GPower31 BRM Paper

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G*Power (Faul, Erdfelder, Lang, & Buchner, 2007) isa stand-alone power analysis program for many statisticaltests commonly used in the social, behavioral, and bio-medical sciences. It is available free of charge via the In-ternet for both Windows and Mac OS X platforms (see theConcluding Remarks section for details). In this article, we

present extensions and improvements of G*Power 3 in the

domain of correlation and regression analyses. G*Powernow covers (1) one-sample correlation tests based on thetetrachoric correlation model, in addition to the bivari-ate normal and point biserial models already available inG*Power 3, (2) statistical tests comparing both dependentand independent Pearson correlations, and statistical testsfor (3) simple linear regression coefficients, (4) multiplelinear regression coefficients for both the fixed- andrandom-predictors models, (5) logistic regression coef-ficients, and (6) Poisson regression coefficients. Thus,in addition to the generic power analysis procedures forthez, t,F, 2, and binomial tests, and those for tests ofmeans, mean vectors, variances, and proportions that have

already been available in G*Power 3 (Faul et al., 2007),the new version, G*Power 3.1, now includes statisticalpower analyses for six correlation and nine regression testproblems, as summarized in Table 1.

As usual in G*Power 3, five types of power analysis areavailable for each of the newly available tests (for morethorough discussions of these types of power analyses, seeErdfelder, Faul, Buchner, & Cpper, in press; Faul et al.,2007):

1.A priori analysis (see Bredenkamp, 1969; Cohen,1988). The necessary sample size is computed as a func-tion of user-specified values for the required significancelevel , the desired statistical power 12, and the to-be-detected population effect size.

2. Compromise analysis (see Erdfelder, 1984). The sta-tistical decision criterion (critical value) and the asso-

ciated and values are computed as a function of thedesired error probability ratio /, the sample size, andthe population effect size.

3. Criterion analysis (see Cohen, 1988; Faul et al.,2007). The required significance level is computed asa function of power, sample size, and population effectsize.

4.Post hoc analysis (see Cohen, 1988). Statistical power12 is computed as a function of significance level ,sample size, and population effect size.

5. Sensitivity analysis (see Cohen, 1988; Erdfelder,Faul, & Buchner, 2005). The required population effectsize is computed as a function of significance level, sta-

tistical power 12, and sample size.As already detailed and illustrated by Faul et al. (2007),G*Power provides for both numerical and graphical out-

put options. In addition, any of four parameters, 12,sample size, and effect sizecan be plotted as a functionof each of the other three, controlling for the values of theremaining two parameters.

Below, we briefly describe the scope, the statistical back-ground, and the handling of the correlation and regression

1149 2009 The Psychonomic Society, Inc.

Statistical power analyses sing G*Power 3.1:Tests for correlation and regression analyses

Franz Faul

Christian-Albrechts-Universitt, Kiel, Germany

EdgarErdFEldEr

Universitt Mannheim, Mannheim, Germany

and

axElBuchnErandalBErt-gEorglang

Heinrich-Heine-Universitt, Dsseldorf, Germany

G*Power is a free power analysis program for a variety of statistical tests. We present extensions and improve-ments of the version introduced by Faul, Erdfelder, Lang, and Buchner (2007) in the domain of correlationand regression analyses. In the new version, we have added procedures to analyze the power of tests based on

(1) single-sample tetrachoric correlations, (2) comparisons of dependent correlations, (3) bivariate linear regres-sion, (4) multiple linear regression based on the random predictor model, (5) logistic regression, and (6) Poissonregression. We describe these new features and provide a brief introduction to their scope and handling.

Behavior Research Methods2009, 41 (4), 1149-1160doi:10.3758/BRM.41.4.1149

E. Erdfelder, [email protected]


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1150 Faul, ErdFEldEr, BuchnEr, and lang

have not been defined for tetrachoric correlations. How-ever, Cohens (1988) conventions for correlations in theframework of the bivariate normal model may serve asrough reference points.

Options. Clicking on the Options button opens awindow in which users may choose between the exact ap-

proach of Brown and Benedetti (1977) (default option) oran approximation suggested by Bonett and Price (2005).

Inpt and otpt parameters. Irrespective of themethod chosen in the options window, the power of the tet-rachoric correlationztest depends not only on the values of under H0 and H1 but also on the marginal distributions ofXand Y. For post hoc power analyses, one therefore needsto provide the following input in the lower left field of themain window: The number of tails of the test (Tail(s):one vs. two), the tetrachoric correlation under H1 (H1 corr), the error probability, the Total sample sizeN, thetetrachoric correlation under H0 (H0 corr), and themarginal probabilities ofX5 1 (Marginal prob x) andY5 1 (Marginal prob y)that is, the proportions of val-ues exceeding the two criteria used for dichotomization.The output parameters include the Critical z requiredfor deciding between H0 and H1 and the Power (12 err

prob). In addition, critical values for the sample tetra-choric correlation r(Critical r upr and Critical r lwr)and the standard error se(r) ofr(Std err r) under H0 arealso provided. Hence, if the Waldzstatistic W5 (r20)/se(r) is unavailable, G*Power users can base their statisti-cal decision on the sample tetrachoric rdirectly. For a two-tailed test, H0 is retained wheneverris not less than Criti-cal r lwr and not larger than Critical r upr; otherwise H0is rejected. For one-tailed tests, in contrast, Critical r lwrand Critical r upr are identical; H0 is rejected if and onlyifrexceeds this critical value.

Illstrative example. Bonett and Price (2005, Exam-ple 1) reported the following yes (5 1) and no (5 2)

power analysis procedures that are new in G*Power 3.1.Further technical details about the tests described here,as well as information on those tests in Table 1 that werealready available in the previous version of G*Power, can

be found on the G*Power Web site (see the ConcludingRemarks section). We describe the new tests in the ordershown in Table 1 (omitting the procedures previously de-scribed by Faul et al., 2007), which corresponds to theirorder in the Tests Correlation and regression drop-down menu of G*Power 3.1 (see Figure 1).

1. Te Tetracoric Correlation Model

The Correlation: Tetrachoric model procedure refersto samples of two dichotomous random variablesXand Yas typically represented by 23 2 contingency tables. Thetetrachoric correlation model is based on the assumptionthat these variables arise from dichotomizing each of twostandardized continuous random variables following a bi-variate normal distribution with correlation in the under-lying population. This latent correlation is called the tet-rachoric correlation. G*Power 3.1 provides power analysis

procedures for tests of H0: 50 against H1: 0 (orthe corresponding one-tailed hypotheses) based on (1) a

precise method developed by Brown and Benedetti (1977)and (2) an approximation suggested by Bonett and Price(2005). The procedure refers to the Waldzstatistic W5(r20)/se0(r), where se0(r) is the standard error of thesample tetrachoric correlation runder H0: 50. Wfol-lows a standard normal distribution under H0.

