sample size justification example qualitative research

• Research article
• Open access
• Published: 21 November 2018

Characterising and justifying sample size sufficiency in interview-based studies: systematic analysis of qualitative health research over a 15-year period

• Konstantina Vasileiou   ORCID: orcid.org/0000-0001-5047-3920 1 ,
• Julie Barnett 1 ,
• Susan Thorpe 2 &
• Terry Young 3  

BMC Medical Research Methodology volume  18 , Article number:  148 ( 2018 ) Cite this article

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Choosing a suitable sample size in qualitative research is an area of conceptual debate and practical uncertainty. That sample size principles, guidelines and tools have been developed to enable researchers to set, and justify the acceptability of, their sample size is an indication that the issue constitutes an important marker of the quality of qualitative research. Nevertheless, research shows that sample size sufficiency reporting is often poor, if not absent, across a range of disciplinary fields.

A systematic analysis of single-interview-per-participant designs within three health-related journals from the disciplines of psychology, sociology and medicine, over a 15-year period, was conducted to examine whether and how sample sizes were justified and how sample size was characterised and discussed by authors. Data pertinent to sample size were extracted and analysed using qualitative and quantitative analytic techniques.

Our findings demonstrate that provision of sample size justifications in qualitative health research is limited; is not contingent on the number of interviews; and relates to the journal of publication. Defence of sample size was most frequently supported across all three journals with reference to the principle of saturation and to pragmatic considerations. Qualitative sample sizes were predominantly – and often without justification – characterised as insufficient (i.e., ‘small’) and discussed in the context of study limitations. Sample size insufficiency was seen to threaten the validity and generalizability of studies’ results, with the latter being frequently conceived in nomothetic terms.

Conclusions

We recommend, firstly, that qualitative health researchers be more transparent about evaluations of their sample size sufficiency, situating these within broader and more encompassing assessments of data adequacy . Secondly, we invite researchers critically to consider how saturation parameters found in prior methodological studies and sample size community norms might best inform, and apply to, their own project and encourage that data adequacy is best appraised with reference to features that are intrinsic to the study at hand. Finally, those reviewing papers have a vital role in supporting and encouraging transparent study-specific reporting.

Peer Review reports

Sample adequacy in qualitative inquiry pertains to the appropriateness of the sample composition and size . It is an important consideration in evaluations of the quality and trustworthiness of much qualitative research [ 1 ] and is implicated – particularly for research that is situated within a post-positivist tradition and retains a degree of commitment to realist ontological premises – in appraisals of validity and generalizability [ 2 , 3 , 4 , 5 ].

Samples in qualitative research tend to be small in order to support the depth of case-oriented analysis that is fundamental to this mode of inquiry [ 5 ]. Additionally, qualitative samples are purposive, that is, selected by virtue of their capacity to provide richly-textured information, relevant to the phenomenon under investigation. As a result, purposive sampling [ 6 , 7 ] – as opposed to probability sampling employed in quantitative research – selects ‘information-rich’ cases [ 8 ]. Indeed, recent research demonstrates the greater efficiency of purposive sampling compared to random sampling in qualitative studies [ 9 ], supporting related assertions long put forward by qualitative methodologists.

Sample size in qualitative research has been the subject of enduring discussions [ 4 , 10 , 11 ]. Whilst the quantitative research community has established relatively straightforward statistics-based rules to set sample sizes precisely, the intricacies of qualitative sample size determination and assessment arise from the methodological, theoretical, epistemological, and ideological pluralism that characterises qualitative inquiry (for a discussion focused on the discipline of psychology see [ 12 ]). This mitigates against clear-cut guidelines, invariably applied. Despite these challenges, various conceptual developments have sought to address this issue, with guidance and principles [ 4 , 10 , 11 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ], and more recently, an evidence-based approach to sample size determination seeks to ground the discussion empirically [ 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 ].

Focusing on single-interview-per-participant qualitative designs, the present study aims to further contribute to the dialogue of sample size in qualitative research by offering empirical evidence around justification practices associated with sample size. We next review the existing conceptual and empirical literature on sample size determination.

Sample size in qualitative research: Conceptual developments and empirical investigations

Qualitative research experts argue that there is no straightforward answer to the question of ‘how many’ and that sample size is contingent on a number of factors relating to epistemological, methodological and practical issues [ 36 ]. Sandelowski [ 4 ] recommends that qualitative sample sizes are large enough to allow the unfolding of a ‘new and richly textured understanding’ of the phenomenon under study, but small enough so that the ‘deep, case-oriented analysis’ (p. 183) of qualitative data is not precluded. Morse [ 11 ] posits that the more useable data are collected from each person, the fewer participants are needed. She invites researchers to take into account parameters, such as the scope of study, the nature of topic (i.e. complexity, accessibility), the quality of data, and the study design. Indeed, the level of structure of questions in qualitative interviewing has been found to influence the richness of data generated [ 37 ], and so, requires attention; empirical research shows that open questions, which are asked later on in the interview, tend to produce richer data [ 37 ].

Beyond such guidance, specific numerical recommendations have also been proffered, often based on experts’ experience of qualitative research. For example, Green and Thorogood [ 38 ] maintain that the experience of most qualitative researchers conducting an interview-based study with a fairly specific research question is that little new information is generated after interviewing 20 people or so belonging to one analytically relevant participant ‘category’ (pp. 102–104). Ritchie et al. [ 39 ] suggest that studies employing individual interviews conduct no more than 50 interviews so that researchers are able to manage the complexity of the analytic task. Similarly, Britten [ 40 ] notes that large interview studies will often comprise of 50 to 60 people. Experts have also offered numerical guidelines tailored to different theoretical and methodological traditions and specific research approaches, e.g. grounded theory, phenomenology [ 11 , 41 ]. More recently, a quantitative tool was proposed [ 42 ] to support a priori sample size determination based on estimates of the prevalence of themes in the population. Nevertheless, this more formulaic approach raised criticisms relating to assumptions about the conceptual [ 43 ] and ontological status of ‘themes’ [ 44 ] and the linearity ascribed to the processes of sampling, data collection and data analysis [ 45 ].

In terms of principles, Lincoln and Guba [ 17 ] proposed that sample size determination be guided by the criterion of informational redundancy , that is, sampling can be terminated when no new information is elicited by sampling more units. Following the logic of informational comprehensiveness Malterud et al. [ 18 ] introduced the concept of information power as a pragmatic guiding principle, suggesting that the more information power the sample provides, the smaller the sample size needs to be, and vice versa.

Undoubtedly, the most widely used principle for determining sample size and evaluating its sufficiency is that of saturation . The notion of saturation originates in grounded theory [ 15 ] – a qualitative methodological approach explicitly concerned with empirically-derived theory development – and is inextricably linked to theoretical sampling. Theoretical sampling describes an iterative process of data collection, data analysis and theory development whereby data collection is governed by emerging theory rather than predefined characteristics of the population. Grounded theory saturation (often called theoretical saturation) concerns the theoretical categories – as opposed to data – that are being developed and becomes evident when ‘gathering fresh data no longer sparks new theoretical insights, nor reveals new properties of your core theoretical categories’ [ 46 p. 113]. Saturation in grounded theory, therefore, does not equate to the more common focus on data repetition and moves beyond a singular focus on sample size as the justification of sampling adequacy [ 46 , 47 ]. Sample size in grounded theory cannot be determined a priori as it is contingent on the evolving theoretical categories.

Saturation – often under the terms of ‘data’ or ‘thematic’ saturation – has diffused into several qualitative communities beyond its origins in grounded theory. Alongside the expansion of its meaning, being variously equated with ‘no new data’, ‘no new themes’, and ‘no new codes’, saturation has emerged as the ‘gold standard’ in qualitative inquiry [ 2 , 26 ]. Nevertheless, and as Morse [ 48 ] asserts, whilst saturation is the most frequently invoked ‘guarantee of qualitative rigor’, ‘it is the one we know least about’ (p. 587). Certainly researchers caution that saturation is less applicable to, or appropriate for, particular types of qualitative research (e.g. conversation analysis, [ 49 ]; phenomenological research, [ 50 ]) whilst others reject the concept altogether [ 19 , 51 ].

Methodological studies in this area aim to provide guidance about saturation and develop a practical application of processes that ‘operationalise’ and evidence saturation. Guest, Bunce, and Johnson [ 26 ] analysed 60 interviews and found that saturation of themes was reached by the twelfth interview. They noted that their sample was relatively homogeneous, their research aims focused, so studies of more heterogeneous samples and with a broader scope would be likely to need a larger size to achieve saturation. Extending the enquiry to multi-site, cross-cultural research, Hagaman and Wutich [ 28 ] showed that sample sizes of 20 to 40 interviews were required to achieve data saturation of meta-themes that cut across research sites. In a theory-driven content analysis, Francis et al. [ 25 ] reached data saturation at the 17th interview for all their pre-determined theoretical constructs. The authors further proposed two main principles upon which specification of saturation be based: (a) researchers should a priori specify an initial analysis sample (e.g. 10 interviews) which will be used for the first round of analysis and (b) a stopping criterion , that is, a number of interviews (e.g. 3) that needs to be further conducted, the analysis of which will not yield any new themes or ideas. For greater transparency, Francis et al. [ 25 ] recommend that researchers present cumulative frequency graphs supporting their judgment that saturation was achieved. A comparative method for themes saturation (CoMeTS) has also been suggested [ 23 ] whereby the findings of each new interview are compared with those that have already emerged and if it does not yield any new theme, the ‘saturated terrain’ is assumed to have been established. Because the order in which interviews are analysed can influence saturation thresholds depending on the richness of the data, Constantinou et al. [ 23 ] recommend reordering and re-analysing interviews to confirm saturation. Hennink, Kaiser and Marconi’s [ 29 ] methodological study sheds further light on the problem of specifying and demonstrating saturation. Their analysis of interview data showed that code saturation (i.e. the point at which no additional issues are identified) was achieved at 9 interviews, but meaning saturation (i.e. the point at which no further dimensions, nuances, or insights of issues are identified) required 16–24 interviews. Although breadth can be achieved relatively soon, especially for high-prevalence and concrete codes, depth requires additional data, especially for codes of a more conceptual nature.

Critiquing the concept of saturation, Nelson [ 19 ] proposes five conceptual depth criteria in grounded theory projects to assess the robustness of the developing theory: (a) theoretical concepts should be supported by a wide range of evidence drawn from the data; (b) be demonstrably part of a network of inter-connected concepts; (c) demonstrate subtlety; (d) resonate with existing literature; and (e) can be successfully submitted to tests of external validity.

Other work has sought to examine practices of sample size reporting and sufficiency assessment across a range of disciplinary fields and research domains, from nutrition [ 34 ] and health education [ 32 ], to education and the health sciences [ 22 , 27 ], information systems [ 30 ], organisation and workplace studies [ 33 ], human computer interaction [ 21 ], and accounting studies [ 24 ]. Others investigated PhD qualitative studies [ 31 ] and grounded theory studies [ 35 ]. Incomplete and imprecise sample size reporting is commonly pinpointed by these investigations whilst assessment and justifications of sample size sufficiency are even more sporadic.

Sobal [ 34 ] examined the sample size of qualitative studies published in the Journal of Nutrition Education over a period of 30 years. Studies that employed individual interviews ( n  = 30) had an average sample size of 45 individuals and none of these explicitly reported whether their sample size sought and/or attained saturation. A minority of articles discussed how sample-related limitations (with the latter most often concerning the type of sample, rather than the size) limited generalizability. A further systematic analysis [ 32 ] of health education research over 20 years demonstrated that interview-based studies averaged 104 participants (range 2 to 720 interviewees). However, 40% did not report the number of participants. An examination of 83 qualitative interview studies in leading information systems journals [ 30 ] indicated little defence of sample sizes on the basis of recommendations by qualitative methodologists, prior relevant work, or the criterion of saturation. Rather, sample size seemed to correlate with factors such as the journal of publication or the region of study (US vs Europe vs Asia). These results led the authors to call for more rigor in determining and reporting sample size in qualitative information systems research and to recommend optimal sample size ranges for grounded theory (i.e. 20–30 interviews) and single case (i.e. 15–30 interviews) projects.

Similarly, fewer than 10% of articles in organisation and workplace studies provided a sample size justification relating to existing recommendations by methodologists, prior relevant work, or saturation [ 33 ], whilst only 17% of focus groups studies in health-related journals provided an explanation of sample size (i.e. number of focus groups), with saturation being the most frequently invoked argument, followed by published sample size recommendations and practical reasons [ 22 ]. The notion of saturation was also invoked by 11 out of the 51 most highly cited studies that Guetterman [ 27 ] reviewed in the fields of education and health sciences, of which six were grounded theory studies, four phenomenological and one a narrative inquiry. Finally, analysing 641 interview-based articles in accounting, Dai et al. [ 24 ] called for more rigor since a significant minority of studies did not report precise sample size.

Despite increasing attention to rigor in qualitative research (e.g. [ 52 ]) and more extensive methodological and analytical disclosures that seek to validate qualitative work [ 24 ], sample size reporting and sufficiency assessment remain inconsistent and partial, if not absent, across a range of research domains.

Objectives of the present study

The present study sought to enrich existing systematic analyses of the customs and practices of sample size reporting and justification by focusing on qualitative research relating to health. Additionally, this study attempted to expand previous empirical investigations by examining how qualitative sample sizes are characterised and discussed in academic narratives. Qualitative health research is an inter-disciplinary field that due to its affiliation with medical sciences, often faces views and positions reflective of a quantitative ethos. Thus qualitative health research constitutes an emblematic case that may help to unfold underlying philosophical and methodological differences across the scientific community that are crystallised in considerations of sample size. The present research, therefore, incorporates a comparative element on the basis of three different disciplines engaging with qualitative health research: medicine, psychology, and sociology. We chose to focus our analysis on single-per-participant-interview designs as this not only presents a popular and widespread methodological choice in qualitative health research, but also as the method where consideration of sample size – defined as the number of interviewees – is particularly salient.

Study design

A structured search for articles reporting cross-sectional, interview-based qualitative studies was carried out and eligible reports were systematically reviewed and analysed employing both quantitative and qualitative analytic techniques.

We selected journals which (a) follow a peer review process, (b) are considered high quality and influential in their field as reflected in journal metrics, and (c) are receptive to, and publish, qualitative research (Additional File  1 presents the journals’ editorial positions in relation to qualitative research and sample considerations where available). Three health-related journals were chosen, each representing a different disciplinary field; the British Medical Journal (BMJ) representing medicine, the British Journal of Health Psychology (BJHP) representing psychology, and the Sociology of Health & Illness (SHI) representing sociology.

Search strategy to identify studies

Employing the search function of each individual journal, we used the terms ‘interview*’ AND ‘qualitative’ and limited the results to articles published between 1 January 2003 and 22 September 2017 (i.e. a 15-year review period).

Eligibility criteria

To be eligible for inclusion in the review, the article had to report a cross-sectional study design. Longitudinal studies were thus excluded whilst studies conducted within a broader research programme (e.g. interview studies nested in a trial, as part of a broader ethnography, as part of a longitudinal research) were included if they reported only single-time qualitative interviews. The method of data collection had to be individual, synchronous qualitative interviews (i.e. group interviews, structured interviews and e-mail interviews over a period of time were excluded), and the data had to be analysed qualitatively (i.e. studies that quantified their qualitative data were excluded). Mixed method studies and articles reporting more than one qualitative method of data collection (e.g. individual interviews and focus groups) were excluded. Figure  1 , a PRISMA flow diagram [ 53 ], shows the number of: articles obtained from the searches and screened; papers assessed for eligibility; and articles included in the review (Additional File  2 provides the full list of articles included in the review and their unique identifying code – e.g. BMJ01, BJHP02, SHI03). One review author (KV) assessed the eligibility of all papers identified from the searches. When in doubt, discussions about retaining or excluding articles were held between KV and JB in regular meetings, and decisions were jointly made.

PRISMA flow diagram

Data extraction and analysis

A data extraction form was developed (see Additional File  3 ) recording three areas of information: (a) information about the article (e.g. authors, title, journal, year of publication etc.); (b) information about the aims of the study, the sample size and any justification for this, the participant characteristics, the sampling technique and any sample-related observations or comments made by the authors; and (c) information about the method or technique(s) of data analysis, the number of researchers involved in the analysis, the potential use of software, and any discussion around epistemological considerations. The Abstract, Methods and Discussion (and/or Conclusion) sections of each article were examined by one author (KV) who extracted all the relevant information. This was directly copied from the articles and, when appropriate, comments, notes and initial thoughts were written down.

To examine the kinds of sample size justifications provided by articles, an inductive content analysis [ 54 ] was initially conducted. On the basis of this analysis, the categories that expressed qualitatively different sample size justifications were developed.

We also extracted or coded quantitative data regarding the following aspects:

Journal and year of publication

Number of interviews

Number of participants

Presence of sample size justification(s) (Yes/No)

Presence of a particular sample size justification category (Yes/No), and

Number of sample size justifications provided

Descriptive and inferential statistical analyses were used to explore these data.

A thematic analysis [ 55 ] was then performed on all scientific narratives that discussed or commented on the sample size of the study. These narratives were evident both in papers that justified their sample size and those that did not. To identify these narratives, in addition to the methods sections, the discussion sections of the reviewed articles were also examined and relevant data were extracted and analysed.

In total, 214 articles – 21 in the BMJ, 53 in the BJHP and 140 in the SHI – were eligible for inclusion in the review. Table  1 provides basic information about the sample sizes – measured in number of interviews – of the studies reviewed across the three journals. Figure  2 depicts the number of eligible articles published each year per journal.

The publication of qualitative studies in the BMJ was significantly reduced from 2012 onwards and this appears to coincide with the initiation of the BMJ Open to which qualitative studies were possibly directed.

Pairwise comparisons following a significant Kruskal-Wallis Footnote 2 test indicated that the studies published in the BJHP had significantly ( p  < .001) smaller samples sizes than those published either in the BMJ or the SHI. Sample sizes of BMJ and SHI articles did not differ significantly from each other.

Sample size justifications: Results from the quantitative and qualitative content analysis

Ten (47.6%) of the 21 BMJ studies, 26 (49.1%) of the 53 BJHP papers and 24 (17.1%) of the 140 SHI articles provided some sort of sample size justification. As shown in Table  2 , the majority of articles which justified their sample size provided one justification (70% of articles); fourteen studies (25%) provided two distinct justifications; one study (1.7%) gave three justifications and two studies (3.3%) expressed four distinct justifications.

