though these terms were meant to indicate whether the researcher was examining possible moderation of a significant mediated effect (i.e., moderated mediation) or investigating whether a variable mediated a significant moderation effect (i.e., mediated moderation), in practice these terms have been used interchangeably because they can be used to describe identical statistical models. Since both are just special cases of conditional process models, we suggest that psychologists are better off referring to all models that contain both moderators and mediators as conditional process models because this requires that the researcher explain in detail the specific model being estimated, which is clearer all around.

Numerous authors have described how to test conditional process model hypotheses using the multiple regression framework (e.g., Hayes, 2013 ). These methods work quite well and significant interactions can be probed in much the same way as previously described for traditional regression models. When conditional process models become complex and at least one of the moderator variables is categorical, however, a better way to test for moderation is to use a multiple-group structural equation model. In the conditional process model case, a multiple-group SEM can be used to simultaneously test the mediation model across groups and directly test for differences in the mediation process between groups. For example, in M plus (Muthén & Muthén, 2015 ), it is possible to formally test the difference between the mediated effects when the moderator variable is dichotomous. This direct testing of group differences makes this method superior to methods that conduct the same analysis separately for each group (e.g., for males and then for females) and indirectly compare the results for differences.

Current Challenges

By definition, moderator variables illustrate the extent to which relations between variables are dependent on other factors including characteristics related to personality, environment, and context. Identifying moderation effects is particularly important for psychologists not only to better understand how mental processes are related to behaviors, but also to ensure that, in the effort to help, harm is not accidently caused to specific groups of individuals. Therefore, a comprehensive plan to examine all potential moderator variables should be an integral piece of any research study in psychology. Determining if a variable moderates the relation between two other variables poses several challenges to researchers, however, including the need to identify when a treatment causes harm to specific individuals, ensuring adequate statistical power to detect a moderation effect, and the difficulty in probing and interpreting complex moderation effects correctly. In this section, these issues are discussed, along with potential strategies for limiting their impact.

Treatment Interactions

As discussed previously, one of the key reasons psychologists should be interested in moderating variables is that they provide information on how the effect of a treatment, such as a CBT or behavioral prevention intervention, may function differently for groups of individuals. The effect of a treatment can vary depending on a number of different moderator variables, including demographic variables such as gender or ethnicity (Judd, McClelland, & Smith, 1996 ), a participant’s aptitude, called an aptitude by treatment interaction (Cronbach & Snow, 1977 ), or a participant’s pre-treatment level of an outcome or mediator variable, called a baseline by treatment interaction (Fritz et al., 2005 ). When present, these effects provide information that may then be used to tailor a treatment to be more effective for specific at-risk individuals. More important than improving the effectiveness of a treatment, however, is making sure there are no iatrogenic effects of the treatment. An iatrogenic effect occurs when a treatment causes an unplanned, harmful effect. For example, consider an intervention designed to prevent teenagers from using marijuana that actually increases marijuana use for some individuals. Iatrogenic effects are easily missed when they occur in only a small percentage of a sample, but ethically these effects need to be identified. Therefore, it is crucial that all theoretically relevant variables that may moderate the effect of a treatment be measured and tested.

Statistical Power

Theoretical moderating variables are not always supported by empirical research, however (e.g., Zedeck, 1971 ). When we fail to reject a null hypothesis of no moderating effect, there are two potential reasons why: either the null hypothesis is true and the variable truly does not moderate the effect, or the null hypothesis is false but it was not detected by the statistical test conducted (i.e., a Type II error occurred). To prevent incorrect conclusions about moderation effects, the probability of detecting a true effect, or statistical power , must be high. The single biggest issue with detecting moderation, other than ensuring that potential moderator variables are measured and tested in the first place, is that interaction effects tend to explain much less variance than main effects (McClelland & Judd, 1993 ). Hence, even studies that are adequately powered to find main effects are likely to be woefully unpowered when it comes to detecting moderator variables. Some of the factors that result in the under-powering of studies in psychology are beyond control—when studying a rare disorder, it may be impossible to adequately power a study simply by increasing the sample size. But there are other ways to increase statistical power for detecting moderation effects. For example, McClelland ( 2000 ) discusses several methods for increasing the statistical power of a study without increasing the sample size, such as using more reliable measures. And McClelland and Judd ( 1993 ) show that oversampling extreme cases can increase the statistical power for tests of moderation.

Part of the cause of these underpowered studies, however, is that psychological theories are rarely specific enough to include hypotheses about effect sizes for main effects, let alone interactions. A larger concern is the conflation of the size of an effect with the theoretical importance of an effect. Too many psychologists interpret Cohen’s ( 1988 ) small, medium, and large designations of effect sizes as being a measure of an effect’s theoretical importance. Cohen did not intend for large to mean important and small to mean unimportant. Instead, these categories were presented as examples of effect sizes found in a very specific area (abnormal social psychology) that needed to be recalibrated for each area of psychology and set of variables. Therefore, an effect that explains 9% of the variance in a variable (a medium effect using Cohen’s designations) may explain so little variance as to be completely disregarded by one area of psychology, yet so large as to be unobtainable in another area. Regardless of the cause, the consequences of under-powering studies to find moderation are the same: an inability to provide context for effects, resulting in a poorer understanding of the world.

Multicollinearity

Another issue that must be considered when testing interactions is multicollinearity between the variables and the interaction terms. Multicollinearity occurs when predictors in a multiple regression are highly correlated with one another and can cause excessively large standard errors, reducing the statistical power to detect an interaction even further. Since the interaction terms are just the product of the predictors, it is not surprising that the individual predictors and the interaction terms can be highly correlated. Aiken and West ( 1991 ) show that centering the predictors prior to creating an interaction term can decrease the correlation between the predictors and the interaction term by removing the nonessential multicollinearity , which is an artificial relation caused by the scaling of the predictors, while leaving the real relation, called essential multicollinearity . Others (e.g., Hayes, 2013 ) have questioned whether multicollinearity is an issue with interactions and whether centering actually addresses multicollinearity because the highest-order term, in this case the interaction term, is unaffected by centering of the lower-order terms.

Too Many Variables

When all theoretically hypothesized moderators are measured and we have adequate power to test the effect of each moderator, we run into a new problem: too many variables. It is easy to see how nearly every variable in a regression model could be moderated by every other variable in the model. But including too many interaction terms can result in an increased risk of making a Type I error, along with extremely large standard errors and potential computational difficulties. In addition, moderating relationships can be difficult to disentangle from multicollinearity and curvilinear relationships between other variables (Ganzach, 1997 ). Multicollinearity between independent variables can lead to a significant interaction term when the true interaction is not significant (Busemeyer & Jones, 1983 ; Lubinski & Humphreys, 1990 ) or may cause the interaction term to have a curvilinear appearance although the true interaction is not curvilinear. A moderating effect may also be erroneously found when there is a curvilinear relationship between the dependent and independent variables, but the model is mis-specified by excluding curvilinear terms. Lubinski and Humphreys ( 1990 ) illustrate the difficulty of distinguishing between an interaction model and a model with a curvilinear effect in which two variables are highly correlated.

The problem of too many variables is compounded when we consider that the effect of a moderator variable on the relation between an independent and dependent variable may not just differ depending on values of a second moderator variable (i.e., a three-way interaction), but also on a fourth or fifth moderator variable. Returning to the Revelle et al. ( 1980 ) example, suppose that the moderation effect of time of day on the two-way interaction between caffeine and personality type was itself different for gender (a four-way interaction). And suppose the four-way interaction between caffeine, personality type, time of day, and gender was moderated by whether the participant routinely drank highly caffeinated beverages such as coffee and soda (a five-way interaction). While four-way and higher interactions may be of interest to a researcher, an added complexity inherent to higher-order interactions is that, as described before, to properly specify a model with higher-order interactions, all lower-order interaction terms must be included in the model (Cohen, 1978 ; Cohen et al., 2003 ). For example, in an ANOVA with five factors, to correctly estimate the five-way interaction between all five factors, all possible four-way (five with five factors), three-way (nine with five factors), and two-way interactions (ten with five factors), as well as the main effects of the five factors must be included, for a total of 30 effects!

A final concern is that interactions that involve more than three variables can become very difficult to interpret in any meaningful way. This is particularly problematic in ANOVA models with large numbers of factors since many software programs automatically include all possible interactions between the factors. While failing to include an interaction term in a model is equivalent to explicitly saying the interaction effect is exactly zero, taking a kitchen-sink approach and testing all possible interactions is generally a poor strategy. Instead, researchers should test all moderation effects hypothesized by the underlying theory being studied and use diagnostic tools such as plots of residuals to determine if specific unhypothesized interactions may exist in the data, making sure to note that these additional analyses are exploratory.

Conclusions

Moderation and moderator variables are one of the most common analyses in the psychological, social, and behavioral sciences. Regardless of the phenomenon being studied, it is helpful to more fully understand for whom and in what context an effect occurs. Moderation variables help researchers test hypotheses about how the strength and/or direction of the relation between two variables may differ between individuals. Though the basic methods for analyzing moderation effects have not changed dramatically in the past 25 years, new tools have been developed to aid researchers in probing and interpreting significant interactions. The challenge for psychologists today is to include moderator variables in their theories, then plan studies that not only measure these potential moderator variables, but also are adequately powered to find moderation effects.

A majority of the interaction models and probing of significant interaction terms described here can be conducted using any general statistical software package. For psychology, popular general statistical software packages to examine moderation include:

SPSS , SAS , Stata ; and R .

While many of these more general statistical programs can also be used to test for moderation in multilevel and SEM models, specialized software may be preferred. For multilevel models, HLM is often used. For SEM models, especially those that include latent variables, Mplus , LISREL , Amos , EQS , or R may be preferred. For power analyses, two excellent programs are G-Power and Optimal Design .

Acknowledgments

This research was supported in part by a grant from the National Institute on Drug Abuse (DA 009757).

Software Resources

• Arbuckle, J. L. (2014). Amos (Version 23.0) [computer software]. Chicago: IBM SPSS.
• Bentler, P. M. (2014). EQS (Version 6.2) [computer software]. Los Angeles, CA: MVSoft, Inc.
• Faul, F. , Erdfelder, E. , Buchner, A. , & Lang, A.-G. (2014). G-Power (version 3.1.9.2) [computer software].
• IBM . (2016). SPSS Statistics . (Version 23.0) [computer software]. Armonk, NY: IBM Corp.
• Joreskog, K. G. , & Sorbom, D. (2016). LISREL (Version 8.8) [computer software]. Skokie, IL: Scientific Software International, Inc.
• Muthén, L. K. , & Muthén, B. O. (2016). Mplus (Version 7.4) [computer software]. Los Angeles: Muthén & Muthén.
• R Core Development Team . (2016). R (Version 3.3) [computer software]. Vienna, Austria: R Foundation for Statistical Computing.
• Raudenbush, S. W. , Bryk, A. S. , & Congdon, R. (2016). HLM (Version 7) [computer software]. Skokie, IL: Scientific Software International, Inc.
• SAS Institute . (2016). SAS (Version 9.4) [computer software]. Cary, NC: SAS Institute Inc.
• Spybrook, J. , Bloom, H. , Congdon, R. , Hill, C. , Martinez, A. , & Raudenbush, S. (2011) Optimal Design [computer software].
• StataCorp . (2015). Stata Statistical Software (Version 14) [computer software]. College Station, TX: StataCorp LP.

