G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences

Transcription

1 Behavior Reearch Method 007, 39 (), 75-9 G*Power 3: A flexible tatitical power analyi program for the ocial, behavioral, and biomedical cience FRAZ FAUL Chritian-Albrecht-Univerität Kiel, Kiel, Germany EDGAR ERDFELDER Univerität Mannheim, Mannheim, Germany AD ALBERT-GEORG LAG AD AXEL BUCHER Heinrich-Heine-Univerität Düeldorf, Düeldorf, Germany G*Power (Erdfelder, Faul, & Buchner, 996) wa deigned a a general tand-alone power analyi program for tatitical tet commonly ued in ocial and behavioral reearch. G*Power 3 i a major extenion of, and improvement over, the previou verion. It run on widely ued computer platform (i.e., Window XP, Window Vita, and Mac OS X 0.4) and cover many different tatitical tet of the t, F, and tet familie. In addition, it include power analye for z tet and ome exact tet. G*Power 3 provide improved effect ize calculator and graphic option, upport both ditribution-baed and deign-baed input mode, and offer all type of power analye in which uer might be intereted. Like it predeceor, G*Power 3 i free. Statitic textbook in the ocial, behavioral, and biomedical cience typically tre the importance of power analye. By definition, the power of a tatitical tet i the probability that it null hypothei (H 0 ) will be rejected given that it i in fact fale. Obviouly, ignificance tet that lack tatitical power are of limited ue becaue they cannot reliably dicriminate between H 0 and the alternative hypothei (H ) of interet. However, although power analye are indipenable for rational tatitical deciion, it wa not until the late 980 that power chart (ee, e.g., Scheffé, 959) and power table (ee, e.g., Cohen, 988) were upplemented by more efficient, precie, and eay-to-ue power analyi program for peronal computer (Goldtein, 989). G*Power (Erdfelder, Faul, & Buchner, 996) can be een a a econdgeneration power analyi program deigned a a tandalone application to handle everal type of tatitical tet commonly ued in ocial and behavioral reearch. In the pat 0 year, thi program ha been found ueful not only in the ocial and behavioral cience but alo in many other dicipline that routinely apply tatitical tet, including biology (Baeza & Stotz, 003), genetic (Akkad et al., 006), ecology (Sheppard, 999), foret and wildlife reearch (Mellina, Hinch, Donaldon, & Pearon, 005), the geocience (Bubey, 999), pharmacology (Quednow et al., 004), and medical reearch (Gleiner, Clumann, Saen, Elger, & Helmtaedter, 006). G*Power wa evaluated poitively in the review of which we are aware (Kornbrot, 997; Orteifen, Bruckner, Burke, & Kieer, 997; Thoma & Kreb, 997). It ha been ued in everal power tutorial (e.g., Buchner, Erdfelder, & Faul, 996, 997; Erdfelder, Buchner, Faul, & Brandt, 004; Levin, 997; Sheppard, 999) and in tatitic textbook (e.g., Field, 005; Keppel & Wicken, 004; Myer & Well, 003; Rach, Friee, Hofmann, & aumann, 006a, 006b). everthele, the uer feedback that we received coincided with our own experience in howing ome limitation and weaknee of G*Power that required a major extenion and reviion. In the preent article, we decribe G*Power 3, a program that wa deigned to addre the problem of G*Power. We begin with an outline of the major improvement in G*Power 3 and then dicu the type of power analye covered by thi program. ext, we decribe program handling and the type of tatitical tet to which it can be applied. We then dicu the tatitical algorithm of G*Power 3 and their accuracy. Finally, program availability and ome Internet reource upporting uer of G*Power 3 are decribed. IMPROVEMETS I G*POWER 3 I COMPARISO WITH G*POWER G*Power 3 i an improvement over G*Power in five major repect. Firt, wherea G*Power require the E. Erdfelder, erdfelder@pychologie.uni-mannheim.de 75 Copyright 007 Pychonomic Society, Inc.

2 76 FAUL, ERDFELDER, LAG, AD BUCHER DOS and Mac OS 7 9 operating ytem that were common in the 990 but are now outdated, G*Power 3 run on the peronal computer platform currently in widet ue: Window XP, Window Vita, and Mac OS X 0.4. The Window and Mac verion of the program are eentially equivalent. They ue the ame computational routine and hare very imilar uer interface. For thi reaon, we will not differentiate between thee verion in what follow; uer imply have to make ure to download the verion appropriate for their operating ytem. Second, wherea G*Power i limited to three type of power analye, G*Power 3 upport five different way to ae tatitical power. In addition to the a priori, pot hoc, and compromie power analye that were already covered by G*Power, the new program offer enitivity analye and criterion analye. Third, G*Power 3 provide dedicated power analyi option for a variety of frequently ued t, F, z,, and exact tet in addition to the tandard tet covered by G*Power. The tet captured by G*Power 3 and their effect ize parameter are decribed in the Program Handling ection. Importantly, uer are not limited to thee tet becaue G*Power 3 alo offer power analye for generic t, F, z,, and binomial tet for which the noncentrality parameter of the ditribution under H may be entered directly. In thi way, uer are provided with a flexible tool for computing the power of baically any tatitical tet that ue t, F, z,, or binomial reference ditribution. Fourth, tatitical tet can be pecified in G*Power 3 uing two different approache: the ditribution-baed approach and the deign-baed approach. In the ditributionbaed approach, uer elect the family of the tet tatitic (t, F, z,, or exact tet) and the particular tet within that family. Thi i how power analye were pecified in G*Power. In addition, a eparate menu in G*Power 3 provide acce to power analye via the deign-baed approach: Uer elect () the parameter cla to which the tatitical tet refer (correlation, mean, proportion, regreion coefficient, variance) and () the deign of the tudy (e.g., number of group, independent v. dependent ample). On the bai of the feedback we received about G*Power, we expect that ome uer might find the deign-baed input mode more intuitive and eaier to ue. Fifth, G*Power 3 upport uer with enhanced graphic feature. The detail of thee feature will be outlined in the Program Handling ection. TYPES OF STATISTICAL POWER AALYSES The power ( ) of a tatitical tet i the complement of, which denote the Type II or beta error probability of falely retaining an incorrect H 0. Statitical power depend on three clae of parameter: () the ignificance level (i.e., the Type I error probability) of the tet, () the ize() of the ample() ued for the tet, and (3) an effect ize parameter defining H and thu indexing the degree of deviation from H 0 in the underlying population. Depending on the available reource, the actual phae of the reearch proce, and the pecific reearch quetion, five different type of power analyi can be reaonable (cf. Erdfelder et al., 004; Erdfelder, Faul, & Buchner, 005). We decribe thee method and their ue in turn. A Priori Power Analye In a priori power analye (Cohen, 988), ample ize i computed a a function of the required power level ( ), the prepecified ignificance level, and the population effect ize to be detected with probability. A priori analye provide an efficient method of controlling tatitical power before a tudy i actually conducted (ee, e.g., Bredenkamp, 969; Hager, 006) and can be recommended whenever reource uch a the time and money required for data collection are not critical. Pot Hoc Power Analye In contrat to a priori power analye, pot hoc power analye (Cohen, 988) often make ene after a tudy ha already been conducted. In pot hoc analye, i computed a a function of, the population effect ize parameter, and the ample ize() ued in a tudy. It thu become poible to ae whether or not a publihed tatitical tet in fact had a fair chance of rejecting an incorrect H 0. Importantly, pot hoc analye, like a priori analye, require an H effect ize pecification for the underlying population. Pot hoc power analye hould not be confued with o-called retropective power analye, in which the effect ize i etimated from ample data and ued to calculate the oberved power, a ample etimate of the true power. Retropective power analye are baed on the highly quetionable aumption that the ample effect ize i eentially identical to the effect ize in the population from which it wa drawn (Zumbo & Hubley, 998). Obviouly, thi aumption i likely to be fale, and the more o the maller the ample. In addition, ample effect ize are typically biaed etimate of their population counterpart (Richardon, 996). For thee reaon, we agree with other critic of retropective power analye (e.g., Gerard, Smith, & Weerakkody, 998; Hoenig & Heiey, 00; Kromrey & Hogarty, 000; Lenth, 00; Steidl, Haye, & Schauber, 997). Rather than ue retropective power analye, reearcher hould pecify population effect ize on a priori ground. To pecify the effect ize imply mean to define the minimum degree of violation of H 0 a reearcher would like to detect with a probability not le than. Cohen definition of mall, medium, and large effect can be helpful in uch effect ize pecification (ee, e.g., Smith & Bayen, 005). However, reearcher hould be aware of the fact that thee convention may have different meaning for different tet (cf. Erdfelder et al., 005). Compromie Power Analye In compromie power analye (Erdfelder, 984; Erdfelder et al., 996; Müller, Manz, & Hoyer, 00), both and are computed a function of the effect ize,, and the error probability ratio q /. To illutrate, etting q to would mean that the reearcher prefer balanced Type I and Type II error rik ( ),