Effect size measre. The tetrachoric correlation underH1, 1, serves as an effect size measure. Using the effectsize drawer (i.e., a subordinate window that slides outfrom the main window after clicking on the Determine

button), it can be calculated from the four probabilities of

the 23 2 contingency tables that define the joint distribu-tion ofXand Y. To our knowledge, effect size conventions

Table 1Smmary of te Correlation and

Regression Test Problems Covered by G*Power 3.1

Correlation Problems Referring to One Correlation

Comparison of a correlation with a constant0 (bivariate normal model)Comparison of a correlation with 0 (point biserial model)Comparison of a correlation with a constant0 (tetrachoric correlation model)

Correlation Problems Referring to Two Correlations

Comparison of two dependent correlationsjkandjh (common index)Comparison of two dependent correlationsjkandhm (no common index)Comparison of two independent correlations 1 and2 (two samples)

Linear Regression Problems, One Predictor (Simple Linear Regression)

Comparison of a slope b with a constant b0Comparison of two independent intercepts a1 and a2 (two samples)Comparison of two independent slopes b1 and b2 (two samples)

Linear Regression Problems, Several Predictors (Multiple Linear Regression)

Deviation of a squared multiple correlation2 from zero (Ftest, fixed model)Deviation of a subset of linear regression coefficients from zero (Ftest, fixed model)Deviation of a single linear regression coefficient bjfrom zero (ttest, fixed model)Deviation of a squared multiple correlation2 from constant (random model)

Generalized Linear Regression Problems

Logistic regressionPoisson regression


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g*PowEr3.1: corrElationand rEgrEssion 1151

correlation coefficient estimated from the data) as H1corr and press Calculate. Using the exact calculationmethod, this results in a sample tetrachoric correlationr5 .334 and marginal proportionspx5 .602 andpy5.582. We then click on Calculate and transfer to mainwindow in the effect size drawer. This copies the calcu-lated parameters to the corresponding input fields in themain window.

For a replication study, say we want to know the sample

size required for detecting deviations from H0:5 0 con-sistent with the above H1 scenario using a one-tailed test

answer frequencies of 930 respondents to two questionsin a personality inventory: f115 203, f125 186, f215167, f225 374. The option From C.I. calculated fromobserved freq in the effect size drawer offers the possi-

bility to use the (12) confidence interval for the sampletetrachoric correlation ras a guideline for the choice ofthe correlation under H1. To use this option, we insertthe observed frequencies in the corresponding fields. Ifwe assume, for example, that the population tetrachoric

correlation under H1 matches the sample tetrachoric cor-relation r, we should choose the center of the C.I. (i.e., the

Figre 1. Te main window of G*Power, sowing te contents of te TestsCorrelation and regression drop-down men.


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.17 with visuospatial working memory (VWM) and au-ditory working memory (AWM), respectively. The cor-relation of the two working memory measures wasr(VWM, AWM) 5 .17. Assume we would like to knowwhether VWM is more strongly correlated with age

than AWM in the underlying population. In other words,H0: (Age, VWM) #(Age, AWM) is tested against theone-tailed H1: (Age, VWM) .(Age, AWM). Assum-ing that the true population correlations correspond to theresults reported by Tsujimoto et al., what is the samplesize required to detect such a correlation difference witha power of 125 .95 and 5 .05? To find the answer,we choose the a priori type of power analysis alongwith Correlations: Two dependent Pearson rs (com-mon index) and insert the above parameters (ac5 .27,ab5 .17, bc5 .17) in the corresponding input fields.Clicking on Calculate provides us with N5 1,663 asthe required sample size. Note that thisNwould drop to

N5 408 if we assumed(VWM, AWM)5 .80 rather than(VWM, AWM)5 .17, other parameters being equal. Ob-viously, the third correlation bc has a strong impact onstatistical power, although it does not affect whether H0 orH1 holds in the underlying population.

2.2. Comparison of Two DependentCorrelations ab andcd (No Common Index)

The Correlations: Two dependent Pearson rs (no com-mon index) procedure is very similar to the proceduredescribed in the preceding section. The single difference isthat H0:ab5cd is now contrasted against H1:abcd;that is, the two dependent correlations here do notshare

a common index. G*Powers power analysis proceduresfor this scenario refer to Steigers (1980, Equation 12)Z2* test statistic. As with Z1*, which is actually a specialcase ofZ2*, the Z2* statistic is asymptoticallyzdistributedunder H0 given a multivariate normal distribution of thefour random variablesXa,Xb,Xc, andXd involved in abandcd (see alsoDunn & Clark, 1969).

Effect size measre. The effect size specification isidentical to that for Correlations: Two dependent Pearsonrs (common index), except that all six pairwise correla-tions of the random variablesXa,Xb,Xc, andXd under H1need to be specified here. As a consequence,ac,ad,bc,andbd are required as input parameters in addition to ab

andcd.Inpt and otpt parameters. By implication,the input parameters include Corr_ac, Corr_ad,Corr_bc, and Corr_bd in addition to the correla-tions H1 corr_cd and H0 corr_ab to which thehypotheses refer. There are no other differences from the

procedure described in the previous section.Illstrative example. Nosek and Smyth (2007) re-

ported a multitraitmultimethod validation using twoattitude measurement methods, the Implicit AssociationTest (IAT) and self-report (SR). IAT and SR measuresof attitudes toward Democrats versus Republicans (DR)were correlated at r(IAT-DR, SR-DR) 5 .51 5rcd. In

contrast, when measuring attitudes toward whites versusblacks (WB), the correlation between both methods wasonly r(IAT-WB, SR-WB) 5 .12 5rab, probably because

and a power (12)5 .95, given 5 .05. If we choose thea priori type of power analysis and insert the correspond-ing input parameters, clicking on Calculate providesus with the result Total sample size 5 229, along withCritical z 5 1.644854 for theztest based on the exact

method of Brown and Benedetti (1977).2. Correlation Problems Referring

to Two Dependent Correlations

This section refers toztests comparing two dependentPearson correlations that either share (Section 2.1) or donot share (Section 2.2) a common index.

2.1. Comparison of Two DependentCorrelations ab andac (Common Index)

The Correlations: Two dependent Pearson rs (commonindex) procedure provides power analyses for tests of thenull hypothesis that two dependent Pearson correlations

ab andac are identical (H0:ab5ac). Two correlationsare dependent if they are defined for the same population.Correspondingly, their sample estimates, rab and rac, areobserved for the same sample ofNobservations of threecontinuous random variablesXa,Xb, andXc. The two cor-relations share a common index because one of the threerandom variables, Xa, is involved in both correlations.Assuming thatXa,Xb, andXc are multivariate normallydistributed, Steigers (1980, Equation 11) Z1* statistic fol-lows a standard normal distribution under H0 (see alsoDunn & Clark, 1969).G*Powers power calculations fordependent correlations sharing a common index refer tothis test.

Effect size measre. To specify the effect size, bothcorrelationsab and ac are required as input parameters.Alternatively, clicking on Determine opens the effectsize drawer, which can be used to compute ac from aband Cohens (1988, p. 109) effect size measure q, the dif-ference between the Fisherr-to-ztransforms ofab andac. Cohen suggested calling effects of sizes q5 .1, .3,and .5 small, medium, and large, respectively. Note,however, that Cohen developed his q effect size conven-tions for comparisons between independentcorrelationsin different populations. Depending on the size of the thirdcorrelation involved, bc, a specific value ofq can havevery different meanings, resulting in huge effects on sta-

tistical power (see the example below). As a consequence,bc is required as a further input parameter.Inpt and otpt parameters. Apart from the num-

ber of Tail(s) of theztest, the post hoc power analysisprocedure requiresac (i.e., H1 Corr_ac), the signifi-cance level err prob, the Sample SizeN, and the tworemaining correlations H0 Corr_ab and Corr_bcas input parameters in the lower left field of the main win-dow. To the right, the Critical z criterion value for the

ztest and the Power (12 err prob) are displayed asoutput parameters.

