Statistics textbooks in the social, behavioral, and biomed- ical sciences typically stress the importance of power analy- ses. By definition, the power of a statistical test is the prob- ability that its null hypothesis (H0) will be rejected given that it is in fact false. Obviously, significance tests that lack sta- tistical power are of limited use because they cannot reliably discriminate between H0 and the alternative hypothesis (H1) of interest. However, although power analyses are indispens- able for rational statistical decisions, it was not until the late 1980s that power charts (see, e.g., Scheff��, 1959) and power tables (see, e.g., Cohen, 1988) were supplemented by more efficient, precise, and easy-to-use power analysis programs for personal computers (Goldstein, 1989). G*Power 2 (Erd- felder, Faul, & Buchner, 1996) can be seen as a second- generation power analysis program designed as a stand- alone application to handle several types of statistical tests commonly used in social and behavioral research. In the past 10 years, this program has been found useful not only in the social and behavioral sciences but also in many other disci- plines that routinely apply statistical tests, including biology (Baeza & Stotz, 2003), genetics (Akkad et al., 2006), ecol- ogy (Sheppard, 1999), forest and wildlife research (Mellina, Hinch, Donaldson, & Pearson, 2005), the geosciences (Bus- bey, 1999), pharmacology (Quednow et al., 2004), and med- ical research (Gleissner, Clusmann, Sassen, Elger, & Helm- staedter, 2006). G*Power 2 was evaluated positively in the reviews of which we are aware (Kornbrot, 1997 Ortseifen, Bruckner, Burke, & Kieser, 1997 Thomas & Krebs, 1997). It has been used in several power tutorials (e.g., Buchner, Erdfelder, & Faul, 1996, 1997 Erdfelder, Buchner, Faul, & Brandt, 2004 Levin, 1997 Sheppard, 1999) and in statis- tics textbooks (e.g., Field, 2005 Keppel & Wickens, 2004 Myers & Well, 2003 Rasch, Friese, Hofmann, & Naumann, 2006a, 2006b). Nevertheless, the user feedback that we re- ceived coincided with our own experience in showing some limitations and weaknesses of G*Power 2 that required a major extension and revision. In the present article, we describe G*Power 3, a program that was designed to address the problems of G*Power 2. We begin with an outline of the major improvements in G*Power 3 and then discuss the types of power analyses cov- ered by this program. Next, we describe program handling and the types of statistical tests to which it can be applied. We then discuss the statistical algorithms of G*Power 3 and their accuracy. Finally, program availability and some Inter- net resources supporting users of G*Power 3 are described. IMPROVEMENTS IN G*POWER 3 IN COMPARISON WITH G*POWER 2 G*Power 3 is an improvement over G*Power 2 in five major respects. First, whereas G*Power 2 requires the 175 Copyright 2007 Psychonomic Society, Inc. G*Power 3:A flexible statistical power analysis program for the social, behavioral, and biomedical sciences FRANZ FAUL Christian-Albrechts-Universit��t Kiel, Kiel, Germany EDGAR ERDFELDER Universit��t Mannheim, Mannheim, Germany AND ALBERT-GEORG LANG AND AXEL BUCHNER Heinrich-Heine-Universit��t D��sseldorf, D��sseldorf, Germany G*Power (Erdfelder, Faul, & Buchner, 1996) was designed as a general stand-alone power analysis program for statistical tests commonly used in social and behavioral research. G*Power 3 is a major extension of, and improvement over, the previous versions. It runs on widely used computer platforms (i.e., Windows XP, Win- dows Vista, and Mac OS X 10.4) and covers many different statistical tests of the t, F, and n2 test families. In addition, it includes power analyses for z tests and some exact tests. G*Power 3 provides improved effect size calculators and graphic options, supports both distribution-based and design-based input modes, and offers all types of power analyses in which users might be interested. Like its predecessors, G*Power 3 is free. Behavior Research Methods 2007, 39 (2), 175-191 E. Erdfelder, erdfelder@psychologie.uni-mannheim.de

176 FAUL, ERDFELDER, LANG, AND BUCHNER DOS and Mac OS 7���9 operating systems that were com- mon in the 1990s but are now outdated, G*Power 3 runs on the personal computer platforms currently in widest use: Windows XP, Windows Vista, and Mac OS X 10.4. The Windows and Mac versions of the program are es- sentially equivalent. They use the same computational routines and share very similar user interfaces. For this reason, we will not differentiate between these versions in what follows users simply have to make sure to download the version appropriate for their operating system. Second, whereas G*Power 2 is limited to three types of power analyses, G*Power 3 supports five different ways to assess statistical power. In addition to the a pri- ori, post hoc, and compromise power analyses that were already covered by G*Power 2, the new program offers sensitivity analyses and criterion analyses. Third, G*Power 3 provides dedicated power analysis options for a variety of frequently used t, F, z, n2, and exact tests in addition to the standard tests covered by G*Power 2. The tests captured by G*Power 3 and their effect size parameters are described in the Program Han- dling section. Importantly, users are not limited to these tests because G*Power 3 also offers power analyses for generic t, F, z, n2, and binomial tests for which the non- centrality parameter of the distribution under H1 may be entered directly. In this way, users are provided