3 G*POWER 3 77 wherea a q of 4 would imply that 4 (cf. Cohen, 988). Compromie power analye can be ueful both before and after data collection. For example, an a priori power analyi might reult in a ample ize that exceed the available reource. In uch a ituation, a reearcher could pecify the maximum affordable ample ize and, uing a compromie power analyi, compute and aociated with, ay, q / 4. Alternatively, if a tudy ha already been conducted but ha not yet been analyzed, a reearcher could ak for a reaonable deciion criterion that guarantee perfectly balanced error rik (i.e., ) given the ize of the ample and the critical effect ize in which he or he i intereted. Of coure, compromie power analye can eaily reult in unconventional ignificance level greater than.05 (in the cae of mall ample or effect ize) or le than.00 (in the cae of large ample or effect ize). However, we believe that the benefit of balanced Type I and Type II error rik often offet the cot of violating ignificance level convention (cf. Gigerenzer, Krau, & Vitouch, 004). Senitivity Analye In enitivity analye, the critical population effect ize i computed a a function of,, and. Senitivity analye may be particularly ueful for evaluating publihed reearch. They provide anwer to quetion uch a What effect ize wa a tudy able to detect with a power of.80 given it ample ize and a pecified by the author? In other word, what i the minimum effect ize to which the tet wa ufficiently enitive? In addition, it may be ueful to perform enitivity analye before conducting a tudy to ee whether, given a limited, the ize of the effect that can be detected i at all realitic (or, for intance, much too large to be expected realitically). Criterion Analye Finally, criterion analye compute (and the aociated deciion criterion) a a function of, the effect ize, and a given ample ize. Criterion analye are alternative to pot hoc power analye. They may be reaonable whenever the control of i le important than the control of. In cae of goodne-of-fit tet for tatitical model, for example, it i mot important to minimize the rik of wrong deciion in favor of the model (H 0 ). Reearcher could thu ue criterion analye to compute the ignificance level which i compatible with.05 for a mall effect ize. Wherea G*Power wa limited to the firt three type of power analyi, G*Power 3 cover all five type. On the bai of the feedback we received from G*Power uer, we believe that any quetion related to tatitical power that arie in reearch practice can be categorized under one of thee analyi type. PROGRAM HADLIG Uing G*Power 3 typically involve the following four tep: () Select the tatitical tet appropriate for the problem, () chooe one of the five type of power analye defined in the previou ection, (3) provide the input parameter required for the analyi, and (4) click on Calculate to obtain the reult. In the firt tep, the tatitical tet i choen uing the ditribution-baed or the deign-baed approach. G*Power uer probably have adapted to the ditributionbaed approach: One firt elect the family of the tet tatitic (t, F, z,, or exact tet) uing the Tet family menu in the main window. The Statitical tet menu adapt accordingly, howing a lit of all tet available for the tet family. For the two-group t tet, for example, one would firt elect the t family of ditribution and then Mean: Difference between two independent mean (two group) in the Statitical tet menu (ee Figure ). Alternatively, one might ue the deign-baed approach of tet election. With the Tet pull-down menu in the top row, it i poible to elect () the parameter cla to which the tatitical tet refer (i.e., correlation and regreion, mean, proportion, variance, or generic) and () the deign of the tudy (e.g., number of group, independent v. dependent ample). For example, a reearcher would elect Mean followed by Two independent group to pecify the two-group t tet (ee Figure ). The deignbaed approach ha the advantage that tet option referring to the ame parameter cla (e.g., mean) are located in cloe proximity, wherea in the ditribution-baed approach they may be cattered acro different ditribution familie. In the econd tep, the Type of power analyi menu in the center of the main window hould be ued to chooe the appropriate analyi type. In the third tep, the power analyi input parameter are pecified in the lower left of the main window. To illutrate, an a priori power analyi for a two-group t tet would require a deciion between a one-tailed and a two-tailed tet, a pecification of Cohen (988) effect ize meaure (d) under H, the ignificance level, the required power ( ) of the tet, and the preferred group ize allocation ratio n /n. The final tep conit of clicking on Calculate to obtain the output in the lower right of the main window. For intance, input parameter pecifying a one-tailed t tet, a medium effect ize of d 0.5,.05,.95, and an allocation ratio of n /n would reult in a total ample ize of 76 (88 obervation unit in each group; ee Figure and ). The noncentrality parameter defining the t ditribution under H, the deciion criterion to be ued (i.e., the critical value of the t tatitic), the degree of freedom of the t tet, and the actual power value are alo diplayed. ote that the actual power will often be lightly larger than the prepecified power in a priori power analye. The reaon i that noninteger ample ize are alway rounded up by G*Power to obtain integer value conitent with a power level not lower than the prepecified one. In addition to the numerical output, G*Power 3 diplay the central (H 0 ) and the noncentral (H ) tet tatitic ditribution along with the deciion criterion and the aociated error probabilitie in the upper part of the main window (ee Figure ). 3 Thi upport undertanding of the effect of the input parameter and i likely to be a

4 78 FAUL, ERDFELDER, LAG, AD BUCHER Figure. The ditribution-baed approach of tet pecification in G*Power 3.0. ueful viualization tool in the teaching of, or the learning about, inferential tatitic. The ditribution plot can be printed, aved, or copied by clicking on the right moue button inide the plot area. The input and output of each power calculation in a G*Power eion i automatically written to a protocol that can be diplayed by electing the Protocol of power analye tab in the main window. It i poible to clear the protocol or to print, ave, and copy the protocol in the ame way a the ditribution plot. Becaue Cohen (988) book on power analyi appear to be well-known in the ocial and behavioral cience, we made ue of hi effect ize meaure whenever poible. Reearcher unfamiliar with thee meaure and uer who prefer to compute Cohen meaure from more baic parameter can click on the Determine button to the left of the Effect ize input field (ee Figure and ). A drawer will open next to the main window and provide acce to an effect ize calculator tailored to the elected tet (ee Figure ). For the two-group t tet, for example, uer can pecify the mean (, ) and the common SD () in the population underlying the group to calculate Cohen d /. Clicking on the Calculate and tranfer to main window button copie the computed effect ize to the appropriate field in the main window. Another ueful option i the Power Plot window (ee Figure 3), which i opened by clicking on X Y plot for a range of value on the lower right ide of the main window (ee Figure and ). By electing the appropriate parameter for the y- and x-axe, one parameter (,, effect ize, or ample ize) can be plotted a a function of any other parameter. Of the remaining two parameter, one can be choen to draw a family of graph, wherea the fourth parameter i kept contant. For intance, ample ize can be drawn a a function of the power for everal different population effect ize while i kept at a particular value. The plot may be printed, aved, or copied by clicking on the right moue button inide the plot area. Selecting the Table tab reveal the data underlying the plot; they may be copied to other application. The Power Plot window inherit all input parameter of the analyi that i active when the X Y plot for a range of

5 G*POWER 3 79 Figure. The deign-baed approach of tet pecification in G*Power 3.0 and the Effect ize drawer. Figure 3. The Power Plot window of G*Power 3.0.