Illstrative example. Tsujimoto, Kuwajima, andSawaguchi (2007, p. 34, Table 2) studied correlations

between age and several continuous measures of work-ing memory and executive functioning in children. In8- to 9-year-olds, they found age correlations of .27 and


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is that power can be computed as a function of the slopevalues under H0 and under H1 directly (Dupont & Plum-mer, 1998).

Effect size measre. The slope b assumed under H1,labeled Slope H1, is used as the effect size measure. Note

that the power does not depend only on the difference be-tween Slope H1 and Slope H0, the latter of which is thevalue b5b0 specified by H0. The population standard de-viations of the predictor and criterion values, Std dev _xand Std dev _y, are also required. The effect size drawercan be used to calculate Slope H1 from other basic pa-rameters such as the correlation assumed under H1.

Inpt and otpt parameters. The number ofTail(s) of the ttest, Slope H1, err prob, Totalsample size, Slope H0, and the standard deviations(Std dev _x and Std dev _y) need to be specifiedas input parameters for a post hoc power analysis. Power(12 err prob) is displayed as an output parameter, in

addition to the Critical t decision criterion and the pa-rameters defining the noncentral tdistribution implied byH1 (the noncentrality parameter and the dfof the test).

Illstrative example. Assume that we would like toassess whether the standardized regression coefficient of a bivariate linear regression ofYonXis consistent withH0: $ .40 or H1: , .40. Assuming that 5 .20 actu-ally holds in the underlying population, how large must thesample sizeNofXYpairs be to obtain a power of 125.95 given 5 .05? After choosing Linear bivariate re-gression: One group, size of slope and the a priori typeof power analysis, we specify the above input parameters,making sure that Tail(s)5one, Slope H15 .20, Slope

H05

.40, and Std dev _x5

Std dev _y5

1, be-cause we want to refer to regression coefficients forstan-dardizedvariables. Clicking on Calculate provides uswith the result Total sample size5 262.

3.2. Comparison of Two IndependentIntercepts a1 and a2 (Two Samples)

The Linear bivariate regression: Two groups, differencebetween intercepts procedure is based on the assumptionthat the standard bivariate linear model described aboveholds within each of two different populations with thesame slope b and possibly different intercepts a1 and a2. Itcomputes the power of the two-tailed ttest of H0: a15a2

against H1: a1

a2 and for the corresponding one-tailedt test, as described in Armitage, Berry, and Matthews(2002, ch. 11).

Effect size measre.The absolute value of the differ-ence between the intercepts, |D intercept| 5 |a12a2|, isused as an effect size measure. In addition to |D intercept|,the significance level, and the sizes n1 and n2 of the twosamples, the power depends on the means and standarddeviations of the criterion and the predictor variable.

Inpt and otpt parameters. The number ofTail(s) of the ttest, the effect size |D intercept|, the err prob, the sample sizes in both groups, the standarddeviation of the error variableEij (Std dev residual ),

the means (Mean m_x1, Mean m_x2), and the standarddeviations (Std dev _x1, Std dev _x2) are requiredas input parameters for a Post hoc power analysis. The

SR measures of attitudes toward blacks are more stronglybiased by social desirability influences. Assuming that(1) these correlations correspond to the true population cor-relations under H1 and (2) the other four between-attitudecorrelations (IAT-WB, IAT-DR), (IAT-WB, SR-DR),

(SR-WB, IAT-DR), and (SR-WB, SR-DR) are zero,how large must the sample be to make sure that this devia-tion from H0:(IAT-DR, SR-DR)5(IAT-WB, SR-WB)is detected with a power of 125 .95 using a one-tailedtest and 5 .05? An a priori power analysis for Corre-lations: Two dependent Pearson rs (no common index)computesN5 114 as the required sample size.

The assumption that the four additional correlationsare zero is tantamount to assuming that the two correla-tions under test are statistically independent (thus, the

procedure in G*Power for independent correlations couldalternatively have been used). If we instead assume that(IAT-WB, IAT-DR)5ac5 .6 and(SR-WB, SR-DR)5

bd5 .7, we arrive at a considerably lower sample size ofN5 56. If our resources were sufficient for recruiting notmore thanN5 40 participants and we wanted to makesure that the / ratio equals 1 (i.e., balanced error riskswith 5), a compromise power analysis for the lat-ter case computes Critical z 5 1.385675 as the optimalstatistical decision criterion, corresponding to 55.082923.

3. Linear Regression Problems,One Predictor (Simple Linear Regression)

This section summarizes power analysis proceduresaddressing regression coefficients in the bivariate linear

standard model Yi5

a1

bXi1

Ei, where Yi andXi repre-sent the criterion and the predictor variable, respectively,a and b the regression coefficients of interest, and Ei anerror term that is independently and identically distrib-uted and follows a normal distribution with expectation 0and homogeneous variance 2 for each observation unit i.Section 3.1 describes a one-sample procedure for tests ad-dressing b, whereas Sections 3.2 and 3.3 refer to hypoth-eses on differences in a and b between two different un-derlying populations. Formally, the tests considered hereare special cases of the multiple linear regression proce-dures described in Section 4. However, the procedures forthe special case provide a more convenient interface that

may be easier to use and interpret if users are interestedin bivariate regression problems only (see also Dupont &Plummer, 1998).

3.1. Comparison of a Slope bWit a Constant b0The Linear bivariate regression: One group, size of

slope procedure computes the power of the t test ofH0: b5b0 against H1: bb0, where b0 is any real-valuedconstant. Note that this test is equivalent to the standard

bivariate regression ttest of H0: b*5 0 against H1: b* 0if we refer to the modified model Yi*5a 1 b*Xi1Ei,with Yi*:5Yi2b0Xi (Rindskopf, 1984). Hence, powercould also be assessed by referring to the standard regres-

sion ttest (or globalFtest) using Y*

rather than Yas acriterion variable. The main advantage of the Linear bi-variate regression: One group, size of slope procedure


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of H0: b1#b2 against H1: b1.b2,given 5 .05? To an-swer this question, we select Linear bivariate regression:Two groups, difference between slopes along with the

post hoc type of power analysis. We then provide the ap-propriate input parameters [Tail(s)5 one, |D slope|5

.57, error prob 5 .05, Sample size group 1 5Sample size group 25 30, Std dev. residual 5 .80,and Std dev _X1 5 Std dev _X2 5 1] and clickon Calculate. We obtain Power (12 err prob) 5.860165.

4. Linear Regression Problems, SeveralPredictors (Mltiple Linear Regression)

In multiple linear regression, the linear relation betweena criterion variable Yand m predictorsX5 (X1, . . . ,Xm)is studied. G*Power 3.1 now provides power analysis pro-cedures for both the conditional (or f ixed-predictors) andthe unconditional (or random-predictors) models of mul-

tiple regression (Gatsonis & Sampson, 1989; Sampson,1974). In the fixed-predictors model underlying previousversions of G*Power, the predictorsXare assumed to befixed and known. In the random-predictors model, by con-trast, they are assumed to be random variables with valuessampled from an underlying multivariate normal distribu-tion. Whereas the fixed-predictors model is often moreappropriate in experimental research (where known pre-dictor values are typically assignedto participants by anexperimenter), the random-predictors model more closelyresembles the design of observational studies (where par-ticipants and their associated predictor values are sam-

pledfrom an underlying population). The test procedures

and the maximum likelihood estimates of the regressionweights are identical for both models. However, the mod-els differ with respect to statistical power.