Further Reading

• Aiken, L. S. , & West, S. G. (1991). Multiple regression: Testing and interpreting interactions . Newbury Park, NJ: SAGE.
• Baron, R. M. , & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology , 51 , 1173–1182.
• Cohen, J. , Cohen, P. , West, S. G. , & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3d ed.). Mahwah, NJ: Lawrence Erlbaum.
• Dawson, J. F. , & Richter, A. W. (2006). Probing three‐way interactions in moderated multiple regression: Development and application of a slope difference test. Journal of Applied Psychology , 91 (4), 917–926.
• Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach . New York: Guilford Press.
• Hoffman, L. (2015). Between-person analysis and interpretation of interactions. In L. Hoffman (Ed.), Longitudinal analysis: Modeling within-person fluctuation and change (pp. 29–78). New York: Routledge.
• Jaccard, J. (1997). Interaction effects in factorial analysis of variance . Thousand Oaks, CA: SAGE.
• Jaccard, J. , & Turrisi, R. (2003). Interaction effects in multiple regression (2d ed.). Thousand Oaks, CA: SAGE.
• Keppel, G. , & Wickens, T. D. (2004). Design and analysis (4th ed.). Upper Saddle River, NJ: Pearson.
• Preacher, K. J. , Curran, P. J. , & Bauer, D. J. (2006). Computational tools for probing interactions in multiple linear regression, multilevel modeling, and latent curve analysis. Journal of Educational and Behavioral Statistics , 31 (4), 437–448.
• Ajzen, I. (1991). The theory of planned behavior. Organizational behavior and human decision processes , 50 , 179–211.
• Angoff, W. H. (1993). Perspectives on differential item functioning methodology. In P. W. Holland & H. Wainer (Eds.), Differential item functioning (pp. 3–23). Hillsdale, NJ: Erlbaum.
• Avolio, B. J. , Mhatre, K. , Norman, S. M. , & Lester, P. (2009). The moderating effect of gender on leadership intervention impact. Journal of Leadership & Organizational Studies , 15 , 325–341.
• Blalock, H. M. (1965). Theory building and the statistical concept of interaction. American Sociological Review , 30 (3), 374–380.
• Bollen, K. A. (1995). Structural equation models that are nonlinear in latent variables: A least-squares estimator. Sociological Methodology , 25 , 223–252.
• Busemeyer, J. R. , & Jones, L. E. (1983). Analysis of multiplicative combination rules when the causal variables are measured with error. Psychological Bulletin , 93 , 549–562.
• Cohen, J. (1968). Multiple regression as a general data-analytic system. Psychological Bulletin , 70 , 426–443.
• Cohen, J. (1978). Partialed products are interactions; Partialed powers are curve components. Psychological Bulletin , 85 , 858–866.
• Cohen, J. (1988). Statistical power analyses for the behavioral sciences (2d ed.). Mahwah, NJ: Lawrence Erlbaum.
• Collins, L. M. , & Lanza, S. T. (2010). Latent class and latent transition analysis: With applications in the social, behavioral, and health sciences . Hoboken, NJ: Wiley.
• Cronbach, L. , & Snow, R. (1977). Aptitudes and instructional methods: A handbook for research on interactions . New York: Irvington.
• Davis, J. (1966). The campus as a frog pond: An application of the theory of relative deprivation to career decisions for college men. American Journal of Sociology , 72 , 17–31.
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1. For illustrative purposes, we are drawing the details for the example from Figure 3 of Revelle et al. ( 1980 ), which combines results across multiple studies. Though the results presented here approximate those of Revelle et al., they are not based on the actual data, so the reader is encouraged to read Revelle et al.’s thoughtful and much more thorough discussion of the actual results.

2. As with the Revelle et al. ( 1980 ) example, only part of the overall Sommet et al. ( 2015 ) study is used for illustration, and the reader is encouraged to read the original paper for a complete discussion of the results.

3. Gender was not found to be a significant moderator in Long and Christian ( 2015 ), it is being used here only for illustrative purposes

4. In the original Seaton et al. ( 2010 ) paper, a third level (country) was included in the model but has been removed here for simplicity.

Related Articles

• Goal Setting Theory: Causal Relationships, Mediators, and Moderators
• Mediator Variables

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A Guide on Data Analysis

17 moderation.

• Spotlight Analysis: Compare the mean of the dependent of the two groups (treatment and control) at every value ( Simple Slopes Analysis )
• Floodlight Analysis: is spotlight analysis on the whole range of the moderator ( Johnson-Neyman intervals )

Other Resources:

BANOVAL : floodlight analysis on Bayesian ANOVA models

cSEM : doFloodlightAnalysis in SEM model

( Spiller et al. 2013 )

Terminology:

Main effects (slopes): coefficients that do no involve interaction terms

Simple slope: when a continuous independent variable interact with a moderating variable, its slope at a particular level of the moderating variable

Simple effect: when a categorical independent variable interacts with a moderating variable, its effect at a particular level of the moderating variable.

\[ Y = \beta_0 + \beta_1 X + \beta_2 M + \beta_3 X \times M \]

\(\beta_0\) = intercept

\(\beta_1\) = simple effect (slope) of \(X\) (independent variable)

\(\beta_2\) = simple effect (slope) of \(M\) (moderating variable)

\(\beta_3\) = interaction of \(X\) and \(M\)

Three types of interactions:

• Continuous by continuous
• Continuous by categorical
• Categorical by categorical

When interpreting the three-way interactions, one can use the slope difference test ( Dawson and Richter 2006 )

17.1 emmeans package

Data set is from UCLA seminar where gender and prog are categorical

17.1.1 Continuous by continuous

Simple slopes for a continuous by continuous model

Spotlight analysis ( Aiken and West 2005 ) : usually pick 3 values of moderating variable:

Mean Moderating Variable + \(\sigma \times\) (Moderating variable)

Mean Moderating Variable

Mean Moderating Variable - \(\sigma \times\) (Moderating variable)

The 3 p-values are the same as the interaction term.

For publication, we use

17.1.2 Continuous by categorical

Get simple slopes by each level of the categorical moderator

17.1.3 Categorical by categorical

Simple effects

17.2 probmod package

• Not recommend: package has serious problem with subscript.

17.3 interactions package

17.3.1 continuous interaction.

• (at least one of the two variables is continuous)

For continuous moderator, the three values chosen are:

-1 SD above the mean

-1 SD below the mean

To include weights from the regression inn the plot

Partial Effect Plot

Check linearity assumption in the model

Plot the lines based on the subsample (red line), and whole sample (black line)

17.3.1.1 Simple Slopes Analysis

continuous by continuous variable interaction (still work for binary)

conditional slope of the variable of interest (i.e., the slope of \(X\) when we hold \(M\) constant at a value)

Using sim_slopes it will

mean-center all variables except the variable of interest

For moderator that is

Continuous, it will pick mean, and plus/minus 1 SD

Categorical, it will use all factor

sim_slopes requires

A regression model with an interaction term)

Variable of interest ( pred = )

Moderator: ( modx = )

17.3.1.2 Johnson-Neyman intervals

To know all the values of the moderator for which the slope of the variable of interest will be statistically significant, we can use the Johnson-Neyman interval ( P. O. Johnson and Neyman 1936 )

Even though we kind of know that the alpha level when implementing the Johnson-Neyman interval is not correct ( Bauer and Curran 2005 ) , not until recently that there is a correction for the type I and II errors ( Esarey and Sumner 2018 ) .

Since Johnson-Neyman inflates the type I error (comparisons across all values of the moderator)

For plotting, we can use johnson_neyman

• y-axis is the conditional slope of the variable of interest

17.3.1.3 3-way interaction

Johnson-Neyman 3-way interaction

17.3.2 Categorical interaction

17.4 interactionR package

• For publication purposes

( Knol and VanderWeele 2012 ) for presentation

( Hosmer and Lemeshow 1992 ) for confidence intervals based on the delta method

( Zou 2008 ) for variance recovery “mover” method

( Assmann et al. 1996 ) for bootstrapping

17.5 sjPlot package

For publication purposes (recommend, but more advanced)

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Section 7.3: Moderation Models, Assumptions, Interpretation, and Write Up

Learning Objectives

At the end of this section you should be able to answer the following questions:

• What are some basic assumptions behind moderation?
• What are the key components of a write up of moderation analysis?

Moderation Models 

Difference between mediation & moderation.

The main difference between a simple interaction, like in ANOVA models or in moderation models, is that mediation implies that there is a causal sequence. In this case, we know that stress causes ill effects on health, so that would be the causal factor.

Some predictor variables interact in a sequence, rather than impacting the outcome variable singly or as a group (like regression).

Moderation and mediation is a form of regression that allows researchers to analyse how a third variable effects the relationship of the predictor and outcome variable.

Moderation analyses imply an interaction on the different levels of M

PowerPoint: Basic Moderation Model

Consider the below model:

• Chapter Seven – Basic Moderation Model

Would the muscle percentage be the same for young, middle-aged, and older participants after training? We know that it is harder to build muscle as we age, so would training have a lower effect on muscle growth in older people?

Example Research Question:

Does cyberbullying moderate the relationship between perceived stress and mental distress?

Moderation Assumptions

• The dependent and independent variables should be measured on a continuous scale.
• There should be a moderator variable that is a nominal variable with at least two groups.
• The variables of interest (the dependent variable and the independent and moderator variables) should have a linear relationship, which you can check with a scatterplot.
• The data must not show multicollinearity (see Multiple Regression).
• There should be no significant outliers, and the distribution of the variables should be approximately normal.

Moderation Interpretation

PowerPoint: Moderation menu, results and output

Please have a look at the following link for the Moderation Menu and Output:

• Chapter Seven – Moderation Output

Interpretation

The effects of cyberbullying can be seen in blue, with the perceived stress in green. These are the main effects of the X and M variable on the outcome variable (Y). The interaction effect can be seen in purple. This will tell us if perceived stress is effecting mental distress equally for average, lower than average or higher than average levels of cyberbullying. If this is significant, then there is a difference in that effect. As can be seen in yellow and grey, cyberbullying has an effect on mental distress, but the effect is stronger for those who report higher levels of cyberbullying (see graph).

Moderation Write Up

The following text represents a moderation write up:

A moderation test was run, with perceived stress as the predictor, mental distress as the dependant, and cyberbullying as a moderator.  There was a significant main effect found between perceived stress and mental distress, b = -1.23, BCa CI [1.11, 1.34], z =21.38 , p <.001, and nonsignificant main effect of cyberbullying on mental distress b = 1.05, BCa CI [0.72, 1.38], z=6.28, p < .001. There was a significant interaction found by cyberbullying on perceived stress and mental distress, b = -0.05, BCa CI [0.01, 0.09], z=2.16, p =.031. It was found that participants who reported higher than average levels of cyberbullying experienced a greater effect of perceived stress on mental distress ( b = 1.35, BCa CI [1.19, 1.50], z=17.1, p < .001), when compared to average or lower than average levels of cyberbullying ( b = 1.23, BCa CI [1.11, 1.34], z=21.3, p < .001, b = 1.11, BCa CI [0.95, 1.27], z=13.8, p < .001, respectively). From these results, it can be concluded that the effect of perceived stress on mental distress is partially moderated by cyberbullying.

Statistics for Research Students Copyright © 2022 by University of Southern Queensland is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Moderating Variable (or Moderator)

Types of Variables >

Types of Moderating Variable

A moderating variable can be qualitative (non-numerical values like race, socioeconomic class or sex) or quantitative (numerical values like weight, reward level or age). For example:

• According to the American Psychological Association , stress has a bigger impact on men than women. Sex is a qualitative variable that moderates the strength of an effect between stress and health status.
• There may be a relationship between socioeconomic status and how often women perform self-exams on their breasts. Age is possibly a numerical moderating variable: the relationship for socioeconomic status and breast self-exams might be weaker in younger women and stronger in older women.

In correlation studies, the moderating variable is defined as a third variable — z — that affects the correlation between two variables x and y. A statistically significant moderating variable can amplify or weaken the correlation between x and y.

Finding a Moderating Variable

The moderating variable is technically another predictor variable, so you would run multiple regression analysis to find the moderating variables.

Note: some software calls the analysis — run within regression — “moderator analysis.”

• In SPSS: Laerd.com has thorough steps for the analysis, with step by step images.
• In Stata: the process is a little lengthy, but you can find a guideline on the UCLA website .

: American Psychological Association (undated). Stress and Gender. Retrieved May 12, 2016 from here . Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182. Hayes, A.F. (2013) Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach. New York, NY: Guilford Press

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Moderating Variable – Definition, Analysis Methods and Examples

Table of Contents

Moderating Variable

Definition:

A moderating variable is a variable that affects the strength or direction of the relationship between two other variables. It is also referred to as an interactive variable or a moderator.

In social science research, a moderating variable is often used to understand how the relationship between two variables changes depending on the level of a third variable. For example, in a study examining the relationship between stress and job performance, age might be a moderating variable. The relationship between stress and job performance may be stronger for younger workers than for older workers, meaning that age is influencing the relationship between stress and job performance.

Moderating Variable Analysis Methods

Moderating Variable Analysis Methods are as follows:

Regression Analysis

Regression analysis is a statistical technique that examines the relationship between a dependent variable and one or more independent variables. In the case of a moderating variable, a regression analysis can be used to examine the interaction between the independent and moderating variables in predicting the dependent variable. This can be done using a simple regression or multiple regression analysis, depending on the number of variables involved.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of two or more groups. In the case of a moderating variable, ANOVA can be used to compare the mean differences between groups based on different levels of the moderating variable. For example, if age is a moderating variable, ANOVA can be used to compare the mean differences in job performance between younger and older workers at different levels of stress.

Multiple Regression Analysis

Multiple regression analysis is a statistical technique used to predict the value of a dependent variable based on two or more independent variables. In the case of a moderating variable, multiple regression analysis can be used to examine the interaction between the independent variables and the moderating variable in predicting the dependent variable.

