# GEN_ETA2: A SAS ® Macro for Computing the Generalized Eta-Squared Effect Size Associated with Analysis of Variance Models

**ABSTRACT** Measures of effect size are recommended to communicate information on the strength of relationships between variables. Such information supplements the reject / fail-to-reject decision obtained in statistical hypothesis testing. The choice of an effect size for ANOVA models can be confusing because indices may differ depending on the research design as well as the magnitude of the effect. Olejnik and Algina (2003) proposed the generalized eta-squared effect size which is comparable across a wide variety of research designs. This paper provides a SAS macro for computing the generalized eta-squared effect size associated with analysis of variance models by utilizing data from PROC GLM ODS tables. The paper provides the macro programming language, as well as results from an executed example of the macro.

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**ABSTRACT:**Analyzes and clarifies the differences between eta-squared and partial eta-squared in fixed factor analysis of variance (ANOVA) designs. The formulas are presented and discussed, and an example is presented along with the appropriate use and meaning of the 2 coefficients. Finally, a general discussion of the use of eta-squared and partial eta-squared is provided. (PsycINFO Database Record (c) 2012 APA, all rights reserved)Educational and Psychological Measurement 01/1973; · 1.07 Impact Factor

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Paper 223-2013

GEN_ETA2: A SAS® Macro for Computing the Generalized Eta-Squared

Effect Size Associated with Analysis of Variance Models

Patricia Rodríguez de Gil, Than Pham, Patrice Rasmussen, Anh Kellermann,

Jeanine Romano, Yi-Hsin Chen, Jeffrey D. Kromrey

University of South Florida, Tampa, FL

ABSTRACT

Measures of effect size are recommended to communicate information on the strength of relationships between

variables. Such information supplements the reject / fail-to-reject decision obtained in statistical hypothesis testing.

The choice of an effect size for ANOVA models can be confusing because indices may differ depending on the

research design as well as the magnitude of the effect. Olejnik and Algina (2003) proposed the generalized eta-

squared effect size which is comparable across a wide variety of research designs. This paper provides a SAS macro

for computing the generalized eta-squared effect size associated with analysis of variance models by utilizing data

from PROC GLM ODS tables. The paper provides the macro programming language, as well as results from an

executed example of the macro.

Keywords: GENERALIZED ETA-SQUARED, EFFECT SIZES, ANOVA, BASE SAS, SAS/STAT

INTRODUCTION

Long gone are the days when social and behavioral science researchers should simply report obtained test statistics

(i.e. t, F, χ2) and their corresponding p-values. Over the years, interpreting the importance of scientific research based

on the dichotomous reject or fail-to-reject decision has become less popular among some disciplines such as

psychology and education. Instead, researchers are encouraged to supplement hypothesis test results with measures

of effect magnitude. In fact, according to Thompson (2007), 24 peer-reviewed journals had explicit editorial policies

that required authors to include effect sizes or other measures of effect magnitude. A large part of this “cultural

change” began over a decade ago when the APA Task Force on Statistical Inference issued their statement that

researchers should regularly report effect sizes, calculate confidence intervals, and use graphics to better

communicate the nature of their findings for all primary outcomes (Wilkinson & APA Task Force on Statistical

Inference, 1999). Unlike p-values that are used to determine if an observed effect or relationship is real or due to

chance or sampling variability, effect sizes are used to estimate how large the effect or relationship is. Thus, when

used together, not only can researchers make statements about the statistical significance of their findings, but they

can also report on the practical significance of their findings.

EFFECT SIZES COMMONLY USED WITH ANALYSIS OF VARIANCE MODELS

Commonly reported effect sizes for ANOVA models include h2, partial h2, w2, and partial w2. In general, all four of

these effect sizes represent measures of association. However, there are important differences among them. First,

both h2 and partial h2 are sample effect size estimates, representing the proportion of sample variability in the

dependent variable that is associated with variability in an independent variable. These statistics, however, are

positively biased as point estimates of the population effect size. The effect sizes w2, and partial w2, are adjusted to

provide better population estimates.