There was no association between the number of interviews (i.e. sample size) conducted and the provision of a justification (rpb = .054, p  = .433). Within journals, Mann-Whitney tests indicated that sample sizes of ‘justifying’ and ‘non-justifying’ articles in the BMJ and SHI did not differ significantly from each other. In the BJHP, ‘justifying’ articles ( Mean rank  = 31.3) had significantly larger sample sizes than ‘non-justifying’ studies ( Mean rank  = 22.7; U = 237.000, p  < .05).

There was a significant association between the journal a paper was published in and the provision of a justification (χ 2 (2) = 23.83, p  < .001). BJHP studies provided a sample size justification significantly more often than would be expected ( z  = 2.9); SHI studies significantly less often ( z  = − 2.4). If an article was published in the BJHP, the odds of providing a justification were 4.8 times higher than if published in the SHI. Similarly if published in the BMJ, the odds of a study justifying its sample size were 4.5 times higher than in the SHI.

The qualitative content analysis of the scientific narratives identified eleven different sample size justifications. These are described below and illustrated with excerpts from relevant articles. By way of a summary, the frequency with which these were deployed across the three journals is indicated in Table  3 .

Saturation was the most commonly invoked principle (55.4% of all justifications) deployed by studies across all three journals to justify the sufficiency of their sample size. In the BMJ, two studies claimed that they achieved data saturation (BMJ17; BMJ18) and one article referred descriptively to achieving saturation without explicitly using the term (BMJ13). Interestingly, BMJ13 included data in the analysis beyond the point of saturation in search of ‘unusual/deviant observations’ and with a view to establishing findings consistency.

Thirty three women were approached to take part in the interview study. Twenty seven agreed and 21 (aged 21–64, median 40) were interviewed before data saturation was reached (one tape failure meant that 20 interviews were available for analysis). (BMJ17). No new topics were identified following analysis of approximately two thirds of the interviews; however, all interviews were coded in order to develop a better understanding of how characteristic the views and reported behaviours were, and also to collect further examples of unusual/deviant observations. (BMJ13).

Two articles reported pre-determining their sample size with a view to achieving data saturation (BMJ08 – see extract in section In line with existing research ; BMJ15 – see extract in section Pragmatic considerations ) without further specifying if this was achieved. One paper claimed theoretical saturation (BMJ06) conceived as being when “no further recurring themes emerging from the analysis” whilst another study argued that although the analytic categories were highly saturated, it was not possible to determine whether theoretical saturation had been achieved (BMJ04). One article (BMJ18) cited a reference to support its position on saturation.

In the BJHP, six articles claimed that they achieved data saturation (BJHP21; BJHP32; BJHP39; BJHP48; BJHP49; BJHP52) and one article stated that, given their sample size and the guidelines for achieving data saturation, it anticipated that saturation would be attained (BJHP50).

Recruitment continued until data saturation was reached, defined as the point at which no new themes emerged. (BJHP48). It has previously been recommended that qualitative studies require a minimum sample size of at least 12 to reach data saturation (Clarke & Braun, 2013; Fugard & Potts, 2014; Guest, Bunce, & Johnson, 2006) Therefore, a sample of 13 was deemed sufficient for the qualitative analysis and scale of this study. (BJHP50).

Two studies argued that they achieved thematic saturation (BJHP28 – see extract in section Sample size guidelines ; BJHP31) and one (BJHP30) article, explicitly concerned with theory development and deploying theoretical sampling, claimed both theoretical and data saturation.

The final sample size was determined by thematic saturation, the point at which new data appears to no longer contribute to the findings due to repetition of themes and comments by participants (Morse, 1995). At this point, data generation was terminated. (BJHP31).

Five studies argued that they achieved (BJHP05; BJHP33; BJHP40; BJHP13 – see extract in section Pragmatic considerations ) or anticipated (BJHP46) saturation without any further specification of the term. BJHP17 referred descriptively to a state of achieved saturation without specifically using the term. Saturation of coding , but not saturation of themes, was claimed to have been reached by one article (BJHP18). Two articles explicitly stated that they did not achieve saturation; instead claiming a level of theme completeness (BJHP27) or that themes being replicated (BJHP53) were arguments for sufficiency of their sample size.

Furthermore, data collection ceased on pragmatic grounds rather than at the point when saturation point was reached. Despite this, although nuances within sub-themes were still emerging towards the end of data analysis, the themes themselves were being replicated indicating a level of completeness. (BJHP27).

Finally, one article criticised and explicitly renounced the notion of data saturation claiming that, on the contrary, the criterion of theoretical sufficiency determined its sample size (BJHP16).

According to the original Grounded Theory texts, data collection should continue until there are no new discoveries ( i.e. , ‘data saturation’; Glaser & Strauss, 1967). However, recent revisions of this process have discussed how it is rare that data collection is an exhaustive process and researchers should rely on how well their data are able to create a sufficient theoretical account or ‘theoretical sufficiency’ (Dey, 1999). For this study, it was decided that theoretical sufficiency would guide recruitment, rather than looking for data saturation. (BJHP16).

Ten out of the 20 BJHP articles that employed the argument of saturation used one or more citations relating to this principle.

In the SHI, one article (SHI01) claimed that it achieved category saturation based on authors’ judgment.

This number was not fixed in advance, but was guided by the sampling strategy and the judgement, based on the analysis of the data, of the point at which ‘category saturation’ was achieved. (SHI01).

Three articles described a state of achieved saturation without using the term or specifying what sort of saturation they had achieved (i.e. data, theoretical, thematic saturation) (SHI04; SHI13; SHI30) whilst another four articles explicitly stated that they achieved saturation (SHI100; SHI125; SHI136; SHI137). Two papers stated that they achieved data saturation (SHI73 – see extract in section Sample size guidelines ; SHI113), two claimed theoretical saturation (SHI78; SHI115) and two referred to achieving thematic saturation (SHI87; SHI139) or to saturated themes (SHI29; SHI50).

Recruitment and analysis ceased once theoretical saturation was reached in the categories described below (Lincoln and Guba 1985). (SHI115). The respondents’ quotes drawn on below were chosen as representative, and illustrate saturated themes. (SHI50).

One article stated that thematic saturation was anticipated with its sample size (SHI94). Briefly referring to the difficulty in pinpointing achievement of theoretical saturation, SHI32 (see extract in section Richness and volume of data ) defended the sufficiency of its sample size on the basis of “the high degree of consensus [that] had begun to emerge among those interviewed”, suggesting that information from interviews was being replicated. Finally, SHI112 (see extract in section Further sampling to check findings consistency ) argued that it achieved saturation of discursive patterns . Seven of the 19 SHI articles cited references to support their position on saturation (see Additional File  4 for the full list of citations used by articles to support their position on saturation across the three journals).

Overall, it is clear that the concept of saturation encompassed a wide range of variants expressed in terms such as saturation, data saturation, thematic saturation, theoretical saturation, category saturation, saturation of coding, saturation of discursive themes, theme completeness. It is noteworthy, however, that although these various claims were sometimes supported with reference to the literature, they were not evidenced in relation to the study at hand.

Pragmatic considerations

The determination of sample size on the basis of pragmatic considerations was the second most frequently invoked argument (9.6% of all justifications) appearing in all three journals. In the BMJ, one article (BMJ15) appealed to pragmatic reasons, relating to time constraints and the difficulty to access certain study populations, to justify the determination of its sample size.

On the basis of the researchers’ previous experience and the literature, [30, 31] we estimated that recruitment of 15–20 patients at each site would achieve data saturation when data from each site were analysed separately. We set a target of seven to 10 caregivers per site because of time constraints and the anticipated difficulty of accessing caregivers at some home based care services. This gave a target sample of 75–100 patients and 35–50 caregivers overall. (BMJ15).

In the BJHP, four articles mentioned pragmatic considerations relating to time or financial constraints (BJHP27 – see extract in section Saturation ; BJHP53), the participant response rate (BJHP13), and the fixed (and thus limited) size of the participant pool from which interviewees were sampled (BJHP18).

We had aimed to continue interviewing until we had reached saturation, a point whereby further data collection would yield no further themes. In practice, the number of individuals volunteering to participate dictated when recruitment into the study ceased (15 young people, 15 parents). Nonetheless, by the last few interviews, significant repetition of concepts was occurring, suggesting ample sampling. (BJHP13).

Finally, three SHI articles explained their sample size with reference to practical aspects: time constraints and project manageability (SHI56), limited availability of respondents and project resources (SHI131), and time constraints (SHI113).

The size of the sample was largely determined by the availability of respondents and resources to complete the study. Its composition reflected, as far as practicable, our interest in how contextual factors (for example, gender relations and ethnicity) mediated the illness experience. (SHI131).

Qualities of the analysis

This sample size justification (8.4% of all justifications) was mainly employed by BJHP articles and referred to an intensive, idiographic and/or latently focused analysis, i.e. that moved beyond description. More specifically, six articles defended their sample size on the basis of an intensive analysis of transcripts and/or the idiographic focus of the study/analysis. Four of these papers (BJHP02; BJHP19; BJHP24; BJHP47) adopted an Interpretative Phenomenological Analysis (IPA) approach.

The current study employed a sample of 10 in keeping with the aim of exploring each participant’s account (Smith et al. , 1999). (BJHP19).

BJHP47 explicitly renounced the notion of saturation within an IPA approach. The other two BJHP articles conducted thematic analysis (BJHP34; BJHP38). The level of analysis – i.e. latent as opposed to a more superficial descriptive analysis – was also invoked as a justification by BJHP38 alongside the argument of an intensive analysis of individual transcripts

The resulting sample size was at the lower end of the range of sample sizes employed in thematic analysis (Braun & Clarke, 2013). This was in order to enable significant reflection, dialogue, and time on each transcript and was in line with the more latent level of analysis employed, to identify underlying ideas, rather than a more superficial descriptive analysis (Braun & Clarke, 2006). (BJHP38).

Finally, one BMJ paper (BMJ21) defended its sample size with reference to the complexity of the analytic task.

We stopped recruitment when we reached 30–35 interviews, owing to the depth and duration of interviews, richness of data, and complexity of the analytical task. (BMJ21).

Meet sampling requirements

Meeting sampling requirements (7.2% of all justifications) was another argument employed by two BMJ and four SHI articles to explain their sample size. Achieving maximum variation sampling in terms of specific interviewee characteristics determined and explained the sample size of two BMJ studies (BMJ02; BMJ16 – see extract in section Meet research design requirements ).

Recruitment continued until sampling frame requirements were met for diversity in age, sex, ethnicity, frequency of attendance, and health status. (BMJ02).

Regarding the SHI articles, two papers explained their numbers on the basis of their sampling strategy (SHI01- see extract in section Saturation ; SHI23) whilst sampling requirements that would help attain sample heterogeneity in terms of a particular characteristic of interest was cited by one paper (SHI127).

The combination of matching the recruitment sites for the quantitative research and the additional purposive criteria led to 104 phase 2 interviews (Internet (OLC): 21; Internet (FTF): 20); Gyms (FTF): 23; HIV testing (FTF): 20; HIV treatment (FTF): 20.) (SHI23). Of the fifty interviews conducted, thirty were translated from Spanish into English. These thirty, from which we draw our findings, were chosen for translation based on heterogeneity in depressive symptomology and educational attainment. (SHI127).

Finally, the pre-determination of sample size on the basis of sampling requirements was stated by one article though this was not used to justify the number of interviews (SHI10).

Sample size guidelines

Five BJHP articles (BJHP28; BJHP38 – see extract in section Qualities of the analysis ; BJHP46; BJHP47; BJHP50 – see extract in section Saturation ) and one SHI paper (SHI73) relied on citing existing sample size guidelines or norms within research traditions to determine and subsequently defend their sample size (7.2% of all justifications).

Sample size guidelines suggested a range between 20 and 30 interviews to be adequate (Creswell, 1998). Interviewer and note taker agreed that thematic saturation, the point at which no new concepts emerge from subsequent interviews (Patton, 2002), was achieved following completion of 20 interviews. (BJHP28). Interviewing continued until we deemed data saturation to have been reached (the point at which no new themes were emerging). Researchers have proposed 30 as an approximate or working number of interviews at which one could expect to be reaching theoretical saturation when using a semi-structured interview approach (Morse 2000), although this can vary depending on the heterogeneity of respondents interviewed and complexity of the issues explored. (SHI73).

In line with existing research

Sample sizes of published literature in the area of the subject matter under investigation (3.5% of all justifications) were used by 2 BMJ articles as guidance and a precedent for determining and defending their own sample size (BMJ08; BMJ15 – see extract in section Pragmatic considerations ).

We drew participants from a list of prisoners who were scheduled for release each week, sampling them until we reached the target of 35 cases, with a view to achieving data saturation within the scope of the study and sufficient follow-up interviews and in line with recent studies [8–10]. (BMJ08).

Similarly, BJHP38 (see extract in section Qualities of the analysis ) claimed that its sample size was within the range of sample sizes of published studies that use its analytic approach.

Richness and volume of data

BMJ21 (see extract in section Qualities of the analysis ) and SHI32 referred to the richness, detailed nature, and volume of data collected (2.3% of all justifications) to justify the sufficiency of their sample size.

Although there were more potential interviewees from those contacted by postcode selection, it was decided to stop recruitment after the 10th interview and focus on analysis of this sample. The material collected was considerable and, given the focused nature of the study, extremely detailed. Moreover, a high degree of consensus had begun to emerge among those interviewed, and while it is always difficult to judge at what point ‘theoretical saturation’ has been reached, or how many interviews would be required to uncover exception(s), it was felt the number was sufficient to satisfy the aims of this small in-depth investigation (Strauss and Corbin 1990). (SHI32).

Meet research design requirements

Determination of sample size so that it is in line with, and serves the requirements of, the research design (2.3% of all justifications) that the study adopted was another justification used by 2 BMJ papers (BMJ16; BMJ08 – see extract in section In line with existing research ).

We aimed for diverse, maximum variation samples [20] totalling 80 respondents from different social backgrounds and ethnic groups and those bereaved due to different types of suicide and traumatic death. We could have interviewed a smaller sample at different points in time (a qualitative longitudinal study) but chose instead to seek a broad range of experiences by interviewing those bereaved many years ago and others bereaved more recently; those bereaved in different circumstances and with different relations to the deceased; and people who lived in different parts of the UK; with different support systems and coroners’ procedures (see Tables 1 and 2 for more details). (BMJ16).

Researchers’ previous experience

The researchers’ previous experience (possibly referring to experience with qualitative research) was invoked by BMJ15 (see extract in section Pragmatic considerations ) as a justification for the determination of sample size.

Nature of study

One BJHP paper argued that the sample size was appropriate for the exploratory nature of the study (BJHP38).

A sample of eight participants was deemed appropriate because of the exploratory nature of this research and the focus on identifying underlying ideas about the topic. (BJHP38).

Further sampling to check findings consistency

Finally, SHI112 argued that once it had achieved saturation of discursive patterns, further sampling was decided and conducted to check for consistency of the findings.

Within each of the age-stratified groups, interviews were randomly sampled until saturation of discursive patterns was achieved. This resulted in a sample of 67 interviews. Once this sample had been analysed, one further interview from each age-stratified group was randomly chosen to check for consistency of the findings. Using this approach it was possible to more carefully explore children’s discourse about the ‘I’, agency, relationality and power in the thematic areas, revealing the subtle discursive variations described in this article. (SHI112).

Thematic analysis of passages discussing sample size

This analysis resulted in two overarching thematic areas; the first concerned the variation in the characterisation of sample size sufficiency, and the second related to the perceived threats deriving from sample size insufficiency.

Characterisations of sample size sufficiency

The analysis showed that there were three main characterisations of the sample size in the articles that provided relevant comments and discussion: (a) the vast majority of these qualitative studies ( n  = 42) considered their sample size as ‘small’ and this was seen and discussed as a limitation; only two articles viewed their small sample size as desirable and appropriate (b) a minority of articles ( n  = 4) proclaimed that their achieved sample size was ‘sufficient’; and (c) finally, a small group of studies ( n  = 5) characterised their sample size as ‘large’. Whilst achieving a ‘large’ sample size was sometimes viewed positively because it led to richer results, there were also occasions when a large sample size was problematic rather than desirable.

‘Small’ but why and for whom?

A number of articles which characterised their sample size as ‘small’ did so against an implicit or explicit quantitative framework of reference. Interestingly, three studies that claimed to have achieved data saturation or ‘theoretical sufficiency’ with their sample size, discussed or noted as a limitation in their discussion their ‘small’ sample size, raising the question of why, or for whom, the sample size was considered small given that the qualitative criterion of saturation had been satisfied.

The current study has a number of limitations. The sample size was small (n = 11) and, however, large enough for no new themes to emerge. (BJHP39). The study has two principal limitations. The first of these relates to the small number of respondents who took part in the study. (SHI73).

Other articles appeared to accept and acknowledge that their sample was flawed because of its small size (as well as other compositional ‘deficits’ e.g. non-representativeness, biases, self-selection) or anticipated that they might be criticized for their small sample size. It seemed that the imagined audience – perhaps reviewer or reader – was one inclined to hold the tenets of quantitative research, and certainly one to whom it was important to indicate the recognition that small samples were likely to be problematic. That one’s sample might be thought small was often construed as a limitation couched in a discourse of regret or apology.

Very occasionally, the articulation of the small size as a limitation was explicitly aligned against an espoused positivist framework and quantitative research.

This study has some limitations. Firstly, the 100 incidents sample represents a small number of the total number of serious incidents that occurs every year. 26 We sent out a nationwide invitation and do not know why more people did not volunteer for the study. Our lack of epidemiological knowledge about healthcare incidents, however, means that determining an appropriate sample size continues to be difficult. (BMJ20).

Indicative of an apparent oscillation of qualitative researchers between the different requirements and protocols demarcating the quantitative and qualitative worlds, there were a few instances of articles which briefly recognised their ‘small’ sample size as a limitation, but then defended their study on more qualitative grounds, such as their ability and success at capturing the complexity of experience and delving into the idiographic, and at generating particularly rich data.