6 80 FAUL, ERDFELDER, LAG, AD BUCHER value button i clicked. Only ome of thee parameter can be directly manipulated in the Power Plot window. For intance, witching from a plot of a two-tailed tet to a plot of a one-tailed tet require chooing the Tail(): One option in the main window and then clicking on the X Y plot for a range of value button. TYPES OF STATISTICAL TESTS G*Power 3 provide power analye for tet tatitic following t, F,, or tandard normal ditribution under H 0 (either exact or aymptotic) and noncentral ditribution of the ame tet familie under H. In addition, it include power analye for ome exact tet. In Table 9, we briefly decribe the tet currently covered by G*Power 3. Table lit the ymbol ued in Table 9 and their meaning. Tet for Correlation and Regreion Table ummarize the procedure upported for teting hypothee on correlation and regreion. One-ample tet are provided for the point bierial model that i, the model for correlation between a binary variable and a continuou variable and for correlation between two normally ditributed variable (Cohen, 988, chap. 3). 4 The latter tet ue the exact ample correlation coefficient ditribution (Barabei & Greco, 00) or, optionally, a large-ample approximation baed on Fiher r-to-z tranformation. The two-ample tet for difference between two correlation ue Cohen (988, chap. 4) effect ize q and i baed on Fiher r-to-z tranformation. Cohen define q of 0.0, 0.30, and 0.50 a mall, medium, and large effect, repectively. The two procedure available for the multiple regreion model handle the cae of () a tet of an overall effect that i, the hypothei that the population value of R i different from zero and () a tet of the hypothei that adding more predictor increae the value of R (Cohen, 988, chap. 9). According to Cohen criteria, effect ize ( f ) of 0.0, 0.5, and 0.35 are conidered mall, medium, and large, repectively. Tet for Mean (Univariate Cae) Table 3 ummarize the power analyi procedure for tet on mean. G*Power 3 upport all cae of the t tet for mean decribed by Cohen (988, chap. ): the tet for independent mean, the tet of the null hypothei that the population mean equal ome pecified value (one ample cae), and the tet on the mean of two dependent ample (matched pair). Cohen d and d z are ued a effect ize indice. Cohen define d of 0., 0.5, and 0.8 a mall, medium, and large effect, repectively. Effect ize dialog are available to compute the appropriate effect ize parameter from mean and SD. For example, aume we want to compare viual earch time for target embedded in rare veru frequent local context in a within-ubject deign (cf. Hoffmann & Sebald, 005, Experiment ). It i expected that the mean earch time for target in rare context (e.g., 600 mec) hould decreae by at leat 0 mec (i.e., to 590 mec) in frequent context a a conequence of local contextual cuing. If prior evidence ugget population SD of, ay, 5 mec in each of the condition and a correlation of.70 between earch time in the two condition, we can ue the Effect ize drawer of G*Power 3 for the matched pair t tet to calculate the effect ize d z 0.56 (ee the econd row of Table 3 for the formula). By electing a pot hoc power analyi for one-tailed matched pair t tet, we eaily ee that for d z 0.56,.05, and 6 participant, the power ( ) i only.47. Thu, provided that the aumption outlined above are appropriate, the nonignificant tatitic [t(5).475] obtained by Hoffmann and Sebald (005, Experiment, p. 34) might in fact be due to a Type II error. Thi interpretation would be conitent with the fact that Hoffmann and Sebald ob- Table Symbol and Their Meaning A Ued in the Table Symbol Meaning ( i ) population mean (in group i) ( i) vector of population mean (in group i) xy population mean of the difference total ample ize n i ample ize in group i tandard deviation in the population tandard deviation of the effect xy tandard deviation of the difference noncentrality parameter of the noncentral F and ditribution noncentrality parameter of the noncentral t ditribution df degree of freedom df, df numerator and denominator degree of freedom, repectively ( i ) population correlation (in group i) RYA, RYA,B quared multiple correlation coefficient, correponding to the proportion of Y variance that can be accounted for by multiple regreion on the et of predictor variable A and AB, repectively population variance covariance matrix M matrix of regreion parameter (population mean) C contrat matrix (contrat between row of M) A contrat matrix (contrat between column of M) ( i ) probability of ucce (in group i)

7 G*POWER 3 8 Table Tet for Correlation and Regreion Tet ull oncentrality Parameter Tet Family Hypothei Effect Size Other Parameter and Degree of Freedom Difference from t tet 0 zero: point bierial model Difference from contant (bivariate normal) Inequality of two correlation coefficient Multiple regreion: deviation of R from zero Multiple regreion: increae of R exact tet c Contant correlation c z tet q z z i z i ln F tet R Y A 0 F tet R Y A,B R Y A f f R R Y A Y A R R Y A, B R Y A i Y A, B umber of predictor p (#A) Total number of predictor p (#A #B) umber of teted predictor q (#B) df m q n n 6 n 3 n 3 f df p df p f df q df p erved ignificant local contextual cuing effect in each of the other four experiment they reported. The procedure provided by G*Power 3 to tet effect in between-ubject deign with more than two group (i.e., one-way AOVA deign and general main effect and interaction in factorial AOVA deign of any order) are identical to thoe in G*Power (Erdfelder et al., 996). In all thee cae, the effect ize f a defined by Cohen (988) i ued. In a one-way AOVA, the Effect ize drawer can be ued to compute f from the mean and group ize of k group and an SD common to all group. For tet of effect in factorial deign, the Effect ize drawer offer the poibility of computing effect ize f from the variance explained by the teted effect and the error variance. Cohen define f of 0., 0.5, and 0.4 a mall, medium, and large effect, repectively. ew in G*Power 3 are procedure for analyzing main effect and interaction for A B mixed deign, where A i a between-ubject factor (or an enumeration of the group generated by cro-claification of everal between-ubject factor) and B i a within-ubject factor (or an enumeration of the repeated meaure generated by cro-claification of everal within-ubject factor). Both the univariate and the multivariate approache to repeated meaure (O Brien & Kaier, 985) are upported. The multivariate approach will be dicued below. The univariate approach i baed on the phericity aumption. Thi aumption i correct if (in the population) all variance of the repeated meaurement are equal and all correlation between pair of repeated meaurement are equal. If all the ditributional aumption are met, then the univariate approach i the mot powerful method (Muller & Barton, 989; O Brien & Kaier, 985). Unfortunately, the aumption of equal correlation i violated quite often, which can lead to very mileading reult. In order to compenate for uch advere effect in tet of within effect or between within interaction, the noncentrality parameter and the degree of freedom of the F ditribution can be multiplied by a correction factor (Geier & Greenhoue, 958; Huynh & Feldt, 970). if the phericity aumption i met and approache /(m ) with increaing degree of violation of phericity, where m denote the number of repeated meaurement. G*Power provide three eparate yet very imilar routine to calculate power in the univariate approach for between effect, within effect, and interaction. If the tobe-detected effect ize f i known, thee procedure are very eay to apply. To illutrate, Berti, Münzer, Schröger, and Pechmann (006) compared the pitch dicrimination ability of 0 muician and 0 control ubject (betweenubject factor A) for 0 different interference condition (within-ubject factor B). Auming that A, B, and A B effect of medium ize ( f 0.5; ee Cohen, 988; Table 3 of the preent article) hould be detected given a correlation of.50 between repeated meaure and a ignificance level of.05, the power value of the F tet for the A main effect, the B main effect, and the A B interaction are eaily computed a.30,.95, and.95, repectively, by inerting f 0.5,.05, the total ample ize (0), the number of group (), the number of repetition (0), and.50 into the appropriate input field of the procedure deigned for thee tet. If the to-be-detected effect ize f i unknown, it mut be computed from more baic parameter characterizing the expected population cenario under H. To demontrate the general procedure, we will how how to do pot hoc power analye in the cenario illutrated in Figure 4 auming the variance and correlation tructure defined in matrix SR. We firt conider the power of the within effect: We elect the F tet family, the Repeated mea-

8 8 FAUL, ERDFELDER, LAG, AD BUCHER Table 3 Tet for Mean (Univariate Cae) Tet ull oncentrality Parameter Tet Family Hypothei Effect Size Other Parameter and Degree of Freedom Difference from t tet c c d d contant (oneample df cae) Inequality of two dependent mean (matched pair) Inequality of two independent mean AOVA, fixed effect, one way: inequality of multiple mean AOVA, fixed effect, multifactor deign, and planned comparion t tet xy 0 x y d z x y t tet d F tet i 0 i,..., k F tet i 0 i,..., k x y x y x y f k n j i i f umber of group k Total number of cell in the deign k Degree of freedom of the teted effect q d z df n n d n n df f df k df k f df q df k AOVA: repeated meaure, between effect AOVA: repeated meaure, within effect AOVA: repeated meaure, between within interaction F tet i 0 i,..., k F tet i 0 i,..., m F tet ij i... j 0 i,..., k j,..., m f f f Level of between factor k Level of repeated meaure factor m f u u m ( m ) df k df k f u u m Population correlation df (m ) among repeated df ( k)(m ) meaure f u u m For within and within between df (k )(m ) interaction: df ( k)(m ) onphericity correction ure: Within factor, AOVA-approach tet, and pot hoc a the type of power analyi. Both the umber of group and Repetition field are et to 3. Total ample ize i et to 90 and error probability to.05. Referring to matrix SR, we inert.3 in the Corr among rep meaure input field and ince phericity obviouly hold in thi cae et nonphericity correction to. To determine effect ize f, we firt calculate, the variance of the Time Time Time 3 i n i Group Group Group j SR SR Figure 4. Sample 3 3 repeated meaure deign. Three group are repeatedly meaured at three different time. The haded portion of the table i the potulated matrix M of population mean ij. The lat column of the table contain the ample ize of each group. The ymmetric matrice SR i pecify two different covariance tructure between meaurement taken at different time: The main diagonal contain the SD of the meaurement at each time, and the off-diagonal element contain the correlation between pair of meaurement taken at different time.