with a flexible tool for computing the power of basically any statistical test that uses t, F, z, n2, or binomial reference distributions. Fourth, statistical tests can be specified in G*Power 3 using two different approaches: the distribution-based ap- proach and the design-based approach. In the distribution- based approach, users select the family of the test statistic (t, F, z, n2, or exact test) and the particular test within that family. This is how power analyses were specified in G*Power 2. In addition, a separate menu in G*Power 3 provides access to power analyses via the design-based approach: Users select (1) the parameter class to which the statistical test refers (correlations, means, proportions, regression coefficients, variances) and (2) the design of the study (e.g., number of groups, independent vs. depen- dent samples). On the basis of the feedback we received about G*Power 2, we expect that some users might find the design-based input mode more intuitive and easier to use. Fifth, G*Power 3 supports users with enhanced graph- ics features. The details of these features will be outlined in the Program Handling section. TYPES OF STATISTICAL POWER ANALYSES The power (1 1 Z) of a statistical test is the complement of Z, which denotes the Type II or beta error probability of falsely retaining an incorrect H0. Statistical power de- pends on three classes of parameters: (1) the significance level (i.e., the Type I error probability) G of the test, (2) the size(s) of the sample(s) used for the test, and (3) an effect size parameter defining H1 and thus indexing the degree of deviation from H0 in the underlying population. De- pending on the available resources, the actual phase of the research process, and the specific research question, five different types of power analysis can be reasonable (cf. Erdfelder et al., 2004 Erdfelder, Faul, & Buchner, 2005). We describe these methods and their uses in turn. A Priori Power Analyses In a priori power analyses (Cohen, 1988), sample size N is computed as a function of the required power level (1 1 Z), the prespecified significance level G, and the population effect size to be detected with probability 1 1 Z. A priori analyses provide an efficient method of controlling statistical power before a study is actually con- ducted (see, e.g., Bredenkamp, 1969 Hager, 2006) and can be recommended whenever resources such as the time and money required for data collection are not critical. Post Hoc Power Analyses In contrast to a priori power analyses, post hoc power analyses (Cohen, 1988) often make sense after a study has already been conducted. In post hoc analyses, 1 1 Z is computed as a function of G, the population effect size parameter, and the sample size(s) used in a study. It thus becomes possible to assess whether or not a published statistical test in fact had a fair chance of rejecting an in- correct H0. Importantly, post hoc analyses, like a priori analyses, require an H1 effect size specification for the underlying population. Post hoc power analyses should not be confused with so-called retrospective power anal- yses, in which the effect size is estimated from sample data and used to calculate the observed power, a sample estimate of the true power.1 Retrospective power analy- ses are based on the highly questionable assumption that the sample effect size is essentially identical to the effect size in the population from which it was drawn (Zumbo & Hubley, 1998). Obviously, this assumption is likely to be false, and the more so the smaller the sample. In addition, sample effect sizes are typically biased estimates of their population counterparts (Richardson, 1996). For these reasons, we agree with other critics of retrospective power analyses (e.g., Gerard, Smith, & Weerakkody, 1998 Hoe- nig & Heisey, 2001 Kromrey & Hogarty, 2000 Lenth, 2001 Steidl, Hayes, & Schauber, 1997). Rather than use retrospective power analyses, researchers should specify population effect sizes on a priori grounds. To specify the effect size simply means to define the minimum degree of violation of H0 a researcher would like to detect with a probability not less than 1 1 Z. Cohen���s definitions of small, medium, and large effects can be helpful in such effect size specifications (see, e.g., Smith & Bayen, 2005). However, researchers should be aware of the fact that these conventions may have different meanings for differ- ent tests (cf. Erdfelder et al., 2005). Compromise Power Analyses In compromise power analyses (Erdfelder, 1984 Erdfelder et al., 1996 M��ller, Manz, & Hoyer, 2002), both G and 1 1 Z are computed as functions of the ef- fect size, N, and the error probability ratio q 4 Z/G. To illustrate, setting q to 1 would mean that the researcher prefers balanced Type I and Type II error risks (G 4 Z),