This research, while limited in size, has sought to capture some of the complexity attached to men’s attitudes and experiences concerning incomes and material circumstances. (SHI35). Our numbers are small because negotiating access to social networks was slow and labour intensive, but our methods generated exceptionally rich data. (BMJ21). This study could be criticised for using a small and unrepresentative sample. Given that older adults have been ignored in the research concerning suntanning, fair-skinned older adults are the most likely to experience skin cancer, and women privilege appearance over health when it comes to sunbathing practices, our study offers depth and richness of data in a demographic group much in need of research attention. (SHI57).

‘Good enough’ sample sizes

Only four articles expressed some degree of confidence that their achieved sample size was sufficient. For example, SHI139, in line with the justification of thematic saturation that it offered, expressed trust in its sample size sufficiency despite the poor response rate. Similarly, BJHP04, which did not provide a sample size justification, argued that it targeted a larger sample size in order to eventually recruit a sufficient number of interviewees, due to anticipated low response rate.

Twenty-three people with type I diabetes from the target population of 133 ( i.e. 17.3%) consented to participate but four did not then respond to further contacts (total N = 19). The relatively low response rate was anticipated, due to the busy life-styles of young people in the age range, the geographical constraints, and the time required to participate in a semi-structured interview, so a larger target sample allowed a sufficient number of participants to be recruited. (BJHP04).

Two other articles (BJHP35; SHI32) linked the claimed sufficiency to the scope (i.e. ‘small, in-depth investigation’), aims and nature (i.e. ‘exploratory’) of their studies, thus anchoring their numbers to the particular context of their research. Nevertheless, claims of sample size sufficiency were sometimes undermined when they were juxtaposed with an acknowledgement that a larger sample size would be more scientifically productive.

Although our sample size was sufficient for this exploratory study, a more diverse sample including participants with lower socioeconomic status and more ethnic variation would be informative. A larger sample could also ensure inclusion of a more representative range of apps operating on a wider range of platforms. (BJHP35).

‘Large’ sample sizes - Promise or peril?

Three articles (BMJ13; BJHP05; BJHP48) which all provided the justification of saturation, characterised their sample size as ‘large’ and narrated this oversufficiency in positive terms as it allowed richer data and findings and enhanced the potential for generalisation. The type of generalisation aspired to (BJHP48) was not further specified however.

This study used rich data provided by a relatively large sample of expert informants on an important but under-researched topic. (BMJ13). Qualitative research provides a unique opportunity to understand a clinical problem from the patient’s perspective. This study had a large diverse sample, recruited through a range of locations and used in-depth interviews which enhance the richness and generalizability of the results. (BJHP48).

And whilst a ‘large’ sample size was endorsed and valued by some qualitative researchers, within the psychological tradition of IPA, a ‘large’ sample size was counter-normative and therefore needed to be justified. Four BJHP studies, all adopting IPA, expressed the appropriateness or desirability of ‘small’ sample sizes (BJHP41; BJHP45) or hastened to explain why they included a larger than typical sample size (BJHP32; BJHP47). For example, BJHP32 below provides a rationale for how an IPA study can accommodate a large sample size and how this was indeed suitable for the purposes of the particular research. To strengthen the explanation for choosing a non-normative sample size, previous IPA research citing a similar sample size approach is used as a precedent.

Small scale IPA studies allow in-depth analysis which would not be possible with larger samples (Smith et al. , 2009). (BJHP41). Although IPA generally involves intense scrutiny of a small number of transcripts, it was decided to recruit a larger diverse sample as this is the first qualitative study of this population in the United Kingdom (as far as we know) and we wanted to gain an overview. Indeed, Smith, Flowers, and Larkin (2009) agree that IPA is suitable for larger groups. However, the emphasis changes from an in-depth individualistic analysis to one in which common themes from shared experiences of a group of people can be elicited and used to understand the network of relationships between themes that emerge from the interviews. This large-scale format of IPA has been used by other researchers in the field of false-positive research. Baillie, Smith, Hewison, and Mason (2000) conducted an IPA study, with 24 participants, of ultrasound screening for chromosomal abnormality; they found that this larger number of participants enabled them to produce a more refined and cohesive account. (BJHP32).

The IPA articles found in the BJHP were the only instances where a ‘small’ sample size was advocated and a ‘large’ sample size problematized and defended. These IPA studies illustrate that the characterisation of sample size sufficiency can be a function of researchers’ theoretical and epistemological commitments rather than the result of an ‘objective’ sample size assessment.

Threats from sample size insufficiency

As shown above, the majority of articles that commented on their sample size, simultaneously characterized it as small and problematic. On those occasions that authors did not simply cite their ‘small’ sample size as a study limitation but rather continued and provided an account of how and why a small sample size was problematic, two important scientific qualities of the research seemed to be threatened: the generalizability and validity of results.

Generalizability

Those who characterised their sample as ‘small’ connected this to the limited potential for generalization of the results. Other features related to the sample – often some kind of compositional particularity – were also linked to limited potential for generalisation. Though not always explicitly articulated to what form of generalisation the articles referred to (see BJHP09), generalisation was mostly conceived in nomothetic terms, that is, it concerned the potential to draw inferences from the sample to the broader study population (‘representational generalisation’ – see BJHP31) and less often to other populations or cultures.

It must be noted that samples are small and whilst in both groups the majority of those women eligible participated, generalizability cannot be assumed. (BJHP09). The study’s limitations should be acknowledged: Data are presented from interviews with a relatively small group of participants, and thus, the views are not necessarily generalizable to all patients and clinicians. In particular, patients were only recruited from secondary care services where COFP diagnoses are typically confirmed. The sample therefore is unlikely to represent the full spectrum of patients, particularly those who are not referred to, or who have been discharged from dental services. (BJHP31).

Without explicitly using the term generalisation, two SHI articles noted how their ‘small’ sample size imposed limits on ‘the extent that we can extrapolate from these participants’ accounts’ (SHI114) or to the possibility ‘to draw far-reaching conclusions from the results’ (SHI124).

Interestingly, only a minority of articles alluded to, or invoked, a type of generalisation that is aligned with qualitative research, that is, idiographic generalisation (i.e. generalisation that can be made from and about cases [ 5 ]). These articles, all published in the discipline of sociology, defended their findings in terms of the possibility of drawing logical and conceptual inferences to other contexts and of generating understanding that has the potential to advance knowledge, despite their ‘small’ size. One article (SHI139) clearly contrasted nomothetic (statistical) generalisation to idiographic generalisation, arguing that the lack of statistical generalizability does not nullify the ability of qualitative research to still be relevant beyond the sample studied.

Further, these data do not need to be statistically generalisable for us to draw inferences that may advance medicalisation analyses (Charmaz 2014). These data may be seen as an opportunity to generate further hypotheses and are a unique application of the medicalisation framework. (SHI139). Although a small-scale qualitative study related to school counselling, this analysis can be usefully regarded as a case study of the successful utilisation of mental health-related resources by adolescents. As many of the issues explored are of relevance to mental health stigma more generally, it may also provide insights into adult engagement in services. It shows how a sociological analysis, which uses positioning theory to examine how people negotiate, partially accept and simultaneously resist stigmatisation in relation to mental health concerns, can contribute to an elucidation of the social processes and narrative constructions which may maintain as well as bridge the mental health service gap. (SHI103).

Only one article (SHI30) used the term transferability to argue for the potential of wider relevance of the results which was thought to be more the product of the composition of the sample (i.e. diverse sample), rather than the sample size.

The second major concern that arose from a ‘small’ sample size pertained to the internal validity of findings (i.e. here the term is used to denote the ‘truth’ or credibility of research findings). Authors expressed uncertainty about the degree of confidence in particular aspects or patterns of their results, primarily those that concerned some form of differentiation on the basis of relevant participant characteristics.

The information source preferred seemed to vary according to parents’ education; however, the sample size is too small to draw conclusions about such patterns. (SHI80). Although our numbers were too small to demonstrate gender differences with any certainty, it does seem that the biomedical and erotic scripts may be more common in the accounts of men and the relational script more common in the accounts of women. (SHI81).

In other instances, articles expressed uncertainty about whether their results accounted for the full spectrum and variation of the phenomenon under investigation. In other words, a ‘small’ sample size (alongside compositional ‘deficits’ such as a not statistically representative sample) was seen to threaten the ‘content validity’ of the results which in turn led to constructions of the study conclusions as tentative.

Data collection ceased on pragmatic grounds rather than when no new information appeared to be obtained ( i.e. , saturation point). As such, care should be taken not to overstate the findings. Whilst the themes from the initial interviews seemed to be replicated in the later interviews, further interviews may have identified additional themes or provided more nuanced explanations. (BJHP53). …it should be acknowledged that this study was based on a small sample of self-selected couples in enduring marriages who were not broadly representative of the population. Thus, participants may not be representative of couples that experience postnatal PTSD. It is therefore unlikely that all the key themes have been identified and explored. For example, couples who were excluded from the study because the male partner declined to participate may have been experiencing greater interpersonal difficulties. (BJHP03).

In other instances, articles attempted to preserve a degree of credibility of their results, despite the recognition that the sample size was ‘small’. Clarity and sharpness of emerging themes and alignment with previous relevant work were the arguments employed to warrant the validity of the results.

This study focused on British Chinese carers of patients with affective disorders, using a qualitative methodology to synthesise the sociocultural representations of illness within this community. Despite the small sample size, clear themes emerged from the narratives that were sufficient for this exploratory investigation. (SHI98).

The present study sought to examine how qualitative sample sizes in health-related research are characterised and justified. In line with previous studies [ 22 , 30 , 33 , 34 ] the findings demonstrate that reporting of sample size sufficiency is limited; just over 50% of articles in the BMJ and BJHP and 82% in the SHI did not provide any sample size justification. Providing a sample size justification was not related to the number of interviews conducted, but it was associated with the journal that the article was published in, indicating the influence of disciplinary or publishing norms, also reported in prior research [ 30 ]. This lack of transparency about sample size sufficiency is problematic given that most qualitative researchers would agree that it is an important marker of quality [ 56 , 57 ]. Moreover, and with the rise of qualitative research in social sciences, efforts to synthesise existing evidence and assess its quality are obstructed by poor reporting [ 58 , 59 ].

When authors justified their sample size, our findings indicate that sufficiency was mostly appraised with reference to features that were intrinsic to the study, in agreement with general advice on sample size determination [ 4 , 11 , 36 ]. The principle of saturation was the most commonly invoked argument [ 22 ] accounting for 55% of all justifications. A wide range of variants of saturation was evident corroborating the proliferation of the meaning of the term [ 49 ] and reflecting different underlying conceptualisations or models of saturation [ 20 ]. Nevertheless, claims of saturation were never substantiated in relation to procedures conducted in the study itself, endorsing similar observations in the literature [ 25 , 30 , 47 ]. Claims of saturation were sometimes supported with citations of other literature, suggesting a removal of the concept away from the characteristics of the study at hand. Pragmatic considerations, such as resource constraints or participant response rate and availability, was the second most frequently used argument accounting for approximately 10% of justifications and another 23% of justifications also represented intrinsic-to-the-study characteristics (i.e. qualities of the analysis, meeting sampling or research design requirements, richness and volume of the data obtained, nature of study, further sampling to check findings consistency).

Only, 12% of mentions of sample size justification pertained to arguments that were external to the study at hand, in the form of existing sample size guidelines and prior research that sets precedents. Whilst community norms and prior research can establish useful rules of thumb for estimating sample sizes [ 60 ] – and reveal what sizes are more likely to be acceptable within research communities – researchers should avoid adopting these norms uncritically, especially when such guidelines [e.g. 30 , 35 ], might be based on research that does not provide adequate evidence of sample size sufficiency. Similarly, whilst methodological research that seeks to demonstrate the achievement of saturation is invaluable since it explicates the parameters upon which saturation is contingent and indicates when a research project is likely to require a smaller or a larger sample [e.g. 29 ], specific numbers at which saturation was achieved within these projects cannot be routinely extrapolated for other projects. We concur with existing views [ 11 , 36 ] that the consideration of the characteristics of the study at hand, such as the epistemological and theoretical approach, the nature of the phenomenon under investigation, the aims and scope of the study, the quality and richness of data, or the researcher’s experience and skills of conducting qualitative research, should be the primary guide in determining sample size and assessing its sufficiency.

Moreover, although numbers in qualitative research are not unimportant [ 61 ], sample size should not be considered alone but be embedded in the more encompassing examination of data adequacy [ 56 , 57 ]. Erickson’s [ 62 ] dimensions of ‘evidentiary adequacy’ are useful here. He explains the concept in terms of adequate amounts of evidence, adequate variety in kinds of evidence, adequate interpretive status of evidence, adequate disconfirming evidence, and adequate discrepant case analysis. All dimensions might not be relevant across all qualitative research designs, but this illustrates the thickness of the concept of data adequacy, taking it beyond sample size.

The present research also demonstrated that sample sizes were commonly seen as ‘small’ and insufficient and discussed as limitation. Often unjustified (and in two cases incongruent with their own claims of saturation) these findings imply that sample size in qualitative health research is often adversely judged (or expected to be judged) against an implicit, yet omnipresent, quasi-quantitative standpoint. Indeed there were a few instances in our data where authors appeared, possibly in response to reviewers, to resist to some sort of quantification of their results. This implicit reference point became more apparent when authors discussed the threats deriving from an insufficient sample size. Whilst the concerns about internal validity might be legitimate to the extent that qualitative research projects, which are broadly related to realism, are set to examine phenomena in sufficient breadth and depth, the concerns around generalizability revealed a conceptualisation that is not compatible with purposive sampling. The limited potential for generalisation, as a result of a small sample size, was often discussed in nomothetic, statistical terms. Only occasionally was analytic or idiographic generalisation invoked to warrant the value of the study’s findings [ 5 , 17 ].

Strengths and limitations of the present study

We note, first, the limited number of health-related journals reviewed, so that only a ‘snapshot’ of qualitative health research has been captured. Examining additional disciplines (e.g. nursing sciences) as well as inter-disciplinary journals would add to the findings of this analysis. Nevertheless, our study is the first to provide some comparative insights on the basis of disciplines that are differently attached to the legacy of positivism and analysed literature published over a lengthy period of time (15 years). Guetterman [ 27 ] also examined health-related literature but this analysis was restricted to 26 most highly cited articles published over a period of five years whilst Carlsen and Glenton’s [ 22 ] study concentrated on focus groups health research. Moreover, although it was our intention to examine sample size justification in relation to the epistemological and theoretical positions of articles, this proved to be challenging largely due to absence of relevant information, or the difficulty into discerning clearly articles’ positions [ 63 ] and classifying them under specific approaches (e.g. studies often combined elements from different theoretical and epistemological traditions). We believe that such an analysis would yield useful insights as it links the methodological issue of sample size to the broader philosophical stance of the research. Despite these limitations, the analysis of the characterisation of sample size and of the threats seen to accrue from insufficient sample size, enriches our understanding of sample size (in)sufficiency argumentation by linking it to other features of the research. As the peer-review process becomes increasingly public, future research could usefully examine how reporting around sample size sufficiency and data adequacy might be influenced by the interactions between authors and reviewers.

The past decade has seen a growing appetite in qualitative research for an evidence-based approach to sample size determination and to evaluations of the sufficiency of sample size. Despite the conceptual and methodological developments in the area, the findings of the present study confirm previous studies in concluding that appraisals of sample size sufficiency are either absent or poorly substantiated. To ensure and maintain high quality research that will encourage greater appreciation of qualitative work in health-related sciences [ 64 ], we argue that qualitative researchers should be more transparent and thorough in their evaluation of sample size as part of their appraisal of data adequacy. We would encourage the practice of appraising sample size sufficiency with close reference to the study at hand and would thus caution against responding to the growing methodological research in this area with a decontextualised application of sample size numerical guidelines, norms and principles. Although researchers might find sample size community norms serve as useful rules of thumb, we recommend methodological knowledge is used to critically consider how saturation and other parameters that affect sample size sufficiency pertain to the specifics of the particular project. Those reviewing papers have a vital role in encouraging transparent study-specific reporting. The review process should support authors to exercise nuanced judgments in decisions about sample size determination in the context of the range of factors that influence sample size sufficiency and the specifics of a particular study. In light of the growing methodological evidence in the area, transparent presentation of such evidence-based judgement is crucial and in time should surely obviate the seemingly routine practice of citing the ‘small’ size of qualitative samples among the study limitations.

A non-parametric test of difference for independent samples was performed since the variable number of interviews violated assumptions of normality according to the standardized scores of skewness and kurtosis (BMJ: z skewness = 3.23, z kurtosis = 1.52; BJHP: z skewness = 4.73, z kurtosis = 4.85; SHI: z skewness = 12.04, z kurtosis = 21.72) and the Shapiro-Wilk test of normality ( p  < .001).

Abbreviations

British Journal of Health Psychology

British Medical Journal

Interpretative Phenomenological Analysis

Sociology of Health & Illness

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Acknowledgments

We would like to thank Dr. Paula Smith and Katharine Lee for their comments on a previous draft of this paper as well as Natalie Ann Mitchell and Meron Teferra for assisting us with data extraction.

This research was initially conceived of and partly conducted with financial support from the Multidisciplinary Assessment of Technology Centre for Healthcare (MATCH) programme (EP/F063822/1 and EP/G012393/1). The research continued and was completed independent of any support. The funding body did not have any role in the study design, the collection, analysis and interpretation of the data, in the writing of the paper, and in the decision to submit the manuscript for publication. The views expressed are those of the authors alone.

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JB and TY conceived the study; KV, JB, and TY designed the study; KV identified the articles and extracted the data; KV and JB assessed eligibility of articles; KV, JB, ST, and TY contributed to the analysis of the data, discussed the findings and early drafts of the paper; KV developed the final manuscript; KV, JB, ST, and TY read and approved the manuscript.

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Terry Young is an academic who undertakes research and occasional consultancy in the areas of health technology assessment, information systems, and service design. He is unaware of any direct conflict of interest with respect to this paper. All other authors have no competing interests to declare.

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Editorial positions on qualitative research and sample considerations (where available). (DOCX 12 kb)

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List of eligible articles included in the review ( N  = 214). (DOCX 38 kb)

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Citations used by articles to support their position on saturation. (DOCX 14 kb)

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Vasileiou, K., Barnett, J., Thorpe, S. et al. Characterising and justifying sample size sufficiency in interview-based studies: systematic analysis of qualitative health research over a 15-year period. BMC Med Res Methodol 18 , 148 (2018). https://doi.org/10.1186/s12874-018-0594-7

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Qualitative Market Research

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Article publication date: 12 September 2016

Qualitative researchers have been criticised for not justifying sample size decisions in their research. This short paper addresses the issue of which sample sizes are appropriate and valid within different approaches to qualitative research.