Moderating Variable Examples

Here are a few examples of moderating variables:

• Age as a moderating variable : Suppose a study examines the relationship between exercise and heart health. Age may act as a moderating variable, influencing the relationship between exercise and heart health. For example, the relationship between exercise and heart health may be stronger for younger adults compared to older adults.
• Gender as a moderating variable: Consider a study examining the relationship between salary and job satisfaction. Gender may act as a moderating variable, influencing the relationship between salary and job satisfaction. For example, the relationship between salary and job satisfaction may be stronger for men than for women.
• Social support as a moderating variable: Suppose a study examines the relationship between stress and mental health. Social support may act as a moderating variable, influencing the relationship between stress and mental health. For example, the relationship between stress and mental health may be stronger for individuals with low social support compared to those with high social support.
• Education level as a moderating variable: Consider a study examining the relationship between technology use and academic performance. Education level may act as a moderating variable, influencing the relationship between technology use and academic performance. For example, the relationship between technology use and academic performance may be stronger for individuals with higher education levels compared to those with lower education levels.

Applications of Moderating Variable

• Market research: Moderating variables are often used in market research to identify the factors that influence consumer behavior. For example, age, income, and education level can be moderating variables that affect the relationship between advertising and consumer purchasing behavior.
• Psychology : In psychology, moderating variables can help explain the relationship between variables such as personality traits and job performance. For example, a person’s level of conscientiousness may moderate the relationship between their job performance and job satisfaction.
• Education: In education, moderating variables can help explain the relationship between teaching methods and student learning outcomes. For example, the level of student engagement may moderate the relationship between a teacher’s teaching style and student learning outcomes.
• Health : In health research, moderating variables can help explain the relationship between risk factors and health outcomes. For example, gender may moderate the relationship between smoking and lung cancer.
• Social sciences: In the social sciences, moderating variables can help explain the relationship between variables such as income and happiness. For example, the level of social support may moderate the relationship between income and happiness.

Purpose of Moderating Variable

The purpose of a moderating variable is to identify the conditions under which the relationship between two other variables changes or becomes stronger or weaker. In other words, a moderating variable helps to explain the context in which a particular relationship exists.

For example, let’s consider the relationship between stress and job performance. The relationship may be different depending on the level of social support that an individual receives. In this case, social support is the moderating variable. If an individual has high levels of social support, the negative impact of stress on job performance may be reduced. On the other hand, if an individual has low levels of social support, the negative impact of stress on job performance may be amplified.

The purpose of identifying moderating variables is to help researchers better understand the complex relationships between variables and to provide more accurate predictions of outcomes in specific situations. By identifying the conditions under which a relationship exists or changes, researchers can develop more effective interventions and treatments. Moderating variables can also help to identify subgroups of individuals who may benefit more or less from a particular intervention or treatment.

When to use Moderating Variable

Here are some scenarios where using a moderating variable can be helpful:

• When there is a complex relationship: In situations where the relationship between two variables is complex, a moderating variable can help to clarify the relationship. For example, the relationship between stress and job performance may be influenced by a variety of factors such as job demands, social support, and coping mechanisms.
• When there is a subgroup effect : In situations where the effect of one variable on another is stronger or weaker for certain subgroups of individuals, a moderating variable can be helpful. For example, the relationship between exercise and weight loss may be stronger for individuals who are obese compared to individuals who are not obese.
• When there is a need for tailored interventions: In situations where the effect of one variable on another is different for different individuals, a moderating variable can be useful for developing tailored interventions. For example, the relationship between diet and weight loss may be influenced by individual differences in genetics, metabolism, and lifestyle.

Characteristics of Moderating Variable

The following are some key characteristics of moderating variables:

• Interact with other variables : Moderating variables interact with other variables in a statistical relationship, influencing the strength or direction of the relationship between two other variables.
• Independent variable: Moderating variables are independent variables in a statistical analysis, meaning that they are not influenced by any of the other variables in the analysis.
• Categorical or continuous: Moderating variables can be either categorical or continuous. Categorical moderating variables have distinct categories or levels (e.g., gender), while continuous moderating variables can take on any value within a range (e.g., age).
• Can be identified through statistical analysis: Moderating variables can be identified through statistical analysis using regression analysis or ANOVA. Researchers can examine the interaction between the independent and moderating variables in predicting the dependent variable to determine if the moderating variable has a significant impact.
• Influence the relationship between other variables : The impact of a moderating variable on the relationship between other variables can be positive, negative, or null. It depends on the specific research question and the data analyzed.
• Provide insight into underlying mechanisms: Moderating variables can provide insight into underlying mechanisms driving the relationship between other variables, providing a more nuanced understanding of the relationship.

Advantages of Moderating Variable

There are several advantages of using a moderating variable in research:

• Provides a more nuanced understanding of relationships: By identifying the conditions under which a particular relationship exists or changes, a moderating variable provides a more nuanced understanding of the relationship between two variables. This can help researchers to better understand complex relationships and to develop more effective interventions.
• Improves accuracy of predictions: By identifying the conditions under which a relationship exists or changes, a moderating variable can improve the accuracy of predictions about outcomes in specific situations. This can help researchers to develop more effective interventions and treatments.
• Identifies subgroups of individuals : Moderating variables can help to identify subgroups of individuals who may benefit more or less from a particular intervention or treatment. This can help researchers to develop more tailored interventions that are more effective for specific groups of individuals.
• Increases generalizability: By identifying the conditions under which a relationship exists or changes, a moderating variable can increase the generalizability of findings. This can help researchers to apply findings from one study to other populations and contexts.
• Provides more complete understanding of phenomena : By considering the role of a moderating variable, researchers can gain a more complete understanding of the phenomena they are studying. This can help to identify areas for future research and to generate new hypotheses.

Disadvantages of Moderating Variable

Disadvantages of Moderating Variable are as follows:

• Complexity: The use of moderating variables can make research more complex and challenging to design, analyze, and interpret. This can require more resources and expertise than simpler research designs.
• Increased risk of Type I errors : When using a moderating variable, there is an increased risk of Type I errors, or false positives. This can occur when a relationship is identified that appears significant, but is actually due to chance.
• Reduced generalizability: Moderating variables can limit the generalizability of findings to other populations and contexts. This is because the relationship between two variables may be influenced by different moderating variables in different contexts.
• Limited explanatory power: While moderating variables can help to identify conditions under which a relationship exists, they may not provide a complete explanation of why the relationship exists. Other variables may also play a role in the relationship.
• Data requirements: Using moderating variables often requires larger sample sizes and more data than simpler research designs. This can increase the time and resources required to conduct the research.

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The Three Most Common Types of Hypotheses

In this post, I discuss three of the most common hypotheses in psychology research, and what statistics are often used to test them.

• Post author By sean
• Post date September 28, 2013
• 37 Comments on The Three Most Common Types of Hypotheses

Simple main effects (i.e., X leads to Y) are usually not going to get you published. Main effects can be exciting in the early stages of research to show the existence of a new effect, but as a field matures the types of questions that scientists are trying to answer tend to become more nuanced and specific.  In this post, I’ll briefly describe the three most common kinds of hypotheses that expand upon simple main effects – at least, the most common ones I’ve seen in my research career in psychology – as well as providing some resources to help you learn about how to test these hypotheses using statistics.

Incremental Validity

“Can X predict Y over and above other important predictors?”

This is probably the simplest of the three hypotheses I propose. Basically, you attempt to rule out potential confounding variables by controlling for them in your analysis.  We do this because (in many cases) our predictor variables are correlated with each other. This is undesirable from a statistical perspective, but is common with real data. The idea is that we want to see if X can predict unique variance in Y over and above the other variables you include.

In terms of analysis, you are probably going to use some variation of multiple regression or partial correlations.  For example, in my own work I’ve shown in the past that friendship intimacy as coded from autobiographical narratives can predict concern for the next generation over and above numerous other variables, such as optimism, depression, and relationship status ( Mackinnon et al., 2011 ).

“Under what conditions does X lead to Y?”

Of the three techniques I describe, moderation is probably the most tricky to understand.  Essentially, it proposes that the size of a relationship between two variables changes depending upon the value of a third variable, known as a “moderator.”  For example, in the diagram below you might find a simple main effect that is moderated by sex. That is, the relationship is stronger for women than for men:

With moderation, it is important to note that the moderating variable can be a category (e.g., sex) or it can be a continuous variable (e.g., scores on a personality questionnaire).  When a moderator is continuous, usually you’re making statements like: “As the value of the moderator increases, the relationship between X and Y also increases.”

“Does X predict M, which in turn predicts Y?”

We might know that X leads to Y, but a mediation hypothesis proposes a mediating, or intervening variable. That is, X leads to M, which in turn leads to Y.  In the diagram below I use a different way of visually representing things consistent with how people typically report things when using path analysis.

I use mediation a lot in my own research. For example, I’ve published data suggesting the relationship between perfectionism and depression is mediated by relationship conflict ( Mackinnon et al., 2012 ). That is, perfectionism leads to increased conflict, which in turn leads to heightened depression. Another way of saying this is that perfectionism has an indirect effect on depression through conflict.

Helpful links to get you started testing these hypotheses

Depending on the nature of your data, there are multiple ways to address each of these hypotheses using statistics. They can also be combined together (e.g., mediated moderation). Nonetheless, a core understanding of these three hypotheses and how to analyze them using statistics is essential for any researcher in the social or health sciences.  Below are a few links that might help you get started:

Are you a little rusty with multiple regression? The basics of this technique are required for most common tests of these hypotheses. You might check out this guide as a helpful resource:

https://statistics.laerd.com/spss-tutorials/multiple-regression-using-spss-statistics.php

David Kenny’s Mediation Website provides an excellent overview of mediation and moderation for the beginner.

http://davidakenny.net/cm/mediate.htm

http://davidakenny.net/cm/moderation.htm

Preacher and Haye’s INDIRECT Macro is a great, easy way to implement mediation in SPSS software, and their MODPROBE macro is a useful tool for testing moderation.

http://afhayes.com/spss-sas-and-mplus-macros-and-code.html

If you want to graph the results of your moderation analyses, the excel calculators provided on Jeremy Dawson’s webpage are fantastic, easy-to-use tools:

http://www.jeremydawson.co.uk/slopes.htm

• Tags mediation , moderation , regression , tutorial

37 replies on “The Three Most Common Types of Hypotheses”

I want to see clearly the three types of hypothesis

Thanks for your information. I really like this

Thank you so much, writing up my masters project now and wasn’t sure whether one of my variables was mediating or moderating….Much clearer now.

Thank you for simplified presentation. It is clearer to me now than ever before.

Thank you. Concise and clear

hello there

I would like to ask about mediation relationship: If I have three variables( X-M-Y)how many hypotheses should I write down? Should I have 2 or 3? In other words, should I have hypotheses for the mediating relationship? What about questions and objectives? Should be 3? Thank you.

Hi Osama. It’s really a stylistic thing. You could write it out as 3 separate hypotheses (X -> Y; X -> M; M -> Y) or you could just write out one mediation hypotheses “X will have an indirect effect on Y through M.” Usually, I’d write just the 1 because it conserves space, but either would be appropriate.

Hi Sean, according to the three steps model (Dudley, Benuzillo and Carrico, 2004; Pardo and Román, 2013)., we can test hypothesis of mediator variable in three steps: (X -> Y; X -> M; X and M -> Y). Then, we must use the Sobel test to make sure that the effect is significant after using the mediator variable.

Yes, but this is older advice. Best practice now is to calculate an indirect effect and use bootstrapping, rather than the causal steps approach and the more out-dated Sobel test. I’d recommend reading Hayes (2018) book for more info:

Hayes, A. F. (2018). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach (2nd ed). Guilford Publications.

Hi! It’s been really helpful but I still don’t know how to formulate the hypothesis with my mediating variable.

I have one dependent variable DV which is formed by DV1 and DV2, then I have MV (mediating variable), and then 2 independent variables IV1, and IV2.

How many hypothesis should I write? I hope you can help me 🙂

Thank you so much!!

If I’m understanding you correctly, I guess 2 mediation hypotheses:

IV1 –> Med –> DV1&2 IV2 –> Med –> DV1&2

Thank you so much for your quick answer! ^^

Could you help me formulate my research question? English is not my mother language and I have trouble choosing the right words. My x = psychopathy y = aggression m = deficis in emotion recognition

thank you in advance

I have mediator and moderator how should I make my hypothesis

Can you have a negative partial effect? IV – M – DV. That is my M will have negative effect on the DV – e.g Social media usage (M) will partial negative mediate the relationship between father status (IV) and social connectedness (DV)?

Thanks in advance

Hi Ashley. Yes, this is possible, but often it means you have a condition known as “inconsistent mediation” which isn’t usually desirable. See this entry on David Kenny’s page:

Or look up “inconsistent mediation” in this reference:

MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology, 58, 593-614.

This is very interesting presentation. i love it.

This is very interesting and educative. I love it.

Hello, you mentioned that for the moderator, it changes the relationship between iv and dv depending on its strength. How would one describe a situation where if the iv is high iv and dv relationship is opposite from when iv is low. And then a 3rd variable maybe the moderator increases dv when iv is low and decreases dv when iv is high.