Sections 4.1, 4.2, and 4.3 describe procedures relatedtoFand ttests in the fixed-predictors model of multiplelinear regression (cf. Cohen, 1988, ch. 9), whereas Sec-tion 4.4 describes the procedure for the random-predictorsmodel. The procedures for the fixed-predictors model are

based on the general linear model (GLM), which includesthe bivariate linear model described in Sections 3.13.3as a special case (Cohen, Cohen, West, & Aiken, 2003).In other words, the following procedures, unlike thosein the previous section, have the advantage that they are

not limited to a single predictor variable. However, theirdisadvantage is that effect size specifications cannot bemade in terms of regression coefficients under H0 and H1directly. Rather, variance proportions, or ratios of variance

proportions, are used to define H0 and H1 (cf. Cohen, 1988,ch. 9). For this reason, we decided to include both bivariateand multiple linear regression procedures in G*Power 3.1.The latter set of procedures is recommended whenever sta-tistical hypotheses are defined in terms of proportions ofexplained variance or whenever they can easily be trans-formed into hypotheses referring to such proportions.

4.1. Deviation of a Sqared Mltiple Correlation

2

From Zero (Fixed Model)The Linear multiple regression: Fixed model, R2 de-viation from zero procedure provides power analyses for

Power (12 err prob) is displayed as an output param-eter in addition to the Critical t decision criterion andthe parameters defining the noncentral tdistribution im-

plied by H1 (the noncentrality parameter and the dfofthe test).

3.3. Comparison of Two IndependentSlopes b1 and b2 (Two Samples)

The linear model used in the previous two sections alsounderlies the Linear bivariate regression: Two groups, dif-ferences between slopes procedure. Two independent sam-

ples are assumed to be drawn from two different popula-tions, each consistent with the model Yij5aj1bjXi1Ei,where Yij,Xij, andEij respectively represent the criterion,the predictor, and the normally distributed error variablefor observation unit i in populationj. Parameters ajand bjdenote the regression coefficients in populationj,j5 1, 2.The procedure provides power analyses for two-tailed ttests

of the null hypothesis that the slopes in the two populationsare equal (H0: b15b2) versus different (H1: b1b2) andfor the corresponding one-tailed ttest (Armitage et al.,2002, ch. 11, Equations 11.1811.20).

Effect size measre. The absolute difference betweenslopes, |D slope|5 |b12b2|, is used as an effect size mea-sure. Statistical power depends not only on |D slope|,, andthe two sample sizes n1and n2. Specifically, the standarddeviations of the error variableEij (Std dev residual ),the predictor variable (Std dev _X), and the criterionvariable (Std dev _Y) in both groups are required tofully specify the effect size.

Inpt and otpt parameters. The input and output

parameters are similar to those for the two procedures de-scribed in Sections 3.1 and 3.2. In addition to |D slope| andthe standard deviations, the number of Tail(s) of the test,the err prob, and the two sample sizes are required inthe Input Parameters fields for the post hoc type of poweranalysis. In the Output Parameters fields, the Noncen-trality parameter of the tdistribution under H1, the deci-sion criterion (Critical t), the degrees of freedom of thettest (Df ), and the Power (12 err prob) implied bythe input parameters are provided.

Illstrative application. Perugini, OGorman, andPrestwich (2007, Study 1) hypothesized that the criterionvalidity of the IAT depends on the degree of self-activation

in the test situation. To test this hypothesis, they asked 60participants to circle certain words while reading a shortstory printed on a sheet of paper. Half of the participantswere asked to circle the words the and a (control con-dition), whereas the remaining 30 participants were askedto circle I, me, my, and myself (self-activationcondition). Subsequently, attitudes toward alcohol versussoft drinks were measured using the IAT (predictorX).In addition, actual alcohol consumption rate was assessedusing the self-report (criterion Y). Consistent with theirhypothesis, they found standardized YXregression coef-ficients of5 .48 and52.09 in the self-activation andcontrol conditions, respectively. Assuming that (1) these

coefficients correspond to the actual coefficients under H1in the underlying populations and (2) the error standard de-viation is .80, how large is the power of the one-tailed ttest


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H0:2Y.X1,...,Xm52Y.X1,...,Xk(i.e., Set B does notincrease the

proportion of explained variance) versus H1: 2Y.X1,...,Xm.2Y.X1,...,Xk(i.e., Set B increases the proportion of explainedvariance). Note that H0 is tantamount to claiming that allm2kregression coefficients of Set B are zero.

As shown by Rindskopf (1984), special F tests canalso be used to assess various constraints on regressioncoefficients in linear models. For example, ifYi5b01b1X1i1 . . . 1bmXmi1Ei is the full linear model andYi5b01bX1i1 . . . 1bXmi1Ei is a restricted H0model claiming that all m regression coefficients are equal(H0: b15b25 . . .5bm5b), we can define a new predic-tor variableXi:5X1i1X2i1 . . .1Xmi and consider therestricted model Yi5b01bXi1Ei.Because this modelis equivalent to the H0 model, we can compare 2Y.X1,...,Xmand2Y.Xwith a specialFtest to test H0.

Effect size measre. Cohens (1988)f2 is again usedas an effect size measure. However, here we are interested

in the proportion of variance explained by predictors fromSet B only, so that f25 (2Y.X1,...,Xm22Y.X1,...,Xk) / (1 2

2Y.X1,...,Xm) serves as an effect size measure. The effect sizedrawer can be used to calculatef2 from the variance ex-

plained by Set B (i.e.,2Y.X1,...,Xm22Y.X1,...,Xk) and the error

variance (i.e., 122Y.X1,...,Xm). Alternatively,f2 can also becomputed as a function of the partial correlation squaredof Set B predictors (Cohen, 1988, ch. 9).

Cohen (1988) suggested the same effect size conven-tions as in the case of global Ftests (see Section 4.1).However, we believe that researchers should reflect thefact that a certain effect size, sayf25 .15, may have verydifferent substantive meanings depending on the propor-

tion of variance explained by Set A (i.e.,2Y.X1,...,Xk).Inpt and otpt parameters. The inputs and out-

puts match those of the Linear multiple regression:Fixed model, R2 deviation from zero procedure (see Sec-tion 4.1), with the exception that the Number of tested

predictors :5m2kthat is, the number of predictors inSet Bis required as an additional input parameter.

Illstrative example. In Section 3.3, we presented apower analysis for differences in regression slopes asanalyzed by Perugini et al. (2007, Study 1). Multiplelinear regression provides an alternative method to ad-dress the same problem. In addition to the self-report ofalcohol use (criterion Y) and the IAT attitude measure

(predictorX), two additional predictors are requiredfor this purpose: a binary dummy variable Grepresent-ing the experimental condition (G5 0, control group;G5 1, self-activation group) and, most importantly, a

product variable GXrepresenting the interaction of theIAT measure and the experimental conditional. Differ-ences in YX regression slopes in the two groups willshow up as a significant effect of the GXinteraction ina regression model using Yas the criterion andX, G, andGXas m5 3 predictors.

