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Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

Power & Sample Size Calculator

Use this advanced sample size calculator to calculate the sample size required for a one-sample statistic, or for differences between two proportions or means (two independent samples). More than two groups supported for binomial data. Calculate power given sample size, alpha, and the minimum detectable effect (MDE, minimum effect of interest).

Experimental design

Data parameters

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• Using the power & sample size calculator

Parameters for sample size and power calculations

Calculator output.

• Why is sample size determination important?
• What is statistical power?

Post-hoc power (Observed power)

• Sample size formula
• Types of null and alternative hypotheses in significance tests
• Absolute versus relative difference and why it matters for sample size determination

    Using the power & sample size calculator

This calculator allows the evaluation of different statistical designs when planning an experiment (trial, test) which utilizes a Null-Hypothesis Statistical Test to make inferences. It can be used both as a sample size calculator and as a statistical power calculator . Usually one would determine the sample size required given a particular power requirement, but in cases where there is a predetermined sample size one can instead calculate the power for a given effect size of interest.

1. Number of test groups. The sample size calculator supports experiments in which one is gathering data on a single sample in order to compare it to a general population or known reference value (one-sample), as well as ones where a control group is compared to one or more treatment groups ( two-sample, k-sample ) in order to detect differences between them. For comparing more than one treatment group to a control group the sample size adjustments based on the Dunnett's correction are applied. These are only approximately accurate and subject to the assumption of about equal effect size in all k groups, and can only support equal sample sizes in all groups and the control. Power calculations are not currently supported for more than one treatment group due to their complexity.

2. Type of outcome . The outcome of interest can be the absolute difference of two proportions (binomial data, e.g. conversion rate or event rate), the absolute difference of two means (continuous data, e.g. height, weight, speed, time, revenue, etc.), or the relative difference between two proportions or two means (percent difference, percent change, etc.). See Absolute versus relative difference for additional information. One can also calculate power and sample size for the mean of just a single group. The sample size and power calculator uses the Z-distribution (normal distribution) .

3. Baseline The baseline mean (mean under H 0 ) is the number one would expect to see if all experiment participants were assigned to the control group. It is the mean one expects to observe if the treatment has no effect whatsoever.

4. Minimum Detectable Effect . The minimum effect of interest, which is often called the minimum detectable effect ( MDE , but more accurately: MRDE, minimum reliably detectable effect) should be a difference one would not like to miss , if it existed. It can be entered as a proportion (e.g. 0.10) or as percentage (e.g. 10%). It is always relative to the mean/proportion under H 0 ± the superiority/non-inferiority or equivalence margin. For example, if the baseline mean is 10 and there is a superiority alternative hypothesis with a superiority margin of 1 and the minimum effect of interest relative to the baseline is 3, then enter an MDE of 2 , since the MDE plus the superiority margin will equal exactly 3. In this case the MDE (MRDE) is calculated relative to the baseline plus the superiority margin, as it is usually more intuitive to be interested in that value.

If entering means data, one needs to specify the mean under the null hypothesis (worst-case scenario for a composite null) and the standard deviation of the data (for a known population or estimated from a sample).

5. Type of alternative hypothesis . The calculator supports superiority , non-inferiority and equivalence alternative hypotheses. When the superiority or non-inferiority margin is zero, it becomes a classical left or right sided hypothesis, if it is larger than zero then it becomes a true superiority / non-inferiority design. The equivalence margin cannot be zero. See Types of null and alternative hypothesis below for an in-depth explanation.

6. Acceptable error rates . The type I error rate, α , should always be provided. Power, calculated as 1 - β , where β is the type II error rate, is only required when determining sample size. For an in-depth explanation of power see What is statistical power below. The type I error rate is equivalent to the significance threshold if one is doing p-value calculations and to the confidence level if using confidence intervals.

The sample size calculator will output the sample size of the single group or of all groups, as well as the total sample size required. If used to solve for power it will output the power as a proportion and as a percentage.

    Why is sample size determination important?

While this online software provides the means to determine the sample size of a test, it is of great importance to understand the context of the question, the "why" of it all.