This isn’t problematic for moderation. Moderation just proposes that the magnitude of the relationship changes as levels of the moderator changes. If the sign flips, probably the original relationship was small. Sometimes people call this a “cross-over” effect, but really, it’s nothing special and can happen in any moderation analysis.

i want to use an independent variable as moderator after this i will have 3 independent variable and 1 dependent variable…. my confusion is do i need to have some past evidence of the X variable moderate the relationship of Y independent variable and Z dependent variable.

Dear Sean It is really helpful as my research model will use mediation. Because I still face difficulty in developing hyphothesis, can you give examples ? Thank you

Hi! is it possible to have all three pathways negative? My regression analysis showed significant negative relationships between x to y, x to m and m to y.

Hi, I have 1 independent variable, 1 dependent variable and 4 mediating variable May I know how many hypothesis should I develop?

Hello I have 4 IV , 1 mediating Variable and 1 DV

My model says that 4 IVs when mediated by 1MV leads to 1 Dv

Pls tell me how to set the hypothesis for mediation

Hi I have 4 IVs ,2 Mediating Variables , 1DV and 3 Outcomes (criterion variables).

Pls can u tell me how many hypotheses to set.

Thankyou in advance

I am in fact happy to read this webpage posts which carries tons of useful information, thanks for providing such data.

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what if the hypothesis and moderator significant in regrestion and insgificant in moderation?

Thank you so much!! Your slide on the mediator variable let me understand!

Very informative material. The author has used very clear language and I would recommend this for any student of research/

Hi Sean, thanks for the nice material. I have a question: for the second type of hypothesis, you state “That is, the relationship is stronger for men than for women”. Based on the illustration, wouldn’t the opposite be true?

Yes, your right! I updated the post to fix the typo, thank you!

I have 3 independent variable one mediator and 2 dependant variable how many hypothesis I have 2 write?

Sounds like 6 mediation hypotheses total:

X1 -> M -> Y1 X2 -> M -> Y1 X3 -> M -> Y1 X1 -> M -> Y2 X2 -> M -> Y2 X3 -> M -> Y2

Clear explanation! Thanks!

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Moderation Analysis in SPSS

Discover Moderation Analysis in SPSS ! Learn how to perform, understand SPSS output , and report results in APA style. Check out this simple, easy-to-follow guide below for a quick read!

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Introduction

Moderation analysis is a valuable tool in research, allowing researchers to understand how the relationship between two variables changes depending on a third variable, known as the moderator. This analysis is crucial for gaining insights into complex relationships and identifying conditions under which certain effects occur. As the field of data analysis grows, the ability to perform and interpret moderation analysis becomes increasingly important.

Using SPSS, a widely-used statistical software, can simplify the process of conducting moderation analysis. This blog post aims to provide a comprehensive guide on performing moderation analysis in SPSS. We will cover the fundamental concepts, differentiate between mediation and moderation, and outline the steps and assumptions involved in testing moderation. Additionally, we will explore practical examples, interpret SPSS output, and provide guidance on reporting results in APA format.

PS: This post explains the traditional regression method in SPSS for moderation analysis. If you prefer to use the Hayes PROCESS Macro, please visit our guide on “ Moderation Analysis with Hayes PROCESS Macro in SPSS .”

What is Moderation Analysis?

Moderation analysis examines how the relationship between an independent variable (X) and a dependent variable (Y) changes as a function of a third variable, called the moderator (M). The moderator can either strengthen, weaken, or reverse the effect of the independent variable on the dependent variable. By including a moderator, researchers can capture more nuanced relationships and better understand the conditions under which certain effects are stronger or weaker.

This type of analysis is particularly useful in social sciences, where the impact of one variable on another often depends on additional contextual factors. For instance, the effect of stress on performance might vary depending on levels of social support. This helps researchers identify these cofounding effects, providing deeper insights into the dynamics of the studied relationships.

What are Steps in Testing Moderation?

• Center the Moderator and Independent Variable: Mean-center the independent variable and the moderator to reduce multicollinearity and simplify the interpretation of the interaction term.
• Create Interaction Term: Multiply the centered independent variable and the centered moderator to create an interaction term.
• Run Regression Analysis: Enter the independent variable, moderator, and interaction term into a multiple regression model predicting the dependent variable.
• Plot Interaction: Plot the interaction to visualise how the relationship between the independent variable and the dependent variable changes at different levels of the moderator.

Which is the Method better: Using Hayes PROCESS Macro or Traditional Regression for Moderation Analysis?

Choosing between Hayes PROCESS Macro and traditional regression for moderation analysis depends on your research needs and statistical expertise. The Hayes PROCESS Macro offers a user-friendly interface, automating many steps of the moderation analysis and providing bootstrap confidence intervals for the interaction effects. This method reduces human error and enhances result reliability, making it a preferred choice for those who seek convenience and precision.

In contrast, traditional regression requires manual computation of interaction terms and more steps in the analysis process. While it offers flexibility and a deeper understanding of the moderation process, it demands a higher level of statistical knowledge. The regression might be better suited for researchers who prefer customising their analyses and exploring the underlying data in more detail. Both methods have their advantages, and the choice ultimately depends on the research context and the user’s familiarity with statistical tools.

In this blog, we will give details about regression for moderation analysis, but you can visit the Hayes PROCESS post to see details of the method.

What are the Assumptions of Moderation Analysis?

• Linearity: The relationships between the independent variable, moderator, and dependent variable must be linear.
• Independence of Errors: The error terms in the regression equations should be independent of each other.
• No Multicollinearity: The independent variable, moderator, and their interaction term should not be highly correlated with each other.
• Homoscedasticity: The variance of the error terms should be constant across all levels of the independent variable and the moderator.
• Normality: The residuals of the regression equations should be normally distributed.
• Measurement without Error: The variables involved in the moderation analysis should be measured accurately without error.

What is the Hypothesis of Moderation Analysis?

The primary hypothesis in moderation analysis posits that the strength or direction of the relationship between an independent variable (X) and a dependent variable (Y) depends on the level of a third variable, the moderator (M).

• H0 (The null hypothesis): The interaction term does not significantly predict the dependent variable (meaning there is no moderation effect.)
• H1 (The alternative hypothesis): the interaction term significantly predicts the dependent variable. (indicating the presence of a moderation effect.)

Testing these hypotheses involves examining the interaction term in the regression model to determine if the moderation effect is statistically significant.

An Example of Moderation Analysis

Consider a study examining the impact of work stress (X) on job performance (Y) and how this relationship is moderated by social support (M). The hypothesis posits that the negative effect of work stress on job performance will be weaker for employees with high social support compared to those with low social support. To test this, researchers would first mean-center the variables of work stress and social support.

Next, researchers would create an interaction term by multiplying the centered work stress and social support variables. By entering work stress, social support, and the interaction term into a regression model predicting job performance, researchers can assess the main effects and the interaction effect. If the interaction term is significant, it indicates that social support moderates the relationship between work stress and job performance.

How to Perform Moderation Analysis in SPSS

Step by Step: Running Moderation Analysis in SPSS Statistics

Let’s embark on a step-by-step guide on performing the Moderation Analysis using SPSS

– Open your dataset in SPSS, ensuring it includes the independent variable (X), dependent variable (Y), and moderator (M).

Center the Variables

– Compute the mean of the independent variable and the moderator, then subtract these means from their respective variables to create centered variables.

Create Interaction Term

– Multiply the centered independent variable by the centered moderator to create an interaction term.

Run Regression Analysis

– Navigate to ` Analyze > Regression > Linear `.

– Enter the dependent variable (Y) into the Dependent box.

– Move the centered independent variable (X), centered moderator (M), then click Next ” for block 2 enter the interaction term into the Independent box.

– Click OK to run the regression analysis.

Interpret the Output

– Examine the coefficients table to assess the significance of the independent variable, moderator, and interaction term.

– Significant interaction term indicates moderation.

Note: Conducting Moderation Analysis in SPSS provides a robust foundation for understanding the key features of your data. Always ensure that you consult the documentation corresponding to your SPSS version, as steps might slightly differ based on the software version in use. This guide is tailored for SPSS version 25 , and for any variations, it’s recommended to refer to the software’s documentation for accurate and updated instructions.

SPSS Output for Moderation Analysis

Spss output 1, spss output 2.

How to Interpret SPSS Output of Moderation Analysis

When interpreting the SPSS output of your moderation analysis, focus on three key tables: Model Summary, ANOVA, and Coefficients.

Model Summary Table:

• R: This represents the correlation between the observed and predicted values of the dependent variable. Higher values indicate a stronger relationship.
• R Square (R²): This value indicates the proportion of variance in the dependent variable explained by the independent, moderator, and interaction variables. An R² value closer to 1 suggests a better fit.
• Adjusted R Square: Adjusts the R² value for the number of predictors in the model. This value is useful for comparing models with different numbers of predictors.

ANOVA Table:

• F-Statistic: This tests the overall significance of the model. A significant F-value (p < 0.05) indicates that the model significantly predicts the dependent variable.
• (p-value): If the p-value is less than 0.05, the model is considered statistically significant, meaning the independent and mediator variables together significantly predict the dependent variable.

Coefficients Table:

• Unstandardized Coefficients (B): Coefficient of variable
• Constant (Intercept): The expected value of the dependent variable when all predictors are zero.
• Standardized Coefficients (Beta): These coefficients are useful for comparing the relative strength of each predictor in the model.
• t-Statistic and Sig. (p-value): Indicates whether each predictor is significantly contributing to the model. If the p-value is less than 0.05, the predictor is considered statistically significant.

By focusing on these tables, you can effectively interpret the results of your mediation analysis in SPSS, identifying the direct and indirect effects, as well as the overall model significance.

How to Report Results of Moderation Analysis in APA

Reporting the results of moderation analysis in APA (American Psychological Association) format requires a structured presentation. Here’s a step-by-step guide in list format:

• Introduction : Briefly describe the purpose of the moderation analysis and the variables involved.
• Descriptive Statistics : Report the means and standard deviations of the independent variable, moderator, and dependent variable.
• Main Effects : Provide the regression coefficients, standard errors, and p-values for the independent variable and moderator.
• Interaction Effect : Report the regression coefficient, standard error, and p-value for the interaction term.
• Model Summary : Include R² and adjusted R² values to indicate the model fit.
• Significance Tests : Present the results of the F-test and the significance levels for the overall model.
• Plot Interaction : Include a plot illustrating the interaction effect, showing how the relationship between the independent variable and the dependent variable changes at different levels of the moderator.
• Figures and Tables : Provide tables and figures to visually represent the statistical results and interaction effects.
• Conclusion : Summarise the key results and suggest directions for future research.

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What is a Moderating Variable? Definition & Example

A moderating variable is a type of variable that affects the relationship between a dependent variable and an independent variable .

When performing regression analysis , we’re often interested in understanding how changes in an independent variable affect a dependent variable. However, sometimes a moderating variable can affect this relationship.

For example, suppose we want to fit a regression model in which we use the independent variable hours spent exercising each week to predict the dependent variable resting heart rate . 

We suspect that more hours spent exercising is associated with a lower resting heart rate. However, this relationship could be affected by a moderating variable such as gender .

It’s possible that each extra hour of exercise causes resting heart rate to drop more for men compared to women.

Another example of a moderating variable could be  age . It’s likely that each extra hour of exercise causes resting heart rate to drop more for younger people compared to older people.

Properties of Moderating Variables

Moderating variables have the following properties:

1. Moderating variables can be qualitative or quantitative .

Qualitative variables are variables that take on names or labels. Examples include:

• Gender (Male or Female)
• Education Level (High School Degree, Bachelor’s Degree, Master’s Degree, etc.)
• Marital Status (Single, Married, Divorced)

Quantitative variables are variables that take on numerical values. Examples include:

• Square Footage
• Population Size

In the previous examples,  gender was a qualitative variable that could affect the relationship between hours studied and resting heart rate while  age was a quantitative variable that could potentially affect the relationship.

2. Moderating variables can affect the relationship between an independent and dependent variable in a variety of ways.

Moderating variables can have the following effects:

• Strengthen the relationship between two variables.
• Weaken the relationship between two variables.
• Negate the relationship between two variables.

Depending on the situation, a moderating variable can  moderate the relationship between two variables in many different ways.

How to Test for Moderating Variables

If X is an independent variable (sometimes called a “predictor” variable) and  Y is a dependent variable (sometimes called a “response” variable), then we could write a regression equation to describe the relationship between the two variables as follows:

Y = β 0 + β 1 X

If we suspect that some other variable, Z , is a moderator variable, then we could fit the following regression model:

Y = β 0 + β 1 X 1 + β 2 Z   + β 3 XZ

In this equation, the term  XZ is known as an  interaction term .

If the p-value for the coefficient of XZ in the regression output is statistically significant, then this indicates that there is a significant interaction between  X and  Z and  Z should be included in the regression model as a moderator variable.