9 G*POWER 3 83 within effect. From the three column mean j of matrix M and the grand mean, we get (0.889) (3.889) ( ) Clicking on the Determine button next to the Effect ize label open the Effect ize drawer. We chooe the From variance option and et Variance explained by pecial effect to and Variance within group to 9 8. Clicking on the Calculate and tranfer to main window button calculate an effect ize f 0.57 and tranfer f to the effect ize field in the main window. Clicking on Calculate yield the reult: The power i.997, the critical F value with df and df 74 i 3.048, and the noncentrality parameter i 5.5. The procedure for tet of between within interaction effect ( Repeated meaure: Within between interaction, AOVA-approach ) i almot identical to that jut decribed. The only difference i in how the effect ize f i computed. Here, we firt calculate the variance of the reidual value ij i j of matrix M: ( )... ( ) Uing the Effect ize drawer in the ame way a above, we get an effect ize f 0.53, which reult in a power of.653. To tet between effect, we chooe Repeated meaure: Between factor, AOVA-approach and et all parameter to the ame value a before. ote that in thi cae we do not need to pecify no correction i neceary becaue tet of between factor do not require the phericity aumption. To calculate the effect ize, we ue Effect ize from mean in the Effect ize drawer. We elect three group, et SD within each group to 9, and inert for each group the correponding row mean i of M (5,.3333,.3333) and an equal group ize of 30. Effect ize f i calculated, and the reulting power i.488. ote that G*Power 3 can eaily handle pure repeated meaure deign without any between-ubject factor (ee, e.g., Fring & Wentura, 005; Schwarz & Müller, 006) by chooing the Repeated meaure: Within factor, AOVA-approach procedure and etting the number of group to. Tet for Mean Vector (Multivariate Cae) G*Power 3 contain everal procedure for performing power analye in multivariate deign (ee Table 4). All thee tet belong to the F tet family. The Hotelling T tet are extenion of univariate t tet to the multivariate cae, in which more than one dependent variable i meaured: Intead of two ingle mean, two mean vector are compared, and intead of a ingle variance, a variance covariance matrix i conidered (Rencher, 998). In the one-ample cae, H 0 poit that the vector of population mean i identical to a pecified contant mean vector. The Effect ize drawer can be ued to calculate the effect ize from the difference c and the expected variance covariance matrix under H. For example, aume that we have two variable, a difference vector c (.88,.88) under H, variance 56.79, 9.8, and a covariance of.98 (Rencher, 998, p. 06). To perform a pot hoc power analyi, chooe F tet, then Multivariate: Hotelling T, one group and et the analyi type to Pot hoc. Enter in the Repone variable field and then click on the Determine button next to the Effect ize label. In the Effect ize drawer, at Input method: Mean and..., chooe Variance covariance matrix and click on Specify/edit input value. Under the Mean tab, inert.88 in both input field; under the Cov igma tab, inert and 9.8 in the main diagonal and.98 a the off-diagonal element in the lower left cell. Clicking on the Calculate and tranfer to main window button initiate the calculation of the effect ize (0.380) and tranfer it to the main window. For thi effect ize,.05, and a total ample ize of 00, the power amount to.98. The procedure in the two-group cae i exactly the ame, with the following exception. Firt, in the Effect ize drawer two mean vector have to be pecified. Second, the group ize may differ. The MAOVA tet in G*Power 3 refer to the multivariate general linear model (O Brien & Muller, 993; O Brien & Shieh, 999): Y XB, where Y i p of rank p, X i r of rank r, and the r p matrix B contain fixed coefficient. The row of are taken to be independent p-variate normal random vector with mean 0 and p p poitive definite covariance matrix. The multivariate general linear hypothei i H 0 : CBA 0, where C i c r with full row rank and A i p a with full column rank (in G*Power 3, 0 i aumed to be zero). H 0 ha df a c degree of freedom. All tet of the hypothei H 0 refer to the matrice and H H T CBU T T C X WX C 0 CBU 0 * T E U U r, where Ẍ i a q q eence model matrix, W i a q q diagonal matrix containing weight w j n j /, and X T X (Ẍ T WẌ) (ee O Brien & Shieh, 999, p. 4). Let { *,..., * } be the min(a,c) eigenvalue of E H * and {,..., } the eigenvalue of E H/( r) that i, i i* /( r). G*Power 3 offer power analye for the multivariate model following either the approach outlined in Muller and Peteron (984; Muller, LaVange, Landeman-Ramey, & Ramey, 99) or, alternatively, the approach of O Brien and Shieh (999; Shieh, 003). Both approache approximate the exact ditribution of Wilk U (Rao, 95), the Hotelling Lawley T (Pillai & Samon, 959), the Hotelling Lawley

10 84 FAUL, ERDFELDER, LAG, AD BUCHER Table 4 Tet for Mean Vector (Multivariate Cae) Tet ull oncentrality Parameter Tet Family Hypothei Effect Size Other Parameter and Degree of Freedom T v umber of difference from v c repone df k contant mean variable k df k vector Hotelling T : F tet c v Hotelling T : difference between two mean vector MAOVA: global effect MAOVA: pecial effect MAOVA: repeated meaure, between effect MAOVA: repeated meaure, within effect MAOVA: repeated meaure, between within interaction F tet F tet CM 0 Mean matrix M Contrat matrix C F tet F tet CMA 0 Mean matrix M Between contrat F tet matrix C Within contrat matrix A F tet v T v v Effect ize f mult depend on the tet tatitic: Wilk U Hotelling Lawley T Hotelling Lawley T Pillai V and algorithm: Muller & Peteron (984) O Brien & Shieh (999) umber of repone variable k umber of group g umber of repone variable k umber of group g umber of predictor p umber of repone variable k Level of between factor k Level of repeated meaure factor m n n n n df k df k oncentrality parameter and degree of freedom depend on the tet tatitic and algorithm ued (ee Effect Size column and Table 5). T (McKeon, 974), and Pillai V (Pillai & Mijare, 959) by F ditribution and are aymptotically equivalent. Table 5 outline detail of both approximation. The type of tatitic (U, T, T, V ) and the approach (Muller & Peteron, 984, or O Brien & Shieh, 999) can be elected in an Option dialog that can be evoked by clicking on the Option button at the bottom of the main window. The approach of Muller and Peteron (984) ha found widepread ue; for intance, it ha been adopted in the SPSS oftware package. We neverthele recommend the approach of O Brien and Shieh (999) becaue it ha a number of advantage: () Unlike the method of Muller and Peteron, it provide the exact noncentral F ditribution whenever the hypothei involve at mot poitive eigenvalue; () it approximation for eigenvalue are almot alway more accurate than thoe of Muller and Peteron method (which ytematically underetimate power); and (3) it provide a impler form of the noncentrality parameter that i, *, where * i not a function of the total ample ize. G*Power 3 provide procedure to calculate the power for global effect in a one-way MAOVA and for pecial effect and interaction in factorial MAOVA deign. Thee procedure are the direct multivariate analogue of the AOVA routine decribed above. Table 5 ummarize information that i needed in addition to the formula given above to calculate effect ize f from hypotheized value for mean matrix M (correponding to matrix B in the model), covariance matrix, and contrat matrix C, which decribe the effect under crutiny. The Effect ize drawer can be ued to calculate f from known value of the tatitic U, T, T, or V. ote, however, that the tranformation of T to f depend on the ample ize. Thu, thi tet tatitic eem not very well uited for a priori analye. In line with Bredenkamp and Erdfelder (985), we recommend V a the multivariate tet tatitic. Another group of procedure in G*Power 3 upport the multivariate approach to power analye of repeated meaure deign. G*Power provide eparate but very imilar routine for the analyi of between effect, within effect, and interaction in imple A B deign, where A i a between-ubject factor and B a within-ubject factor. To illutrate the general procedure, we decribe in ome detail a pot hoc analyi of the within effect for