Estimating the required sample size before running an experiment that will be judged by a statistical test (a test of significance, confidence interval, etc.) allows one to:

• determine the sample size needed to detect an effect of a given size with a given probability
• be aware of the magnitude of the effect that can be detected with a certain sample size and power
• calculate the power for a given sample size and effect size of interest

This is crucial information with regards to making the test cost-efficient. Having a proper sample size can even mean the difference between conducting the experiment or postponing it for when one can afford a sample of size that is large enough to ensure a high probability to detect an effect of practical significance.

For example, if a medical trial has low power, say less than 80% (β = 0.2) for a given minimum effect of interest, then it might be unethical to conduct it due to its low probability of rejecting the null hypothesis and establishing the effectiveness of the treatment. Similarly, for experiments in physics, psychology, economics, marketing, conversion rate optimization, etc. Balancing the risks and rewards and assuring the cost-effectiveness of an experiment is a task that requires juggling with the interests of many stakeholders which is well beyond the scope of this text.

    What is statistical power?

Statistical power is the probability of rejecting a false null hypothesis with a given level of statistical significance , against a particular alternative hypothesis. Alternatively, it can be said to be the probability to detect with a given level of significance a true effect of a certain magnitude. This is what one gets when using the tool in "power calculator" mode. Power is closely related with the type II error rate: β, and it is always equal to (1 - β). In a probability notation the type two error for a given point alternative can be expressed as [1] :

β(T α ; μ 1 ) = P(d(X) ≤ c α ; μ = μ 1 )

It should be understood that the type II error rate is calculated at a given point, signified by the presence of a parameter for the function of beta. Similarly, such a parameter is present in the expression for power since POW = 1 - β [1] :

POW(T α ; μ 1 ) = P(d(X) > c α ; μ = μ 1 )

In the equations above c α represents the critical value for rejecting the null (significance threshold), d(X) is a statistical function of the parameter of interest - usually a transformation to a standardized score, and μ 1 is a specific value from the space of the alternative hypothesis.

One can also calculate and plot the whole power function, getting an estimate of the power for many different alternative hypotheses. Due to the S-shape of the function, power quickly rises to nearly 100% for larger effect sizes, while it decreases more gradually to zero for smaller effect sizes. Such a power function plot is not yet supported by our statistical software, but one can calculate the power at a few key points (e.g. 10%, 20% ... 90%, 100%) and connect them for a rough approximation.

Statistical power is directly and inversely related to the significance threshold. At the zero effect point for a simple superiority alternative hypothesis power is exactly 1 - α as can be easily demonstrated with our power calculator. At the same time power is positively related to the number of observations, so increasing the sample size will increase the power for a given effect size, assuming all other parameters remain the same.

Power calculations can be useful even after a test has been completed since failing to reject the null can be used as an argument for the null and against particular alternative hypotheses to the extent to which the test had power to reject them. This is more explicitly defined in the severe testing concept proposed by Mayo & Spanos (2006).

Computing observed power is only useful if there was no rejection of the null hypothesis and one is interested in estimating how probative the test was towards the null . It is absolutely useless to compute post-hoc power for a test which resulted in a statistically significant effect being found [5] . If the effect is significant, then the test had enough power to detect it. In fact, there is a 1 to 1 inverse relationship between observed power and statistical significance, so one gains nothing from calculating post-hoc power, e.g. a test planned for α = 0.05 that passed with a p-value of just 0.0499 will have exactly 50% observed power (observed β = 0.5).

I strongly encourage using this power and sample size calculator to compute observed power in the former case, and strongly discourage it in the latter.

    Sample size formula

The formula for calculating the sample size of a test group in a one-sided test of absolute difference is:

where Z 1-α is the Z-score corresponding to the selected statistical significance threshold α , Z 1-β is the Z-score corresponding to the selected statistical power 1-β , σ is the known or estimated standard deviation, and δ is the minimum effect size of interest. The standard deviation is estimated analytically in calculations for proportions, and empirically from the raw data for other types of means.