Second, partial and non-partial estimates represent different measures of association. For example, consider a two-

way, balanced factorial ANOVA with independent variables of student gender and grade level, and a dependent

variable of mathematics achievement. The analysis of variance will provide sums-of-squares (and mean squares) for

each of the two main effects (main effects for gender and grade level), the interaction effect, and the within-cell

residual or error. The total sums-of-square, representing all of the sample variability in mathematics achievement, is

simply the sum of these four components. The formula for

associated with the gender main effect is

2

In contrast, the formula for partial

2

for this effect is

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As is evident in the first formula

that is associated with gender. In contrast, partial

associated with gender, after the effects of grade level and the interaction between gender and grade level have been

removed (note the different denominators for

and partial

sample effect sizes for the other two sources of systematic variance in this design – grade level and the interaction

between gender and grade level.

represents the proportion of the total sample variance in math achievement

represents the proportion of variance in math achievement

). Analogous formulas are used to calculate

Next, to obtain a relatively unbiased estimate of the variance explained in the population by an independent variable,

omega-squared can be calculated. Using the same example as above, the formula for

2

is presented below.

where k – 1 = the degrees of freedom for the independent variable.

In this formula,

gender. However, unlike eta-squared, omega-squared takes random error ( ) into account. Through this

process, omega-squared values will be smaller than eta-squared values, with more noticeable differences occurring

with smaller samples and/or research designs that include more independent variables.

represents the proportion of the population variance in math achievement associated with

Similarly, partial omega-squared represents an unbiased estimate of the population proportion of variance in math

achievement associated with gender, after the effects of grade level and the interaction between gender and grade

level have been removed (note the different denominators for

and partial

).

Unfortunately, all four of these effect sizes can be influenced by the research design that underlies the ANOVA model

(see, for example, Cohen, 1973; Fleis, 1969; Wampold & Serlin, 2000). Olejnik and Algina (2003) proposed the

generalized eta-squared effect size, whose value is not influenced by the study design and consequently is

comparable across designs. The generalized eta-squared statistic is estimated using

2

G

effect

SS

effectMeasK

MeasK

SS

SSSS

where is the sum of squares for the effect of interest,

if the effect is a manipulated factor (and is zero otherwise),

the are the sums of squares for all sources of variance that involve measured factors (rather than

manipulated factors) but do not include subjects, and

the are the sums of squares for all sources of variance that involve subjects.

SOFTWARE LIMITATIONS

SAS does not provide the generalized eta-squared effect size in either PROC ANOVA or PROC GLM (with the

exception of single factor models, for which this statistic is equivalent to eta-squared). Thus, in order for SAS users to

follow the APA Task Force recommendations for reporting effect sizes, extra work is required. For example, using the

output from PROC ANOVA or PROC GLM, researchers can calculate these values either by hand or by outputting

the ModelANOVA and OverallANOVA ODS tables and calculating the effect sizes through a data step. Unfortunately,

the computational formulas for the generalized eta-squared statistic can be confusing with many ANOVA designs.

Thus, in an effort to reduce the burden of calculating this effect size to accompany analysis of variance hypothesis

test results, this paper provides a SAS macro to calculate generalized eta-squared for a variety of ANOVA research

designs.

Generalized eta squared (

For its estimation, it is important to understand the layout of the study factors, which helps shape and guide not only

the analysis of variance but also the estimation of the generalized eta-squared effect size. One important distinction

needs to be kept in mind when computing

individual differences (e.g., demographic variables) and “assigned”, manipulated factors by the researcher. That is,

“variability in the data arise due to two sources of variance: manipulated factors and individual differences” (Olejnick

& Algina, 2003; p. 436), or due to within-subjects and between-subjects factors.

is intended for designs in which there is at least one categorical independent variable.

. Research designs will involve “found”, measured factors which are

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Table 1 presents the ANOVA models addressed in this paper and for which

is computed.

Within-Subjects Factors

0 1 2

Between-

Subjects

Factors

0

N-A

1

2

> 2

Table 1. ANOVA Models For Which

Is Computed.

MACRO GEN_ETA2 DETAILS

The Gen_Eta2 SAS macro computes the generalized eta-squared effect size (

comparable across a wide variety of ANOVA research designs, utilizing data from PROC GLM ODS tables. The

arguments to the macro include the name of the SAS dataset and the model ANOVA elements to be specified

according to the research design (i.e., class, model, repeated, measured), as well as the specification of the

dependent variable Y.