We would write the final model as:

Y = β 0 + β 1 X + β 2 Z   + β 3 XZ

If the p-value for the coefficient of  XZ in the regression output is not statistically significant, then  Z is not a moderator variable.

However it’s possible that the coefficient for Z could still be statistically significant. In this case, we would simply include Z as another independent variable in the regression model.

We would then write the final model as:

Y = β 0 + β 1 X + β 2 Z

Additional Resources

How to Read and Interpret a Regression Table How to Use Dummy Variables in Regression Analysis Introduction to Confounding Variables

When to Use a Chi-Square Test (With Examples)

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• Regression with a moderator 101

What is a moderator?

The linear regression modeling as illustrated in chapter 3 has different variations, just like ANOVA. The mathematics behind various regression models can be messy; however, the main objective of this book is to how to apply it. Regression with a moderator is an advance regression model to test for interaction, but what is a moderator in the first place?

A moderator was defined in Hayes (2017), “The effect of X on some variable Y is moderated by W if its size, sign, or strength depends on or can be predicted by W. In that case, W is said to be a moderator of X’s effect on Y, or that W and X interact in their influence on Y (220).” Now, let’s translate it into English. Let’s consider the following hypothetical scenarios.

You live in a city A or a literally utopian city with extremely low crime rate. Everyone in the city has abundant resources. There are many designer shops including Gucci, Louis Vuitton, Montblanc, etc. In the city, people with higher income tend to buy designer products to exhibit their socioeconomic status.

Now, it is reasonable to assume there is a positive significant correlation between the income and the amount of money spent on designer products. Let’s assume that beta is 20, and p-value is .01.

You live in a city B town where the crime rate has been skyrocketing in the past decade. Everyone in the city is afraid of being robbed. Therefore, designer shops are not quite as popular as they are in the miracle town. In city B, people with higher income do not really buy designer products to exhibit their socioeconomic status. Instead, they hire personal guards or install high-tech security system at home.

Now, consider the correlation between the income and the money spent on designer products. It vanishes. There is no significant correlation anymore. Let’s assume that beta is .02, and p-value is .82.

If you were a researcher, you collect samples from both city A and B. Your research population is residents from both city A and B, and you found that income is not a significant predictor variable of the money spent on designer products. Is your finding accurate? The answer is obvious NO. What is happening here? What makes the difference?

The main difference between city A and B is the crime rate. The crime rate influences people’s perception of safety, or how safe people think it is to exhibit their wealth by wearing designer products. People’s perception of safety here is a moderator. Put it back to the definition. People’s perception about safety is a moderator of income’s effect on the amount of money they spend on designer products, or people’s perception of safe and their income interacts with the amount of money they spend on designer products.

The other way to think how a moderator works is to look at beta or coefficient. In the first scenario when the moderator’s size is high, meaning that people have a high sense of security, the corresponding coefficient is 20. When the moderator’s size is low, meaning that people have a low sense of security, the corresponding coefficient is 0.02 which is a lot lower.

Figure 5.1.a                             Illustrating the moderator

Look at the figure 5.1.a, the moderator variable is color-coded by blue and orange. Both lines are  the linear regression line where income predicts the money spent on designer products. The orange one is in the dangerous city where people do not feel safe showing off their designer products while the blue one is in the safe city where people feel safe doing so. It is logically plausible.

Regression with a moderator?

What is the statistical way to test for an interaction? How can we test if there is a significant moderator? First, we need to think if it logically makes sense. When one variable can change the direction of another variable, it is a potential candidate for the moderator.

Let’s consider the following hypothesis. Students tend to develop a more positive attitude towards the renewable energy policy if he/she knows more about climate change. Assume that we do not find a correlation between attitude towards the renewable energy policy and the amount of knowledge about climate change. We can stop here concluding that the data does not support enough evidences to reject the null.

If we do not know anything about moderators, we would definitely stop here, but since we know the concept of the moderator. Maybe, political stand or belief about climate change might be a moderator. If a person believes that climate change is fake and created as a political tool by the leftist, it is possible that he would only read and accumulate knowledge about why climate change is fake due to the conformation bias. Therefore, climate change deniers might indeed spend more time reading and have a great volume of knowledge about the invalidity of climate change. Therefore, for people holding this belief, there might be a negative correlation instead of a positive correlation. In fact, then, if tested to be significant, the belief about whether climate change is real is the moderator here!

The formula for regression with a moderator is 

Y= b 1 X 1 + b 2 X 2 + b 3 X 1 X 2 +C                                                                                             (5.1)

By testing this model, three possible coefficients and p values would be given. Then, b1 and b2 is the coefficient for direct effect while B3 is the interaction. P value for b3 would indicate whether the correlation is significant.

Applying to Gender Report Data

Are there any interaction in the World Bank Gender Report Database (2017)? Off course, there are. Let’s look at one closer. Let’s define binary qualitative variable X (independent variable) to be whether law requires equal gender hiring (1=yes; 0=no). Then, define a quantitative continuous variable Y (dependent variable) to be expected years of schooling for girls in the country. Now, we define our moderator M to be whether men and married women have equal ownership rights to property.

Our research hypothesis is legislation about whether men and married women, and legislation about whether law requires equal gender hiring interacts in their influence to expected years of schooling for girls in a country.

The result is significant. Let’s take a look at the figure 5.1 (b).

Figure 5.1(b)

It is obvious that marriage with or without equal ownership of property changes the coefficient. For countries with legislation requiring equal ownership within marriage, there is a positive correlation between law requiring equal gender hiring and expected years of schooling for girls. It means that girls receive longer education if the countries have laws requiring equal gender hiring.

However, if the countries do not have legislation requiring equal ownership, there is a negative correlation meaning girls receive less education if the countries have laws requiring equal gender hiring. It is plausible logically. If there is no hope for females because once they are married, their husbands own everything, there is no point for them to go to school even if there is law requiring equal gender hiring. Why even work if all money females make belong to their husbands.

To run regression with a moderator, you can either code the moderator by multiplying your independent variable and moderator. Otherwise, you can use Andrew F. Hayes’s process in SPSS. It also gives you conditional effects of the focal predictor at values of the moderator. Please read Andrew F. Hayes’s book, Introduction to Mediation, Moderation, and Conditional Process Analysis , if you want to learn more about moderation and conditional process analysis.

The interaction modeling can help us explain a lot of real-life scenarios. Sometimes, even if two variables are not correlated significant, there would still something to discover.

Practice and Homework

Hayes, A. F. (2017). Introduction to Mediation, Moderation, and Conditional Process Analysis, Second Edition: A Regression-Based Approach. Guilford Publications.

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Chapter 10 moderating, mediating, and confounding variables, 10.1 more than the iv and the dv.

In this section, we’ll expand our understanding of variables in the study. So far, we have discussed three types of variables:

Independent variable (IV): The variable that is implied (quasi-experiment, non-experiment) or demonstrated to be (experiment) the cause of an effect. When there is a manipulation, the variable that is manipulated is the IV.

Dependent variable (DV): The variable that is implied or demonstrated to be the outcome.

Confounding variable: Also called a nuisance variable or third variable. This is a third variable that causes a change in both the IV and the DV at the same time. To borrow an example, we might observe a correlation between ice cream consumption and snake bites. We might wonder if eating ice cream causes snake bites on the basis of this result. Although this seems ridiculous, it’s easier for us to make these sorts of conclusions when the variables are psychological constructs (for example, personality and job outcomes). In this example, the weather is a confounding variable. When the weather is warmer, ice cream consumption (it’s warm) and snake bites (people go on hikes) increase.

From this example, you might wonder if other factors matter, such as the location (regions with lots of snakes versus regions with fewer). Some of these other variables may affect the DV in other ways, such as by weakening the relationship between the IV and the DV. Therefore, confounding variables are one type of extraneous variable. Extraneous variables include anything we have not included in our study.

Some extraneous variables are not likely to affect anything. In this example, the gender of people buying ice cream probably does not affect their likelihood of snake bites. Other extraneous variables can affect the relationship we are trying to observe in our study. Whenever you design a study, an important step is to stop and consider “what else could be affecting this relationship?” When you do this, you will brainstorm a list of possible confounding and extraneous variables. Then, you’ll decide if the variables are likely to affect the relationship of interest. If they are, then usually you can redesign your study to avoid them.

To summarize: A study is essentially a search to identify and explain relationships between IVs and DVs. When claims about the relationship between an IV and DV are true, the claim has internal validity.

Next, we will explore two more complex relationships between variables that develop when we add a second IV to our model.

10.2 Moderating Variables: Interaction Effects

Interactions are also called moderated relationships or moderation. An interaction occurs when the effect of one variable depends on the value of another variable. ** For example, how do you increase the sweetness of coffee? Imagine that sweetness is the DV, and the two variables are stirring (yes vs no) and adding a sugar cube (yes vs no).

We diagram a moderated relationship using this notation:

Diagram of a moderated relationship with IV 2 and IV 1 interacting to affect DV

And, when we have group means for every condition, we can see the impact of these two factors (factor is a fancy word for IV) in a table:

When is the coffee sweet? Stirring alone does not change the taste of the coffee. Adding a sugar cube alone also doesn’t change the taste of the coffee, since the sugar will just sink to the bottom. It’s only when sugar is added, and the coffee is stirred that it tastes sweet.

We can say there is an interaction between adding sugar and stirring coffee. The effect of the stirring depends on the value of another variable (whether or not sugar is added).

10.3 Some Terminology

When more than one IV is included in a model, we are using a factorial design. Factorial designs include 2 or more factors (or IVs) with 2 or more levels each. In the coffee example, our design has two factors (stirring and adding sugar), each with two levels.

In factorial designs (i.e., studies that manipulate two or more factors), participants are observed at each level of each factor. Because every possible combination of each IV is included, the effects of each factor alone can be observed. We also get to see how these factors impact each other. We say this design is fully crossed because every possible combination of levels is included.

10.4 Main Effects

A main effect is the effect of one factor. There is one potential main effect for each factor.

In this example, the potential main effects are stirring and adding sugar. To find the main effects, find the mean of each column (i.e., add the two numbers and divide by 2). If there are differences in these means, there is a significant main effect for one of the factors. Next, find the mean of each row (add going across and divide by 2). If there are differences in these row means, then there is a main effect for the other factor.

In our example, we see two main effects. Adding a sugar cube (mean of 50) differs from not adding sugar (mean of 0). That’s the first main effect. The second is stirring; stirring (mean of 50) differs from not stirring (mean of 0).

10.5 Simple Effects

When an interaction effect is present, each part of an interaction is called a simple effect. To examine the simple effects, compare each cell to every other cell in the same row. Next, compare each cell to ever other cell in the same column. Simple effects are never diagonal from each other.

In our example, we see a simple effect as we go from Stir+Sugar to NoStir+Sugar. There is no simple effect between Stir+NoSugar and NoStir+NoSugar (both are 0). What makes this an interaction effect is that these two simple effects are different from one another.

On the vertical, there is a simple effect from Stir+Sugar to Stir+NoSugar. There is no simple effect from NoStir+Sugar to NoStir+NoSugar (both are 0). Again, this is an interaction effect because these two simple effects are different.

10.6 Interaction Effect

When there is at least one (significant) simple effect that differs across levels of one of the IVs (as demonstrated above), then you can say there is an interaction between the two factors. In a two-way ANOVA, there is one possible interaction effect. We sometimes show this with a multiplication symbol: Sugar*Stir. In our example, there is an interaction between sugar and stirring.

In summary: An interaction effect is when the impact of one variable depends on the level of another variable.

Interaction effects are important in psychology because they let us explain the circumstances under which an effect occurs. Anytime we say that an effect depends on something else, we are describing an interaction effect.

10.7 Mediators and Mediated Relationships

A mediated relationship is a chain reaction; one variable causes another variable (the mediator), which then causes the DV. Please forgive another silly example; I am including it to keep the example as simple as possible. Here is how we diagram it:

This is a totally different situation that the previous one. The first variable is a preference for sweetness; do you like sweet foods and beverages? If participants prefer sweetness, then they will add more sugar. If they don’t prefer sugar in their coffee, then they will add less (or no) sugar. Thus, preference for sweetness is an IV that causes a change in the mediator, adding sugar. Finally, adding sugar is what causes the coffee to taste sweet. Any time we can string together three variables in a causal chain, we are describing a mediated relationship.

In summary: A mediated relationship occurs when one variable affects another (the mediator), and that variable (the mediator), affects something else.

Mediated relationships are important in psychology because they let us explain why or how an effect happens. The mediator is the how or the why. Why do participants who prefer sweetness end up with sweeter coffee? It is because they added sugar.

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Integrating Mediators and Moderators in Research Design

David p. mackinnon.