Design/methodology/approach

The sparse literature on sample sizes in qualitative research is reviewed and discussed. This examination is informed by the personal experience of the author in terms of assessing, as an editor, reviewer comments as they relate to sample size in qualitative research. Also, the discussion is informed by the author’s own experience of undertaking commercial and academic qualitative research over the last 31 years.

In qualitative research, the determination of sample size is contextual and partially dependent upon the scientific paradigm under which investigation is taking place. For example, qualitative research which is oriented towards positivism, will require larger samples than in-depth qualitative research does, so that a representative picture of the whole population under review can be gained. Nonetheless, the paper also concludes that sample sizes involving one single case can be highly informative and meaningful as demonstrated in examples from management and medical research. Unique examples of research using a single sample or case but involving new areas or findings that are potentially highly relevant, can be worthy of publication. Theoretical saturation can also be useful as a guide in designing qualitative research, with practical research illustrating that samples of 12 may be cases where data saturation occurs among a relatively homogeneous population.

Practical implications

Sample sizes as low as one can be justified. Researchers and reviewers may find the discussion in this paper to be a useful guide to determining and critiquing sample size in qualitative research.

Originality/value

Sample size in qualitative research is always mentioned by reviewers of qualitative papers but discussion tends to be simplistic and relatively uninformed. The current paper draws attention to how sample sizes, at both ends of the size continuum, can be justified by researchers. This will also aid reviewers in their making of comments about the appropriateness of sample sizes in qualitative research.

• Qualitative research
• Qualitative methodology
• Case studies
• Sample size

Boddy, C.R. (2016), "Sample size for qualitative research", Qualitative Market Research , Vol. 19 No. 4, pp. 426-432. https://doi.org/10.1108/QMR-06-2016-0053

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Answers to the most commonly asked questions here

I was asked to help justify the sample size for a qualitative research study. When the goals of a study are qualitative, the sample size justification is also qualitative.

Qualitative studies have a level of flexibility that would cause trouble for a quantitative study and you need to get comfortable with this flexibility. Frequently sampling is done on an ongoing basis and the sampling strategy is often modified as the qualitative summaries of the early results occur. You might change the size, location, and composition of your focus groups for example, to try to explore more efficiently some of the provisional findings from the earlier focus groups. In a study of nursing work environment, for example, there was a large, vocal, and unexpected commentary about generational differences between nurses. This led to formation of new focus groups with only Baby Boomer nurses and only Generation X nurses. Sampling typically ends in a qualitative study when saturation occurs. Saturation is when you start hearing the same themes over and over again and no new themes are emerging.

That's all well and good, but you have to submit a grant and the budget for the grant before any qualitative research is done. What you have to do is to rely on sample sizes used in similar qualitative studies, project a reasonable upper bound, and then say that sampling will continue until saturation occurs or the upper bound is reached. You might want to also toss in something along the lines of "the sampling approach will be modified if the upper bound for sample size is being approached without any evidence of convergence of themes."

8   Sample Size Justification

You can listen to an audio recording of this chapter here . You can download a translation of this chapter in Chinese here

Scientists perform empirical studies to collect data that helps to answer a research question. The more data that is collected, the more informative the study will be with respect to its inferential goals. A sample size justification should consider how informative the data will be given an inferential goal, such as estimating an effect size, or testing a hypothesis. Even though a sample size justification is sometimes requested in manuscript submission guidelines, when submitting a grant to a funder, or submitting a proposal to an ethical review board, the number of observations is often simply stated , but not justified . This makes it difficult to evaluate how informative a study will be. To prevent such concerns from emerging when it is too late (e.g., after a non-significant hypothesis test has been observed), researchers should carefully justify their sample size before data is collected. In this chapter, which is largely identical to Lakens ( 2022 ) , we will explore in detail how to justify your sample size.

8.1 Six Approaches to Justify Sample Sizes

Researchers often find it difficult to justify their sample size (i.e., a number of participants, observations, or any combination thereof). In this review article six possible approaches are discussed that can be used to justify the sample size in a quantitative study (see Table  8.1 ). This is not an exhaustive overview, but it includes the most common and applicable approaches for single studies. The topic of power analysis for meta-analyses is outside the scope of this chapter, but see Hedges & Pigott ( 2001 ) and Valentine et al. ( 2010 ) . The first justification is that data from (almost) the entire population has been collected. The second justification centers on resource constraints, which are almost always present, but rarely explicitly evaluated. The third and fourth justifications are based on a desired statistical power or a desired accuracy. The fifth justification relies on heuristics, and finally, researchers can choose a sample size without any justification. Each of these justifications can be stronger or weaker depending on which conclusions researchers want to draw from the data they plan to collect.

All of these approaches to the justification of sample sizes, even the ‘no justification’ approach, give others insight into the reasons that led to the decision for a sample size in a study. It should not be surprising that the ‘heuristics’ and ‘no justification’ approaches are often unlikely to impress peers. However, it is important to note that the value of the information that is collected depends on the extent to which the final sample size allows a researcher to achieve their inferential goals, and not on the sample size justification that is chosen.

The extent to which these approaches make other researchers judge the data that is collected as informative depends on the details of the question a researcher aimed to answer and the parameters they chose when determining the sample size for their study. For example, a badly performed a-priori power analysis can quickly lead to a study with very low informational value. These six justifications are not mutually exclusive, and multiple approaches can be considered when designing a study.

8.2 Six Ways to Evaluate Which Effect Sizes are Interesting

The informativeness of the data that is collected depends on the inferential goals a researcher has, or in some cases, the inferential goals scientific peers will have. A shared feature of the different inferential goals considered in this review article is the question which effect sizes a researcher considers meaningful to distinguish. This implies that researchers need to evaluate which effect sizes they consider interesting. These evaluations rely on a combination of statistical properties and domain knowledge. In Table  8.2 six possibly useful considerations are provided. This is not intended to be an exhaustive overview, but it presents common and useful approaches that can be applied in practice. Not all evaluations are equally relevant for all types of sample size justifications. The online Shiny app accompanying Lakens ( 2022 ) provides researchers with an interactive form that guides researchers through the considerations for a sample size justification. These considerations often rely on the same information (e.g., effect sizes, the number of observations, the standard deviation, etc.) so these six considerations should be seen as a set of complementary approaches that can be used to evaluate which effect sizes are of interest.

To start, researchers should consider what their smallest effect size of interest is. Second, although only relevant when performing a hypothesis test, researchers should consider which effect sizes could be statistically significant given a choice of an alpha level and sample size. Third, it is important to consider the (range of) effect sizes that are expected. This requires a careful consideration of the source of this expectation and the presence of possible biases in these expectations. Fourth, it is useful to consider the width of the confidence interval around possible values of the effect size in the population, and whether we can expect this confidence interval to reject effects we considered a-priori plausible. Fifth, it is worth evaluating the power of the test across a wide range of possible effect sizes in a sensitivity power analysis. Sixth, a researcher can consider the effect size distribution of related studies in the literature.

8.3 The Value of Information

Since all scientists are faced with resource limitations, they need to balance the cost of collecting each additional datapoint against the increase in information that datapoint provides. This is referred to as the value of information ( Eckermann et al., 2010 ) . Calculating the value of information is notoriously difficult ( Detsky, 1990 ) . Researchers need to specify the cost of collecting data, and weigh the costs of data collection against the increase in utility that having access to the data provides. From a value of information perspective not every data point that can be collected is equally valuable ( J. Halpern et al., 2001 ; Wilson, 2015 ) . Whenever additional observations do not change inferences in a meaningful way, the costs of data collection can outweigh the benefits.

The value of additional information will in most cases be a non-monotonic function, especially when it depends on multiple inferential goals. A researcher might be interested in comparing an effect against a previously observed large effect in the literature, a theoretically predicted medium effect, and the smallest effect that would be practically relevant. In such a situation the expected value of sampling information will lead to different optimal sample sizes for each inferential goal. It could be valuable to collect informative data about a large effect, with additional data having less (or even a negative) marginal utility, up to a point where the data becomes increasingly informative about a medium effect size, with the value of sampling additional information decreasing once more until the study becomes increasingly informative about the presence or absence of a smallest effect of interest.

Because of the difficulty of quantifying the value of information, scientists typically use less formal approaches to justify the amount of data they set out to collect in a study. Even though the cost-benefit analysis is not always made explicit in reported sample size justifications, the value of information perspective is almost always implicitly the underlying framework that sample size justifications are based on. Throughout the subsequent discussion of sample size justifications, the importance of considering the value of information given inferential goals will repeatedly be highlighted.

8.4 Measuring (Almost) the Entire Population

In some instances it might be possible to collect data from (almost) the entire population under investigation. For example, researchers might use census data, are able to collect data from all employees at a firm or study a small population of top athletes. Whenever it is possible to measure the entire population, the sample size justification becomes straightforward: the researcher used all the data that is available.

8.5 Resource Constraints

A common reason for the number of observations in a study is that resource constraints limit the amount of data that can be collected at a reasonable cost ( Lenth, 2001 ) . In practice, sample sizes are always limited by the resources that are available. Researchers practically always have resource limitations, and therefore even when resource constraints are not the primary justification for the sample size in a study, it is always a secondary justification.

Despite the omnipresence of resource limitations, the topic often receives little attention in texts on experimental design (for an example of an exception, see Bulus & Dong ( 2021 ) ). This might make it feel like acknowledging resource constraints is not appropriate, but the opposite is true: Because resource limitations always play a role, a responsible scientist carefully evaluates resource constraints when designing a study. Resource constraint justifications are based on a trade-off between the costs of data collection, and the value of having access to the information the data provides. Even if researchers do not explicitly quantify this trade-off, it is revealed in their actions. For example, researchers rarely spend all the resources they have on a single study. Given resource constraints, researchers are confronted with an optimization problem of how to spend resources across multiple research questions.

Time and money are two resource limitations all scientists face. A PhD student has a certain time to complete a PhD thesis, and is typically expected to complete multiple research lines in this time. In addition to time limitations, researchers have limited financial resources that often directly influence how much data can be collected. A third limitation in some research lines is that there might simply be a very small number of individuals from whom data can be collected, such as when studying patients with a rare disease. A resource constraint justification puts limited resources at the center of the justification for the sample size that will be collected, and starts with the resources a scientist has available. These resources are translated into an expected number of observations ( N ) that a researcher expects they will be able to collect with an amount of money in a given time. The challenge is to evaluate whether collecting N observations is worthwhile. How do we decide if a study will be informative, and when should we conclude that data collection is not worthwhile?

When evaluating whether resource constraints make data collection uninformative, researchers need to explicitly consider which inferential goals they have when collecting data ( Parker & Berman, 2003 ) . Having data always provides more knowledge about the research question than not having data, so in an absolute sense, all data that is collected has value. However, it is possible that the benefits of collecting the data are outweighed by the costs of data collection.

It is most straightforward to evaluate whether data collection has value when we know for certain that someone will make a decision, with or without data. In such situations any additional data will reduce the error rates of a well-calibrated decision process, even if only ever so slightly. For example, without data we will not perform better than a coin flip if we guess which of two conditions has a higher true mean score on a measure. With some data, we can perform better than a coin flip by picking the condition that has the highest mean. With a small amount of data we would still very likely make a mistake, but the error rate is smaller than without any data. In these cases, the value of information might be positive, as long as the reduction in error rates is more beneficial than the cost of data collection.

Another way in which a small dataset can be valuable is if its existence eventually makes it possible to perform a meta-analysis ( Maxwell & Kelley, 2011 ) . This argument in favor of collecting a small dataset requires 1) that researchers share the data in a way that a future meta-analyst can find it, and 2) that there is a decent probability that someone will perform a high-quality meta-analysis that will include this data in the future ( S. D. Halpern et al., 2002 ) . The uncertainty about whether there will ever be such a meta-analysis should be weighed against the costs of data collection.

One way to increase the probability of a future meta-analysis is if researchers commit to performing this meta-analysis themselves, by combining several studies they have performed into a small-scale meta-analysis ( Cumming, 2014 ) . For example, a researcher might plan to repeat a study for the next 12 years in a class they teach, with the expectation that after 12 years a meta-analysis of 12 studies would be sufficient to draw informative inferences (but see ter Schure & Grünwald ( 2019 ) ). If it is not plausible that a researcher will collect all the required data by themselves, they can attempt to set up a collaboration where fellow researchers in their field commit to collecting similar data with identical measures. If it is not likely that sufficient data will emerge over time to reach the inferential goals, there might be no value in collecting the data.

Even if a researcher believes it is worth collecting data because a future meta-analysis will be performed, they will most likely perform a statistical test on the data. To make sure their expectations about the results of such a test are well-calibrated, it is important to consider which effect sizes are of interest, and to perform a sensitivity power analysis to evaluate the probability of a Type II error for effects of interest. From the six ways to evaluate which effect sizes are interesting that will be discussed in the second part of this review, it is useful to consider the smallest effect size that can be statistically significant, the expected width of the confidence interval around the effect size, and effects that can be expected in a specific research area, and to evaluate the power for these effect sizes in a sensitivity power analysis. If a decision or claim is made, a compromise power analysis is worthwhile to consider when deciding upon the error rates while planning the study. When reporting a resource constraints sample size justification it is recommended to address the five considerations in Table  8.3 ). Addressing these points explicitly facilitates evaluating if the data is worthwhile to collect. To make it easier to address all relevant points explicitly, an interactive form to implement the recommendations in this chapter can be found at https://shiny.ieis.tue.nl/sample_size_justification/.

8.6 A-priori Power Analysis

When designing a study where the goal is to test whether a statistically significant effect is present, researchers often want to make sure their sample size is large enough to prevent erroneous conclusions for a range of effect sizes they care about. In this approach to justifying a sample size, the value of information is to collect observations up to the point that the probability of an erroneous inference is, in the long run, not larger than a desired value. If a researcher performs a hypothesis test, there are four possible outcomes:

• A false positive (or Type I error), determined by the \(\alpha\) level. A test yields a significant result, even though the null hypothesis is true.
• A false negative (or Type II error), determined by \(\beta\) , or 1 - power. A test yields a non-significant result, even though the alternative hypothesis is true.
• A true negative, determined by 1- \(\alpha\) . A test yields a non-significant result when the null hypothesis is true.
• A true positive, determined by 1- \(\beta\) . A test yields a significant result when the alternative hypothesis is true.

Given a specified effect size, alpha level, and power, an a-priori power analysis can be used to calculate the number of observations required to achieve the desired error rates, given the effect size. Power analyses can be performed based on standardized effect sizes or effect sizes expressed on the original scale. It is important to know the standard deviation of the effect (see the ‘Know Your Measure’ section) but I find it slightly more convenient to talk about standardized effects in the context of sample size justifications. Figure  8.1 illustrates how the statistical power increases as the number of observations (per group) increases in an independent t test with a two-sided alpha level of 0.05. If we are interested in detecting an effect of d = 0.5, a sample size of 90 per condition would give us more than 90% power. Statistical power can be computed to determine the number of participants, or the number of items ( Westfall et al., 2014 ) but can also be performed for single case studies ( Ferron & Onghena, 1996 ; McIntosh & Rittmo, 2021 ) .

Although it is common to set the Type I error rate to 5% and aim for 80% power, error rates should be justified ( Lakens, Adolfi, et al., 2018 ) . As explained in the section on compromise power analysis, the default recommendation to aim for 80% power lacks a solid justification. In general, the lower the error rates (and thus the higher the power), the more informative a study will be, but the more resources are required. Researchers should carefully weigh the costs of increasing the sample size against the benefits of lower error rates, which would probably make studies designed to achieve a power of 90% or 95% more common for articles reporting a single study. An additional consideration is whether the researcher plans to publish an article consisting of a set of replication and extension studies, in which case the probability of observing multiple Type I errors will be very low, but the probability of observing mixed results even when there is a true effect increases ( Lakens & Etz, 2017 ) , which would also be a reason to aim for studies with low Type II error rates, perhaps even by slightly increasing the alpha level for each individual study.

Figure  8.2 visualizes two distributions. The left distribution (dashed line) is centered at 0. This is a model for the null hypothesis. If the null hypothesis is true a statistically significant result will be observed if the effect size is extreme enough (in a two-sided test either in the positive or negative direction), but any significant result would be a Type I error (the dark grey areas under the curve). If there is no true effect, formally statistical power for a null hypothesis significance test is undefined. Any significant effects observed if the null hypothesis is true are Type I errors, or false positives, which occur at the chosen alpha level. The right distribution (solid line) is centered on an effect of d = 0.5. This is the specified model for the alternative hypothesis in this study, illustrating the expectation of an effect of d = 0.5 if the alternative hypothesis is true. Even though there is a true effect, studies will not always find a statistically significant result. This happens when, due to random variation, the observed effect size is too close to 0 to be statistically significant. Such results are false negatives (the light grey area under the curve on the right). To increase power, we can collect a larger sample size. As the sample size increases, the distributions become more narrow, reducing the probability of a Type II error. These figures can be reproduced and adapted in an online shiny app .

It is important to highlight that the goal of an a-priori power analysis is not to achieve sufficient power for the true effect size. The true effect size is unknown. The goal of an a-priori power analysis is to achieve sufficient power, given a specific assumption of the effect size a researcher wants to detect. Just like a Type I error rate is the maximum probability of making a Type I error conditional on the assumption that the null hypothesis is true, an a-priori power analysis is computed under the assumption of a specific effect size. It is unknown if this assumption is correct. All a researcher can do is to make sure their assumptions are well justified. Statistical inferences based on a test where the Type II error rate is controlled are conditional on the assumption of a specific effect size. They allow the inference that, assuming the true effect size is at least as large as that used in the a-priori power analysis, the maximum Type II error rate in a study is not larger than a desired value.