The formula applies to single sample tests as well as to tests of absolute difference between two samples. A proprietary modification is employed when calculating the required sample size in a test of relative difference . This modification has been extensively tested under a variety of scenarios through simulations.

    Types of null and alternative hypotheses in significance tests

When doing sample size calculations, it is important that the null hypothesis (H 0 , the hypothesis being tested) and the alternative hypothesis is (H 1 ) are well thought out. The test can reject the null or it can fail to reject it. Strictly logically speaking it cannot lead to acceptance of the null or to acceptance of the alternative hypothesis. A null hypothesis can be a point one - hypothesizing that the true value is an exact point from the possible values, or a composite one: covering many possible values, usually from -∞ to some value or from some value to +∞. The alternative hypothesis can also be a point one or a composite one.

In a Neyman-Pearson framework of NHST (Null-Hypothesis Statistical Test) the alternative should exhaust all values that do not belong to the null, so it is usually composite. Below is an illustration of some possible combinations of null and alternative statistical hypotheses: superiority, non-inferiority, strong superiority (margin > 0), equivalence.

All of these are supported in our power and sample size calculator.

Careful consideration has to be made when deciding on a non-inferiority margin, superiority margin or an equivalence margin . Equivalence trials are sometimes used in clinical trials where a drug can be performing equally (within some bounds) to an existing drug but can still be preferred due to less or less severe side effects, cheaper manufacturing, or other benefits, however, non-inferiority designs are more common. Similar cases exist in disciplines such as conversion rate optimization [2] and other business applications where benefits not measured by the primary outcome of interest can influence the adoption of a given solution. For equivalence tests it is assumed that they will be evaluated using a two one-sided t-tests (TOST) or z-tests, or confidence intervals.

Note that our calculator does not support the schoolbook case of a point null and a point alternative, nor a point null and an alternative that covers all the remaining values. This is since such cases are non-existent in experimental practice [3][4] . The only two-sided calculation is for the equivalence alternative hypothesis, all other calculations are one-sided (one-tailed) .

    Absolute versus relative difference and why it matters for sample size determination

When using a sample size calculator it is important to know what kind of inference one is looking to make: about the absolute or about the relative difference, often called percent effect, percentage effect, relative change, percent lift, etc. Where the fist is μ 1 - μ the second is μ 1 -μ / μ or μ 1 -μ / μ x 100 (%). The division by μ is what adds more variance to such an estimate, since μ is just another variable with random error, therefore a test for relative difference will require larger sample size than a test for absolute difference. Consequently, if sample size is fixed, there will be less power for the relative change equivalent to any given absolute change.

For the above reason it is important to know and state beforehand if one is going to be interested in percentage change or if absolute change is of primary interest. Then it is just a matter of fliping a radio button.

    References

1 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science . The Netherlands: Elsevier.

2 Georgiev G.Z. (2017) "The Case for Non-Inferiority A/B Tests", [online] https://blog.analytics-toolkit.com/2017/case-non-inferiority-designs-ab-testing/ (accessed May 7, 2018)

3 Georgiev G.Z. (2017) "One-tailed vs Two-tailed Tests of Significance in A/B Testing", [online] https://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ (accessed May 7, 2018)

4 Hyun-Chul Cho Shuzo Abe (2013) "Is two-tailed testing for directional research hypotheses tests legitimate?", Journal of Business Research 66:1261-1266

5 Lakens D. (2014) "Observed power, and what to do if your editor asks for post-hoc power analyses" [online] http://daniellakens.blogspot.bg/2014/12/observed-power-and-what-to-do-if-your.html (accessed May 7, 2018)

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Sample Size Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/power-sample-size-calculator.php URL [Accessed Date: 17 Apr, 2024].

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Statistical Power Calculator using the t-distribution*

*plot adapted from behavioral research data analysis with r.

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Power of a Hypothesis Test

The probability of not committing a Type II error is called the power of a hypothesis test.

Effect Size

To compute the power of the test, one offers an alternative view about the "true" value of the population parameter, assuming that the null hypothesis is false. The effect size is the difference between the true value and the value specified in the null hypothesis.