Olejnick & Algina, 2003), which is

%macro GEN_ETA2 (data = _last_, class = , model = NONE, repeated = NONE, measured =

NONE, dependent = Y);

* This loop executes if we have one or more measured factors;

%if &measured ne NONE %then %do;

data Vnames;

noblank = 0; numchars = 0; vseq = 0;

varnames = symget('measured');

do i = 1 to length(varnames);

if noblank = 0 then do;

if substr(varnames,i,1) = ' ' then noblank = 0;

if substr(varnames,i,1) ^= ' ' then noblank = i;

end;

if noblank ^= 0 then do;

if substr(varnames,i,1) = ' ' | i = length(varnames) then do;

numchars = i - noblank;

if i = length(varnames) then numchars = numchars + 1;

varname = substr(varnames,noblank,numchars);

vseq = vseq + 1;

noblank = 0;

output;

end;

end;

end;

data Vnames; set Vnames; keep vseq varname;

data _null_; set Vnames end=lastrec;

if lastrec then do;

call symput('N_Measured',_n_);

end;

run;

data Measured_variables;

Merge_Var = 1;

run;

%do jk = 1 %to &N_Measured;

data build;

set Vnames; if _n_ = &jk; measured_var&jk = varname; run;

data Measured_variables; merge Measured_variables build; run;

%end;

%end; * End the measured variables loop;

* This loop extracts names and number of levels of the within-subjects factor(s);

%if &repeated ne NONE %then %do;

data Vnames;

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data Vnames; set Vnames;

do i = 1 to length(varname);

end;

record_num = _N_;

keep vseq varname record_num;

data _null_; set Vnames end=lastrec;

if lastrec then do;call symput('N_Repeated',_n_);end; run;

data Repeated_variables; Merge_Var = 1; %PUT &N_Repeated; run;

%do jk = 1 %to &N_Repeated;

data build;

set Vnames;

if _n_ = &jk;

if ROUND(record_num/2) ^= record_num/2 then do; * Odd numbered records;

Repeated_var&jk = varname;

end;

if ROUND(record_num/2) = record_num/2 then do; * Even numbered records;

Levels_var&jk = varname;

end;

data Repeated_variables; merge Repeated_variables build; run;

%end;

%end; * End the repeated factors loop;

** This executes if we have no within-subjects factors;

%if &repeated = NONE %then %do;

proc glm data = &data;

class &class;

model &dependent = &model / ss3;

ods output ModelANOVA = two OverallANOVA = three ;

run;

data threeA;

set three; if Source = 'Error'; SS_W = SS; keep SS_W;

** This loop executes if we have one or more measured factors;

%if &measured ne NONE %then %do;

data two; set two; Merge_Var = 1;

data twoA; merge Measured_variables two; by Merge_Var; Measured_Var = 0;

%do jk = 1 %to &N_Measured;

if INDEX(Source,TRIM(measured_var&jk)) ^= 0 then Measured_Var = 1;

%end;

delta = 1 - Measured_Var;

proc sort data = twoA; by HypothesisType Measured_Var;

proc means noprint data = twoA; by HypothesisType Measured_Var; var SS;

output out = TwoB Sum = SS_Measured;

data TwoB; set TwoB; if Measured_Var = 1; drop Measured_Var;

data two_back; merge TwoA TwoB; by HypothesisType;

noblank = 0; numchars = 0; vseq = 0;

varnames = symget('repeated');

do i = 1 to length(varnames);

if noblank = 0 then do;

if substr(varnames,i,1) = ' ' then noblank = 0;

if substr(varnames,i,1) ^= ' ' then noblank = i;

end;

if noblank ^= 0 then do;

if substr(varnames,i,1) = ' ' | i = length(varnames) then do;

numchars = i - noblank;

if i = length(varnames) then numchars = numchars + 1;

varname = substr(varnames,noblank,numchars);

vseq = vseq + 1;

noblank = 0;

output;

end;

end;

end;

if substr(varname,i,1) = ',' then substr(varname,i,1) = '';

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%end; * End of measured factors loop;

%if &measured = NONE %then %do; * This executes if we have no measured factors;

data final; if _N_ = 1 then set threeA; retain SS_W; set two; delta = 1;

Generalized_Eta2 = SS/(delta*SS + SS_W);

%end; * End of no measured factors loop;

%end; * End of no within-subjects factors loop;

%if &repeated ^= NONE %then %do; * This executes if we have a within-subjects factor;

%LET Model_Type = %INDEX(&model,NONE);