This point is perhaps best illustrated if we consider a study where an a-priori power analysis is performed both for a test of the presence of an effect, as for a test of the absence of an effect. When designing a study, it essential to consider the possibility that there is no effect (e.g., a mean difference of zero). An a-priori power analysis can be performed both for a null hypothesis significance test, as for a test of the absence of a meaningful effect, such as an equivalence test that can statistically provide support for the null hypothesis by rejecting the presence of effects that are large enough to matter ( Lakens, 2017 ; Meyners, 2012 ; Rogers et al., 1993 ) . When multiple primary tests will be performed based on the same sample, each analysis requires a dedicated sample size justification. If possible, a sample size is collected that guarantees that all tests are informative, which means that the collected sample size is based on the largest sample size returned by any of the a-priori power analyses.

For example, if the goal of a study is to detect or reject an effect size of d = 0.4 with 90% power, and the alpha level is set to 0.05 for a two-sided independent t test, a researcher would need to collect 133 participants in each condition for an informative null hypothesis test, and 136 participants in each condition for an informative equivalence test. Therefore, the researcher should aim to collect 272 (that is, 136 participants in each condition) participants in total for an informative result for both tests that are planned. This does not guarantee a study has sufficient power for the true effect size (which can never be known), but it guarantees the study has sufficient power given an assumption of the effect a researcher is interested in detecting or rejecting. Therefore, an a-priori power analysis is useful, as long as a researcher can justify the effect sizes they are interested in.

If researchers correct the alpha level when testing multiple hypotheses, the a-priori power analysis should be based on this corrected alpha level. For example, if four tests are performed, an overall Type I error rate of 5% is desired, and a Bonferroni correction is used, the a-priori power analysis should be based on a corrected alpha level of .0125.

An a-priori power analysis can be performed analytically or by performing computer simulations. Analytic solutions are faster but less flexible. A common challenge researchers face when attempting to perform power analyses for more complex or uncommon tests is that available software does not offer analytic solutions. In these cases simulations can provide a flexible solution to perform power analyses for any test ( Morris et al., 2019 ) . The following code is an example of a power analysis in R based on 10000 simulations for a one-sample t test against zero for a sample size of 20, assuming a true effect of d = 0.5. All simulations consist of first randomly generating data based on assumptions of the data generating mechanism (e.g., a normal distribution with a mean of 0.5 and a standard deviation of 1), followed by a test performed on the data. By computing the percentage of significant results, power can be computed for any design.

There is a wide range of tools available to perform power analyses. Whichever tool a researcher decides to use, it will take time to learn how to use the software correctly to perform a meaningful a-priori power analysis. Resources to educate psychologists about power analysis consist of book-length treatments ( Aberson, 2019 ; Cohen, 1988 ; Julious, 2004 ; Murphy et al., 2014 ) , general introductions ( Baguley, 2004 ; Brysbaert, 2019 ; Faul et al., 2007 ; Maxwell et al., 2008 ; Perugini et al., 2018 ) , and an increasing number of applied tutorials for specific tests ( Brysbaert & Stevens, 2018 ; DeBruine & Barr, 2021 ; P. Green & MacLeod, 2016 ; Kruschke, 2013 ; Lakens & Caldwell, 2021 ; Schoemann et al., 2017 ; Westfall et al., 2014 ) . It is important to be trained in the basics of power analysis, and it can be extremely beneficial to learn how to perform simulation-based power analyses. At the same time, it is often recommended to enlist the help of an expert, especially when a researcher lacks experience with a power analysis for a specific test.

When reporting an a-priori power analysis, make sure that the power analysis is completely reproducible. If power analyses are performed in R it is possible to share the analysis script and information about the version of the package. In many software packages it is possible to export the power analysis that is performed as a PDF file. For example, in G*Power analyses can be exported under the ‘protocol of power analysis’ tab. If the software package provides no way to export the analysis, add a screenshot of the power analysis to the supplementary files.

The reproducible report needs to be accompanied by justifications for the choices that were made with respect to the values used in the power analysis. If the effect size used in the power analysis is based on previous research, the factors presented in Table  8.5 (if the effect size is based on a meta-analysis) or Table  8.6 (if the effect size is based on a single study) should be discussed. If an effect size estimate is based on the existing literature, provide a full citation, and preferably a direct quote from the article where the effect size estimate is reported. If the effect size is based on a smallest effect size of interest, this value should not just be stated, but justified (e.g., based on theoretical predictions or practical implications, see Lakens, Scheel, et al. ( 2018 ) ). For an overview of all aspects that should be reported when describing an a-priori power analysis, see Table  8.4 .

8.7 Planning for Precision

Some researchers have suggested to justify sample sizes based on a desired level of precision of the estimate ( Cumming & Calin-Jageman, 2016 ; Kruschke, 2018 ; Maxwell et al., 2008 ) . The goal when justifying a sample size based on precision is to collect data to achieve a desired width of the confidence interval around a parameter estimate. The width of the confidence interval around the parameter estimate depends on the standard deviation and the number of observations. The only aspect a researcher needs to justify for a sample size justification based on accuracy is the desired width of the confidence interval with respect to their inferential goal, and their assumption about the population standard deviation of the measure.

If a researcher has determined the desired accuracy, and has a good estimate of the true standard deviation of the measure, it is straightforward to calculate the sample size needed for a desired level of accuracy. For example, when measuring the IQ of a group of individuals a researcher might desire to estimate the IQ score within an error range of 2 IQ points for 95% of the observed means, in the long run. The required sample size to achieve this desired level of accuracy (assuming normally distributed data) can be computed by:

\[N = \left(\frac{z \cdot sd}{error}\right)^2\]

where N is the number of observations, z is the critical value related to the desired confidence interval, sd is the standard deviation of IQ scores in the population, and error is the width of the confidence interval within which the mean should fall, with the desired error rate. In this example, (1.96 × 15 / 2)^2 = 216.1 observations. If a researcher desires 95% of the means to fall within a 2 IQ point range around the true population mean, 217 observations should be collected. If a desired accuracy for a non-zero mean difference is computed, accuracy is based on a non-central t -distribution. For these calculations, an expected effect size estimate needs to be provided, but it has relatively little influence on the required sample size ( Maxwell et al., 2008 ) . It is also possible to incorporate uncertainty about the observed effect size in the sample size calculation, known as assurance ( Kelley & Rausch, 2006 ) . The MBESS package in R provides functions to compute sample sizes for a wide range of tests ( Kelley, 2007 ) .

What is less straightforward is to justify how a desired level of accuracy is related to inferential goals. There is no literature that helps researchers to choose a desired width of the confidence interval. Morey ( 2020 ) convincingly argues that most practical use-cases of planning for precision involve an inferential goal of distinguishing an observed effect from other effect sizes (for a Bayesian perspective, see Kruschke ( 2018 ) ). For example, a researcher might expect an effect size of r = 0.4 and would treat observed correlations that differ more than 0.2 (i.e., 0.2 < r < 0.6) differently, in that effects of r = 0.6 or larger are considered too large to be caused by the assumed underlying mechanism ( Hilgard, 2021 ) , while effects smaller than r = 0.2 are considered too small to support the theoretical prediction. If the goal is indeed to get an effect size estimate that is precise enough so that two effects can be differentiated with high probability, the inferential goal is actually a hypothesis test, which requires designing a study with sufficient power to reject effects (e.g., testing a range prediction of correlations between 0.2 and 0.6).

If researchers do not want to test a hypothesis, for example because they prefer an estimation approach over a testing approach, then in the absence of clear guidelines that help researchers to justify a desired level of precision, one solution might be to rely on a generally accepted norm of precision. This norm could be based on ideas about a certain resolution below which measurements in a research area no longer lead to noticeably different inferences. Just as researchers normatively use an alpha level of 0.05, they could plan studies to achieve a desired confidence interval width around the observed effect that is determined by a norm. Future work is needed to help researchers choose a confidence interval width when planning for accuracy (see also the section on which confidence interval to use in Bayesian tests of range predictions ).

8.8 Heuristics

When a researcher uses a heuristic, they are not able to justify their sample size themselves, but they trust in a sample size recommended by some authority. When I started as a PhD student in 2005 it was common to collect 15 participants in each between subject condition. When asked why this was a common practice, no one was really sure, but people trusted that there was a justification somewhere in the literature. Now, I realize there was no justification for the heuristics we used. As Berkeley ( 1735 ) already observed: “Men learn the elements of science from others: And every learner hath a deference more or less to authority, especially the young learners, few of that kind caring to dwell long upon principles, but inclining rather to take them upon trust: And things early admitted by repetition become familiar: And this familiarity at length passeth for evidence.”

Some papers provide researchers with simple rules of thumb about the sample size that should be collected. Such papers clearly fill a need, and are cited a lot, even when the advice in these articles is flawed. For example, Wilson VanVoorhis & Morgan ( 2007 ) translate an absolute minimum of 50+8 observations for regression analyses suggested by a rule of thumb examined in S. B. Green ( 1991 ) into the recommendation to collect ~50 observations. Green actually concludes in his article that “In summary, no specific minimum number of subjects or minimum ratio of subjects-to-predictors was supported”. He does discuss how a general rule of thumb of N = 50 + 8 provided an accurate minimum number of observations for the ‘typical’ study in the social sciences because these have a ‘medium’ effect size, as Green claims by citing Cohen (1988). Cohen actually didn’t claim that the typical study in the social sciences has a ‘medium’ effect size, and instead said (1988, p. 13): “Many effects sought in personality, social, and clinical-psychological research are likely to be small effects as here defined”. We see how a string of mis-citations eventually leads to a misleading rule of thumb.

Rules of thumb seem to primarily emerge due to mis-citations and/or overly simplistic recommendations. Simonsohn, Nelson, and Simmons ( 2011 ) recommended that “Authors must collect at least 20 observations per cell”. A later recommendation by the same authors presented at a conference suggested to use n > 50, unless you study large effects ( Simmons et al., 2013 ) . Regrettably, this advice is now often mis-cited as a justification to collect no more than 50 observations per condition without considering the expected effect size. If authors justify a specific sample size (e.g., n = 50) based on a general recommendation in another paper, either they are mis-citing the paper, or the paper they are citing is flawed.

Another common heuristic is to collect the same number of observations as were collected in a previous study. This strategy is not recommended in scientific disciplines with widespread publication bias, and/or where novel and surprising findings from largely exploratory single studies are published. Using the same sample size as a previous study is only a valid approach if the sample size justification in the previous study also applies to the current study. Instead of stating that you intend to collect the same sample size as an earlier study, repeat the sample size justification, and update it in light of any new information (such as the effect size in the earlier study, see Table  8.6 ).

Peer reviewers and editors should carefully scrutinize rules of thumb sample size justifications, because they can make it seem like a study has high informational value for an inferential goal even when the study will yield uninformative results. Whenever one encounters a sample size justification based on a heuristic, ask yourself: ‘Why is this heuristic used?’ It is important to know what the logic behind a heuristic is to determine whether the heuristic is valid for a specific situation. In most cases, heuristics are based on weak logic, and not widely applicable. That said, it might be possible that fields develop valid heuristics for sample size justifications. For example, it is possible that a research area reaches widespread agreement that effects smaller than d = 0.3 are too small to be of interest, and all studies in a field use sequential designs (see below) that have 90% power to detect a d = 0.3. Alternatively, it is possible that a field agrees that data should be collected with a desired level of accuracy, irrespective of the true effect size. In these cases, valid heuristics would exist based on generally agreed goals of data collection. For example, Simonsohn ( 2015 ) suggests to design replication studies that have 2.5 times as large sample sizes as the original study, as this provides 80% power for an equivalence test against an equivalence bound set to the effect the original study had 33% power to detect, assuming the true effect size is 0. As original authors typically do not specify which effect size would falsify their hypothesis, the heuristic underlying this ‘small telescopes’ approach is a good starting point for a replication study with the inferential goal to reject the presence of an effect as large as was described in an earlier publication. It is the responsibility of researchers to gain the knowledge to distinguish valid heuristics from mindless heuristics, and to be able to evaluate whether a heuristic will yield an informative result given the inferential goal of the researchers in a specific study, or not.

8.9 No Justification

It might sound like a contradictio in terminis , but it is useful to distinguish a final category where researchers explicitly state they do not have a justification for their sample size. Perhaps the resources were available to collect more data, but they were not used. A researcher could have performed a power analysis, or planned for precision, but they did not. In those cases, instead of pretending there was a justification for the sample size, honesty requires you to state there is no sample size justification. This is not necessarily bad. It is still possible to discuss the smallest effect size of interest, the minimal statistically detectable effect, the width of the confidence interval around the effect size, and to plot a sensitivity power analysis, in relation to the sample size that was collected. If a researcher truly had no specific inferential goals when collecting the data, such an evaluation can perhaps be performed based on reasonable inferential goals peers would have when they learn about the existence of the collected data.

Do not try to spin a story where it looks like a study was highly informative when it was not. Instead, transparently evaluate how informative the study was given effect sizes that were of interest, and make sure that the conclusions follow from the data. The lack of a sample size justification might not be problematic, but it might mean that a study was not informative for most effect sizes of interest, which makes it especially difficult to interpret non-significant effects, or estimates with large uncertainty.

8.10 What is Your Inferential Goal?

The inferential goal of data collection is often in some way related to the size of an effect. Therefore, to design an informative study, researchers will want to think about which effect sizes are interesting. First, it is useful to consider three effect sizes when determining the sample size. The first is the smallest effect size a researcher is interested in, the second is the smallest effect size that can be statistically significant (only in studies where a significance test will be performed), and the third is the effect size that is expected. Beyond considering these three effect sizes, it can be useful to evaluate ranges of effect sizes. This can be done by computing the width of the expected confidence interval around an effect size of interest (for example, an effect size of zero), and examine which effects could be rejected. Similarly, it can be useful to plot a sensitivity curve and evaluate the range of effect sizes the design has decent power to detect, as well as to consider the range of effects for which the design has low power. Finally, there are situations where it is useful to consider a range of effects that is likely to be observed in a specific research area.

8.11 What is the Smallest Effect Size of Interest?

The strongest possible sample size justification is based on an explicit statement of the smallest effect size that is considered interesting. The smallest effect size of interest can be based on theoretical predictions or practical considerations. For a review of approaches that can be used to determine the smallest effect size of interest in randomized controlled trials, see Cook et al. ( 2014 ) and Keefe et al. ( 2013 ) , for reviews of different methods to determine a smallest effect size of interest, see King ( 2011 ) and Copay et al. ( 2007 ) , and for a discussion focused on psychological research, see Lakens, Scheel, et al. ( 2018 ) .

It can be challenging to determine the smallest effect size of interest whenever theories are not very developed, or when the research question is far removed from practical applications, but it is still worth thinking about which effects would be too small to matter. A first step forward is to discuss which effect sizes are considered meaningful in a specific research line with your peers. Researchers will differ in the effect sizes they consider large enough to be worthwhile ( Murphy et al., 2014 ) . Just as not every scientist will find every research question interesting enough to study, not every scientist will consider the same effect sizes interesting enough to study, and different stakeholders will differ in which effect sizes are considered meaningful ( Kelley & Preacher, 2012 ) .

Even though it might be challenging, there are important benefits of being able to specify the smallest effect size of interest. The population effect size is always uncertain (indeed, estimating this is typically one of the goals of the study), and therefore whenever a study is powered for an expected effect size, there is considerable uncertainty about whether the statistical power is high enough to detect the true effect in the population. However, if the smallest effect size of interest can be specified and agreed upon after careful deliberation, it becomes possible to design a study that has sufficient power (given the inferential goal to detect or reject the smallest effect size of interest with a certain error rate). Put differently, although the smallest effect of interest may be subjective (one researcher might find effect sizes smaller than d = 0.3 meaningless, while another researcher might still be interested in effects smaller than d = 0.1), and there might be uncertainty about the parameters required to specify the smallest effect size of interest (e.g., when performing a cost-benefit analysis), once researchers determine the smallest effect size of interest, a study can be designed with a known Type II error rate to detect or reject this value. For this reason an a-priori power based on a smallest effect size of interest is generally preferred, whenever researchers are able to specify one ( Aberson, 2019 ; Albers & Lakens, 2018 ; Brown, 1983 ; Cascio & Zedeck, 1983 ; Dienes, 2014 ; Lenth, 2001 ) .

8.12 The Minimal Statistically Detectable Effect

The minimal statistically detectable effect is the smallest effect size that, if observed, would yield a statistically significant p -value ( Cook et al., 2014 ) . In Figure  8.4 , the distribution of Cohen’s d is plotted for 15 participants per group when the true effect size is either d = 0 or d = 0.5. This figure is similar to Figure  8.2 , with the addition that the critical d is indicated. We see that with such a small number of observations in each group only observed effects larger than d = 0.75 will be statistically significant. Whether such effect sizes are interesting, and can realistically be expected, should be carefully considered and justified.

Computing a minimal statistically detectable effect is useful for a study where no a-priori power analysis is performed, both for studies in the published literature that do not report a sample size justification ( Lakens, Scheel, et al., 2018 ) , as for researchers who rely on heuristics for their sample size justification.

It can be informative to ask yourself whether the critical effect size for a study design is within the range of effect sizes that can realistically be expected. If not, then whenever a significant effect is observed in a published study, either the effect size is surprisingly larger than expected, or more likely, it is an upwardly biased effect size estimate. In the latter case, given publication bias, published studies will lead to biased effect size estimates. If it is still possible to increase the sample size, for example by ignoring rules of thumb and instead performing an a-priori power analysis, then do so. If it is not possible to increase the sample size, for example due to resource constraints, then reflecting on the minimal statistically detectable effect should make it clear that an analysis of the data should not focus on p values, but on the effect size and the confidence interval (see Table  8.3 ).

It is also useful to compute the minimal statistically detectable effect if an ‘optimistic’ power analysis is performed. For example, if you believe a best case scenario for the true effect size is d = 0.57 and use this optimistic expectation in an a-priori power analysis, effects smaller than d = 0.4 will not be statistically significant when you collect 50 observations in a two independent group design. If your worst case scenario for the alternative hypothesis is a true effect size of d = 0.35 your design would not allow you to declare a significant effect if effect size estimates close to the worst case scenario are observed. Taking into account the minimal statistically detectable effect size should make you reflect on whether a hypothesis test will yield an informative answer, and whether your current approach to sample size justification (e.g., the use of rules of thumb, or letting resource constraints determine the sample size) leads to an informative study, or not.