11 G*POWER 3 85 Table 5 Approximating Univariate Statitic for Multivariate Hypothee Effect Size and Statitic Formula umerator df oncentrality Parameter Wilk U MP Wilk U OS Pillai V MP Pillai V OS Hotelling Lawley T MP Hotelling Lawley T OS Hotelling Lawley T MP Hotelling Lawley T OS U k U k k k * V k / k k * * V k / k k T k k T * k k T k k T * k k df g( g ) g g r a c g ca ca 3 g ( ca) 4 ca 4 c a 5 df ( r a ) df ( r a ) df 4 (ca )g ( r) ( r) g g g ( r) g g g c a a g c a g 3 a(a 3) g 4 a 3 df h r a 4 3 f ( U ) U / U f (U) df f ( U ) U / U g f (U) f ( V ) / g g / g g V ( V ) f (V) df f ( V ) V ( V ) f (V) f (T) T/ f (T) df f (T) T/ f (T) f (T) T/h f (T) df f (T) T/h h f (T) ote MP, Muller Peteron algorithm; OS, O Brien and Shieh algorithm. and * are eigenvalue of the effect ize matrix (for detail and the meaning of the variable a, c, r, and, ee text on p. 83). the cenario illutrated in Figure 4, auming the variance and correlation tructure defined in matrix SR. We firt chooe F tet, then Repeated meaure: Within factor, MAOVA-approach. In the Type of power analyi menu, we chooe Pot hoc. We click on the Option button to open a dialog in which we deelect the Ue mean correlation in effect ize calculation option. We chooe Pillai V tatitic and the O Brien and Shieh algorithm. Back at the main window, we et both number of group and repetition to 3, total ample ize to 90, and error probability to.05. To compute the effect ize f(v ) for the Pillai tatitic, we open the Effect ize drawer by clicking on the Determine button next to the Effect ize label. In the Effect ize drawer, elect, a procedure, Effect ize from mean and variance covariance matrix and, a input method, SD and correlation matrix. Clicking on Specify/edit matrice open another window, in which we pecify the hypotheized parameter. Under the Mean tab, we inert our mean matrix M; under the Cov igma tab, we chooe SD and correlation and inert the value of SR. Becaue thi matrix i alway ymmetric, it uffice to pecify the lower diagonal value. After cloing the dialog and clicking on Calculate and tranfer to main window, we get a value of 0.79 for Pillai V and the effect ize f(v ) Clicking on Calculate how that the power i.980. The analye of between effect and interaction effect are performed analogouly. Tet for Proportion The upport for tet on proportion ha been greatly enhanced in G*Power 3. Table 6 ummarize the tet that are currently implemented. In particular, all tet on proportion conidered by Cohen (988) are now available, including the ign tet (chap. 5), the z tet for the difference between two proportion (chap. 6), and the tet for goodne-of-fit and contingency table (chap. 7). The ign tet i implemented a a pecial cae (c.5) of the more general binomial tet (alo available in G*Power 3) that a ingle proportion ha a pecified value c. In both procedure, Cohen (988) effect ize g i ued and exact power value baed on the binomial ditribution are calculated. ote, however, that, due to the dicrete nature of the binomial ditribution, the nominal value of uually cannot be realized. Since the table in chapter 5 of Cohen book ue the value cloet to the nominal value, even if it i higher than the nominal value, the tabulated power value

12 86 FAUL, ERDFELDER, LAG, AD BUCHER Table 6 Tet for Proportion Tet oncentrality Tet Family Hypothei Effect Size Other Parameter Parameter Contingency table and goodne of fit Difference from contant (oneample cae) Inequality of two dependent proportion (Mcemar) Sign tet Inequality of two independent proportion Inequality of two independent proportion (Fiher exact tet) Inequality of two independent proportion (unconditional) Inequality with offet of two independent proportion (unconditional) tet exact tet exact tet exact tet i 0i i,..., k k 0 i i w k i i 0 i 0i c g c contant proportion c / odd ratio / proportion of dicordant pair / g / z tet (A) alternate proportion: (B) h (A) i arcin i exact tet exact tet exact tet (A) null proportion: alternate proportion: null proportion: (A) alternate proportion: (B) difference: (C) rik ratio: / (D) odd ratio: / / c (A) alternate proportion: H (B) difference: H (C) rik ratio: / H (D) odd ratio: ote (A) (D) indicate alternative effect ize meaure. / H H / null proportion: (A) proportion: H0 (B) difference: H0 (C) rik ratio: / H0 (D) odd ratio: (A) null proportion: / H0 H0 / w are ometime larger than thoe calculated by G*Power 3. G*Power 3 alway require the actual not to be larger than the nominal value. umerou procedure have been propoed to tet the null hypothei that two independent proportion are identical (Cohen, 988; D Agotino, Chae, & Belanger, 988; Suia & Shuter, 985; Upton, 98), and G*Power 3 implement everal of them. The implet procedure i a z tet with optional arcin tranformation and optional continuity correction. Beide thee two computational option, one can alo chooe whether Cohen effect ize meaure h or, alternatively, two proportion are ued to pecify the alternate hypothei. With the option Ue continuity correction off and Ue arcin tranform on, the procedure calculate power value cloe to thoe tabulated by Cohen (988, chap. 6). With both Ue continuity correction and Ue arcin tranform off, the uncorrected approximation i computed (Flei, 98); with Ue continuity correction on and Ue arcin tranform off, the corrected approximation i computed (Flei, 98). A econd variant i Fiher exact conditional tet (Haeman, 978). ormally, G*Power 3 calculate the exact unconditional power. However, depite the highly optimized algorithm ued in G*Power 3, long computation time may reult for large ample ize (e.g.,,000). Therefore, a limiting can be pecified in the Option dialog that determine at which ample ize G*Power 3 witche to a large ample approximation. A third variant calculate the exact unconditional power for approximate tet tatitic T (Table 7 ummarize the upported tatitic). The logic underlying thi procedure i to enumerate all poible outcome for the binomial table, given fixed ample ize n, n in the two repective group. Thi i done by chooing, a ucce frequencie x and x in the firt and the econd group, repectively, any combination of the value 0 x n and 0 x n. Given the ucce probabilitie, in the two repective group, the probability of oberving a table X with ucce frequencie x, x i PX, n x x n x n x x n x.