Given a total ofN5 60 participants (30 in each group),5 .05, and a medium sizef25 .15 of the interaction ef-fect in the underlying population, how large is the power

of the specialFtest assessing the increase in explainedvariance due to the interaction? To answer this question,we choose the post hoc power analysis in the Linear mul-

omnibus (or global)Ftests of the null hypothesis thatthe squared multiple correlation between a criterion vari-able Yand a set ofm predictor variablesX1,X2, . . . ,Xm iszero in the underlying population (H0: 2Y.X1,...,Xm5 0) ver-sus larger than zero (H1:2Y.X1,...,Xm. 0). Note that the for-

mer hypothesis is equivalent to the hypothesis that all m re-gression coefficients of the predictors are zero (H0: b15b25 . . . 5bm5 0). By implication, the omnibusFtestcan also be used to test fully specified linear models of thetype Yi5b01c1X1i1 . . .1cmXmi1Ei,where c1, . . . ,cm are user-specified real-valued constants defining H0.To test this fully specified model, simply define a newcriterion variable Yi*:5Yi2c1X1i2 . . . 2cmXmiand

perform a multiple regression ofY* on the m predictorsX1toXm. H0 holds if and only if2Y*.X1,...,Xm5 0that is, if allregression coefficients are zero in the transformed regres-sion equation pertaining to Y* (see Rindskopf, 1984).

Effect size measre. Cohensf2, the ratio of explained

variance and error variance, serves as the effect size mea-sure (Cohen, 1988, ch. 9). Using the effect size drawer,f2can be calculated directly from the squared multiple cor-relation2Y.X1,...,Xm in the underlying population. For omni-

busFtests, the relation betweenf2 and2Y.X1,...,Xm is simplygiven byf252Y.X1,...,Xm / (1 2

2Y.X1,...,Xm). According to

Cohen (ch. 9),f2 values of .02, .15, and .35 can be calledsmall, medium, and large effects, respectively. Al-ternatively, one may compute f2 by specifying a vectoru of correlations between Yand the predictorsXi alongwith the (m3m) matrixB of intercorrelations among the

predictors. Given u andB, it is easy to derive 2Y.X1,...,Xm5uTB21u and, via the relation betweenf2 and2Y.X1,...,Xm de-

scribed above, alsof2

.Inpt and otpt parameters. The post hoc poweranalysis procedure requires the population Effect size f2,the err prob, the Total sample sizeN, and the Totalnumber of predictors m in the regression model as input

parameters. It provides as output parameters the Non-centrality parameter of theFdistribution under H1, thedecision criterion (Critical F), the degrees of freedom(Numerator df, Denominator df), and the power ofthe omnibusFtest [Power (12 err prob)].

Illstrative example. Given three predictor variablesX1,X2, andX3, presumably correlated to Ywith 15 .3,252.1, and 35 .7, and with pairwise correlations of

135

.4 and 125

235

0, we first determine the effectsizef2. Inserting b 5(.3,2.1, .7) and the intercorrelationmatrix of the predictors in the corresponding input dialogin G*Powers effect size drawer, we find that2Y.X1,...,Xm5 .5andf25 1. The results of an a priori analysis with this effectsize reveals that we need a sample size ofN5 22 to achievea power of .95 in a test based on 5 .05.

4.2. Deviation of a Sbset of Linear RegressionCoefficients From Zero (Fixed Model)

Like the previous procedure, this procedure is based onthe GLM, but this one considers the case of two PredictorSets A (includingX1, . . . ,Xk) and B (includingXk11, . . . ,

Xm) that define the full model with m predictors. TheLinear multiple regression: Fixed model, R2 increaseprocedure provides power analyses for the specialFtest of


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tively, the three-momentFapproximation suggested byLee (1972) can also be used.

In addition to the five types of power analysis and thegraphic options that G*Power 3.1 supports for any test,the program provides several other useful procedures for

the random-predictors model: (1) It is possible to calculateexact confidence intervals and confidence bounds for thesquared multiple correlation; (2) the critical values ofR2given in the output may be used in hypothesis testing; and(3) the probability density function, the cumulative distri-

bution function, and the quantiles of the sampling distri-bution of the squared multiple correlation coefficients areavailable via the G*Power calculator.

The implemented procedures provide power analysesfor tests of the null hypothesis that the population squaredmultiple correlation coefficient2 equals20 (H0:

2520)versus a one- or a two-tailed alternative.

Effect size measre. The squared population correla-

tion coefficient2

under the alternative hypothesis (H12

)serves as an effect size measure. To fully specify the ef-fect size, the squared multiple correlation under the nullhypothesis (H02) is also needed.

The effect size drawer offers two ways to calculate theeffect size 2. First, it is possible to choose a certain per-centile of the (12) confidence interval calculated for anobservedR2 as H12. Alternatively, H12 can be obtained

by specifying a vectoru of correlations between criterionYand predictorsXi and the (m3m) matrixB of correla-tions among the predictors. By definition,25uTB21u.

Inpt and otpt parameters. The post hoc poweranalysis procedure requires the type of test (Tail(s): one

vs. two), the population 2

under H1, the population 2

under H0, the error prob, the Total sample size N,and the Number of predictors m in the regression modelas input parameters. It provides the Lower critical R2,the Upper critical R2, and the power of the test Power(12 err prob) as output. For a two-tailed test, H0 is re-tained whenever the sampleR2 lies in the interval defined

by Lower critical R2 and Upper critical R2; otherwise,H0 is rejected. For one-tailed tests, in contrast, Lowercritical R2 and Upper critical R2 are identical; H0 isrejected if and only ifR2 exceeds this critical value.

Illstrative examples. In Section 4.1, we presenteda power analysis for a three-predictor example using the

fixed-predictors model. Using the same input procedurein the effect size drawer (opened by pressing Insert/editmatrix in the From predictor correlations section) de-scribed in Section 4.1, we again find H1 2 5 .5. How-ever, the result of the same a priori analysis (power5 .95,5 .05, 3 predictors) previously computed for the fixedmodel shows that we now need a sample size of at least

N5 26 for the random model, instead of only theN5 22previously found for the fixed model. This difference il-lustrates the fact that sample sizes required for the random-predictors model are always slightly larger than those re-quired for the corresponding fixed-predictors model.

As a further example, we use the From confidence in-

terval procedure in the effect size drawer to determinethe effect size based on the results of a pilot study. Weinsert the values from a previous study (Total sample

tiple regression: Fixed model, R2 increase procedure. Inaddition to Effect size f25 .15, err prob5 .05, andTotal sample size 5 60, we specify Number of tested

predictors 5 1 and Total number of predictors 5 3as input parameters, because we have m5 3 predictors

in the full model and only one of these predictors cap-tures the effect of interestnamely, the interaction ef-fect. Clicking on Calculate provides us with the resultPower (12 err prob)5 .838477.

4.3. Deviation of a Single Linear RegressionCoefficient bj From Zero (tTest, Fixed Model)

Special F tests assessing effects of a single predic-torXj in multiple regression models (hence, numeratordf5 1) are equivalent to two-tailed ttests of H0: bj5 0.The Linear multiple regression: Fixed model, single re-gression coefficient procedure has been designed for thissituation. The main reason for including this procedure

in G*Power 3.1 is that ttests for single regression coef-ficients can take the form of one-tailed tests of, for ex-ample, H0: bj# 0 versus H1: bj. 0. Power analyses forone-tailed tests can be done most conveniently with theLinear multiple regression: Fixed model, single regres-sion coefficient ttest procedure.

Effect size measres, inpt and otpt param-eters. Because two-tailed regression t tests are specialcases of the special Ftests described in Section 4.2, theeffect size measure and the input and output parametersare largely the same for both procedures. One exception isthat Numerator df is not required as an input parameterforttests. Also, the number of Tail(s) of the test (one

vs. two) is required as an additional input parameter in thettest procedure.Illstrative application. See the example described in

Section 4.2. For the reasons outlined above, we would ob-tain the same power results if we were to analyze the sameinput parameters using the Linear multiple regression:Fixed model, single regression coefficient procedurewith Tail(s) 5 two. In contrast, if we were to chooseTail(s)5 one and keep the other parameters unchanged,the power would increase to .906347.