G*POWER 3 177 whereas a q of 4 would imply that Z 4 4G (cf. Cohen, 1988). Compromise power analyses can be useful both before and after data collection. For example, an a priori power analysis might result in a sample size that exceeds the available resources. In such a situation, a researcher could specify the maximum affordable sample size and, using a compromise power analysis, compute G and 1 1 Z associated with, say, q 4 Z/G 4 4. Alternatively, if a study has already been conducted but has not yet been analyzed, a researcher could ask for a reasonable decision criterion that guarantees perfectly balanced error risks (i.e., G 4 Z) given the size of the sample and the critical effect size in which he or she is interested. Of course, compromise power analyses can easily result in unconventional sig- nificance levels greater than G 4 .05 (in the case of small samples or effect sizes) or less than G 4 .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 conven- tions (cf. Gigerenzer, Krauss, & Vitouch, 2004). Sensitivity Analyses In sensitivity analyses, the critical population effect size is computed as a function of G, 1 1 Z, and N. Sensitivity analyses may be particularly useful for evaluating pub- lished research. They provide answers to questions such as ���What effect size was a study able to detect with a power of 1 1 Z 4 .80 given its sample size and G as specified by the author? In other words, what is the minimum ef- fect size to which the test was sufficiently sensitive?��� In addition, it may be useful to perform sensitivity analyses before conducting a study to see whether, given a lim- ited N, the size of the effect that can be detected is at all realistic (or, for instance, much too large to be expected realistically). Criterion Analyses Finally, criterion analyses compute G (and the associ- ated decision criterion) as a function of 1 1 Z, the effect size, and a given sample size. Criterion analyses are alter- natives to post hoc power analyses. They may be reason- able whenever the control of G is less important than the control of Z. In case of goodness-of-fit tests for statistical models, for example, it is most important to minimize the Z risk of wrong decisions in favor of the model (H0). Re- searchers could thus use criterion analyses to compute the significance level G which is compatible with Z 4 .05 for a small effect size. Whereas G*Power 2 was limited to the first three types of power analysis, G*Power 3 covers all five types. On the basis of the feedback we received from G*Power 2 users, we believe that any question related to statistical power that arises in research practice can be categorized under one of these analysis types. PROGRAM HANDLING Using G*Power 3 typically involves the following four steps: (1) Select the statistical test appropriate for the problem, (2) choose one of the five types of power analyses defined in the previous section, (3) provide the input parameters required for the analysis, and (4) click on ���Calculate��� to obtain the results. In the first step, the statistical test is chosen using the distribution-based or the design-based approach. G*Power 2 users probably have adapted to the distribution- based approach: One first selects the family of the test statistic (t, F, z, n2, or exact test) using the ���Test fam- ily��� menu in the main window. The ���Statistical test��� menu adapts accordingly, showing a list of all tests available for the test family. For the two-groups t test, for example, one would first select the t family of distributions and then ���Means: Difference between two independent means (two groups)��� in the ���Statistical test��� menu (see Figure 1). Al- ternatively, one might use the design-based approach of test selection. With the ���Tests��� pull-down menu in the top row, it is possible to select (1) the parameter class to which the statistical test refers (i.e., correlation and regression, means, proportions, variances, or generic) and (2) the de- sign of the study (e.g., number of groups, independent vs. dependent samples). For example, a researcher would select ���Means��� followed by ���Two independent groups��� to specify the two-groups t test (see Figure 2). The design- based approach has the advantage that test options refer- ring to the same parameter class (e.g., means) are located in close proximity, whereas in the distribution-based ap- proach they may be scattered across different distribution families. In the second step, the ���Type of power analysis��� menu in the center of the main window should be used to choose the appropriate analysis type. In the third step, the power analysis input parameters are specified in the lower left of the main window. To illustrate, an a priori power analysis for a two-groups t test would require a decision between a one-tailed and a two-tailed test, a specification of Cohen���s (1988) effect size measure (d ) under H1, the significance level G, the required power (1 1 Z) of the test, and the preferred group size allocation ratio n2/n1. The final step consists of clicking on ���Calculate��� to obtain the output in the lower right of the main window. For instance, input parameters specifying a one-tailed t test, a medium effect size of d 4 0.5, G 4 .05, 1 1 Z 4 .95, and an allocation ratio of n2/n1 4 1 would result in a total sample size of N 4 176 (88 observation units in each group see Figures 1 and 2). The noncentrality pa- rameter x defining the t distribution under H1, the decision criterion to be used (i.e., the critical value of the t statis- tic), the degrees of freedom2 of the t test, and the actual power value are also displayed. Note that the actual power will often be slightly larger than the prespecified power in a priori power analyses. The reason is that noninteger sample sizes are always rounded up by G*Power to obtain integer values consistent with a power level not lower than the prespecified one. In addition to the numerical output, G*Power 3 dis- plays the central (H0) and the noncentral (H1) test statistic distributions along with the decision criterion and the as- sociated error probabilities in the upper part of the main window (see Figure 1).3 This supports understanding of the effects of the input parameters and is likely to be a