13 G*POWER 3 87 Table 7 Tet Statitic Ued in Tet of the Difference Between Two Independent Proportion o. ame Statitic z tet pooled variance z tet pooled variance with continuity correction 3 z tet unpooled variance 4 z tet unpooled variance with continuity correction 5 Mantel Haenzel tet 6 Likelihood ratio (Upton, 98) ˆ ˆ n ˆ z ; ˆ ˆ ˆ n n ; ˆ n ˆ ˆ n n k ˆ ˆ n n z ; ˆ( k ee o.); ˆ ˆ ˆ z ; ˆ ˆ ˆ ˆ ˆ ˆ n n k ˆ ˆ n n z ; ˆ( k ee o. 3); ˆ tx tx t x t x t x E x z V x n x x n n ; E x ; V x ( )... lr... t n t n t x x t x x lower tail upper tail lower tail upper tail x x x x ( ) ; t( x) : xln( x) 7 t tet with df (D Agotino et al., 988) t x x x x nx x nx x ote x i, ucce frequency in group i; n i, ample ize in group i; n n, total ample ize; ˆ i x i /n i. The z tet in the table are more commonly known a tet (the equivalent z tet i ued to provide two-ided tet). To calculate power and the actual Type I error *, the tet tatitic T i computed for each table and compared with the critical value T. If A denote the et of all table X rejected by thi criterion that i, thoe with T T then the power and the level are given by and P X, X A * P X,, X A where denote the ucce probability in both group a aumed in the null hypothei. ote that the actual level can be larger than the nominal level! The preferred input method (proportion, difference, rik ratio, or odd ratio; ee Table 6) and the tet tatitic to ue (ee Table 7) can be changed in the Option dialog. ote that the tet tatitic actually ued to analyze the data mut be choen. For large ample ize, the exact computation may take too much time. Therefore, a limiting can be pecified in the Option dialog that determine at which ample ize G*Power witche to large ample approximation. G*Power 3 alo provide a group of procedure to tet the hypothei that the difference, rik ratio, or odd ratio of a proportion with repect to a pecified reference proportion i different under H from a difference, rik ratio, or odd ratio of the ame reference proportion aumed in H 0. Thee procedure are available in the Exact tet family a Proportion: Inequality (offet), two independent group (unconditional). The enumeration procedure decribed above for the tet on difference between proportion without offet i alo ued in thi cae. In the tet without offet, the different input parameter (e.g., difference, rik ratio) are equivalent way of pecifying two proportion. The pecific choice ha no influence on the reult. In the cae of tet with offet, however, each input method ha a different et of available tet tatitic. The preferred input method (ee Table 6) and the tet tatitic to ue (ee Table 8) can be changed in the Option dialog. A in the other exact procedure, the computation may be time-conuming, and a limiting can be pecified in the Option dialog that determine at which ample ize G*Power witche to large ample approximation. Alo new in G*Power 3 i an exact procedure to calculate the power for the Mcemar tet. The null hypothei of thi tet tate that the proportion of uccee are identical in two dependent ample. Figure 5 how the tructure of the underlying deign: A binary repone i ampled from the ame ubject or a matched pair in a tandard condition and in a treatment condition. The null hypothei, t, i formally equivalent to the hypothei for the odd ratio: OR /. To fully pecify H, we need to pecify not only the odd ratio but alo the proportion of dicordant pair ( D ) that i, the expected proportion of repone that differ in the tandard and the treatment condition. The exact procedure ued in G*Power 3 calculate the unconditional power for the exact conditional tet, which calculate the power conditional on the number of dicordant pair (n D ). Let p(n D i) be the probability that the number of dicordant pair i i. Then, the unconditional power i the um over all i {0,..., } of the conditional power for n D i weighted with p(n D i). Thi procedure i very efficient, but for very large ample ize the exact computation may take too much time. Again, a limiting that determine at which ample ize G*Power witche to a large ample approximation can be pecified in the Option dialog. The large ample approximation calculate

14 88 FAUL, ERDFELDER, LAG, AD BUCHER Table 8 Tet Statitic Ued in Tet of the Difference With Offet Between Two Independent Proportion o. ame Statitic z tet pooled variance z tet pooled variance with continuity correction 3 z tet unpooled variance 4 z tet unpooled variance with continuity correction 5 t tet with df (D Agotino et al., 988) 6 Likelihood core ratio (difference) Miettinen & urminen (985) Farrington & Manning (990) Gart & am (990) 7 Likelihood core ratio (rik ratio) Miettinen & urminen (985) Farrington & Manning (990) Gart & am (988) 8 Likelihood core ratio (odd ratio) Miettinen & urminen (985) ˆ ˆ z ; ˆ ˆ ˆ / n / n ; ˆ ˆ n ˆ nˆ n n ˆ ˆ k / / n / n lower tail z ; ˆ( ee o.); k ˆ upper tail ˆ ˆ z ; ˆ ˆ ˆ / n ˆ ˆ / n ˆ ˆ ˆ k / / n / n lower tail z ; ˆ( ee o. 3); k ˆ upper tail t x n x x x n K; K ( ) / nx x nx x ˆ ˆ z ; ˆ ~ / n / n ˆ ~ ~ ~ K Miettinen & urminen: K /( ); Farrington & Manning: K ~ u co(w) b/(3a); ~ ~ n /n ; a ; b [ ˆ ˆ ( )] c ( ˆ ) ˆ ˆ ; d ˆ ( ) v b 3 /(3a) 3 bc/(6a ) d/(a); w [3.459 co (v/u 3 )]/3 u gn(v) b /(3a) c/(3a) Skewne corrected z (Gart & am, 990); z according to Farrington & Manning: z [ 4( z) ]/; V [ ~ ( ~ )/n ~ ( ~ )/n ] V /3 /6[ ~ ( ~ )( ~ )/n ~ ( ~ )( ~ )/n ] ˆ ˆ z ; ˆ ~ / n / n ˆ ~ ~ ~ K Miettinen & urminen: K /( ); Farrington & Manning: K 4 ~ ~ ; ~ b b x x ; b n x x n Skewne corrected z (Gart & am, 988); z according to Farrington & Manning: z [ 4( z) ]/; V ( ~ )/( ~ n ) ( ~ )/( ~ n ) /(6V /3 )[( ~ )( ~ )/(n ~ ) ( ~ )( ~ )/(n ~ ) ] ˆ ~ / ~ ˆ ~ ~ / ~ ~ z / n ~ ~ / n ~ ~ K Miettinen & urminen: K /( ); Farrington & Manning: K ~ ~ /[ ~ ( )]; ~ [b b 4a(x x )]/(a) a n ( ); b n n (x x )( ) ote x i, ucce frequency in group i; n i, ample ize in group i; n n, total ample ize; ˆ i x i /n i ;, difference between proportion potulated in H 0 ;, rik ratio potulated in H 0 ;, odd ratio potulated in H 0. the power on the bai of an ordinary one-ample binomial tet with Bin( D, 0.5) a the ditribution under H 0 and Bin[ D, OR/( OR)] a the H ditribution. Tet for Variance Table 9 ummarize important propertie of the two procedure for teting hypothee on variance that are currently upported by G*Power 3. In the one-group cae, the null hypothei that the population variance ha a pecified value c i teted. The variance ratio /c i ued a the effect ize. The central and noncentral ditribution, correponding to H 0 and H, repectively, are central ditribution with df (becaue H 0 and H are baed on the ame mean). To compare the variance ditribution under both hypothee, the H ditribution i caled with the value r potulated for the ratio /c in the alternate Standard Treatment Ye o Ye t o t Proportion of dicordant pair: D Hypothei: t or, equivalently, Figure 5. Matched binary repone deign (Mcemar tet).