4.4. Deviation of Mltiple CorrelationSqared 2 From Constant (Random Model)

The random-predictors model of multiple linear regres-sion is based on the assumption that (Y,X1, . . . ,Xm) arerandom variables with a joint multivariate normal distri-

bution. Sampson (1974) showed that choice of the fixedor the random model has no bearing on the test of sig-nificance or on the estimation of the regression weights.However, the choice of model affects the power of thetest. Several programs exist that can be used to assessthe power for random-model tests (e.g., Dunlap, Xin, &Myers, 2004; Mendoza & Stafford, 2001; Shieh & Kung,2007; Steiger & Fouladi, 1992). However, these programseither rely on other software packages (Mathematica,Excel) or provide only a rather inflexible user interface.

The procedures implemented in G*Power use the exactsampling distribution of the squared multiple correlationcoefficient (Benton & Krishnamoorthy, 2003). Alterna-


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5.1. Logistic RegressionLogistic regression models address the relationship

between a binary dependent variable (or criterion) Yandone or more independent variables (or predictors) Xj,with discrete or continuous probability distributions.

In contrast to linear regression models, the logit trans-form ofY, rather than Yitself, serves as the criterion tobe predicted by a linear combination of the independentvariables. More precisely, ify5 1 andy5 0 denote thetwo possible values ofY, with probabilitiesp(y5 1) and

p(y5 0), respectively, so thatp(y5 1)1p(y5 0)5 1,then logit(Y) :5 ln[p(y5 1) / p(y5 0)] is modeled aslogit(Y) 5011X11 . . . 1mXm. A logistic re-gression model is calledsimple ifm5 1. Ifm. 1, wehave a multiple logistic regression model. The imple-mented procedures provide power analyses for the Waldtestz5 j / se( j) assessing the effect of a specific pre-dictorXj (e.g., H0: j5 0 vs. H1: j 0, or H0: j# 0

vs. H1: j. 0) in both simple and multiple logistic re-gression models. In addition, the procedure of Lyles et al.(2007) also supports power analyses for likelihood ratiotests. In the case of multiple logistic regression models, asimple approximation proposed by Hsieh et al. (1998) isused: The sample sizeNis multiplied by (12R2), where

R2 is the squared multiple correlation coefficient whenthe predictor of interest is regressed on the other predic-tors. The following paragraphs refer to the simple modellogit(Y)5011X.

Effect size measre. Given the conditional probabilityp1:5p(Y5 1 |X5 1) under H0, we may define the effectsize either by specifyingp2:5p(Y5 1 | X5 1) under

H1 or by specifying the odds ratio OR:5

[p2/(12

p2)] /[p1/(1 2p1)]. The parameters 0 and 1 are related top1and p2 as follows: 05 ln[p1/(1 2p1)], 15 ln[OR].Under H0,p15p2 orOR5 1.

Inpt and otpt parameters. The post hoc type ofpower analysis for the Logistic regression procedurerequires the following input parameters: (1) the num-

ber of Tail(s) of the test (one vs. two); (2) Pr(Y51 |X 5 1) under H0, corresponding to p1; (3) the effectsize [either Pr(Y 5 1 | X51) under H1 or, option-ally, the Odds ratio specifyingp2]; (4) the err prob;(5) the Total sample size N; and (6) the proportion ofvariance ofXj explained by additional predictors in the

model (R2

other X). Finally, because the power of thetest also depends on the distribution of the predictorX, theX distribution and its parameters need to be specified.Users may choose between six predefined distributions(binomial, exponential, log-normal, normal, Poisson, oruniform) or select a manual input mode. Depending onthis selection, additional parameters (corresponding to thedistribution parameters or, in the manual mode, the vari-ances v0 and v1 of1 under H0 and H1, respectively) must

be specif ied. In the manual mode, sensitivity analysesare not possible. After clicking Calculate, the statisticaldecision criterion (Critical z in the large-sample pro-cedures, Noncentrality parameter, Critical 2, and

df in the enumeration approach) and the power of thetest [Power (12 err prob)] are displayed in the OutputParameters fields.

size 5 50, Number of predictors 5 5, and ObservedR2 5 .3), choose Confidence level (12) 5 .95, andchoose the center of the confidence interval as H1 2[Rel. C.I. pos to use (05left,15right) 5 .5]. PressingCalculate produces the two-sided interval (.0337, .4603)

and the two one-sided intervals (0, .4245) and (.0589, 1),thus confirming the values computed by Shieh and Kung(2007, p. 733). In addition, H1 2 is set to .2470214,the center of the interval. Given this value, an a priorianalysis of the one-sided test of H0:25 0 using 5 .05shows that we need a sample size of at least 71 to achievea power of .95. The output also provides Upper criticalR2 5 .153427. Thus, if in our new study we foundR2 to

be larger than this value, the result here implies that thetest would be significant at the .05 level.

Suppose we were to find a value ofR25 .19. Thenwe could use the G*Power calculator to compute the pvalue of the test statistic: The syntax for the cumulative

distribution function of the sampling distribution ofR2

is mr2cdf(R2, 2, m11, N). Thus, in our case, we needto insert 12mr2cdf(0.19, 0, 511, 71) in the calculator.Pressing Calculate shows thatp5 .0156.

5. Generalized Linear Regression Problems

G*Power 3.1 includes power procedures for logistic andPoisson regression models in which the predictor variableunder test may have one of six different predefined dis-tributions (binary, exponential, log-normal, normal, Pois-son, and uniform). In each case, users can choose betweenthe enumeration approach proposed by Lyles, Lin, andWilliamson (2007), which allows power calculations for

Wald and likelihood ratio tests, and a large-sample ap-proximation of the Wald test based on the work of Demi-denko (2007, 2008) and Whittemore (1981). To allow forcomparisons with published results, we also implementedsimple but less accurate power routines that are in wide-spread use (i.e., the procedures of Hsieh, Bloch, & Lar-sen, 1998, and of Signorini, 1991, for logistic and Poissonregressions, respectively). Problems with Whittemoresand Signorinis procedures have already been discussed byShieh (2001) and need not be reiterated here.

The enumeration procedure of Lyles et al. (2007)is conceptually simple and provides a rather direct,simulation-like approach to power analysis. Its main

disadvantage is that it is rather slow and requires muchcomputer memory for analyses with large sample sizes.The recommended practice is therefore to begin with thelarge-sample approximation and to use the enumerationapproach mainly to validate the results (if the sample sizeis not too large).

Demidenko (2008, pp. 37f ) discussed the relative mer-its of power calculations based on Wald and likelihoodratio tests. A comparison of both approaches using theLyles et al. (2007) enumeration procedure indicated thatin most cases the difference in calculated power is small.In cases in which the results deviated, the simulated powerwas slightly higher for the likelihood ratio test than for

the Wald test procedure, which tended to underestimatethe true power, and the likelihood ratio procedure also ap-peared to be more accurate.


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criterion Y) and one or more independent variables (i.e.,the predictorsXj,j5 1, . . . , m). For a count variable Yindexing the numberY5 y of events of a certain type in afixed amount of time, a Poisson distribution model is oftenreasonable (Hays, 1972). Poisson distributions are defined

by a single parameter, the so-called intensity of the Pois-son process. The larger the, the larger the number of crit-ical events per time, as indexed by Y. Poisson regressionmodels are based on the assumption that the logarithm ofis a linear combination ofm predictorsXj,j5 1, . . . , m. Inother words, ln() 5011X11 . . . 1mXm, with jmeasuring the effect of predictorXjon Y.