*------------------------------

WITHIN-subject factors only

-------------------------------;

%if %EVAL(&Model_Type) ^= 0 %then %do;

data temp; set &data; subject = _N_; merge_var = 1;

data long; merge temp repeated_variables; by merge_var;

%if &N_repeated = 1 or &N_repeated = 2 %then %do;

CALL SYMPUT ('R_Var1', repeated_var1);

%end;

%if &N_repeated > 2 %then %do;

L_var1 = Levels_var2 + 0;

L_var2 = Levels_var4 + 0;

CALL SYMPUT ('R_Var1', repeated_var1);

CALL SYMPUT ('R_Var2', repeated_var3);

%end;

data long; set long; array varlist [*] &dependent; my_index = 1;

%if &N_repeated = 1 or &N_repeated = 2 %then %do;

do &R_var1 = 1 to DIM(varlist);

y = varlist[my_index];

output;

my_index = my_index + 1;

end;

%end;

%if &N_repeated > 2 %then %do;

do &R_Var1 = 1 to L_var1;

do &R_Var2 = 1 to L_var2;

y = varlist[my_index];

output;

my_index = my_index + 1;

end;

end;

%end;

%if &N_repeated = 1 or &N_repeated = 2 %then %do;

proc glm data = long;

class subject &R_Var1;

model y = subject|&R_Var1 / ss3;

ods output ModelANOVA = two;

run;

%end;

%if &N_repeated > 2 %then %do;

proc glm data = long;

class subject &R_Var1 &R_Var2;

model y = subject|&R_Var1|&R_Var2 / ss3;

ods output ModelANOVA = two;

run;

%end;

data two; set two; drop dependent;

data twoA; set two; if INDEX(Source,"subject") ^= 0;

proc means noprint data = twoA; var SS;

output out = Subjects SUM = SS_Subjects;

data final; if _N_ = 1 then set Subjects; retain SS_Subjects;

set two; if INDEX(Source,"subject")= 0;

Dependent = 'WithinSubject'; delta = 1;

data final; if _N_ = 1 then set threeA; retain SS_W; set two_back;

Generalized_Eta2 = SS/(delta*SS + SS_Measured + SS_W);

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%end; * End Model_type ^= 0 loop;

Generalized_Eta2 = SS/(delta*SS + SS_Subjects);

%if %EVAL(&Model_Type) = 0 %then %do; * Both between and within factors;

proc glm data = &data;

class &class;

model &dependent = &model / ss3;

repeated &repeated;

ods output ModelANOVA = two OverallANOVA = three;

run;

data twoA; set two; if INDEX(Source,"Error") ^= 0;

proc means noprint data = twoA; var SS;

output out = Subjects SUM = SS_Subjects;

%if &measured ^= NONE %then %do;

data two; set two; Merge_Var = 1;

data twoB; merge Measured_variables two; by Merge_Var;

if INDEX(Source,"Error") = 0 and (Dependent = 'WithinSubject'

or Dependent = 'BetweenSubjects'); Measured_Var = 0;

%do jk = 1 %to &N_Measured;

if INDEX(Source,TRIM(measured_var&jk)) ^= 0

then Measured_Var = 1;

%end;

delta = 1 - Measured_Var;

proc sort data = twoB; by Measured_Var;

proc means noprint data = twoB; by Measured_Var; var SS;

output out = TwoC Sum = SS_Measured;

data TwoC; set TwoC; if Measured_Var = 1; drop Measured_Var;

data two_back; if _N_ = 1 then set TwoC; retain SS_Measured;

set twoB;

data final; if _N_ = 1 then set Subjects; retain SS_Subjects;

set two_back;

Generalized_Eta2 = SS/(delta*SS + SS_Measured + SS_Subjects);

%end; * End of measured factors loop;

%if &measured = NONE %then %do;

data final; if _N_ = 1 then set Subjects; retain SS_Subjects;

set two;

if INDEX(Source,"Error") = 0 and (Dependent = 'WithinSubject' or

Dependent = 'BetweenSubjects');

delta = 1;

Generalized_Eta2 = SS/(delta*SS + SS_Subjects);

%end; * End of no measured factors loop;

%end; *End of between and within loop;

%end; * End of within-subjects factors loop;