15 G*POWER 3 89 Table 9 Tet for Variance Tet ull Other Tet Family Hypothei Effect Size Parameter oncentrality Parameter Difference from contant (one ample cae) Inequality of two variance tet c Variance ratio 0 r (H : central ditribution, c caled with r) df F tet Variance ratio r 0 (H : central F ditribution, caled with r) df n df n hypothei that i, the noncentral ditribution i r (Otle & Malone, 988). In the two-group cae, H 0 tate that the variance in two population are identical ( / ). A in the one-ample cae, two central F ditribution are compared, the H ditribution being caled by the value of the variance ratio / potulated in H. Generic Tet Beide the pecific routine decribed in Table 9 that cover a coniderable part of the tet commonly ued, G*Power 3 provide generic power analyi routine that may be ued for any tet baed on the t, F,, z, or binomial ditribution. In generic routine, the parameter of the central and noncentral ditribution are pecified directly. To demontrate the ue and limitation of thee generic routine, we will how how to do a two-tailed power analyi for the one-ample t tet uing the generic routine. The reult can be compared with thoe of the pecific routine available in G*Power for that tet. Firt, we elect the t tet family and then Generic t tet (the generic tet option i alway located at the end of the lit of tet). ext, we elect Pot hoc a the type of power analyi. We chooe a two-tailed tet and.05 a error probability. We now need to pecify the noncentrality parameter and the degree of freedom for our tet. We look up the definition for the one-ample tet in Table 3 and find that d and df. Auming a medium effect of d 0.5 and 5, we arrive at and df 4. After inerting thee value and clicking on Calculate, we obtain a power of The critical value t.0639 correpond to the pecified. In thi pot hoc power analyi, the generic routine i almot a imple a the pecific routine. The main diadvantage of the generic routine i, however, that the dependence of the noncentrality parameter on the ample ize i implicit. A a conequence, we cannot perform a priori analye automatically. Rather, we need to iterate by hand until we find an appropriate power value. STATISTICAL METHODS AD UMERICAL ALGORITHMS The ubroutine ued to compute the ditribution function (and the invere) of the noncentral t, F,, z, and binomial ditribution are baed on the C verion of the DCDFLIB (available from which wa lightly modified for our purpoe. G*Power 3 doe not provide the approximate power analye that were available in the peed mode of G*Power. Two argument guided u in upporting exact power calculation only. Firt, four-digit preciion of power calculation may be mandatory in many application. For example, both compromie power analye for very large ample, and error probability adjutment in cae of multiple tet of ignificance may reult in very mall value of or (Wetermann & Hager, 986). Second, a a conequence of improved computer technology, exact calculation have become o fat that the peed gain aociated with approximate power calculation i not even noticeable. Thu, from a computational tandpoint, there i little advantage to uing approximate rather than exact method (cf. Bradley, Ruell, & Reeve, 998). PROGRAM AVAILABILITY AD ITERET SUPPORT To ummarize, G*Power 3 i a major extenion of, and improvement over, G*Power in that it offer eay-toapply power analye for a much larger variety of common tatitical tet. Program handling i more flexible, eaier to undertand, and more intuitive than in G*Power, reducing the rik of erroneou application. The added graphical feature hould be ueful for both reearch and teaching purpoe. Thu, G*Power 3 i likely to become a ueful tool for empirical reearcher and tudent of applied tatitic. Like it predeceor, G*Power 3 i a noncommercial program that can be downloaded free of charge. Copie of the Mac and Window verion are available only at Uer intereted in ditributing the program in another way mut ak for permiion from the author. Commercial ditribution i trictly forbidden. The G*Power 3 Web page offer an expanding Webbaed tutorial decribing how to ue the program, along with example. Uer who let u know their addree will be informed of update. Although coniderable effort ha been put into program development and evaluation, there i no warranty whatoever. Uer are aked to kindly report poible bug and difficultie in program handling to gpower-feedback@uni-dueeldorf.de.

16 90 FAUL, ERDFELDER, LAG, AD BUCHER AUTHOR OTE Manucript preparation wa upported by Grant SFB 504 (Project A) from the Deutche Forchunggemeinchaft and a grant from the tate of Baden-Württemberg, Germany (Landeforchungprogramm Evidenzbaierte Streprävention ). Correpondence concerning thi article hould be addreed to F. Faul, Intitut für Pychologie, Chritian- Albrecht-Univerität, Olhauentr. 40, D-4098 Kiel, Germany, or to E. Erdfelder, Lehrtuhl für Pychologie III, Univerität Mannheim, Schlo Ehrenhof Ot 55, D-683 Mannheim, Germany ( ffaul@ pychologie.uni-kiel.de or erdfelder@pychologie.uni-mannheim.de). REFERECES Akkad, D. A., Jagiello, P., Szyld, P., Goedde, R., Wieczorek, S., Gro, W. L., & Epplen, J. T. (006). Promoter polymorphim r in the MHC cla II tranactivator gene i not aociated with uceptibility for elected autoimmune dieae in German patient group. International Journal of Immunogenetic, 33, Back, M. D., Schmukle, S. C., & Egloff, B. (005). Meauring takwitching ability in the Implicit Aociation Tet. Experimental Pychology, 5, Baeza, J. A., & Stotz, W. (003). Hot-ue and election of differently colored ea anemone by the ymbiotic crab Allopetrolithe pinifron. Journal of Experimental Marine Biology & Ecology, 84, Barabei, L., & Greco, L. (00). A note on the exact computation of the Student t, Snedecor F, and ample correlation coefficient ditribution function. Journal of the Royal Statitical Society, 5D, Berti, S., Münzer, S., Schröger, E., & Pechmann, T. (006). Different interference effect in muician and a control group. Experimental Pychology, 53, -6. Bradley, D. R., Ruell, R. L., & Reeve, C. P. (998). The accuracy of four approximation to noncentral F. Behavior Reearch Method, Intrument, & Computer, 30, Bredenkamp, J. (969). Über die Anwendung von Signifikanztet bei Theorie-tetenden Experimenten [The application of ignificance tet in theory-teting experiment]. Pychologiche Beiträge,, Bredenkamp, J., & Erdfelder, E. (985). Multivariate Varianzanalye nach dem V-Kriterium [Multivariate analyi of variance baed on the V-criterion]. Pychologiche Beiträge, 7, Buchner, A., Erdfelder, E., & Faul, F. (996). Tettärkeanalyen [Power analye]. In E. Erdfelder, R. Maufeld, T. Meier, & G. Rudinger (Ed.), Handbuch Quantitative Methoden [Handbook of quantitative method] (pp. 3-36). Weinheim, Germany: Pychologie Verlag Union. Buchner, A., Erdfelder, E., & Faul, F. (997). How to ue G*Power [Computer manual]. Available at aap/project/gpower/how_to_ue_gpower.html. Bubey, A. B. I. (999). Macintoh hareware/freeware earthcience oftware. Computer & Geocience, 5, Cohen, J. (988). Statitical power analyi for the behavioral cience (nd ed.). Hilldale, J: Erlbaum. D Agotino, R. B., Chae, W., & Belanger, A. (988). The appropriatene of ome common procedure for teting the equality of two independent binomial population. American Statitician, 4, Erdfelder, E. (984). Zur Bedeutung und Kontrolle de -Fehler bei der inferenztatitichen Prüfung log-linearer Modelle [Significance and control of the error in tatitical tet of log-linear model]. Zeitchrift für Sozialpychologie, 5, 8-3. Erdfelder, E., Buchner, A., Faul, F., & Brandt, M. (004). GPOWER: Tettärkeanalyen leicht gemacht [Power analye made eay]. In E. Erdfelder & J. Funke (Ed.), Allgemeine Pychologie und deduktivitiche Methodologie [Experimental pychology and deductive methodology] (pp ). Göttingen: Vandenhoeck & Ruprecht. Erdfelder, E., Faul, F., & Buchner, A. (996). GPOWER: A general power analyi program. Behavior Reearch Method, Intrument, & Computer, 8, -. Erdfelder, E., Faul, F., & Buchner, A. (005). Power analyi for categorical method. In B. S. Everitt & D. C. Howell (Ed.), Encyclopedia of tatitic in behavioral cience (pp ). Chicheter, U.K.: Wiley. Farrington, C. P., & Manning, G. (990). Tet tatitic and ample ize formulae for comparative binomial trial with null hypothei of non-zero rik difference or non-unity relative rik. Statitic in Medicine, 9, Field, A. P. (005). Dicovering tatitic with SPSS (nd ed.). London: Sage. Flei, J. L. (98). Statitical method for rate and proportion (nd ed.). ew York: Wiley. Fring, C., & Wentura, D. (005). egative priming with maked ditractor-only prime trial: Awarene moderate negative priming. Experimental Pychology, 5, Gart, J. J., & am, J. (988). Approximate interval etimation of the ratio in binomial parameter: A review and correction for kewne. Biometric, 44, Gart, J. J., & am, J. (990). Approximate interval etimation of the difference in binomial parameter: Correction for kewne and extenion to multiple table. Biometric, 46, Geier, S., & Greenhoue, S. W. (958). An extenion of Box reult on the ue of the F ditribution in multivariate analyi. Annal of Mathematical Statitic, 9, Gerard, P. D., Smith, D. R., & Weerakkody, G. (998). Limit of retropective power analyi. Journal of Wildlife Management, 6, Gigerenzer, G., Krau, S., & Vitouch, O. (004). The null ritual: What you alway wanted to know about ignificance teting but were afraid to ak. In D. Kaplan (Ed.), The SAGE handbook of quantitative methodology for the ocial cience (pp ). Thouand Oak, CA: Sage. Gleiner, U., Clumann, H., Saen, R., Elger, C. E., & Helmtaedter, C. (006). Poturgical outcome in pediatric patient with epilepy: A comparion of patient with intellectual diabilitie, ubaverage intelligence, and average-range intelligence. Epilepia, 47, Goldtein, R. (989). Power and ample ize via MS/PC-DOS computer. American Statitician, 43, Hager, W. (006). Die Fallibilität empiricher Daten und die otwendigkeit der Kontrolle von falchen Entcheidungen [The fallibility of empirical data and the need for controlling for fale deciion]. Zeitchrift für Pychologie, 4, 0-3. Haeman, J. K. (978). Exact ample ize for ue with the Fiher Irwin tet for table. Biometric, 34, Hoenig, J.., & Heiey, D. M. (00). The abue of power: The pervaive fallacy of power calculation for data analyi. American Statitician, 55, 9-4. Hoffmann, J., & Sebald, A. (005). Local contextual cuing in viual earch. Experimental Pychology, 5, Huynh, H., & Feldt, L. S. (970). Condition under which mean quare ratio in repeated meaurement deign have exact F-ditribution. Journal of the American Statitical Aociation, 65, Keppel, G., & Wicken, T. D. (004). Deign and analyi. A reearcher handbook (4th ed.). Upper Saddle River, J: Pearon Education International. Kornbrot, D. E. (997). Review of tatitical hareware G*Power. Britih Journal of Mathematical & Statitical Pychology, 50, Kromrey, J., & Hogarty, K. Y. (000). Problem with probabilitic hindight: A comparion of method for retropective tatitical power analyi. Multiple Linear Regreion Viewpoint, 6, 7-4. Lenth, R. V. (00). Some practical guideline for effective ample ize determination. American Statitician, 55, Levin, J. R. (997). Overcoming feeling of powerlene in aging reearche: A primer on tatitical power in analyi of variance deign. Pychology & Aging,, McKeon, J. J. (974). F approximation to the ditribution of Hotelling T 0. Biometrika, 6, Mellina, E., Hinch, S. G., Donaldon, E. M., & Pearon, G. (005). Stream habitat and rainbow trout (Oncorhynchu myki) phyiological tre repone to treamide clear-cut logging in Britih Columbia. Canadian Journal of Foret Reearch, 35, Miettinen, O., & urminen, M. (985). Comparative analyi of two rate. Statitic in Medicine, 4, 3-6. Müller, J., Manz, R., & Hoyer, J. (00). Wa tun, wenn die Tettärke zu gering it? Eine praktikable Strategie für Prä Pot-Deign [What to do if tatitical power i low? A practical trategy for pre