8.13 What is the Expected Effect Size?

Although the true population effect size is always unknown, there are situations where researchers have a reasonable expectation of the effect size in a study, and want to use this expected effect size in an a-priori power analysis. Even if expectations for the observed effect size are largely a guess, it is always useful to explicitly consider which effect sizes are expected. A researcher can justify a sample size based on the effect size they expect, even if such a study would not be very informative with respect to the smallest effect size of interest. In such cases a study is informative for one inferential goal (testing whether the expected effect size is present or absent), but not highly informative for the second goal (testing whether the smallest effect size of interest is present or absent).

There are typically three sources for expectations about the population effect size: a meta-analysis, a previous study, or a theoretical model. It is tempting for researchers to be overly optimistic about the expected effect size in an a-priori power analysis, as higher effect size estimates yield lower sample sizes, but being too optimistic increases the probability of observing a false negative result. When reviewing a sample size justification based on an a-priori power analysis, it is important to critically evaluate the justification for the expected effect size used in power analyses.

8.14 Using an Estimate from a Meta-Analysis

In a perfect world effect size estimates from a meta-analysis would provide researchers with the most accurate information about which effect size they could expect. Due to widespread publication bias in science, effect size estimates from meta-analyses are regrettably not always accurate. They can be biased, sometimes substantially so. Furthermore, meta-analyses typically have considerable heterogeneity, which means that the meta-analytic effect size estimate differs for subsets of studies that make up the meta-analysis. So, although it might seem useful to use a meta-analytic effect size estimate of the effect you are studying in your power analysis, you need to take great care before doing so.

If a researcher wants to enter a meta-analytic effect size estimate in an a-priori power analysis, they need to consider three things (see Table  8.5 ). First, the studies included in the meta-analysis should be similar enough to the study they are performing that it is reasonable to expect a similar effect size. In essence, this requires evaluating the generalizability of the effect size estimate to the new study. It is important to carefully consider differences between the meta-analyzed studies and the planned study, with respect to the manipulation, the measure, the population, and any other relevant variables.

Second, researchers should check whether the effect sizes reported in the meta-analysis are homogeneous. If there is substantial heterogeinity in the meta-analytic effect sizes, it means not all included studies can be expected to have the same true effect size estimate. A meta-analytic estimate should be used based on the subset of studies that most closely represent the planned study. Note that heterogeneity remains a possibility (even direct replication studies can show heterogeneity when unmeasured variables moderate the effect size in each sample ( Kenny & Judd, 2019 ; Olsson-Collentine et al., 2020 ) ), so the main goal of selecting similar studies is to use existing data to increase the probability that your expectation is accurate, without guaranteeing it will be.

Third, the meta-analytic effect size estimate should not be biased. Check if the bias detection tests that are reported in the meta-analysis are state-of-the-art, or perform multiple bias detection tests yourself ( Carter et al., 2019 ) , and consider bias corrected effect size estimates (even though these estimates might still be biased, and do not necessarily reflect the true population effect size).

8.15 Using an Estimate from a Previous Study

If a meta-analysis is not available, researchers often rely on an effect size from a previous study in an a-priori power analysis. The first issue that requires careful attention is whether the two studies are sufficiently similar. Just as when using an effect size estimate from a meta-analysis, researchers should consider if there are differences between the studies in terms of the population, the design, the manipulations, the measures, or other factors that should lead one to expect a different effect size. For example, intra-individual reaction time variability increases with age, and therefore a study performed on older participants should expect a smaller standardized effect size than a study performed on younger participants. If an earlier study used a very strong manipulation, and you plan to use a more subtle manipulation, a smaller effect size should be expected. Finally, effect sizes do not generalize to studies with different designs. For example, the effect size for a comparison between two groups is most often not similar to the effect size for an interaction in a follow-up study where a second factor is added to the original design ( Lakens & Caldwell, 2021 ) .

Even if a study is sufficiently similar, statisticians have warned against using effect size estimates from small pilot studies in power analyses. Leon, Davis, and Kraemer ( 2011 ) write:

Contrary to tradition, a pilot study does not provide a meaningful effect size estimate for planning subsequent studies due to the imprecision inherent in data from small samples.

The two main reasons researchers should be careful when using effect sizes from studies in the published literature in power analyses is that effect size estimates from studies can differ from the true population effect size due to random variation, and that publication bias inflates effect sizes. Figure  8.5 shows the distribution for \(\eta_p^2\) for a study with three conditions with 25 participants in each condition when the null hypothesis is true (dotted grey curve) and when there is a ‘medium’ true effect of \(\eta_p^2\) = 0.0588 [solid black curve; Richardson ( 2011 ) ]. As in Figure  8.4 the critical effect size is indicated, which shows observed effects smaller than \(\eta_p^2\) = 0.08 will not be significant with the given sample size. If the null hypothesis is true, effects larger than \(\eta_p^2\) = 0.08 will be a Type I error (the dark grey area), and when the alternative hypothesis is true effects smaller than \(\eta_p^2\) = 0.08 will be a Type II error (light grey area). It is clear all significant effects are larger than the true effect size ( \(\eta_p^2\) = 0.0588), so power analyses based on a significant finding (e.g., because only significant results are published in the literature) will be based on an overestimate of the true effect size, introducing bias.

But even if we had access to all effect sizes (e.g., from pilot studies you have performed yourself) due to random variation the observed effect size will sometimes be quite small. Figure  8.5 shows it is quite likely to observe an effect of \(\eta_p^2\) = 0.01 in a small pilot study, even when the true effect size is 0.0588. Entering an effect size estimate of \(\eta_p^2\) = 0.01 in an a-priori power analysis would suggest a total sample size of 957 observations to achieve 80% power in a follow-up study. If researchers only follow up on pilot studies when they observe an effect size in the pilot study that, when entered into a power analysis, yields a sample size that is feasible to collect for the follow-up study, these effect size estimates will be upwardly biased, and power in the follow-up study will be systematically lower than desired ( Albers & Lakens, 2018 ) .

In essence, the problem with using small studies to estimate the effect size that will be entered into an a-priori power analysis is that due to publication bias or follow-up bias the effect sizes researchers end up using for their power analysis do not come from a full F distribution, but from what is known as a truncated F distribution ( Taylor & Muller, 1996 ) . For example, imagine if there is extreme publication bias in the situation illustrated in Figure  8.5 . The only studies that would be accessible to researchers would come from the part of the distribution where \(\eta_p^2\) > 0.08, and the test result would be statistically significant. It is possible to compute an effect size estimate that, based on certain assumptions, corrects for bias. For example, imagine we observe a result in the literature for a One-Way ANOVA with 3 conditions, reported as F (2, 42) = 0.017, \(\eta_p^2\) = 0.176. If we would take this effect size at face value and enter it as our effect size estimate in an a-priori power analysis, the suggested sample size to achieve 80% power would suggest we need to collect 17 observations in each condition.

However, if we assume bias is present, we can use the BUCSS R package ( Anderson et al., 2017 ) to perform a power analysis that attempts to correct for bias. In the example above, a power analysis that takes bias into account (under a specific model of publication bias, based on a truncated F distribution where only significant results are published) suggests collecting 73 participants in each condition instead. It is possible that the bias corrected estimate of the non-centrality parameter used to compute power is zero, in which case it is not possible to correct for bias using this method. As an alternative to formally modeling a correction for publication bias whenever researchers assume an effect size estimate is biased, researchers can simply use a more conservative effect size estimate, for example by computing power based on the lower limit of a 60% two-sided confidence interval around the effect size estimate, which Perugini et al. ( 2014 ) refer to as safeguard power . Both these approaches lead to a more conservative power analysis, but not necessarily a more accurate power analysis. It is simply not possible to perform an accurate power analysis on the basis of an effect size estimate from a study that might be biased and/or had a small sample size ( Teare et al., 2014 ) . If it is not possible to specify a smallest effect size of interest, and there is great uncertainty about which effect size to expect, it might be more efficient to perform a study with a sequential design (discussed below).

To summarize, an effect size from a previous study in an a-priori power analysis can be used if three conditions are met (see Table  8.6 ). First, the previous study is sufficiently similar to the planned study. Second, there was a low risk of bias (e.g., the effect size estimate comes from a Registered Report, or from an analysis for which results would not have impacted the likelihood of publication). Third, the sample size is large enough to yield a relatively accurate effect size estimate, based on the width of a 95% CI around the observed effect size estimate. There is always uncertainty around the effect size estimate, and entering the upper and lower limit of the 95% CI around the effect size estimate might be informative about the consequences of the uncertainty in the effect size estimate for an a-priori power analysis.

8.16 Using an Estimate from a Theoretical Model

When your theoretical model is sufficiently specific such that you can build a computational model, and you have knowledge about key parameters in your model that are relevant for the data you plan to collect, it is possible to estimate an effect size based on the effect size estimate derived from a computational model. For example, if one had strong ideas about the weights for each feature stimuli share and differ on, it could be possible to compute predicted similarity judgments for pairs of stimuli based on Tversky’s contrast model ( Tversky, 1977 ) , and estimate the predicted effect size for differences between experimental conditions. Although computational models that make point predictions are relatively rare, whenever they are available, they provide a strong justification of the effect size a researcher expects.

8.17 Compute the Width of the Confidence Interval around the Effect Size

If a researcher can estimate the standard deviation of the observations that will be collected, it is possible to compute an a-priori estimate of the width of the 95% confidence interval around an effect size ( Kelley, 2007 ) . Confidence intervals represent a range around an estimate that is wide enough so that in the long run the true population parameter will fall inside the confidence intervals 100 - \(\alpha\) percent of the time. In any single study the true population effect either falls in the confidence interval, or it doesn’t, but in the long run one can act as if the confidence interval includes the true population effect size (while keeping the error rate in mind). Cumming ( 2013 ) calls the difference between the observed effect size and the upper bound of the 95% confidence interval (or the lower bound of the 95% confidence interval) the margin of error.

If we compute the 95% CI for an effect size of d = 0 based on the t statistic and sample size ( Smithson, 2003 ) , we see that with 15 observations in each condition of an independent t test the 95% CI ranges from d = -0.716 to d = 0.716. Confidence intervals around effect sizes can be computed using the MOTE Shiny app: https://www.aggieerin.com/shiny-server/. The margin of error is half the width of the 95% CI, 0.716. A Bayesian estimator who uses an uninformative prior would compute a credible interval with the same (or a very similar) upper and lower bound ( Albers et al., 2018 ; Kruschke, 2011 ) , and might conclude that after collecting the data they would be left with a range of plausible values for the population effect that is too large to be informative. Regardless of the statistical philosophy you plan to rely on when analyzing the data, the evaluation of what we can conclude based on the width of our interval tells us that with 15 observation per group we will not learn a lot.

One useful way of interpreting the width of the confidence interval is based on the effects you would be able to reject if the true effect size is 0. In other words, if there is no effect, which effects would you have been able to reject given the collected data, and which effect sizes would not be rejected, if there was no effect? Effect sizes in the range of d = 0.7 are findings such as “People become aggressive when they are provoked”, “People prefer their own group to other groups”, and “Romantic partners resemble one another in physical attractiveness” ( Richard et al., 2003 ) . The width of the confidence interval tells you that you can only reject the presence of effects that are so large, if they existed, you would probably already have noticed them. If it is true that most effects that you study are realistically much smaller than d = 0.7, there is a good possibility that we do not learn anything we didn’t already know by performing a study with n = 15. Even without data, in most research lines we would not consider certain large effects plausible (although the effect sizes that are plausible differ between fields, as discussed below). On the other hand, in large samples where researchers can for example reject the presence of effects larger than d = 0.2, if the null hypothesis was true, this analysis of the width of the confidence interval would suggest that peers in many research lines would likely consider the study to be informative.

We see that the margin of error is almost, but not exactly, the same as the minimal statistically detectable effect ( d = 0.748). The small variation is due to the fact that the 95% confidence interval is calculated based on the t distribution. If the true effect size is not zero, the confidence interval is calculated based on the non-central t distribution, and the 95% CI is asymmetric. Figure  8.6 visualizes three t distributions, one symmetric at 0, and two asymmetric distributions with a noncentrality parameter (the normalized difference between the means) of 2 and 3. The asymmetry is most clearly visible in very small samples (the distributions in the plot have 5 degrees of freedom) but remains noticeable in larger samples when calculating confidence intervals and statistical power. For example, for a true effect size of d = 0.5 observed with 15 observations per group would yield \(d_s\) = 0.50, 95% CI [-0.23, 1.22]. If we compute the 95% CI around the critical effect size, we would get \(d_s\) = 0.75, 95% CI [0.00, 1.48]. We see the 95% CI ranges from exactly 0 to 1.484, in line with the relation between a confidence interval and a p value, where the 95% CI excludes zero if the test is statistically significant. As noted before, the different approaches recommended here to evaluate how informative a study is are often based on the same information.

8.18 Plot a Sensitivity Power Analysis

A sensitivity power analysis fixes the sample size, desired power, and alpha level, and answers the question which effect size a study could detect with a desired power. A sensitivity power analysis is therefore performed when the sample size is already known. Sometimes data has already been collected to answer a different research question, or the data is retrieved from an existing database, and you want to perform a sensitivity power analysis for a new statistical analysis. Other times, you might not have carefully considered the sample size when you initially collected the data, and want to reflect on the statistical power of the study for (ranges of) effect sizes of interest when analyzing the results. Finally, it is possible that the sample size will be collected in the future, but you know that due to resource constraints the maximum sample size you can collect is limited, and you want to reflect on whether the study has sufficient power for effects that you consider plausible and interesting (such as the smallest effect size of interest, or the effect size that is expected).

Assume a researcher plans to perform a study where 30 observations will be collected in total, 15 in each between participant condition. Figure  8.7 shows how to perform a sensitivity power analysis in G*Power for a study where we have decided to use an alpha level of 5%, and desire 90% power. The sensitivity power analysis reveals the designed study has 90% power to detect effects of at least d = 1.23. Perhaps a researcher believes that a desired power of 90% is quite high, and is of the opinion that it would still be interesting to perform a study if the statistical power was lower. It can then be useful to plot a sensitivity curve across a range of smaller effect sizes.

The two dimensions of interest in a sensitivity power analysis are the effect sizes, and the power to observe a significant effect assuming a specific effect size. Fixing the sample size, these two dimensions can be plotted against each other to create a sensitivity curve. For example, a sensitivity curve can be plotted in G*Power by clicking the ‘X-Y plot for a range of values’ button, as illustrated in Figure  8.8 . Researchers can examine which power they would have for an a-priori plausible range of effect sizes, or they can examine which effect sizes would provide reasonable levels of power. In simulation-based approaches to power analysis, sensitivity curves can be created by performing the power analysis for a range of possible effect sizes. Even if 50% power is deemed acceptable (in which case deciding to act as if the null hypothesis is true after a non-significant result is a relatively noisy decision procedure), Figure  8.8 shows a study design where power is extremely low for a large range of effect sizes that are reasonable to expect in most fields. Thus, a sensitivity power analysis provides an additional approach to evaluate how informative the planned study is, and can inform researchers that a specific design is unlikely to yield a significant effect for a range of effects that one might realistically expect.

If the number of observations per group had been larger, the evaluation might have been more positive. We might not have had any specific effect size in mind, but if we had collected 150 observations per group, a sensitivity analysis could have shown that power was sufficient for a range of effects we believe is most interesting to examine, and we would still have approximately 50% power for quite small effects. For a sensitivity analysis to be meaningful, the sensitivity curve should be compared against a smallest effect size of interest, or a range of effect sizes that are expected. A sensitivity power analysis has no clear cut-offs to examine ( Bacchetti, 2010 ) . Instead, the idea is to make a holistic trade-off between different effect sizes one might observe or care about, and their associated statistical power.

8.19 The Distribution of Effect Sizes in a Research Area

In my personal experience the most commonly entered effect size estimate in an a-priori power analysis for an independent t test is Cohen’s benchmark for a ‘medium’ effect size, because of what is known as the default effect . When you open G*Power, a ‘medium’ effect is the default option for an a-priori power analysis. Cohen’s benchmarks for small, medium, and large effects should not be used in an a-priori power analysis ( Cook et al., 2014 ; Correll et al., 2020 ) , and Cohen regretted having proposed these benchmarks ( Funder & Ozer, 2019 ) . The large variety in research topics means that any ‘default’ or ‘heuristic’ that is used to compute statistical power is not just unlikely to correspond to your actual situation, but it is also likely to lead to a sample size that is substantially misaligned with the question you are trying to answer with the collected data.

Some researchers have wondered what a better default would be, if researchers have no other basis to decide upon an effect size for an a-priori power analysis. Brysbaert ( 2019 ) recommends d = 0.4 as a default in psychology, which is the average observed in replication projects and several meta-analyses. It is impossible to know if this average effect size is realistic, but it is clear there is huge heterogeneity across fields and research questions. Any average effect size will often deviate substantially from the effect size that should be expected in a planned study. Some researchers have suggested to change Cohen’s benchmarks based on the distribution of effect sizes in a specific field ( Bosco et al., 2015 ; Funder & Ozer, 2019 ; Hill et al., 2008 ; Kraft, 2020 ; Lovakov & Agadullina, 2021 ) . As always, when effect size estimates are based on the published literature, one needs to evaluate the possibility that the effect size estimates are inflated due to publication bias. Due to the large variation in effect sizes within a specific research area, there is little use in choosing a large, medium, or small effect size benchmark based on the empirical distribution of effect sizes in a field to perform a power analysis.

Having some knowledge about the distribution of effect sizes in the literature can be useful when interpreting the confidence interval around an effect size. If in a specific research area almost no effects are larger than the value you could reject in an equivalence test (e.g., if the observed effect size is 0, the design would only reject effects larger than for example d = 0.7), then it is a-priori unlikely that collecting the data would tell you something you didn’t already know.

It is more difficult to defend the use of a specific effect size derived from an empirical distribution of effect sizes as a justification for the effect size used in an a-priori power analysis. One might argue that the use of an effect size benchmark based on the distribution of effects in the literature will outperform a wild guess, but this is not a strong enough argument to form the basis of a sample size justification. There is a point where researchers need to admit they are not ready to perform an a-priori power analysis due to a lack of clear expectations ( Scheel et al., 2021 ) . Alternative sample size justifications, such as a justification of the sample size based on resource constraints, perhaps in combination with a sequential study design, might be more in line with the actual inferential goals of a study.