** Generate final report ;

data output; set final; getord = 0;

do i = 1 to length(Source);

if substr(Source,i,1) ^= '*' then getord = getord +1;

else getord = getord +2;

end;

if dependent ne 'BetweenSubjects' and dependent ne 'WithinSubject' then dependent2

= 'BetweenSubjects';

if dependent = 'BetweenSubjects' or dependent = 'WithinSubject' then dependent2

= dependent;

proc sort data = output; by dependent2 HypothesisType getord;

data ord; set output; by dependent2;

file print notitles header =hd;

if first.dependent2 then do; put @1 / dependent2 /; end;

put @5 Source @45 SS 8.4 @60 HypothesisType @85 Generalized_Eta2 8.4;

Return;

hd: put @1 'Generalized Eta-Squared Values' // @1 'Source' @47 'SS'

@60 'Hypothesis Type' @85 'Generalized Eta2';

run;

%MEND GEN_ETA2;

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MACRO GEN_ETA2 EXECUTION

In Example 1 of the use of the macro Gen_Eta2, the SAS data step creates a SAS data set called ONE, containing

eight observations. Each observation has teachers’ student ratings on three scales (cooperation, avoid, and peer),

taken before and after an intervention (i.e., pretest-posttest). The macro Gen_Eta2 is called twice to illustrate the

computation of generalized eta squared,

for one within-subjects and for two within-subjects factors, respectively.

Data ONE;

input student coop1 coop2 avoid1 avoid2 peer1 peer2;

cards;

1 31 33 21 27 28 30

2 31 25 15 25 30 32

3 16 35 26 33 17 32

4 27 26 22 31 21 31

5 32 26 29 32 29 35

6 32 33 16 33 20 27

7 26 32 14 27 19 19

8 19 26 14 29 10 29

;

run;

**run the macro for one WITHIN-subject factor;

%gen_eta2 (data = one, class = , repeated = %str(domain 3),model = NONE, dependent =

coop1 avoid1 peer1 );

**run the macro for two WITHIN-subject factors;

%gen_eta2 (data = one, class = , repeated = %str(time 2, domain 3),model = NONE,

dependent = coop1 avoid1 peer1 coop2 avoid2 peer2);

In Example 2, the SAS data step creates the SAS data set ONE from 10 observations. Each observation records data

corresponding to a score and the participant’s group allocation, where “score” is the dependent variable and “group”

is the between-subjects manipulated factor. The macro Gen_Eta2 is invoked to illustrate the computation of the

generalized eta squared,

for one between-subjects design.

data ONE;

input score group;

SUBJECT = _N_;

cards;

17 1

23 1

20 1

14 1

12 1

18 2

22 2

11 2

21 2

19 2

;

RUN;

** run the macro for one between-subjects factor;

%gen_eta2 (data = one, class = group, model = group, repeated = NONE, measured = none,

dependent = score);

Lastly, in Example 3, the SAS data step creates SAS data set ONE from 12 observations, and it is used to show the

computation of the generalized eta-squared effect size for three between-subjects and one within-subjects design,

where “sex”, is a between-subjects measured factor and “group” and “intensity”” are between-subjects manipulated

factors. X1 – X3 is the repeated measures within-subjects factor.

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data ONE;

input x1 x2 x3 group sex intensity;

SUBJECT = _N_;

cards;

1 2 3 1 1 1

1 1 1 1 1 2

2 2 5 1 2 2

3 3 3 1 2 1

3 2 1 1 2 1

6 5 4 2 1 2

4 5 4 2 2 2

3 6 7 2 1 1

3 4 8 2 2 1

2 6 8 2 2 2

2 5 3 1 1 2

5 4 8 2 2 1

;

run;

** run the macro between-subjects and within-subjects factors;

%gen_eta2 (data = one, class = group sex intensity, model = group|sex|intensity,

repeated = %str(time 3), measured = sex, dependent = X1 – X3);

OUTPUT EXAMPLES OF THE MACRO GEN_ETA2

In data Example 1 the macro was invoked to illustrate the computation of

shows the

“domain” is provided, along with the Hypothesis Type 3 value

for one within-subjects factor. Output 1a

computed for one within-subjects factor main effect. Sums-of-squares for the within-subjects factor

Output1a. Example Output from Macro For One Within-Subject Design

For two within-subjects factors, the macro Gen_Eta2 computed the generalized eta squared effect size for both main

effects (time and domain, in data example 1) and for the interaction. Output 1b shows the output provided by the

macro, which in addition to the effect sizes, it also provides the factors and interaction Sum-of-Squares (SS) and the

Hypothesis Type 3 values.