17 G*POWER 3 9 pot-deign]. Pychotherapie, Pychoomatik, Mediziniche Pychologie, 5, Muller, K. E., & Barton, C.. (989). Approximate power for repeatedmeaure AOVA lacking phericity. Journal of the American Statitical Aociation, 84, Muller, K. E., LaVange, L. M., Landeman-Ramey, S., & Ramey, C. T. (99). Power calculation for general linear multivariate model including repeated meaure application. Journal of the American Statitical Aociation, 87, Muller, K. E., & Peteron, B. L. (984). Practical method for computing power in teting the multivariate general linear hypothei. Computational Statitic & Data Analyi,, Myer, J. L., & Well, A. D. (003). Reearch deign and tatitical analyi (nd ed.). Mahwah, J: Erlbaum. O Brien, R. G., & Kaier, M. K. (985). MAOVA method for analyzing repeated meaure deign: An extenive primer. Pychological Bulletin, 97, O Brien, R. G., & Muller, K. E. (993). Unified power analyi for t-tet through multivariate hypothee. In L. K. Edward (Ed.), Applied analyi of variance in behavioral cience (pp ). ew York: Dekker. O Brien, R. G., & Shieh, G. (999). Pragmatic, unifying algorithm give power probabilitie for common F tet of the multivariate general linear hypothei. Available at Orteifen, C., Bruckner, T., Burke, M., & Kieer, M. (997). An overview of oftware tool for ample ize determination. Informatik, Biometrie & Epidemiologie in Medizin & Biologie, 8, 9-8. Otle, B., & Malone, L. C. (988). Statitic in reearch: Baic concept and technique for reearch worker (4th ed.). Ame: Iowa State Pre. Pillai, K. C. S., & Mijare, T. A. (959). On the moment of the trace of a matrix and approximation to it ditribution. Annal of Mathematical Statitic, 30, Pillai, K. C. S., & Samon, P., Jr. (959). On Hotelling generalization of T. Biometrika, 46, Quednow, B. B., Kühn, K.-U., Stelzenmueller, R., Hoenig, K., Maier, W., & Wagner, M. (004). Effect of erotonergic and noradrenergic antidepreant on auditory tartle repone in patient with major depreion. Pychopharmacology, 75, Rao, C. R. (95). An aymptotic expanion of the ditribution of Wilk criterion. Bulletin of the International Statitical Intitute, 33, Rach, B., Friee, M., Hofmann, W. J., & aumann, E. (006a). Quantitative Methoden : Einführung in die Statitik (. Auflage) [Quantitative method : Introduction to tatitic (nd ed.)]. Heidelberg, Germany: Springer. Rach, B., Friee, M., Hofmann, W. J., & aumann, E. (006b). Quantitative Methoden : Einführung in die Statitik (. Auflage) [Quantitative method : Introduction to tatitic (nd ed.)]. Heidelberg, Germany: Springer. Rencher, A. C. (998). Multivariate tatitical inference and application. ew York: Wiley. Richardon, J. T. E. (996). Meaure of effect ize. Behavior Reearch Method, Intrument, & Computer, 8, -. Scheffé, H. (959). The analyi of variance. ew York: Wiley. Schwarz, W., & Müller, D. (006). Spatial aociation in numberrelated tak: A comparion of manual and pedal repone. Experimental Pychology, 53, 4-5. Sheppard, C. (999). How large hould my ample be? Some quick guide to ample ize and the power of tet. Marine Pollution Bulletin, 38, Shieh, G. (003). A comparative tudy of power and ample ize calculation for multivariate general linear model. Multivariate Behavioral Reearch, 38, Smith, R. E., & Bayen, U. J. (005). The effect of working memory reource availability on propective memory: A formal modeling approach. Experimental Pychology, 5, Steidl, R. J., Haye, J. P., & Schauber, E. (997). Statitical power analyi in wildlife reearch. Journal of Wildlife Management, 6, Suia, S., & Shuter, J. J. (985). Exact unconditional ample ize for binomial trial. Journal of the Royal Statitical Society A, 48, Thoma, L., & Kreb, C. J. (997). A review of tatitical power analyi oftware. Bulletin of the Ecological Society of America, 78, Upton, G. J. G. (98). A comparion of alternative tet for the comparative trial. Journal of the Royal Statitical Society A, 45, Wetermann, R., & Hager, W. (986). Error probabilitie in educational and pychological reearch. Journal of Educational Statitic,, Zumbo, B. D., & Hubley, A. M. (998). A note on miconception concerning propective and retropective power. The Statitician, 47, OTES. The oberved power i reported in many frequently ued computer program (e.g., the MAOVA procedure of SPSS).. We recommend checking the degree of freedom reported by G*Power by comparing them, for example, with thoe reported by the program ued to analyze the ample data. If the degree of freedom do not match, the input provided to G*Power i incorrect and the power calculation do not apply. 3. Plot of the central and noncentral ditribution are hown only for tet baed on the t, F, z,, or binomial ditribution. o plot are hown for tet that involve an enumeration procedure (e.g., the Mcemar tet). 4. We thank Dave Kenny for making u aware of the fact that the t tet (correlation) power analye of G*Power are correct only in the point bierial cae (i.e., for correlation between a binary variable and a continuou variable, the latter being normally ditributed for each value of the binary variable). For correlation between two continuou variable following a bivariate normal ditribution, the t tet (correlation) procedure of G*Power overetimate power. For thi reaon, G*Power 3 offer eparate power analye for point bierial correlation (in the t family of ditribution) and correlation between two normally ditributed variable (in the exact ditribution family). However, power value uually differ only lightly between procedure. To illutrate, aume we are intereted in the power of a two-tailed tet of H 0 :.00 for continuouly ditributed meaure derived from two Implicit Aociation Tet (IAT) differing in content. Aume further that, due to methodpecific variance in both verion of the IAT, the true Pearon correlation i actually.30 (effect ize). Given.05 and 57 (ee Back, Schmukle, & Egloff, 005, p. 73), an exact pot hoc power analyi for Correlation: Difference from contant (one ample cae) reveal the correct power value of.63. Chooing the incorrect Correlation: point bierial model procedure from the t tet family would reult in.65. (Manucript received December 8, 006; accepted for publication January 3, 007.)