Effect size = True value - Hypothesized value

For example, suppose the null hypothesis states that a population mean is equal to 100. A researcher might ask: What is the probability of rejecting the null hypothesis if the true population mean is equal to 90? In this example, the effect size would be 90 - 100, which equals -10.

Factors That Affect Power

The power of a hypothesis test is affected by three factors.

• Sample size ( n ). Other things being equal, the greater the sample size, the greater the power of the test.
• Significance level (α). The lower the significance level, the lower the power of the test. If you reduce the significance level (e.g., from 0.05 to 0.01), the region of acceptance gets bigger. As a result, you are less likely to reject the null hypothesis. This means you are less likely to reject the null hypothesis when it is false, so you are more likely to make a Type II error. In short, the power of the test is reduced when you reduce the significance level; and vice versa.
• The "true" value of the parameter being tested. The greater the difference between the "true" value of a parameter and the value specified in the null hypothesis, the greater the power of the test. That is, the greater the effect size, the greater the power of the test.

Test Your Understanding

Other things being equal, which of the following actions will reduce the power of a hypothesis test?

I. Increasing sample size. II. Changing the significance level from 0.01 to 0.05. III. Increasing beta, the probability of a Type II error.

(A) I only (B) II only (C) III only (D) All of the above (E) None of the above

The correct answer is (C). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Changing the significance level from 0.01 to 0.05 makes the region of acceptance smaller, which makes the hypothesis test more likely to reject the null hypothesis, thus increasing the power of the test. Since, by definition, power is equal to one minus beta, the power of a test will get smaller as beta gets bigger.

Suppose a researcher conducts an experiment to test a hypothesis. If she doubles her sample size, which of the following will increase?

I. The power of the hypothesis test. II. The effect size of the hypothesis test. III. The probability of making a Type II error.

The correct answer is (A). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Thus, it increases the power of the test. The effect size is not affected by sample size. And the probability of making a Type II error gets smaller, not bigger, as sample size increases.

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S.5 power analysis, why is power analysis important section  .

Consider a research experiment where the p -value computed from the data was 0.12. As a result, one would fail to reject the null hypothesis because this p -value is larger than \(\alpha\) = 0.05. However, there still exist two possible cases for which we failed to reject the null hypothesis:

• the null hypothesis is a reasonable conclusion,
• the sample size is not large enough to either accept or reject the null hypothesis, i.e., additional samples might provide additional evidence.

Power analysis is the procedure that researchers can use to determine if the test contains enough power to make a reasonable conclusion. From another perspective power analysis can also be used to calculate the number of samples required to achieve a specified level of power.

Example S.5.1

Let's take a look at an example that illustrates how to compute the power of the test.

Let X denote the height of randomly selected Penn State students. Assume that X is normally distributed with unknown mean \(\mu\) and a standard deviation of 9. Take a random sample of n = 25 students, so that, after setting the probability of committing a Type I error at \(\alpha = 0.05\), we can test the null hypothesis \(H_0: \mu = 170\) against the alternative hypothesis that \(H_A: \mu > 170\).

What is the power of the hypothesis test if the true population mean were \(\mu = 175\)?

\[\begin{align}z&=\frac{\bar{x}-\mu}{\sigma / \sqrt{n}} \\ \bar{x}&= \mu + z \left(\frac{\sigma}{\sqrt{n}}\right) \\ \bar{x}&=170+1.645\left(\frac{9}{\sqrt{25}}\right) \\ &=172.961\\ \end{align}\]

So we should reject the null hypothesis when the observed sample mean is 172.961 or greater:

\[\begin{align}\text{Power}&=P(\bar{x} \ge 172.961 \text{ when } \mu =175)\\ &=P\left(z \ge \frac{172.961-175}{9/\sqrt{25}} \right)\\ &=P(z \ge -1.133)\\ &= 0.8713\\ \end{align}\]

and illustrated below:

In summary, we have determined that we have an 87.13% chance of rejecting the null hypothesis \(H_0: \mu = 170\) in favor of the alternative hypothesis \(H_A: \mu > 170\) if the true unknown population mean is, in reality, \(\mu = 175\).