Output 1b. Example Output from Macro For Two Within-Subjects Factors

Generalized Eta-Squared Values

Source SS Hypothesis Type Generalized Eta2

WithinSubject

domain 214.0833 3 0.2045

Generalized Eta-Squared Values

Source SS Hypothesis Type Generalized Eta2

WithinSubject

time 553.5208 3 0.3196

domain 105.0417 3 0.0818

time*domain 109.2917 3 0.0849

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Output 2. Example Output from Macro For One Between-Subjects Design

Output 2 shows the computed of

Squares and Hypothesis Type 3 values.

for the group, between-subjects factor main effect, as well as the Sum-of

The output for data Example 3 shows the macro results for a design with three between-subjects factors and one

within-subjects factor. The generalized eta squared effect sizes are provided for all main effects and for all interaction

effects. The between-subjects and within-subjects parts of the model are separated in the same way that PROC GLM

separates these effects.

Output 3. Example Output from Macro For Between-Subjects and Within-Subjects Design

time*group*intensity 8.0833 3 0.1424

time*group*sex*intensity 5.0833 3 0.1045

CONCLUSION

Measures of effect size quantify the magnitude of the relationship between variables. Their value resides in the fact

that they are independent of sample size and either as standardized mean differences or as the proportion of

variance explained, effect size measures enhance the interpretability of results of applied research. Within analysis

of variance models, eta squared and omega squared are commonly used measures of effect size; however, these

effect size measures can be misinterpreted or more importantly, their value is limited due to differences in research

designs across studies. In those cases, both eta-squared and omega-squared effect sizes do not make up for the

differences in the sources of variation, such as between-subject and within-subjects variability. Generalized eta-

squared accounts for variability of design factors; however, it is not provided directly in either PROC ANOVA or

PROC GLM. In addition, its hand computation can be confusing. The purpose of this paper was to provide SAS

users with an easy to use SAS Macro for computing the generalized eta-squared effect size for ANOVA designs,

under several design layouts.

Generalized Eta-Squared Values

Source SS Hypothesis Type Generalized Eta2

BetweenSubjects

group 2.5000 3 0.016

Generalized Eta-Squared Values

Source SS Hypothesis Type Generalized Eta2

BetweenSubjects

sex 0.6806 3 0.0140

group 58.6806 3 0.5466

intensity 0.0139 3 0.0003

group*sex 1.1250 3 0.0231

sex*intensity 0.0139 3 0.0003

group*intensity 1.1250 3 0.0226

group*sex*intensity 0.1250 3 0.0026

WithinSubject

time 15.1944 3 0.2379

time*sex 4.1944 3 0.0862

time*group 1.6944 3 0.0336

time*intensity 2.0278 3 0.0400

time*group*sex 2.5833 3 0.0531

time*sex*intensity 9.3611 3 0.1924

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REFERENCES

Cohen, J. (1973). Eta-squared and partial eta-squared in fixed factor ANOVA designs. Educational and Psychological

Measurement, 33, 107-112.

Fleiss, J. L. (1969). Estimating the magnitude of experimental effects. Psychological Bulletin, 81, 273-276.

Olejnick, S., & Algina, J. (2003). Generalized eta and omega squared statistics: Measures of effect size for some

common research designs. Psychological Methods, 8(4), 434-447.

Thompson, B. (2007). Effect sizes, confidence intervals, and confidence intervals for effect sizes. Psychology in the

Schools, 44(5), 423-432.

Wampold, B. E., & Serlin, R. C. (2000). The consequences of ignoring a nested factor on measures of effect size in

analysis of variance. Psychological Methods, 5, 425-433.

Wilkinson, L., & Task Force on Statistical Inference (1999). Statistical methods in psychological journals: Guidelines

and Explanations. American Psychologist, 54, 594-604.

CONTACT INFORMATION

Your comments and questions are valued and encouraged. Contact the first author at:

Patricia Rodríguez de Gil

University of South Florida

4202 East Fowler Ave., EDU 105

Tampa, FL 33620

Email: prodrig6@usf.edu

SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS

Institute Inc. in the USA and other countries. ® indicates USA registration.

Other brand and product names are trademarks of their respective companies.

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