1 Department of Psychology, Arizona State University, Tempe, AZ, USA

The purpose of this article is to describe mediating variables and moderating variables and provide reasons for integrating them in outcome studies. Separate sections describe examples of moderating and mediating variables and the simplest statistical model for investigating each variable. The strengths and limitations of incorporating mediating and moderating variables in a research study are discussed as well as approaches to routinely including these variables in outcome research. The routine inclusion of mediating and moderating variables holds the promise of increasing the amount of information from outcome studies by generating practical information about interventions as well as testing theory. The primary focus is on mediating and moderating variables for intervention research but many issues apply to nonintervention research as well.

It is sufficiently obvious that both analysis and synthesis is necessary in classification and that both splitting and lumping have a place, or, to the extent that the terms involve antithesis, that neither one is correct. It is desirable that all distinguishable groups should be distinguished (although it is not necessary that all enter into formal classification and receive names). It is also desirable that they should all be gathered into larger units of increasing magnitude with grades, each of which has practical value and which are numerous enough to suggest degrees of affinity that can be and need to be specified. ( Simpson, 1945 , p. 23)

Two common questions in intervention outcome research are “How does the intervention work?” and “For which groups does the intervention work?” The first question is a question about mediating variables —variables that describe the process by which the intervention achieves its effects. The second question is about moderating variables —variables for which the intervention has a different effect at different values of the moderating variable. More information can be extracted from research studies if measures of mediating and moderating variables are included in the study design and data-collection plan. Furthermore, including measures of moderating and mediating variables is inexpensive, given their potential for providing information about how interventions work and for whom interventions work. Mediating and moderating variables are important for nonintervention outcome research as well as intervention research. A mediating variable is relevant whenever a researcher wants to understand the process by which two variables are related, such that one variable causes a mediating variable which then causes a dependent variable. Moderating variables are important whenever a researcher wants to assess whether two variables have the same relation across groups.

Third-Variable Effects

Mediating and moderating variables are examples of third variables. Most research focuses on the relation between two variables—an independent variable X and an outcome variable Y . Example statistics for two-variable effects are the correlation coefficient, odds ratio, and regression coefficient. With two variables, there are a limited number of possible causal relations between them: X causes Y , Y causes X , both X and Y are reciprocally related. With three variables, the number of possible relations among the variables increases substantially: X may cause the third variable Z and Z may cause Y ; Y may cause both X and Z , and the relation between X and Y may differ for each value of Z , along with others. Mediation and moderation are names given to two types of third-variable effects. If the third variable Z is intermediate in a causal sequence such that X causes Z and Z causes Y , then Z is a mediating variable; it is in a causal sequence X → Z → Y . If the relation between X and Y is different at different values of Z , then Z is a moderating variable. A primary distinction between mediating and moderating variables is that the mediating variable specifies a causal sequence in that a mediating variable transmits the causal effect of X to Y but the moderating variable does not specify a causal relation, only that the relation between X and Y differs across levels of Z . Diagrams for a mediating variable in Figure 1 and a moderating variable in Figure 2 demonstrate the difference between these two variables where the causal sequence is shown with directed arrows in Figure 1 to demonstrate a mediation relation. For moderation in Figure 2 , there is not an indirect relation of X to Y but there is an interaction XZ that corresponds to a potentially different X to Y relation at values of Z .

Single mediator model.

Single moderator model.

Another important third variable is the confounding variable that causes both X and Y such that failure to adjust for the confounding variable will confound or lead to incorrect conclusions about the relation of X to Y . A confounding variable differs from a mediating variable in that the confounding variable is not in a causal sequence but the confounding variable is related to both X and Y . A confounder differs from a moderating variable because the relation of X to Y may not differ across values of the confounding variable. Mediating and moderating variables are the focus of this article. More on these different types of third-variable effects are described elsewhere ( Greenland & Morgenstern, 2001 ; MacKinnon, 2008 ; MacKinnon, Krull, & Lockwood, 2000 ).

As you might expect, there are many more possible combinations of relations among four variables and as more variables are added, the number of possible relations among variables soon grows very complex. In this case with many variables, researchers typically often focus on third-variable effects such as moderating and mediating variables even in the most complex models. It is useful to remember that even though I focus on the simplest moderating and mediating model in this article, as the number of variables increases the number of possible models increases dramatically. Typically, the complexity of multivariable models is addressed with specific theoretical comparisons, specific types of variables, randomization, and specific tests based on prior research.

Mediating Variables

A single mediator model represents the addition of a third variable to an X → Y relation so that the causal sequence is modeled such that X causes the mediator, M , and M causes Y , that is, X → M → Y . Mediating variables are central to many fields because they are used to understand the process by which two variables are related. There are two overlapping ways in which mediating variables have been used in prior research: (a) mediation for design where interventions are designed to change a mediating variable and (b) mediation for explanation where mediators are selected after an effect of X to Y has been demonstrated to explain the mediating process by which X affects Y ( MacKinnon, 2008 , Chap. 2). The use of mediating variables for design is central to interventions designed to affect behavior. Intervention studies are based on theory for how the intervention is expected to change mediating variables and the change in the mediating variables is hypothesized to be what causes changes in an outcome variable. If the theory that the mediating constructs are causally related to the outcome is correct, then an intervention that changes the outcome will change the mediator. In an intervention to prevent sexually transmitted diseases, the intervention may be designed to change mediators of abstinence and condom use. In drug prevention research, mediating variables such as social norms, social competence skills, and expectations about drug use are targeted in order to change drug use. Many researchers have stressed the importance of assessing mediation in intervention research ( Baranowski, Anderson, & Carmack, 1998 ; Fraser & Galinsky, 2010 ; Judd & Kenny, 1981a , 1981b ; Kazdin, 2009 ; Kraemer, Wilson, Fairburn, & Agras, 2002 ; MacKinnon, 1994 ; Weiss, 1997 ).

The other major application of mediating variables is after an effect is observed and researchers investigate how this effect occurred. Mediation for explanation has a long history starting with the work of Lazarsfeld and others ( Hyman, 1955 ; Lazarsfeld, 1955 ) whereby observed relations between two variables are elaborated by considering a third variable and one explanation of why the two variables are related is because of mediation. Evaluating mediation to explain an observed effect is probably more susceptible to chance findings than evaluating mediation by design because the mediators in the mediation for design studies are selected before the study and mediators for explanation are usually selected after the study. Most programs of research employ both mediation by design and mediation for explanation approaches ( MacKinnon, 2008 , Chap. 2).

Reasons for including mediating variables in a research study

There are many overlapping reasons for including mediating variables in a research study. Seven reasons are listed below for the case of an intervention study as described elsewhere ( MacKinnon, 1994 , 2008 ; MacKinnon & Luecken, 2011 ).

• Manipulation check: Mediation analyses provide a check on whether the intervention produced a change in the mediating variables it was designed to change (e.g., if the intervention was designed to engender an antitobacco norm, then program effects on norms should be observed). If the program did not change the mediating variable, it is unlikely to have the desired effects on the targeted outcome. Failure to significantly change the mediator may occur because the intervention was unsuccessful, the measurement of the mediating variable was inadequate, or by chance statistical fluctuations.
• Program improvement: Mediation analyses generate information to identify successful and unsuccessful intervention components. If an intervention component did not change the mediating variable, then the actions selected to change the mediating variable need improvement. For example, if no program effects on social norms are found, the intervention may need to reconsider the intervention components used to change norms. If the program increases norms but norms do not affect the outcome, the norms component of the program may be ineffective and/or unnecessary and new mediators may need to be included.
• Measurement improvement: A lack of an intervention effect on a mediator can suggest that the measures of the mediator were not reliable or valid enough to detect changes. If no program effects are found on norms, for example, it may be that the method used to measure norms is not reliable or valid.
• Possibility of delayed program effects: If the intervention does not have the desired effect on the outcome variable but does significantly affect theorized mediating variables, it is possible that effects on outcomes will emerge later after the effects of the mediating variable have accumulated over time. For example, the effects of a norm change intervention to reduce smoking onset among young children may not be evident until the children are older and have more opportunities to smoke.
• Evaluating the process of change: Mediation analysis provides information on the processes by which the intervention achieved its effects on an outcome measure. For example, it is possible to study whether the changes in mediators like norms or another mediator were responsible for intervention effects on smoking.
• Building and refining theory: One of the greatest strengths of including mediating variables is the ability to test the theories upon which intervention programs were based. Many theories are based on results of cross-sectional relations with little or no randomized experimental manipulation. Mediation analysis in the randomized design is ideal for testing theories because it improves causal inference. Competing theories for smoking onset and addiction, for example, may suggest alternative mediating variables that can be tested in an experimental design.
• Practical implications: The majority of intervention programs have limited resources to accomplish their goals. Intervention programs will cost less and provide greater benefits if the critical ingredients of interventions can be identified because critical components can be retained and ineffective components removed. Mediation analyses can help decide whether to discontinue an intervention approach or not by providing information about whether it was a failure of the intervention to change the mediator, called action theory or whether it was a failure of a significant relation of the mediator to the outcome, called conceptual theory, or both.

How to include mediating variables in a research study

There are two major aspects to adding mediating variables to a research study. First is during the planning stage where the theoretical framework of the study and testing theory is considered and often specified in a logic model. In many cases, the development of a logic model may take considerable time and resources because it forces researchers to carefully consider how the intervention components could be reasonably expected to change an outcome. In fact, the most important aspect of considering mediating variables in a research study may be that it forces researchers to think in a concrete manner about how the intervention could be expected to work both in terms of action theory for how the intervention affects the mediators and conceptual theory for which mediators are related to the outcome. The second aspect to adding mediating variables is deciding how to measure theoretical mediating variables. Typically, this requires adding measures to a questionnaire or some other measurement procedure. In many cases, there may not be existing measures of relevant mediating constructs and psychometric work must be done to develop measures of mediating variables. Furthermore, the addition of measures of mediating variables can add to the respondent burden on a questionnaire. Nevertheless, the addition of mediating variable measures may generate practical and theoretical information from a research study. It is important to measure mediating variables in both intervention and control conditions before and after the intervention to ascertain change in the measures and for statistical mediation analysis.

Mediation Regression Equations

The ideas regarding mediating variables can be translated into equations suitable for estimating mediated effects and conducting statistical tests as for the single mediator model for X, M , and Y shown in Figure 1 and defined in Equations 2 and 3 below. Equation 1 is also shown because it provides information for mediation relations and corresponds to the overall intervention effect:

Where X is the independent variable, Y is the outcome variable, and M is the mediating variable; the parameters i 1 , i 2 , and i 3 are intercepts in each equation; and e 1 , e 2 , and e 3 are residuals. In Equation 1 , the coefficient c represents the total effect, that is, the total effect that X can have on Y , the outcome variable. In Equation 2 , the parameter c’ denotes the relation between X and Y controlling for M , representing the direct effect—the effect of X on Y that is adjusted for M , the parameter b denotes the relation between M and Y adjusted for X . Finally, in Equation 3 , the coefficient a denotes the relation between X and M . Equations 2 and 3 are represented in Figure 1 , which shows how the total effect of X on Y is separated into a direct effect relating X to Y and a mediated effect by which X indirectly affects Y through M . Complete mediation is the case where the total effect is completely explained by the mediator, that is, there is no direct effect. In this case, the total effect is equal to the mediated effect (i.e., c = ab ). Partial mediation is the case where the relation between the independent and the outcome variable is not completely accounted for by the mediating variable. There are alternative estimators of the mediated effect including difference in coefficients and product of coefficients, which are based on the regression equations. More on the different approaches to mediation analysis can be found elsewhere including standard errors, confidence limit estimation, multiple mediators, qualitative methods, experimental designs, limitations for causal inference, and categorical outcomes ( MacKinnon, 2008 ).

Assumptions of the Single Mediator Model

Although statistical mediation analysis is straightforward under the assumption that the mediation equations above are correctly specified, the identification of true mediating variables is complicated by several testable and untestable assumptions ( MacKinnon, 2008 ). New developments in mediation analysis address statistical and inferential assumptions of the mediation model. For the estimator of the mediated effect, simultaneous regression analysis assumptions are required, such as the residuals in Equations 2 and 3 are independent and that M and the residual in Equation 2 are independent ( MacKinnon, 2008 ; McDonald, 1997 ). No XM interaction in Equation 2 is assumed, although this can be tested statistically. The temporal order of the variables in the model is also assumed to be correctly specified (see Cheong, MacKinnon, & Khoo, 2003 ; Cole & Maxwell, 2003 ; MacKinnon, 2008 ). The methods assume a self-contained model such that no variables related to the variables in the mediation equations are omitted from the estimated model and that coefficients estimate causal effects ( Holland, 1988 ; Imai, Keele, & Tingley, 2010 ; Lynch, Cary, Gallop, & Ten Have, 2008 ; Ten Have et al., 2007 ; VanderWeele, 2010 ). It is also assumed that the model has minimal errors of measurement ( James & Brett, 1984 ; McDonald, 1997 ).