Calculating Sample Size Section  

If the sample size is fixed, then decreasing Type I error \(\alpha\) will increase Type II error \(\beta\). If one wants both to decrease, then one has to increase the sample size.

To calculate the smallest sample size needed for specified \(\alpha\), \(\beta\), \(\mu_a\), then (\(\mu_a\) is the likely value of \(\mu\) at which you want to evaluate the power.

Let's investigate by returning to our previous example.

Example S.5.2

Let X denote the height of randomly selected Penn State students. Assume that X is normally distributed with unknown mean \(\mu\) and standard deviation 9. We are interested in testing at \(\alpha = 0.05\) level , the null hypothesis \(H_0: \mu = 170\) against the alternative hypothesis that \(H_A: \mu > 170\).

Find the sample size n that is necessary to achieve 0.90 power at the alternative μ = 175.

\[\begin{align}n&= \dfrac{\sigma^2(Z_{\alpha}+Z_{\beta})^2}{(\mu_0−\mu_a)^2}\\ &=\dfrac{9^2 (1.645 + 1.28)^2}{(170-175)^2}\\ &=27.72\\ n&=28\\ \end{align}\]

In summary, you should see how power analysis is very important so that we are able to make the correct decision when the data indicate that one cannot reject the null hypothesis. You should also see how power analysis can also be used to calculate the minimum sample size required to detect a difference that meets the needs of your research.

Statistical Power Calculator

The statistical power is a power of a binary hypothesis test. It is the probability that effectively rejects the null hypothesis value (H 0 ) when the alternative hypothesis value (H 1 ) is true. In this calculator, calculate the statistical power of a test (p = 1 - β) from the beta value.

Null Hypothesis Test

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11.8: Effect Size, Sample Size and Power

• Last updated
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• Page ID 8245

• Danielle Navarro
• University of New South Wales

In previous sections I’ve emphasised the fact that the major design principle behind statistical hypothesis testing is that we try to control our Type I error rate. When we fix α=.05 we are attempting to ensure that only 5% of true null hypotheses are incorrectly rejected. However, this doesn’t mean that we don’t care about Type II errors. In fact, from the researcher’s perspective, the error of failing to reject the null when it is actually false is an extremely annoying one. With that in mind, a secondary goal of hypothesis testing is to try to minimise β, the Type II error rate, although we don’t usually talk in terms of minimising Type II errors. Instead, we talk about maximising the power of the test. Since power is defined as 1−β, this is the same thing.

power function

Let’s take a moment to think about what a Type II error actually is. A Type II error occurs when the alternative hypothesis is true, but we are nevertheless unable to reject the null hypothesis. Ideally, we’d be able to calculate a single number β that tells us the Type II error rate, in the same way that we can set α=.05 for the Type I error rate. Unfortunately, this is a lot trickier to do. To see this, notice that in my ESP study the alternative hypothesis actually corresponds to lots of possible values of θ. In fact, the alternative hypothesis corresponds to every value of θ except 0.5. Let’s suppose that the true probability of someone choosing the correct response is 55% (i.e., θ=.55). If so, then the true sampling distribution for X is not the same one that the null hypothesis predicts: the most likely value for X is now 55 out of 100. Not only that, the whole sampling distribution has now shifted, as shown in Figure 11.4. The critical regions, of course, do not change: by definition, the critical regions are based on what the null hypothesis predicts. What we’re seeing in this figure is the fact that when the null hypothesis is wrong, a much larger proportion of the sampling distribution distribution falls in the critical region. And of course that’s what should happen: the probability of rejecting the null hypothesis is larger when the null hypothesis is actually false! However θ=.55 is not the only possibility consistent with the alternative hypothesis. Let’s instead suppose that the true value of θ is actually 0.7. What happens to the sampling distribution when this occurs? The answer, shown in Figure 11.5, is that almost the entirety of the sampling distribution has now moved into the critical region. Therefore, if θ=0.7 the probability of us correctly rejecting the null hypothesis (i.e., the power of the test) is much larger than if θ=0.55. In short, while θ=.55 and θ=.70 are both part of the alternative hypothesis, the Type II error rate is different.