The procedures currently implemented in G*Powerprovide power analyses for the Wald testz5 j / se( j),which assesses the effect of a specific predictorXj (e.g.,H0: j5 0 vs. H1: j 0 or H0: j# 0 vs. H1: j. 0)in both simple and multiple Poisson regression models.In addition, the procedure of Lyles et al. (2007) provides

power analyses for likelihood ratio tests. In the case ofmultiple Poisson regression models, a simple approxima-tion proposed by Hsieh et al. (1998) is used: The samplesizeNis multiplied by (12R2), whereR2 is the squaredmultiple correlation coefficient when the predictor ofinterest is regressed on the other predictors. In the fol-lowing discussion, we assume the simple model ln() 5011X.

Effect size measre. The ratio R5(H1)/(H0) ofthe intensities of the Poisson processes under H1 and H0,given X15 1, is used as an effect size measure. UnderH0,R5 1. Under H1, in contrast,R5 exp(1).

Inpt and otpt parameters. In addition to

Exp(1), the number of Tail(s) of the test, the errprob, and the Total sample size, the following input pa-rameters are required for post hoc power analysis tests inPoisson regression: (1) The intensity5 exp(0) assumedunder H0 [Base rate exp(0)], (2) the mean exposuretime during which the Yevents are counted (Mean expo-sure), and (3) R2 other X, a factor intended to approxi-mate the influence of additional predictorsXk (if any) onthe predictor of interest. The meaning of the latter fac-tor is identical to the correction factor proposed by Hsiehet al. (1998, Equation 2) for multiple logistic regression(see Section 5.1), so that R2 other X is the proportionof the variance ofXexplained by the other predictors. Fi-

nally, because the power of the test also depends on thedistribution of the predictorX, the X distribution and itsparameters need to be specified. Users may choose fromsix predefined distributions (binomial, exponential, log-normal, normal, Poisson, or uniform) or select a manualinput mode. Depending on this choice, additional param-eters (corresponding to distribution parameters or, in themanual mode, the variances v0 and v1 of1 under H0 andH1, respectively) must be specified. In the manual mode,sensitivity analyses are not possible.

In post hoc power analyses, the statistical decisioncriterion (Critical z in the large-sample procedures,Noncentrality parameter , Critical 2, and df

in the enumeration approach) and the power of the test[Power (12 err prob)] are displayed in the Output Pa-rameters fields.

Illstrative examples. Using multiple logistic regres-sion, Friese, Bluemke, and Wnke (2007) assessed the ef-fects of (1) the intention to vote for a specific political

partyx (5 predictorX1) and (2) the attitude toward partyxas measured with the IAT (5 predictorX2) on actual vot-

ing behavior in a political election (5 criterion Y). BothX1 and Yare binary variables taking on the value 1 if aparticipant intends to vote forx or has actually voted forx,respectively, and the value 0 otherwise. In contrast, X2 isa continuous implicit attitude measure derived from IATresponse time data.

Given the fact that Friese et al. (2007) were able toanalyze data forN5 1,386 participants, what would

be a reasonable statistical decision criterion (cr iti-cal zvalue) to detect effects of size p15 .30 and p25.70 (so that OR5 .7/.3 .7/.3 5 5.44) for predictorX1,given a base rate B5 .10 for the intention to vote forthe Green party, a two-tailed ztest, and balanced and

error risks so that q5/5 1? A compromise poweranalysis helps find the answer to this question. In the ef-fect size drawer, we enter Pr(Y 5 1 | X5 1) H15 .70,Pr(Y 5 1 | X 5 1) H0 5 .30. Clicking Calcu-late and transfer to main window yields an odds ratioof 5.44444 and transfers this value and the value ofPr(Y 5 1 | X 5 1) H0 to the main window. Here wespecify Tail(s) 5 two, / ratio 5 1, Total sam-ple size 5 1,386, X distribution 5 binomial, andx parm 5 .1 in the Input Parameters fields. If we as-sume that the attitude toward the Green party (X2) explains40% of the variance ofX1, we need R2 other X 5 .40as an additional input parameter. In the Options dialog

box, we choose the Demidenko (2007) procedure (withoutvariance correction). Clicking on Calculate provides uswith Critical z 5 3.454879, corresponding to 55.00055. Thus, very small values may be reasonable if thesample size, the effect size, or both are very large, as is thecase in the Friese et al. study.

We now refer to the effect of the attitude toward theGreen party (i.e., predictorX2,assumed to be standardnormally distributed). Assuming that a proportion ofp15.10 of the participants with an average attitude toward theGreens actually vote for them, whereas a proportion of.15 of the participants with an attitude one standard de-viation above the mean would vote for them [thus, OR5

(.15/.85) (.90/.10)5

1.588 and b15

ln(1.588)5

.4625],what is a reasonable criticalzvalue to detect effects of thissize for predictorX2 with a two-tailedztest and balanced and error risks (i.e., q5/5 1)? We set X distribu-tion5 normal, X parm m5 0, and X parm 5 1. Ifall other input parameters remain unchanged, a compro-mise power analysis results in Critical z 5 2.146971,corresponding to 55 .031796. Thus, different de-cision criteria and error probabilities may be reasonablefor different predictors in the same regression model, pro-vided we have reason to expect differences in effect sizesunder H1.

5.2. Poisson RegressionA Poisson regression model describes the relationshipbetween a Poisson-distributed dependent variable (i.e., the


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AuThOR NOTE

Manuscript preparation was supported by Grant SFB 504 (Project A12)from the Deutsche Forschungsgemeinschaft. We thank two anonymousreviewers for valuable comments on a previous version of the manuscript.Correspondence concerning this article should be addressed to E. Erd-

felder, Lehrstuhl fr Psychologie III, Universitt Mann heim, SchlossEhrenhof Ost 255, D-68131 Mannheim, Germany (e-mail: [email protected]) or F. Faul, Institut fr Psychologie,Christian-Albrechts-Universitt, Olshausenstr. 40, D-24098 Kiel, Ger-many (e-mail: [email protected]).

NoteThis article is based on information presented at the 2008meeting of the Society for Computers in Psychology, Chicago .

REFERENCES

Armitage, P., Berry, G., & Matthews, J. N. S. (2002). Statisticalmethods in medical research (4th ed.). Oxford: Blackwell.

Benton, D., & Krishnamoorthy, K. (2003). Computing discrete mix-tures of continuous distributions: Noncentral chisquare, noncentral tand the distribution of the square of the sample multiple correlation

coefficient. Computational Statistics & Data Analysis, 43, 249-267.Bonett, D. G., & Price, R. M. (2005). Inferential methods for the tet-rachoric correlation coefficient.Journal of Educational & BehavioralStatistics, 30, 213-225.

Bredenkamp, J. (1969). ber die Anwendung von Signifikanztests beitheorie-testenden Experimenten [On the use of significance tests intheory-testing experiments].Psychologische Beitrge, 11, 275-285.

Brown, M. B., & Benedetti, J. K. (1977). On the mean and variance ofthe tetrachoric correlation coefficient.Psychometrika, 42, 347-355.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Appliedmultiple regression/correlation analysis for the behavioral sciences(3rd ed.). Mahwah, NJ: Erlbaum.

Demidenko, E. (2007). Sample size determination for logistic regres-sion revisited. Statistics in Medicine, 26, 3385-3397.