Moderating Variables

The strength and form of a relation between two variables may depend on the value of a moderating variable. A moderator is a variable that modifies the form or strength of the relation between an independent and a dependent variable. The examination of moderator effects has a long and important history in a variety of research areas ( Aguinis, 2004 ; Aiken & West, 1991 ). Moderator effects are also called interactions because the third variable interacts with the relation between two other variables. However, theoretically moderator effects differ slightly from interaction effects in that moderators refer to variables that alter an observed relation in the population while interaction effects refer to any situation in which the effect of one variable depends on the level of another variable. As mentioned earlier, the moderator is not part of a causal sequence but qualifies the relation between X and Y . For intervention research, moderator variables may reflect subgroups of persons for which the intervention is more or less effective than for other groups. In general, moderator variables are critical for understanding the generalizability of a research finding to subgroups.

The moderator variable can be a continuous or categorical variable, although interpretation of a categorical moderator is usually easier than a continuous moderator. A moderating variable may be a factor in a randomized manipulation, representing random assignment to levels of the factor. For example, participants may be randomly assigned to a moderator of treatment dosage in addition to type of treatment received in order to test the moderator effect of duration of treatment across the two treatments. Moderator variables can be stable aspects of individuals such as sex, race, age, ethnicity, genetic predispositions, and so on. Moderator variables may also be variables that may not change during the period of a research study, such as socioeconomic status, risk-taking tendency, prior health care utilization, impulsivity, and intelligence. Moderator variables may also be environmental contexts such as type of school and geographic location. Moderator variables may also be baseline measure of an outcome or mediating measure such that intervention effects depend on the starting point for each participant. The values of the moderating variable may be latent such as classes of individuals formed by analysis of repeated measures from participants. The important aspect is that the relation between two variables X and Y depends on the value of the moderator variable, although the type of moderator variable, randomized or not, stable characteristic, or changing characteristic often affects interpretation of a moderation analysis. Moderator variables may be specified before a study as a test of theory or they may be investigated after the study in an exploratory search for different relations across subgroups. Although single moderators are described here referring to the situation where the relation between two variables differs across the levels of a third variable, higher-way interactions involving more than one moderator are also possible.

Reasons for including moderating variables in a research study

There are several overlapping reasons for including moderating variables in a research study.

• Acknowledgment of the complexity of behavior: The investigation of moderating variables acknowledges the complexity of behavior, experiences, and relationships. Individuals are not the same. It would be unusual if there were no differences across individuals. This focus on individual versus group effects is more commonly known as the tendency for researchers to be either lumpers or splitters ( Simpson, 1945 ). Lumpers seek to group individuals and focus on how persons are the same. Splitters, in contrast, look for differences among groups. By making this distinction, I guess I am a splitter. Generally, the search for moderators is a goal of splitters while lumpers would tend not to focus on moderator variables but on general results across all persons. Of course both approaches have problems with splitters yielding smaller and smaller groups until there is one person in each group. Lumpers will fail to observe real subgroups, including subgroups that may have iatrogenic effects or miss predictive relations because of opposite effects in subgroups.
• Manipulation check: If there is an additional experimental factor picked so that an observed relation is differentially observed across subgroups, then the intervention effect is a test of the intervention theory. For example, if dose of an intervention is manipulated as well as intervention or control, then the additional dosages could be considered a moderator and if the intervention effect is successful, the size of the effect should differ across levels of dosage.
• Generalizability of results: Moderation analysis provides a way to test whether an intervention has similar effects across groups. It would be important, for example, to demonstrate that intervention effects are obtained for males and females if the program would be disseminated to a whole group containing males and females. Similarly, the consistency of an intervention effect across subgroups demonstrates important information about the generalizability of an intervention.
• Specificity of effects: In contrast to generalizability, it is important to identify groups for which an intervention has its greatest effects or no effects. This information could then be used to target groups for intervention thereby tailoring of an intervention.
• Identification of iatrogenic effects in subgroups: Moderation analysis can be used to identify subgroups for which effects are counterproductive. It is possible that there will be subgroups for which the intervention causes more negative outcomes.
• Investigation of lack of an intervention effect: If there are two groups that are affected by an intervention in opposite ways, the overall effect will be nonsignificant even if there is a statistically significant intervention effect in both groups, albeit opposite. Without investigation of moderating variables, these types of effects would not be observable. In addition, before abandoning an intervention or area of research it is useful to investigate subgroups for any intervention effect. Of course, this type of exploratory search must consider the possibility of multiplicity where by testing more effects will lead to finding effects by chance alone.
• Moderators as a test of theory: There are situations where intervention effects may be theoretically expected in one group and not another. For example, there may be different social tobacco intervention effects for boys versus girls because reasons for smoking may differ across sex. In this way, mediation and moderation may be combined if it is expected that a theoretical mediating process would be present in one group but not in another group.
• Measurement improvement: Lack of a moderating variable effect may be due to poor measurement of the moderator variable. Many moderator variables have reasonably good reliability such as age, sex, and ethnicity but others may have measurement limitations such as risk-taking propensity or impulsivity.
• Practical implications: If moderator effects are found, then decisions about intervention delivery may depend on this information. If intervention effects are positive at all levels of the moderator, then it is reasonable to deliver the whole program. If intervention effects are observed for one group and not another, it may be useful to deliver the program to the group where it had success and develop a new intervention for other groups. Of course, there are cases where the delivery of an intervention as a function of a moderating variable cannot be realistically or ethically used in the delivery of an intervention. For example, it may be realistic to deliver different programs to different ages and sexes but less realistic to deliver programs based on risk taking, impulsivity, or prior drug use, for example, because of labeling of individuals or practical issues in identifying groups. By grouping persons for intervention, there may also be iatrogenic effects, for example, grouping adolescent drug users together may have harmful effects by enhancing a social norm to take drugs in this group.

How to include moderators in a research study

Moderators such as age, sex, and race are often routinely included in surveys. Demographic characteristics are also often measured including family income, marital status, number of siblings, and so on. Other measures of potential moderators have the same measurement and time demand issues as for mediating variables described earlier; that is, additional measures may increase respondent burden.

Moderation Regression Equations

The single moderating variable effect model is shown in Figure 2 and summarized in Equation 4 .

Where Y is the dependent variable, X is the independent variable, Z is the moderator variable, and XZ is the interaction of the moderator and the independent variable; e 1 is a residual, and c 1 , c 2 , and c 3 represent the relation between the dependent variable and the independent variable, moderator variable, and moderator by independent variable interaction, respectively. The moderating variable XZ is the product of X and Z where X and Z are often centered (centered means that the average is subtracted from each observed value of the variable). If the XZ interaction is statistically significant, the source of the significant interaction is often explored by examining conditional effects with contrasts and plots. More on interaction effects including procedures to plot interactions and test contrasts can be found in Aguinis (2004) , Aiken and West (1991) , Keppel and Wickens (2004) , and Rothman, Greenland, and Lash (2008) .

Assumptions of Moderation Analysis

There are several challenges to accurate identification of moderator effects. For example, interactions are often scale dependent so that changing the measurement scale can remove an interaction effect. The range of values of the moderator may affect whether a moderator effect can be detected. The number of moderators tested may increase the chance of finding a Type I error and the splitting of the total sample into subgroups limits power to detect moderator effects. Several types of interaction effects are possible that reflect different conclusions, for example, an intervention effect may be statistically significant and beneficial in each group but the effect may differ, an intervention effect may be statistically significant in one group but not another, and so on. More on these issues can be found in Aiken and West (1991) and Rothman et al. (2008) and a special issue on subgroup analysis in a forthcoming issue of the journal Prevention Science .

Moderation and Mediation in the Same Analysis

Both moderating and mediating variables can be investigated in the same research project but the interpretation of mediation in the presence of moderation can be complex statistically and conceptually ( Baron & Kenny, 1986 ; Edwards & Lambert, 2007 ; Fairchild & MacKinnon, 2009 ; Hayduk & Wonnacott, 1980 ; James & Brett, 1984 ; Preacher, Rucker, & Hayes, 2007 ). There are two major types of effects that combine moderation and mediation as described in the literature ( Baron & Kenny, 1986 ; Fairchild & MacKinnon, 2009 ): (a) moderation of a mediation effect , where the mediated effect is different at different values of a moderator and (b) mediation of a moderation effect , where the effect of an interaction on a dependent variable is mediated.

An example of moderation of a mediation effect is a case where a mediation process differs for males and females. For example, a program may affect social norms equally for both males and females but social norms only significantly reduce subsequent tobacco use for females not for males. These types of analyses can be used to test homogeneity of action theory across groups and homogeneity of conceptual theory across groups ( MacKinnon, 2008 ). An example of moderation of a mediated effect is a case where social norms mediate the effect of an intervention on drug use but the size of the mediated effect differs as a function of risk-taking propensity. An example of mediation of a moderator effect would occur if the effect of an intervention depends on baseline risk-taking propensity such that the interaction is due to a mediating variable of social norms, which then affects drug use. These types of effects are important because they help specify types of subgroups for whom mediational processes differ and help quantify more complicated hypotheses about mediation and moderation relations. Despite the potential for moderation of a mediation effect and mediation of a moderation effect, few research studies include both mediation and moderation, at least in part because of the difficulty of specifying and interpreting these models. General models that include mediation and moderation have been described that include the single mediator model as a special case and the single moderator model as special cases ( Fairchild & MacKinnon, 2009 ; MacKinnon, 2008 ).

Both mediating variables and moderating variables are ideally specified before the study is conducted. Describing mediation and moderation theory clarifies the purpose of the intervention and forces consideration of alternative interpretations of the results of the study leading to better research design and more information gleaned from the study. Stable characteristic moderator variables such as age and sex are often routinely included in research studies. Often existing studies include some measures of moderating and mediating variables so that mediation and moderation analysis of many existing data sets can be conducted. More information can be obtained from these studies if mediation and moderation analyses are conducted.

There are some limitations of including moderating and mediating variables. First, there is the cost and time spent developing mediation and moderation theory prior to a study. It is likely that consideration of these variables may alter the entire design of a study possibly delaying an important research project. However, it is likely that interventions will be more successful if based on theory and prior research and the application of these analyses inform the next intervention study. Second, there are substantial conceptual and statistical challenges to identifying true moderating and mediating variables that require a program of research. The identification of true mediating processes, for example, requires a program of research with information from many sources. Third, the questions added to a survey to measure mediating and moderating variables must be balanced with the quality of data collected. A longer survey that bores participants or renders some or all of their data inaccurate will not help a research project. Nevertheless, the addition of mediating and moderating variables to any research program reflects the maturation of scientific research to addressing the specifics of how and for whom interventions achieve their effects.

Acknowledgments

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by Public Health Service Grant DA0957.

This article was previously presented at the Stockholm Conference on Outcome Studies of Social, Behavioral, and Educational Interventions, on February 7, 2011. It was invited and accepted at the discretion of the editor.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Research Variables 101

Independent variables, dependent variables, control variables and more

By: Derek Jansen (MBA) | Expert Reviewed By: Kerryn Warren (PhD) | January 2023

If you’re new to the world of research, especially scientific research, you’re bound to run into the concept of variables , sooner or later. If you’re feeling a little confused, don’t worry – you’re not the only one! Independent variables, dependent variables, confounding variables – it’s a lot of jargon. In this post, we’ll unpack the terminology surrounding research variables using straightforward language and loads of examples .

Overview: Variables In Research

What (exactly) is a variable?

The simplest way to understand a variable is as any characteristic or attribute that can experience change or vary over time or context – hence the name “variable”. For example, the dosage of a particular medicine could be classified as a variable, as the amount can vary (i.e., a higher dose or a lower dose). Similarly, gender, age or ethnicity could be considered demographic variables, because each person varies in these respects.

Within research, especially scientific research, variables form the foundation of studies, as researchers are often interested in how one variable impacts another, and the relationships between different variables. For example:

• How someone’s age impacts their sleep quality
• How different teaching methods impact learning outcomes
• How diet impacts weight (gain or loss)

As you can see, variables are often used to explain relationships between different elements and phenomena. In scientific studies, especially experimental studies, the objective is often to understand the causal relationships between variables. In other words, the role of cause and effect between variables. This is achieved by manipulating certain variables while controlling others – and then observing the outcome. But, we’ll get into that a little later…

The “Big 3” Variables

Variables can be a little intimidating for new researchers because there are a wide variety of variables, and oftentimes, there are multiple labels for the same thing. To lay a firm foundation, we’ll first look at the three main types of variables, namely:

• Independent variables (IV)
• Dependant variables (DV)
• Control variables

What is an independent variable?

Simply put, the independent variable is the “ cause ” in the relationship between two (or more) variables. In other words, when the independent variable changes, it has an impact on another variable.

For example:

• Increasing the dosage of a medication (Variable A) could result in better (or worse) health outcomes for a patient (Variable B)
• Changing a teaching method (Variable A) could impact the test scores that students earn in a standardised test (Variable B)
• Varying one’s diet (Variable A) could result in weight loss or gain (Variable B).