What all this means is that the power of a test (i.e., 1−β) depends on the true value of θ. To illustrate this, I’ve calculated the expected probability of rejecting the null hypothesis for all values of θ, and plotted it in Figure 11.6. This plot describes what is usually called the power function of the test. It’s a nice summary of how good the test is, because it actually tells you the power (1−β) for all possible values of θ. As you can see, when the true value of θ is very close to 0.5, the power of the test drops very sharply, but when it is further away, the power is large.

Effect size

Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned with mice when there are tigers abroad – George Box 1976

The plot shown in Figure 11.6 captures a fairly basic point about hypothesis testing. If the true state of the world is very different from what the null hypothesis predicts, then your power will be very high; but if the true state of the world is similar to the null (but not identical) then the power of the test is going to be very low. Therefore, it’s useful to be able to have some way of quantifying how “similar” the true state of the world is to the null hypothesis. A statistic that does this is called a measure of effect size (e.g. Cohen 1988; Ellis 2010). Effect size is defined slightly differently in different contexts, 165 (and so this section just talks in general terms) but the qualitative idea that it tries to capture is always the same: how big is the difference between the true population parameters, and the parameter values that are assumed by the null hypothesis? In our ESP example, if we let θ 0 =0.5 denote the value assumed by the null hypothesis, and let θ denote the true value, then a simple measure of effect size could be something like the difference between the true value and null (i.e., θ−θ 0 ), or possibly just the magnitude of this difference, abs(θ−θ 0 ).

Why calculate effect size? Let’s assume that you’ve run your experiment, collected the data, and gotten a significant effect when you ran your hypothesis test. Isn’t it enough just to say that you’ve gotten a significant effect? Surely that’s the point of hypothesis testing? Well, sort of. Yes, the point of doing a hypothesis test is to try to demonstrate that the null hypothesis is wrong, but that’s hardly the only thing we’re interested in. If the null hypothesis claimed that θ=.5, and we show that it’s wrong, we’ve only really told half of the story. Rejecting the null hypothesis implies that we believe that θ≠.5, but there’s a big difference between θ=.51 and θ=.8. If we find that θ=.8, then not only have we found that the null hypothesis is wrong, it appears to be very wrong. On the other hand, suppose we’ve successfully rejected the null hypothesis, but it looks like the true value of θ is only .51 (this would only be possible with a large study). Sure, the null hypothesis is wrong, but it’s not at all clear that we actually care , because the effect size is so small. In the context of my ESP study we might still care, since any demonstration of real psychic powers would actually be pretty cool 166 , but in other contexts a 1% difference isn’t very interesting, even if it is a real difference. For instance, suppose we’re looking at differences in high school exam scores between males and females, and it turns out that the female scores are 1% higher on average than the males. If I’ve got data from thousands of students, then this difference will almost certainly be statistically significant , but regardless of how small the p value is it’s just not very interesting. You’d hardly want to go around proclaiming a crisis in boys education on the basis of such a tiny difference would you? It’s for this reason that it is becoming more standard (slowly, but surely) to report some kind of standard measure of effect size along with the the results of the hypothesis test. The hypothesis test itself tells you whether you should believe that the effect you have observed is real (i.e., not just due to chance); the effect size tells you whether or not you should care.

Increasing the power of your study

Not surprisingly, scientists are fairly obsessed with maximising the power of their experiments. We want our experiments to work, and so we want to maximise the chance of rejecting the null hypothesis if it is false (and of course we usually want to believe that it is false!) As we’ve seen, one factor that influences power is the effect size. So the first thing you can do to increase your power is to increase the effect size. In practice, what this means is that you want to design your study in such a way that the effect size gets magnified. For instance, in my ESP study I might believe that psychic powers work best in a quiet, darkened room; with fewer distractions to cloud the mind. Therefore I would try to conduct my experiments in just such an environment: if I can strengthen people’s ESP abilities somehow, then the true value of θ will go up 167 and therefore my effect size will be larger. In short, clever experimental design is one way to boost power; because it can alter the effect size.