8.20 Additional Considerations When Designing an Informative Study

So far, the focus has been on justifying the sample size for quantitative studies. There are a number of related topics that can be useful to design an informative study. First, in addition to a-priori or prospective power analysis and sensitivity power analysis, it is important to discuss compromise power analysis (which is useful) and post-hoc or retrospective power analysis (which is not useful, e.g., Zumbo & Hubley ( 1998 ) , Lenth ( 2007 ) ). When sample sizes are justified based on an a-priori power analysis it can be very efficient to collect data in sequential designs where data collection is continued or terminated based on interim analyses of the data. Furthermore, it is worthwhile to consider ways to increase the power of a test without increasing the sample size. An additional point of attention is to have a good understanding of your dependent variable, especially it’s standard deviation. Finally, sample size justification is just as important in qualitative studies, and although there has been much less work on sample size justification in this domain, some proposals exist that researchers can use to design an informative study. Each of these topics is discussed in turn.

8.21 Compromise Power Analysis

In a compromise power analysis the sample size and the effect are fixed, and the error rates of the test are calculated, based on a desired ratio between the Type I and Type II error rate. A compromise power analysis is useful both when a very large number of observations will be collected, as when only a small number of observations can be collected.

In the first situation a researcher might be fortunate enough to be able to collect so many observations that the statistical power for a test is very high for all effect sizes that are deemed interesting. For example, imagine a researcher has access to 2000 employees who are all required to answer questions during a yearly evaluation in a company where they are testing an intervention that should reduce subjectively reported stress levels. You are quite confident that an effect smaller than d = 0.2 is not large enough to be subjectively noticeable for individuals ( Jaeschke et al., 1989 ) . With an alpha level of 0.05 the researcher would have a statistical power of 0.994, or a Type II error rate of 0.006. This means that for the smallest effect size of interest of d = 0.2 the researcher is 8.3 times more likely to make a Type I error than a Type II error.

Although the original idea of designing studies that control Type I and Type II error rates was that researchers would need to justify their error rates ( Neyman & Pearson, 1933 ) , a common heuristic is to set the Type I error rate to 0.05 and the Type II error rate to 0.20, meaning that a Type I error is 4 times as unlikely as a Type II error. This default use of 80% power (or a Type II error rate/ \(\beta\) of 0.20) is based on a personal preference of Cohen ( 1988 ) , who writes:

It is proposed here as a convention that, when the investigator has no other basis for setting the desired power value, the value .80 be used. This means that \(\beta\) is set at .20. This arbitrary but reasonable value is offered for several reasons (Cohen, 1965, pp. 98-99). The chief among them takes into consideration the implicit convention for \(\alpha\) of .05. The \(\beta\) of .20 is chosen with the idea that the general relative seriousness of these two kinds of errors is of the order of .20/.05, i.e., that Type I errors are of the order of four times as serious as Type II errors. This .80 desired power convention is offered with the hope that it will be ignored whenever an investigator can find a basis in his substantive concerns in his specific research investigation to choose a value ad hoc.

We see that conventions are built on conventions: the norm to aim for 80% power is built on the norm to set the alpha level at 5%. What we should take away from Cohen is not that we should aim for 80% power, but that we should justify our error rates based on the relative seriousness of each error. This is where compromise power analysis comes in. If you share Cohen’s belief that a Type I error is 4 times as serious as a Type II error, and building on our earlier study on 2000 employees, it makes sense to adjust the Type I error rate when the Type II error rate is low for all effect sizes of interest ( Cascio & Zedeck, 1983 ) . Indeed, Erdfelder et al. ( 1996 ) created the G*Power software in part to give researchers a tool to perform compromise power analysis.

Figure  8.9 illustrates how a compromise power analysis is performed in G*Power when a Type I error is deemed to be equally costly as a Type II error (that is, when the \(\beta/\alpha\) ratio = 1), which for a study with 1000 observations per condition would lead to a Type I error and a Type II error of 0.0179. As Faul, Erdfelder, Lang, and Buchner ( 2007 ) write:

Of course, compromise power analyses can easily result in unconventional significance levels greater than \(\alpha\) = .05 (in the case of small samples or effect sizes) or less than \(\alpha\) = .001 (in the case of large samples or effect sizes). However, we believe that the benefit of balanced Type I and Type II error risks often offsets the costs of violating significance level conventions.

This brings us to the second situation where a compromise power analysis can be useful, which is when we know the statistical power in our study is low. Although it is highly undesirable to make decisions when error rates are high, if one finds oneself in a situation where a decision must be made based on little information, Winer ( 1962 ) writes:

The frequent use of the .05 and .01 levels of significance is a matter of convention having little scientific or logical basis. When the power of tests is likely to be low under these levels of significance, and when Type I and Type II errors are of approximately equal importance, the .30 and .20 levels of significance may be more appropriate than the .05 and .01 levels.

For example, if we plan to perform a two-sided t test, can feasibly collect at most 50 observations in each independent group, and expect a population effect size of 0.5, we would have 70% power if we set our alpha level to 0.05. Alternatively, using compromise power analysis, we can choose to weigh both types of error equally ( \(\beta/\alpha\) ratio = 1) by setting both the alpha level and Type II error rate to 0.149. Doing so, we would have 85.10% power to detect the expected population effect size of d = 0.5 instead. The choice of \(\alpha\) and \(\beta\) in a compromise power analysis can be extended to take prior probabilities of the null and alternative hypothesis into account ( Maier & Lakens, 2022 ; Miller & Ulrich, 2019 ; Murphy et al., 2014 ) .

A compromise power analysis requires a researcher to specify the sample size. This sample size itself requires a justification, so a compromise power analysis will typically be performed together with a resource constraint justification for a sample size. It is especially important to perform a compromise power analysis if your resource constraint justification is strongly based on the need to make a decision, in which case a researcher should think carefully about the Type I and Type II error rates stakeholders are willing to accept. However, a compromise power analysis also makes sense if the sample size is very large, but a researcher did not have the freedom to set the sample size. This might happen if, for example, data collection is part of a larger international study and the sample size is based on other research questions. In designs where the Type II error rate is very small (and power is very high) some statisticians have also recommended to lower the alpha level to prevent Lindley’s paradox, a situation where a significant effect ( p < \(\alpha\) ) is evidence for the null hypothesis ( Good, 1992 ; Jeffreys, 1939 ) . Lowering the alpha level as a function of the statistical power of the test can prevent this paradox, providing another argument for a compromise power analysis when sample sizes are large ( Maier & Lakens, 2022 ) . Finally, a compromise power analysis needs a justification for the effect size, either based on a smallest effect size of interest or an effect size that is expected. Table  8.7 lists three aspects that should be discussed alongside a reported compromise power analysis.

8.22 What to do if Your Editor Asks for Post-hoc Power?

Post-hoc, retrospective, or observed power is used to describe the statistical power of the test that is computed assuming the effect size that has been estimated from the collected data is the true effect size ( Lenth, 2007 ; Zumbo & Hubley, 1998 ) . Post-hoc power is therefore not performed before looking at the data, based on effect sizes that are deemed interesting, as in an a-priori power analysis, and it is unlike a sensitivity power analysis where a range of interesting effect sizes is evaluated. Because a post-hoc or retrospective power analysis is based on the effect size observed in the data that has been collected, it does not add any information beyond the reported p value, but it presents the same information in a different way. Despite this fact, editors and reviewers often ask authors to perform post-hoc power analysis to interpret non-significant results. This is not a sensible request, and whenever it is made, you should not comply with it. Instead, you should perform a sensitivity power analysis, and discuss the power for the smallest effect size of interest and a realistic range of expected effect sizes.

Post-hoc power is directly related to the p value of the statistical test ( Hoenig & Heisey, 2001 ) . For a z test where the p value is exactly 0.05, post-hoc power is always 50%. The reason for this relationship is that when a p value is observed that equals the alpha level of the test (e.g., 0.05), the observed z score of the test is exactly equal to the critical value of the test (e.g., z = 1.96 in a two-sided test with a 5% alpha level). Whenever the alternative hypothesis is centered on the critical value half the values we expect to observe if this alternative hypothesis is true fall below the critical value, and half fall above the critical value. Therefore, a test where we observed a p value identical to the alpha level will have exactly 50% power in a post-hoc power analysis, as the analysis assumes the observed effect size is true.

For other statistical tests, where the alternative distribution is not symmetric (such as for the t test, where the alternative hypothesis follows a non-central t distribution, see Figure  8.6 ), a p = 0.05 does not directly translate to an observed power of 50%, but by plotting post-hoc power against the observed p value we see that the two statistics are always directly related. As Figure  8.10 shows, if the p value is non-significant (i.e., larger than 0.05) the observed power will be less than approximately 50% in a t test. Lenth ( 2007 ) explains how observed power is also completely determined by the observed p value for F tests, although the statement that a non-significant p value implies a power less than 50% no longer holds.

When editors or reviewers ask researchers to report post-hoc power analyses they would like to be able to distinguish between true negatives (concluding there is no effect, when there is no effect) and false negatives (a Type II error, concluding there is no effect, when there actually is an effect). Since reporting post-hoc power is just a different way of reporting the p value, reporting the post-hoc power will not provide an answer to the question editors are asking ( Hoenig & Heisey, 2001 ; Lenth, 2007 ; Schulz & Grimes, 2005 ; Yuan & Maxwell, 2005 ) . To be able to draw conclusions about the absence of a meaningful effect, one should perform an equivalence test , and design a study with high power to reject the smallest effect size of interest. Alternatively, if no smallest effect size of interest was specified when designing the study, researchers can report a sensitivity power analysis.

8.23 Sequential Analyses

Whenever the sample size is justified based on an a-priori power analysis it can be very efficient to collect data in a sequential design. Sequential designs control error rates across multiple looks at the data (e.g., after 50, 100, and 150 observations have been collected) and can reduce the average expected sample size that is collected compared to a fixed design where data is only analyzed after the maximum sample size is collected ( Proschan et al., 2006 ; Wassmer & Brannath, 2016 ) . Sequential designs have a long history ( Dodge & Romig, 1929 ) , and exist in many variations, such as the Sequential Probability Ratio Test ( Wald, 1945 ) , combining independent statistical tests ( Westberg, 1985 ) , group sequential designs ( Jennison & Turnbull, 2000 ) , sequential Bayes factors ( Schönbrodt et al., 2017 ) , and safe testing ( Grünwald et al., 2019 ) . Of these approaches, the Sequential Probability Ratio Test is most efficient if data can be analyzed after every observation ( Schnuerch & Erdfelder, 2020 ) . Group sequential designs, where data is collected in batches, provide more flexibility in data collection, error control, and corrections for effect size estimates ( Wassmer & Brannath, 2016 ) . Safe tests provide optimal flexibility if there are dependencies between observations ( ter Schure & Grünwald, 2019 ) .

Sequential designs are especially useful when there is considerable uncertainty about the effect size, or when it is plausible that the true effect size is larger than the smallest effect size of interest the study is designed to detect ( Lakens, 2014 ) . In such situations data collection has the possibility to terminate early if the effect size is larger than the smallest effect size of interest, but data collection can continue to the maximum sample size if needed. Sequential designs can prevent waste when testing hypotheses, both by stopping early when the null hypothesis can be rejected, as by stopping early if the presence of a smallest effect size of interest can be rejected (i.e., stopping for futility). Group sequential designs are currently the most widely used approach to sequential analyses, and can be planned and analyzed using rpact or gsDesign . Shiny apps are available for both rpact and gsDesign .

8.24 Increasing Power Without Increasing the Sample Size

The most straightforward approach to increase the informational value of studies is to increase the sample size. Because resources are often limited, it is also worthwhile to explore different approaches to increasing the power of a test without increasing the sample size. The first option is to use directional (one-sided) tests where relevant, instead of two-sided tests. Researchers often make directional predictions, such as ‘we predict X is larger than Y’. The statistical test that logically follows from this prediction is a directional (or one-sided) t test. A directional test moves the Type I error rate to one side of the tail of the distribution, which lowers the critical value, and therefore requires less observations to achieve the same statistical power.

Although there is some discussion about when directional tests are appropriate, they are perfectly defensible from a Neyman-Pearson perspective on hypothesis testing ( Cho & Abe, 2013 ) , which makes a (preregistered) directional test a straightforward approach to both increase the power of a test, as the riskiness of the prediction. However, there might be situations where you do not want to ask a directional question. Sometimes, especially in research with applied consequences, it might be important to examine if a null effect can be rejected, even if the effect is in the opposite direction as predicted. For example, when you are evaluating a recently introduced educational intervention, and you predict the intervention will increase the performance of students, you might want to explore the possibility that students perform worse, to be able to recommend abandoning the new intervention. In such cases it is also possible to distribute the error rate in a ‘lop-sided’ manner, for example assigning a stricter error rate to effects in the negative than in the positive direction ( Rice & Gaines, 1994 ) .

Another approach to increase the power without increasing the sample size, is to increase the alpha level of the test, as explained in the section on compromise power analysis. Obviously, this comes at an increased probability of making a Type I error. The risk of making either type of error should be carefully weighed, which typically requires taking into account the prior probability that the null hypothesis is true ( Cascio & Zedeck, 1983 ; Miller & Ulrich, 2019 ; Mudge et al., 2012 ; Murphy et al., 2014 ) . If you have to make a decision, or want to make a claim, and the data you can feasibly collect is limited, increasing the alpha level is justified, either based on a compromise power analysis, or based on a cost-benefit analysis ( Baguley, 2004 ; Field et al., 2004 ) .

Another widely recommended approach to increase the power of a study is use a within participant design where possible. In almost all cases where a researcher is interested in detecting a difference between groups, a within participant design will require collecting less participants than a between participant design. The reason for the decrease in the sample size is explained by the equation below from Maxwell et al. ( 2017 ) . The number of participants needed in a two group within-participants design (NW) relative to the number of participants needed in a two group between-participants design (NB), assuming normal distributions, is:

\[NW = \frac{NB (1-\rho)}{2}\]

The required number of participants is divided by two because in a within-participants design with two conditions every participant provides two data points. The extent to which this reduces the sample size compared to a between-participants design also depends on the correlation between the dependent variables (e.g., the correlation between the measure collected in a control task and an experimental task), as indicated by the (1- \(\rho\) ) part of the equation. If the correlation is 0, a within-participants design simply needs half as many participants as a between participant design (e.g., 64 instead 128 participants). The higher the correlation, the larger the relative benefit of within-participants designs, and whenever the correlation is negative (up to -1) the relative benefit disappears.

In Figure  8.11 we see two normally distributed scores with a mean difference of 6, where the standard deviation of each mean is 15, and the correlation between the measurements is 0. The standard deviation of the difference score is \(\sqrt{2}\) times as large as the standard deviation in each measurement, and indeed, 15× \(\sqrt{2}\) = 21.21, which is rounded to 21. This situation where the correlation between measurements is zero equals the situation in an independent t -test, where the correlation between measurements is not taken into account.

In Figure  8.12 we can see what happens when the two variables are correlated, for example with r = 0.7. Nothing has changed when we plot the means. The correlation between measurements is now strongly positive, and the important difference is in the standard deviation of the difference scores, which is 11 instead of 21 in the uncorrelated example. Because the standardized effect size is the difference divided by the standard deviation, the effect size (Cohen’s \(d_z\) in within designs) is larger in this test than in the uncorrelated test.

The correlation between dependent variables is an important aspect of within designs. I recommend explicitly reporting the correlation between dependent variables in within designs (e.g., participants responded significantly slower ( M = 390, SD = 44) when they used their feet than when they used their hands ( M = 371, SD = 44, r = .953), t (17) = 5.98, p < 0.001, Hedges’ g = 0.43, \(M_{diff}\) = 19, 95% CI [12; 26]). Since most dependent variables in within designs in psychology are positively correlated, within designs will increase the power you can achieve given the sample size you have available. Use within-designs when possible, but weigh the benefits of higher power against the downsides of order effects or carryover effects that might be problematic in a within-subject design ( Maxwell et al., 2017 ) .

You can use this Shiny app to play around with different means, standard deviations, and correlations, and see the effect of the distribution of the difference scores.

In general, the smaller the variation, the larger the standardized effect size (because we are dividing the raw effect by a smaller standard deviation) and thus the higher the power given the same number of observations. Some additional recommendations are provided in the literature ( Allison et al., 1997 ; Bausell & Li, 2002 ; Hallahan & Rosenthal, 1996 ) , such as:

• Use better ways to screen participants for studies where participants need to be screened before participation.
• Assign participants unequally to conditions (if data in the control condition is much cheaper to collect than data in the experimental condition, for example).
• Use reliable measures that have low error variance ( Williams et al., 1995 ) .
• Smart use of preregistered covariates ( Meyvis & Van Osselaer, 2018 ) .

It is important to consider if these ways to reduce the variation in the data do not come at too large a cost for external validity. For example, in an intention-to-treat analysis in randomized controlled trials participants who do not comply with the protocol are maintained in the analysis such that the effect size from the study accurately represents the effect of implementing the intervention in the population, and not the effect of the intervention only on those people who perfectly follow the protocol ( Gupta, 2011 ) . Similar trade-offs between reducing the variance and external validity exist in other research areas.

8.25 Know Your Measure

Although it is convenient to talk about standardized effect sizes, it is generally preferable if researchers can interpret effects in the raw (unstandardized) scores, and have knowledge about the standard deviation of their measures ( Baguley, 2009 ; Lenth, 2001 ) . To make it possible for a research community to have realistic expectations about the standard deviation of measures they collect, it is beneficial if researchers within a research area use the same validated measures. This provides a reliable knowledge base that makes it easier to plan for a desired accuracy, and to use a smallest effect size of interest on the unstandardized scale in an a-priori power analysis.

In addition to knowledge about the standard deviation it is important to have knowledge about the correlations between dependent variables (for example because Cohen’s d z for a dependent t test relies on the correlation between means). The more complex the model, the more aspects of the data-generating process need to be known to make predictions. For example, in hierarchical models researchers need knowledge about variance components to be able to perform a power analysis ( DeBruine & Barr, 2021 ; Westfall et al., 2014 ) . Finally, it is important to know the reliability of your measure ( Parsons et al., 2019 ) , especially when relying on an effect size from a published study that used a measure with different reliability, or when the same measure is used in different populations, in which case it is possible that measurement reliability differs between populations. With the increasing availability of open data, it will hopefully become easier to estimate these parameters using data from earlier studies.