It’s useful to know that independent variables can go by a few different names, including, explanatory variables (because they explain an event or outcome) and predictor variables (because they predict the value of another variable). Terminology aside though, the most important takeaway is that independent variables are assumed to be the “cause” in any cause-effect relationship. As you can imagine, these types of variables are of major interest to researchers, as many studies seek to understand the causal factors behind a phenomenon.

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What is a dependent variable?

While the independent variable is the “ cause ”, the dependent variable is the “ effect ” – or rather, the affected variable . In other words, the dependent variable is the variable that is assumed to change as a result of a change in the independent variable.

Keeping with the previous example, let’s look at some dependent variables in action:

• Health outcomes (DV) could be impacted by dosage changes of a medication (IV)
• Students’ scores (DV) could be impacted by teaching methods (IV)
• Weight gain or loss (DV) could be impacted by diet (IV)

In scientific studies, researchers will typically pay very close attention to the dependent variable (or variables), carefully measuring any changes in response to hypothesised independent variables. This can be tricky in practice, as it’s not always easy to reliably measure specific phenomena or outcomes – or to be certain that the actual cause of the change is in fact the independent variable.

As the adage goes, correlation is not causation . In other words, just because two variables have a relationship doesn’t mean that it’s a causal relationship – they may just happen to vary together. For example, you could find a correlation between the number of people who own a certain brand of car and the number of people who have a certain type of job. Just because the number of people who own that brand of car and the number of people who have that type of job is correlated, it doesn’t mean that owning that brand of car causes someone to have that type of job or vice versa. The correlation could, for example, be caused by another factor such as income level or age group, which would affect both car ownership and job type.

To confidently establish a causal relationship between an independent variable and a dependent variable (i.e., X causes Y), you’ll typically need an experimental design , where you have complete control over the environmen t and the variables of interest. But even so, this doesn’t always translate into the “real world”. Simply put, what happens in the lab sometimes stays in the lab!

As an alternative to pure experimental research, correlational or “ quasi-experimental ” research (where the researcher cannot manipulate or change variables) can be done on a much larger scale more easily, allowing one to understand specific relationships in the real world. These types of studies also assume some causality between independent and dependent variables, but it’s not always clear. So, if you go this route, you need to be cautious in terms of how you describe the impact and causality between variables and be sure to acknowledge any limitations in your own research.

What is a control variable?

In an experimental design, a control variable (or controlled variable) is a variable that is intentionally held constant to ensure it doesn’t have an influence on any other variables. As a result, this variable remains unchanged throughout the course of the study. In other words, it’s a variable that’s not allowed to vary – tough life 🙂

As we mentioned earlier, one of the major challenges in identifying and measuring causal relationships is that it’s difficult to isolate the impact of variables other than the independent variable. Simply put, there’s always a risk that there are factors beyond the ones you’re specifically looking at that might be impacting the results of your study. So, to minimise the risk of this, researchers will attempt (as best possible) to hold other variables constant . These factors are then considered control variables.

Some examples of variables that you may need to control include:

• Temperature
• Time of day
• Noise or distractions

Which specific variables need to be controlled for will vary tremendously depending on the research project at hand, so there’s no generic list of control variables to consult. As a researcher, you’ll need to think carefully about all the factors that could vary within your research context and then consider how you’ll go about controlling them. A good starting point is to look at previous studies similar to yours and pay close attention to which variables they controlled for.

Of course, you won’t always be able to control every possible variable, and so, in many cases, you’ll just have to acknowledge their potential impact and account for them in the conclusions you draw. Every study has its limitations , so don’t get fixated or discouraged by troublesome variables. Nevertheless, always think carefully about the factors beyond what you’re focusing on – don’t make assumptions!

Other types of variables

As we mentioned, independent, dependent and control variables are the most common variables you’ll come across in your research, but they’re certainly not the only ones you need to be aware of. Next, we’ll look at a few “secondary” variables that you need to keep in mind as you design your research.

• Moderating variables
• Mediating variables
• Confounding variables
• Latent variables

Let’s jump into it…

What is a moderating variable?

A moderating variable is a variable that influences the strength or direction of the relationship between an independent variable and a dependent variable. In other words, moderating variables affect how much (or how little) the IV affects the DV, or whether the IV has a positive or negative relationship with the DV (i.e., moves in the same or opposite direction).

For example, in a study about the effects of sleep deprivation on academic performance, gender could be used as a moderating variable to see if there are any differences in how men and women respond to a lack of sleep. In such a case, one may find that gender has an influence on how much students’ scores suffer when they’re deprived of sleep.

It’s important to note that while moderators can have an influence on outcomes , they don’t necessarily cause them ; rather they modify or “moderate” existing relationships between other variables. This means that it’s possible for two different groups with similar characteristics, but different levels of moderation, to experience very different results from the same experiment or study design.

What is a mediating variable?

Mediating variables are often used to explain the relationship between the independent and dependent variable (s). For example, if you were researching the effects of age on job satisfaction, then education level could be considered a mediating variable, as it may explain why older people have higher job satisfaction than younger people – they may have more experience or better qualifications, which lead to greater job satisfaction.

Mediating variables also help researchers understand how different factors interact with each other to influence outcomes. For instance, if you wanted to study the effect of stress on academic performance, then coping strategies might act as a mediating factor by influencing both stress levels and academic performance simultaneously. For example, students who use effective coping strategies might be less stressed but also perform better academically due to their improved mental state.

In addition, mediating variables can provide insight into causal relationships between two variables by helping researchers determine whether changes in one factor directly cause changes in another – or whether there is an indirect relationship between them mediated by some third factor(s). For instance, if you wanted to investigate the impact of parental involvement on student achievement, you would need to consider family dynamics as a potential mediator, since it could influence both parental involvement and student achievement simultaneously.

What is a confounding variable?

A confounding variable (also known as a third variable or lurking variable ) is an extraneous factor that can influence the relationship between two variables being studied. Specifically, for a variable to be considered a confounding variable, it needs to meet two criteria:

• It must be correlated with the independent variable (this can be causal or not)
• It must have a causal impact on the dependent variable (i.e., influence the DV)

Some common examples of confounding variables include demographic factors such as gender, ethnicity, socioeconomic status, age, education level, and health status. In addition to these, there are also environmental factors to consider. For example, air pollution could confound the impact of the variables of interest in a study investigating health outcomes.

Naturally, it’s important to identify as many confounding variables as possible when conducting your research, as they can heavily distort the results and lead you to draw incorrect conclusions . So, always think carefully about what factors may have a confounding effect on your variables of interest and try to manage these as best you can.

What is a latent variable?

Latent variables are unobservable factors that can influence the behaviour of individuals and explain certain outcomes within a study. They’re also known as hidden or underlying variables , and what makes them rather tricky is that they can’t be directly observed or measured . Instead, latent variables must be inferred from other observable data points such as responses to surveys or experiments.

For example, in a study of mental health, the variable “resilience” could be considered a latent variable. It can’t be directly measured , but it can be inferred from measures of mental health symptoms, stress, and coping mechanisms. The same applies to a lot of concepts we encounter every day – for example:

• Emotional intelligence
• Quality of life
• Business confidence
• Ease of use

One way in which we overcome the challenge of measuring the immeasurable is latent variable models (LVMs). An LVM is a type of statistical model that describes a relationship between observed variables and one or more unobserved (latent) variables. These models allow researchers to uncover patterns in their data which may not have been visible before, thanks to their complexity and interrelatedness with other variables. Those patterns can then inform hypotheses about cause-and-effect relationships among those same variables which were previously unknown prior to running the LVM. Powerful stuff, we say!

Let’s recap

In the world of scientific research, there’s no shortage of variable types, some of which have multiple names and some of which overlap with each other. In this post, we’ve covered some of the popular ones, but remember that this is not an exhaustive list .

To recap, we’ve explored:

• Independent variables (the “cause”)
• Dependent variables (the “effect”)
• Control variables (the variable that’s not allowed to vary)

If you’re still feeling a bit lost and need a helping hand with your research project, check out our 1-on-1 coaching service , where we guide you through each step of the research journey. Also, be sure to check out our free dissertation writing course and our collection of free, fully-editable chapter templates .

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Moderator Variable

A moderator variable , commonly denoted as just M, is a third variable that affects the strength of the relationship between a dependent and independent variable. In correlation , a moderator is a third variable that affects the correlation of two variables.  In a causal relationship, if x is the predictor variable and y is an outcome variable, then z is the moderator that affects the casual relationship of x and y.  Most of the moderator variables measure causal relationship using regression coefficient.  The moderator, if found to be significant, can cause an amplifying or weakening effect between x and y.  In ANOVA , the moderator variable effect is represented by the interaction effect between the dependent variable and the factor variable.

Questions Answered:

Does gender effectively moderate the relationship between desire to marry and attitudes of marriage?

Does Z treatment effect the impact of X drug onto Y symptoms?

Moderated regression analysis

This is a regression based technique that is used to identify the moderator.  To explain how MRA technique works, we can use the following example:

In this equation, if  (the interaction between the independent variable and moderator) is not statistically significant, then Z is not a moderator, it is just an independent variable.  If is statistically significant, then Z will be a moderator, and thus moderation is supported.

Linear vs. non-linear measurement

In a regression equation, when the relationship between the dependent variable and the independent variable is linear, then the dependent variable may change when the value of the moderator changes.  In a linear relationship, the following equation is used to represent the effect:

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In this equation, the relationship is linear and represents the interaction effect of the moderator and the independent variable.  When the relationship is non-linear, the following equation shows the effect of the moderator variable effect:

In this equation, the relationship between the dependent and the independent variable is non-linear, so and shows the interaction effect. In a repeated measure design moderator, the variable can also be used.  In multi-level modeling, if a variable predicts the effect size , that variable is called the moderator.

• Alternative: In a non-linear relationship, a significant value of a moderator does not prove the true moderator effect.  Unless the moderator is a manipulated variable, we cannot say if the moderator variable is a true moderator or if it is just used as a proxy.
• Level of measurement: The moderator is an independent variable that is used to measure the causal relationship.  Like other independent variables, it may be categorized or continuous.

Assumptions

• Causal assumption: When x variable is not randomized, then causation must be assumed.  The moderator can reversely effect the causation, if the causation between x and y is not presumed.
• Causal variable relationship: The moderator variable and independent variable, in principal, should not be related.  No special interpretation can be found between a correlated independent and moderator variable.  However, they should not be too highly correlated, otherwise, estimation problems may occur.  The moderator must be related to the dependent variable.
• Measurement: Usually, the moderation effect is represented by the interaction effect between the the dependent and independent variable.  In a multiple regression equation, the moderator is as follows:

In this equation, the interaction effect between X and Z measures the moderation effect.  Typically, if there is no significant relationship on the dependent variable from the interaction between the moderator and independent variable, moderation is not supported.

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• Multiple Regression
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Impact of audit committee characteristics on risk disclosure: evidence from the banking sector of Pakistan

• Original Article
• Published: 21 August 2024

Cite this article

• Awais Akbar 1 ,
• Shumaila Zeb 1 &
• Hassan Zada   ORCID: orcid.org/0000-0003-3347-2867 1  

Risk disclosures are crucial for a robust corporate governance framework. It permits shareholders to assess the financial health of banks by understanding various risks (credit, market, operational, etc.). This study investigated the impact of audit committee characteristics such as the size, meetings, and expertise of audit committee members on risk disclosures. For this purpose, we collected data from 20 commercial banks over 17 years, from 2006 to 2022, listed on the Pakistan Stock Exchange. We employed panel data analysis by incorporating both period- and firm-fixed effects, providing a clearer picture of risk disclosure across periods and cross sections. We find a positive and significant impact of the expertise of the audit committee members on the levels of transparency and adequacy of risk disclosure. However, the study also reveals that risk disclosure tends to decrease with increased audit committee size. The study also finds that the Bank of Punjab has the highest risk disclosure, while Habib Bank has the lowest disclosure. Additionally, the period effects show that banks disclosed the highest level of risk in 2020, whereas in 2007, banks provided the least risk disclosure. This study enhances the level of risk disclosure in the banking sector of Pakistan. It also reduces information asymmetry between management and shareholders by strengthening the audit committee and explaining changes across risk disclosure. The findings of this study are helpful for bank BODs in formulating and appointing effective audit committee boards in line with the factors that have been shown to impact risk disclosure significantly. Other sectors can also improve risk disclosure practices by enhancing audit committee expertise and managing committee size, leading to better transparency and stakeholder trust.

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Akbar, A., Zeb, S. & Zada, H. Impact of audit committee characteristics on risk disclosure: evidence from the banking sector of Pakistan. Int J Discl Gov (2024). https://doi.org/10.1057/s41310-024-00263-2

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Received : 11 June 2024

Accepted : 12 August 2024

Published : 21 August 2024

DOI : https://doi.org/10.1057/s41310-024-00263-2

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