Demidenko, E. (2008). Sample size and optimal design for logistic re-gression with binary interaction. Statistics in Medicine, 27, 36-46.Dunlap, W. P., Xin, X., & Myers, L. (2004). Computing aspects of

power for multiple regression. Behavior Research Methods, Instru-ments, & Computers, 36, 695-701.

Dunn, O. J., & Clark, V. A. (1969). Correlation coefficients measuredon the same individuals.Journal of the American Statistical Associa-tion, 64, 366-377.

Dupont, W. D., & Plummer, W. D. (1998). Power and sample size cal-culations for studies involving linear regression. Controlled ClinicalTrials, 19, 589-601.

Erdfelder, E. (1984). Zur Bedeutung und Kontrolle des beta-Fehlersbei der inferenzstatistischen Prfung log-linearer Modelle [On sig-nificance and control of the beta error in statistical tests of log-linearmodels]. Zeitschrift fr Sozialpsychologie, 15,18-32.

Erdfelder, E., Faul, F., & Buchner, A. (2005). Power analysis for

categorical methods. In B. S. Everitt & D. C. Howell (Eds.),Encyclo-pedia of statistics in behavioral science (pp. 1565-1570). Chichester,U.K.: Wiley.

Erdfelder, E., Faul, F., Buchner, A., & Cpper, L. (in press).Effektgre und Teststrke [Effect size and power]. In H. Holling &B. Schmitz (Eds.), Handbuch der Psychologischen Methoden und

Evaluation. Gttingen: Hogrefe.Faul, F., Erdfelder, E., Lang, A.G., & Buchner, A. (2007).

G*Power 3: A flexible statistical power analysis program for the so-cial, behavioral, and biomedical sciences. Behavior Research Meth-ods, 39, 175-191.

Friese, M., Bluemke, M., & Wnke, M. (2007). Predicting voting be-havior with implicit attitude measures: The 2002 German parliamen-tary election.Experimental Psychology, 54, 247-255.

Gatsonis, C., & Sampson, A. R. (1989). Multiple correlation: Exact powerand sample size calculations.Psychological Bulletin, 106, 516-524.

Hays, W. L. (1972). Statistics for the social sciences (2nd ed.). NewYork: Holt, Rinehart & Winston.

Illstrative example.We refer to theexample givenin Signorini (1991, p. 449). The number of infections Y(modeled as a Poisson-distributed random variable) ofswimmers (X5 1) versus nonswimmers (X5 0) duringa swimming season (defined as Mean exposure5 1) is

analyzed using a Poisson regression model. How manyparticipants do we need to assess the effect of swimmingversus not swimming (X) on the infection counts (Y)? Weconsider this to be a one-tailed test problem (Tail(s) 5one). The predictorXis assumed to be a binomially dis-tributed random variable with 5 .5 (i.e., equal numbersof swimmers and nonswimmers are sampled), so that weset X distribution to Binomial and X param to .5.The Base rate exp(0)that is, the infection rate innonswimmersis estimated to be .85. We want to knowthe sample size required to detect a 30% increase in in-fection rate in swimmers with a power of .95 at 5 .05.Using the a priori analysis in the Poisson regression

procedure, we insert the values given above in the cor-responding input fields and choose Exp(1) 5 1.3 5(100% 1 30%)/100% as the to-be-detected effect size.In the single-predictor case considered here, we set R2other X5 0. The sample size calculated for these valueswith the Signorini procedure isN5 697. The procedure

based on Demidenko (2007; without variance correc-tion) and the Wald test procedure proposed by Lyles et al.(2007) both result in N5655. A computer simulationwith 150,000 cases revealed mean power values of .952and .962 for sample sizes 655 and 697, respectively, con-firming that Signorinis procedure is less accurate thanthe other two.

6. Conclding Remarks

G*Power 3.1 is a considerable extension of G*Power 3.0,introduced by Faul et al. (2007). The new version providesa priori, compromise, criterion, post hoc, and sensitivity

power analysis procedures for six correlation and nineregression test problems as described above. Readersinterested in obtaining a free copy of G*Power 3.1 maydownload it from the G*Power Web site at www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3/. We recom-mend registering as a G*Power user at the same Web site.Registered G*Power users will be informed about futureupdates and new G*Power versions.

The current version of G*Power 3.1 runs on Windowsplatforms only. However, a version for Mac OS X 10.4 (orlater) is currently in preparation. Registered users will beinformed when the Mac version of G*Power 3.1 is madeavailable for download.

The G*Power site also provides a Web-based manualwith detailed information about the statistical backgroundof the tests, the program handling, the implementation,and the validation methods used to make sure that each ofthe G*Power procedures works correctly and with the ap-

propriate precision. However, although considerable efforthas been put into program evaluation, there is no warrantywhatsoever. Users are kindly asked to report possible bugs

and difficulties in program handling by writing an e-mailto [email protected].


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Shieh, G. (2001). Sample size calculations for logistic and Poisson re-gression models.Biometrika, 88, 1193-1199.

Shieh, G., & Kung, C.F. (2007). Methodological and computationalconsiderations for multiple correlation analysis.Behavior Research

Methods, 39, 731-734.Signorini, D. F.(1991). Sample size for Poisson regression.Biometrika,

78, 446-450.Steiger, J. H. (1980). Tests for comparing elements of a correlationmatrix.Psychological Bulletin, 87, 245-251.

Steiger, J. H., & Fouladi, R. T. (1992). R2: A computer program forinterval estimation, power calculations, sample size estimation, andhypothesis testing in multiple regression. Behavior Research Meth-ods, Instruments, & Computers, 24, 581-582.

Tsujimoto, S., Kuwajima, M., & Sawaguchi, T. (2007). Developmen-tal fractionation of working memory and response inhibition duringchildhood.Experimental Psychology, 54, 30-37.

Whittemore, A. S. (1981). Sample size for logistic regression withsmall response probabilities.Journal of the American Statistical As-

sociation, 76, 27-32.

(Manuscript received December 22, 2008;

revision accepted for publication June 18, 2009.)

Hsieh, F. Y., Bloch, D. A., & Larsen, M. D. (1998). A simple methodof sample size calculation for linear and logistic regression. Statisticsin Medicine, 17, 1623-1634.

Lee, Y.S. (1972). Tables of upper percentage points of the multiple cor-relation coefficient.Biometrika, 59, 175-189.

Lyles, R. H., Lin, H.M., & Williamson, J. M. (2007). A practical ap-

proach to computing power for generalized linear models with nominal,count, or ordinal responses. Statistics in Medicine, 26, 1632-1648.Mendoza, J. L., & Stafford, K. L. (2001). Confidence intervals,

power calculation, and sample size estimation for the squared mul-tiple correlation coefficient under the f ixed and random regressionmodels: A computer program and useful standard tables.Educational& Psychological Measurement, 61, 650-667.

Nosek, B. A., & Smyth, F. L. (2007). A multitraitmultimethod valida-tion of the Implicit Association Test: Implicit and explicit attitudes arerelated but distinct constructs.Experimental Psychology, 54, 14-29.

Perugini, M., OGorman, R., & Prestwich, A. (2007). An ontologicaltest of the IAT: Self-activation can increase predictive validity.Experi-mental Psychology, 54, 134-147.

Rindskopf, D. (1984). Linear equality restrictions in regression and log-linear models.Psychological Bulletin, 96, 597-603.

Sampson, A. R. (1974). A tale of two regressions.Journal of the Ameri-

can Statistical Association, 69, 682-689.

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