If we calculate a standard deviation from a sample, this value is an estimate of the true value in the population. In small samples, our estimate can be quite far off, while due to the law of large numbers, as our sample size increases, we will be measuring the standard deviation more accurately. Since the sample standard deviation is an estimate with uncertainty, we can calculate a confidence interval around the estimate ( Smithson, 2003 ) , and design pilot studies that will yield a sufficiently reliable estimate of the standard deviation. The confidence interval for the variance \(\sigma^2\) is provided in the following formula, and the confidence for the standard deviation is the square root of these limits:

\[(N - 1)s^2/\chi^2_{N-1:\alpha/2},(N - 1)s^2/\chi^2_{N-1:1-\alpha/2}\]

Whenever there is uncertainty about parameters, researchers can use sequential designs to perform an internal pilot study ( Wittes & Brittain, 1990 ) . The idea behind an internal pilot study is that researchers specify a tentative sample size for the study, perform an interim analysis, use the data from the internal pilot study to update parameters such as the variance of the measure, and finally update the final sample size that will be collected. As long as interim looks at the data are blinded (e.g., information about the conditions is not taken into account) the sample size can be adjusted based on an updated estimate of the variance without any practical consequences for the Type I error rate ( Friede & Kieser, 2006 ; Proschan, 2005 ) . Therefore, if researchers are interested in designing an informative study where the Type I and Type II error rates are controlled, but they lack information about the standard deviation, an internal pilot study might be an attractive approach to consider ( Chang, 2016 ) .

8.26 Conventions as meta-heuristics

Even when a researcher might not use a heuristic to directly determine the sample size in a study, there is an indirect way in which heuristics play a role in sample size justifications. Sample size justifications based on inferential goals such as a power analysis, accuracy, or a decision all require researchers to choose values for a desired Type I and Type II error rate, a desired accuracy, or a smallest effect size of interest. Although it is sometimes possible to justify these values as described above (e.g., based on a cost-benefit analysis), a solid justification of these values might require dedicated research lines. Performing such research lines will not always be possible, and these studies might themselves not be worth the costs (e.g., it might require less resources to perform a study with an alpha level that most peers would consider conservatively low, than to collect all the data that would be required to determine the alpha level based on a cost-benefit analysis). In these situations, researchers might use values based on a convention.

When it comes to a desired width of a confidence interval, a desired power, or any other input values required to perform a sample size computation, it is important to transparently report the use of a heuristic or convention (for example by using the accompanying online Shiny app). A convention such as the use of a 5% Type 1 error rate and 80% power practically functions as a lower threshold of the minimum informational value peers are expected to accept without any justification (whereas with a justification, higher error rates can also be deemed acceptable by peers). It is important to realize that none of these values are set in stone. Journals are free to specify that they desire a higher informational value in their author guidelines (e.g., Nature Human Behavior requires Registered Reports to be designed to achieve 95% statistical power, and my own department has required staff to submit ERB proposals where, whenever possible, the study was designed to achieve 90% power). Researchers who choose to design studies with a higher informational value than a conventional minimum should receive credit for doing so.

In the past some fields have changed conventions, such as the 5 sigma threshold now used in physics to declare a discovery instead of a 5% Type I error rate. In other fields such attempts have been unsuccessful (e.g., Johnson ( 2013 ) ). Improved conventions should be context dependent, and it seems sensible to establish them through consensus meetings ( Mullan & Jacoby, 1985 ) . Consensus meetings are common in medical research, and have been used to decide upon a smallest effect size of interest (for an example, see Fried et al. ( 1993 ) ). In many research areas current conventions can be improved. For example, it seems peculiar to have a default alpha level of 5% both for single studies and for meta-analyses, and one could imagine a future where the default alpha level in meta-analyses is much lower than 5%. Hopefully, making the lack of an adequate justification for certain input values in specific situations more transparent will motivate fields to start a discussion about how to improve current conventions. The online Shiny app links to good examples of justifications where possible, and will continue to be updated as better justifications are developed in the future.

8.27 Sample Size Justification in Qualitative Research

A value of information perspective to sample size justification also applies to qualitative research. A sample size justification in qualitative research should be based on the consideration that the cost of collecting data from additional participants does not yield new information that is valuable enough given the inferential goals. One widely used application of this idea is known as saturation and is indicated by the observation that new data replicates earlier observations, without adding new information ( Morse, 1995 ) . For example, let’s imagine we ask people why they have a pet. Interviews might reveal reasons that are grouped into categories, but after interviewing 20 people, no new categories emerge, at which point saturation has been reached. Alternative philosophies to qualitative research exist, and not all value planning for saturation. Regrettably, principled approaches to justify sample sizes have not been developed for these alternative philosophies ( Marshall et al., 2013 ) .

When sampling, the goal is often not to pick a representative sample, but a sample that contains a sufficiently diverse number of subjects such that saturation is reached efficiently. Fugard and Potts ( 2015 ) show how to move towards a more informed justification for the sample size in qualitative research based on 1) the number of codes that exist in the population (e.g., the number of reasons people have pets), 2) the probability a code can be observed in a single information source (e.g., the probability that someone you interview will mention each possible reason for having a pet), and 3) the number of times you want to observe each code. They provide an R formula based on binomial probabilities to compute a required sample size to reach a desired probability of observing codes.

A more advanced approach is used in Rijnsoever ( 2017 ) , which also explores the importance of different sampling strategies. In general, purposefully sampling information from sources you expect will yield novel information is much more efficient than random sampling, but this also requires a good overview of the expected codes, and the sub-populations in which each code can be observed. Sometimes, it is possible to identify information sources that, when interviewed, would at least yield one new code (e.g., based on informal communication before an interview). A good sample size justification in qualitative research is based on 1) an identification of the populations, including any sub-populations, 2) an estimate of the number of codes in the (sub-)population, 3) the probability a code is encountered in an information source, and 4) the sampling strategy that is used.

8.28 Discussion

Providing a coherent sample size justification is an essential step in designing an informative study. There are multiple approaches to justifying the sample size in a study, depending on the goal of the data collection, the resources that are available, and the statistical approach that is used to analyze the data. An overarching principle in all these approaches is that researchers consider the value of the information they collect in relation to their inferential goals.

The process of justifying a sample size when designing a study should sometimes lead to the conclusion that it is not worthwhile to collect the data, because the study does not have sufficient informational value to justify the costs. There will be cases where it is unlikely there will ever be enough data to perform a meta-analysis (for example because of a lack of general interest in the topic), the information will not be used to make a decision or claim, and the statistical tests do not allow you to test a hypothesis with reasonable error rates or to estimate an effect size with sufficient accuracy. If there is no good justification to collect the maximum number of observations that one can feasibly collect, performing the study anyway is a waste of time and/or money ( Brown, 1983 ; Button et al., 2013 ; S. D. Halpern et al., 2002 ) .

The awareness that sample sizes in past studies were often too small to meet any realistic inferential goals is growing among psychologists ( Button et al., 2013 ; Fraley & Vazire, 2014 ; Lindsay, 2015 ; Sedlmeier & Gigerenzer, 1989 ) . As an increasing number of journals start to require sample size justifications, some researchers will realize they need to collect larger samples than they were used to. This means researchers will need to request more money for participant payment in grant proposals, or that researchers will need to increasingly collaborate ( Moshontz et al., 2018 ) . If you believe your research question is important enough to be answered, but you are not able to answer the question with your current resources, one approach to consider is to organize a research collaboration with peers, and pursue an answer to this question collectively.

8.29 Test Yourself

Q1 : A student has at most 2 months to collect data. They need to pay participants for their participation, and their budget is limited to 250 euro. They decide to collect all the participants they can in the amount of time, and with the money they have available. What type of sample size justification is this?

Q2 : What is the goal of an a-priori power analysis?

Q3 : A researcher already knows the sample size they will be able to collect. Given this sample size, they choose to compute equal Type 1 and Type 2 error rates for an effect size of interest. This is known as:

Q4 : Looking at the formula in the section ‘Increasing Power Without Increasing the Sample Size’. which two factors contribute to the fact that within subject designs can have much more power, with the same number of participants, than between subject designs?

Q5 : Which factors determine the minimal statistically detectable effect?

Q6 : All else equal, if you want to perform a study that has the highest possible informational value, which approach to specifying the effect size of interest is the best choice?

Q7 : In an a-priori power analysis based on an empirical estimate of the literature, which 2 issues are important to consider, both when using an estimate from a meta-analysis, as from a single study?

Q8 : Imagine a researcher did not justify their sample size before performing the study, and had no justification for the sample size they choose. After submitting their scientific article to a journal reviewers ask for a justification of the sample size. Of course, honesty requires the authors to write down there was no justification, but how can they still evaluate the informational value of the study for effect sizes of interest?

Q9 : Why can it be useful to consider the effect size distribution of findings in a specific research area when evaluating the informational value of the study you are planning?

Q10 : Why is it nonsensical to ask researchers to perform a post-hoc or retrospective power analysis, where the observed effect size and the collected sample size is used to calculate the statistical power of a test, when a non-significant finding is observed?

Q11 : Researchers should not perform a post-hoc power analysis. There are two solutions, one that can be implemented when designing a study, and one when interpreting a non-significant result after the data is in. Which solution can be implemented when the data is in?

Q12 : What is a way/are ways to increase the statistical power of a test, without increasing the sample size?

8.29.1 Open Questions

Why are resource constraints, if not the primary justification, always a secondary justification (if it is not possible to collect data from the entire population)?

What is the goal of an a-priori power analysis, and why is the goal not to achieve a desired Type 2 error rate for the true effect size?

Which factors determine the Minimal Statistically Detectable Effect, and why can it be useful to compute it for a study you are planning to perform?

What is a benefit of planning for precision, given that the effect size is typically unknown (and might even be 0). Which aspect of the decisions that need to be made when planning for precision is most difficult to justify?

What is a problem of using heuristics as the basis of a sample size justification?

It seems counter-intuitive to have a ‘no justification’ category in a chapter on sample size justification, but why is it important to explicitly state there was no justification?

From all effect sizes that might be related to the inferential goal in a study, which of the 6 categories in Table  8.2 is the best approach (if it can be specified)?

Why can’t you simply take an effect size estimate from a meta-analysis as the basis of an a-priori power analysis for a related study?

Why can’t you simply take an effect size estimate from a single study as the basis of an a-priori power analysis for a related study?

What is the goal in a compromise power analysis?

Why is ‘post-hoc’ or ‘retrospective’ power not a useful way to draw inferences about non-significant results?

When would you perform a sensitivity power analysis?

How can the statistical power of a study be increased, without increasing the sample size?

Why can it be beneficial to use a within-design compared to a between-design (where possible)?

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Adequacy of sample size in a qualitative case study and the dilemma of data saturation: A narrative review

2021, World Journal of Advanced Research and Reviews

The consensus of many researchers on data saturation is that data saturation is a key driver for determining the adequacy of sample size in a qualitative case study. Despite these global consensuses, some researchers described data saturation as complex because the decision to stop data collection is solely dictated by the judgment and experience of researchers. Other researchers claimed that guidelines for determining non-probability sample sizes, used as an indication of data saturation are virtually non-existent, problematic, or controversial. Others claimed that data saturation hitched to sample size is practically weak, because data are never truly saturated, as there are always new data to be discovered. This narrative study highlights the dilemma of data saturation and strategies to adequately determine sample size in a qualitative case study. A narrative review of prior research that focused on the vast works of literature that revealed significant information on data saturation and strategies to adequately determine sample size was adopted. Peer-reviewed articles within the last five years from electronic databases, using some keywords such as "qualitative case study", "sample size in a qualitative case study", "data saturation", etc., were also extracted. Results show that data saturation is very helpful especially at the conceptual stage, but its concept and standard is elusive, because it lacks practical guidance for estimating sample size for a robust research prior to data collection. Findings from this study may encourage researcher on better guidelines for determining non-probability sample sizes.

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Failure to reach data saturation has an impact on the quality of the research conducted and hampers content validity. The aim of a study should include what determines when data saturation is achieved, for a small study will reach saturation more rapidly than a larger study. Data saturation is reached when there is enough information to replicate the study when the ability to obtain additional new information has been attained, and when further coding is no longer feasible. The following article critiques two qualitative studies for data saturation: Wolcott (2004) and Landau and Drori (2008). Failure to reach data saturation has a negative impact on the validity on one’s research. The intended audience is novice student researchers. Keywords: Data Saturation, Triangulation, Interviews, Personal Lens, Bias.

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Jacob Owusu Sarfo , Timothy Debrah , Paul Obeng , Senam Jubey

Qualitative research is currently growing in acceptance, especially within the health research scope. Notwithstanding this positive trend, issues about the adequacy of sample size have been a contention among qualitative and quantitative-based researchers. Our paper seeks to address some of the issues facing popular qualitative designs in human research with this backdrop. Our article explains the five key qualitative designs (case study, narrative inquiry, ethnography, phenomenology, and grounded theory). Based on the existing studies, we reported their respective sample size ranges that supported their data adequacy points. Our paper posits that sample size concerns for qualitative designs revolve around their extensiveness and appropriateness. Therefore, qualitative researchers' judgement for data adequacy for a particular method should not only rely on data saturation or a rule-of-thumb. Instead, they should also be guided by their research goals, sampling approach, and research participants. Furthermore, we recommend that qualitative researchers always verify the quality of data saturation by conducting additional interviews and be more open in reporting their selected methodologies.

Shalini Rawla

Choosing a suitable sample size in qualitative research is an area of conceptual debate and practical uncertainty. Sample size principles, guidelines and tools have been developed to enable researchers to justify the acceptability of their sample size. Nevertheless, research shows that sample size sufficiency reporting is often poor, if not absent, across a range of disciplinary fields. The issue of sample size is accepted as an important marker of the quality of qualitative research. The purpose of this paper is to delineate a standardized framework for qual studies to arrive at a sample size strategy that is transparent and logical about its sample size sufficiency.

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Benjamin Saunders , Julius Sim , Jackie Waterfield

Saturation has attained widespread acceptance as a methodological principle in qualitative research. It is commonly taken to indicate that, on the basis of the data that have been collected or analysed hitherto, further data collection and/or analysis are unnecessary. However, there appears to be uncertainty as to how saturation should be conceptualized, and inconsistencies in its use. In this paper, we look to clarify the nature, purposes and uses of saturation, and in doing so add to theoretical debate on the role of saturation across different methodologies. We identify four distinct approaches to saturation, which differ in terms of the extent to which an inductive or a deductive logic is adopted, and the relative emphasis on data collection, data analysis, and theorizing. We explore the purposes saturation might serve in relation to these different approaches, and the implications for how and when saturation will be sought. In examining these issues, we highlight the uncertain logic underlying saturation— as essentially a predictive statement about the unobserved based on the observed, a judgement that, we argue, results in equivocation, and may in part explain the confusion surrounding its use. We conclude that saturation should be operationalized in a way that is consistent with the research question(s), and the theoretical position and analytic framework adopted, but also that there should be some limit to its scope, so as not to risk saturation losing its coherence and potency if its conceptualization and uses are stretched too widely.

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Determining Sample Size for Qualitative Research: What is the Magical Number?

• August 21, 2018

Article Summary:   Sample sizes in qualitative research are much lower than those needed in quantitative research because of careful audience segmentation, study design, and the principle of data saturation.

Everyone wants to know the magical number for sample size when doing qualitative research .

Of course, when you’re doing quantitative research numbers matter more – especially when you’re aiming for statistically meaningful results. But when doing qualitative research, several factors must be considered before blindly accepting an arbitrary number. Here are some important points that should be kept in mind when thinking about sample size:

Point #1: Quality over Quantity

Whether your company is targeting moms from the Midwest with household incomes of $70k+, or teens who use Facebook more than 8 hours a week, or B2B decision makers in HR – you are most likely looking for insights on your services from a very specific audience type. That being said, the first objective when conducting qualitative market research should be to ensure the right people are recruited for the study.

The right respondents for the study are those who meet each and every criteria line-item identified from quantitative research studies, and also the criteria that the clients have identified through their own research. Thus, a high quality panel includes much more than just members who are pulled from a general population that falls within broad parameters. Only those participants who match the audience specifications and background relevance expressed by the client should be recruited.

Point #2: Choose an appropriate study design

The type of qualitative study is another important factor to consider when choosing sample size. There are various methods that can be used to gather insightful data, but not all methods may be applicable to your study and its aim. In-depth interviews , focus groups , and ethnographic research are the most common methods used in qualitative market research, each method being unique in the information it can provide and the setting it can be used in. Moreover, the types of questions being studied have an equally important role to play in deciding sample size. Thus before choosing a sample size, make sure all the parameters, aims, and controls are clearly outlined.

Point #3: Keep in mind the Principle of Saturation

A sample size should be large enough to sufficiently describe the phenomenon of interest, and address the research question at hand. But at the same time, a large sample size risks having repetitive data. The goal of qualitative research should thus be the attainment of saturation . Saturation occurs when adding more participants to the study does not result in obtaining additional perspectives or information. One can say there is a point of diminishing return with larger samples, as it leads to more data but doesn’t necessarily lead to more information.

The objective of qualitative research is to lessen discovery failure; whilst quantitative research aims to reduce estimation error. As qualitative research works to obtain diverse opinions from a sample size on a client’s product/service/project, saturated data does not serve to do anything. One respondent’s opinion is enough to generate a code, part of the analysis framework.

The goal of a qualitative study should be to have a large enough sample size to uncover a variety of opinions, but to limit the sample size at the point of saturation.

So… is there a magical number to aim for in qualitative research?

After bearing in mind all the factors mentioned above, there is a number you can aim for. Based on research conducted on this very issue, 30 seems to be a good number for the most comprehensive assessment. Some studies have noted having a sample size as little as 10 can be extremely fruitful, and still yield applicable results. This would of course be only after a rigorous recruiting process is put in place.

Our general recommendation for in-depth interviews is to have a sample size of 20-30, if we’re building similar segments within the population. In some cases, a minimum of 10 is acceptable – assuming the population integrity in recruiting.

If you would like to learn more about sample sizes and learn how to set up studies and perfect your moderating and interviewing skills, check out our InterQ Learning Labs training programs.

Let’s discuss your project and the type of sample size that would work best. Request a proposal >

Author Bio : Sakshi Shetty was an intern with InterQ Research, working as a coder for qualitative data. She now works as a product manager.

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