Unfortunately, it’s often the case that even with the best of experimental designs you may have only a small effect. Perhaps, for example, ESP really does exist, but even under the best of conditions it’s very very weak. Under those circumstances, your best bet for increasing power is to increase the sample size. In general, the more observations that you have available, the more likely it is that you can discriminate between two hypotheses. If I ran my ESP experiment with 10 participants, and 7 of them correctly guessed the colour of the hidden card, you wouldn’t be terribly impressed. But if I ran it with 10,000 participants and 7,000 of them got the answer right, you would be much more likely to think I had discovered something. In other words, power increases with the sample size. This is illustrated in Figure 11.7, which shows the power of the test for a true parameter of θ=0.7, for all sample sizes N from 1 to 100, where I’m assuming that the null hypothesis predicts that θ 0 =0.5.

Because power is important, whenever you’re contemplating running an experiment it would be pretty useful to know how much power you’re likely to have. It’s never possible to know for sure, since you can’t possibly know what your effect size is. However, it’s often (well, sometimes) possible to guess how big it should be. If so, you can guess what sample size you need! This idea is called power analysis , and if it’s feasible to do it, then it’s very helpful, since it can tell you something about whether you have enough time or money to be able to run the experiment successfully. It’s increasingly common to see people arguing that power analysis should be a required part of experimental design, so it’s worth knowing about. I don’t discuss power analysis in this book, however. This is partly for a boring reason and partly for a substantive one. The boring reason is that I haven’t had time to write about power analysis yet. The substantive one is that I’m still a little suspicious of power analysis. Speaking as a researcher, I have very rarely found myself in a position to be able to do one – it’s either the case that (a) my experiment is a bit non-standard and I don’t know how to define effect size properly, (b) I literally have so little idea about what the effect size will be that I wouldn’t know how to interpret the answers. Not only that, after extensive conversations with someone who does stats consulting for a living (my wife, as it happens), I can’t help but notice that in practice the only time anyone ever asks her for a power analysis is when she’s helping someone write a grant application. In other words, the only time any scientist ever seems to want a power analysis in real life is when they’re being forced to do it by bureaucratic process. It’s not part of anyone’s day to day work. In short, I’ve always been of the view that while power is an important concept, power analysis is not as useful as people make it sound, except in the rare cases where (a) someone has figured out how to calculate power for your actual experimental design and (b) you have a pretty good idea what the effect size is likely to be. Maybe other people have had better experiences than me, but I’ve personally never been in a situation where both (a) and (b) were true. Maybe I’ll be convinced otherwise in the future, and probably a future version of this book would include a more detailed discussion of power analysis, but for now this is about as much as I’m comfortable saying about the topic.

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Article Contents

• 1 Introduction
• 2 The ensemble Burden test
• 3 Extensions of the ensemble Burden test with auxiliary information about relative effect sizes
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Ensemble methods for testing a global null

Conflict of interest: None declared.

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Yaowu Liu, Zhonghua Liu, Xihong Lin, Ensemble methods for testing a global null, Journal of the Royal Statistical Society Series B: Statistical Methodology , Volume 86, Issue 2, April 2024, Pages 461–486, https://doi.org/10.1093/jrsssb/qkad131

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Testing a global null is a canonical problem in statistics and has a wide range of applications. In view of the fact that no uniformly most powerful test exists, prior and/or domain knowledge are commonly used to focus on a certain class of alternatives to improve the testing power. However, it is generally challenging to develop tests that are particularly powerful against a certain class of alternatives. In this paper, motivated by the success of ensemble learning methods for prediction or classification, we propose an ensemble framework for testing that mimics the spirit of random forests to deal with the challenges. Our ensemble testing framework aggregates a collection of weak base tests to form a final ensemble test that maintains strong and robust power for global nulls. We apply the framework to four problems about global testing in different classes of alternatives arising from whole-genome sequencing (WGS) association studies. Specific ensemble tests are proposed for each of these problems, and their theoretical optimality is established in terms of Bahadur efficiency. Extensive simulations and an analysis of a real WGS dataset are conducted to demonstrate the type I error control and/or power gain of the proposed ensemble tests.

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