Effect size calculations for ANOVAs

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Description

Calculates eta-squared, partial eta-squared and generalized eta-squared

Usage

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EtaSq(x, type = 2, anova = FALSE)

## S3 method for class 'lm'
EtaSq(x, type = 2, anova = FALSE)

## S3 method for class 'aovlist'
EtaSq(x, type = 2, anova = FALSE)

Arguments

x

An analysis of variance (aov, aovlist) object.

type

What type of sum of squares to calculate? EtaSq.aovlist requires type=1.

anova

Should the full ANOVA table be printed out in addition to the effect sizes?

Details

Calculates the eta-squared, partial eta-squared, and generalized eta-squared measures of effect size that are commonly used in analysis of variance. The input x should be the analysis of variance object itself. For between-subjects designs, generalized eta-squared equals partial eta-squared. The reported generalized eta-squared for repeated-measures designs assumes that all factors are manipulated, i.e., that there are no measured factors like gender (see references).

For unbalanced designs, the default in EtaSq is to compute Type II sums of squares (type=2), in keeping with the Anova function in the car package. It is possible to revert to the Type I SS values (type=1) to be consistent with anova, but this rarely tests hypotheses of interest. Type III SS values (type=3) can also be computed. EtaSq.aovlist requires type=1.

Value

If anova=FALSE, the output for EtaSq.lm is an M x 2 matrix, for EtaSq.aovlist it is an M x 3 matrix. Each of the M rows corresponds to one of the terms in the ANOVA (e.g., main effect 1, main effect 2, interaction, etc), and each of the columns corresponds to a different measure of effect size. Column 1 contains the eta-squared values, and column 2 contains partial eta-squared values. Column 3 contains the generalized eta-squared values. If anova=TRUE, the output contains additional columns containing the sums of squares, mean squares, degrees of freedom, F-statistics and p-values. For EtaSq.aovlist, additional columns contain the error sum of squares and error degrees of freedom corresponding to an effect term.

Author(s)

Daniel Navarro <daniel.navarro@adelaide.edu.au>, Daniel Wollschlaeger <dwoll@psychologie.uni-kiel.de>

References

Bakeman, R. (2005). Recommended effect size statistics for repeated measures designs. Behavior Research Methods 37(3), 379-384.

Olejnik, S. and Algina, J. (2003). Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs. Psychological Methods 8(4), 434-447.

See Also

aov, anova, Anova

Examples

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#### Example 1: one-way ANOVA ####

outcome <- c(1.4,2.1,3.0,2.1,3.2,4.7,3.5,4.5,5.4)    # data
treatment1 <- factor(c(1,1,1,2,2,2,3,3,3))           # grouping variable
anova1 <- aov(outcome ~ treatment1)                  # run the ANOVA
summary(anova1)                                      # print the ANOVA table
EtaSq(anova1)                                        # effect size                            

#### Example 2: two-way ANOVA ####

treatment2 <- factor(c(1,2,3,1,2,3,1,2,3))       # second grouping variable
anova2 <- aov(outcome ~ treatment1 + treatment2) # run the ANOVA
summary(anova2)                                  # print the ANOVA table
EtaSq(anova2)                                    # effect size

#### Example 3: two-way ANOVA unbalanced cell sizes ####
#### data from Maxwell & Delaney, 2004              ####
#### Designing experiments and analyzing data       ####

dfMD <- data.frame(IV1=factor(rep(1:3, c(3+5+7, 5+6+4, 5+4+6))),
                   IV2=factor(rep(rep(1:3, 3), c(3,5,7, 5,6,4, 5,4,6))),
                   DV=c(c(41, 43, 50), c(51, 43, 53, 54, 46), c(45, 55, 56, 60, 58, 62, 62),
                        c(56, 47, 45, 46, 49), c(58, 54, 49, 61, 52, 62), c(59, 55, 68, 63),
                        c(43, 56, 48, 46, 47), c(59, 46, 58, 54), c(55, 69, 63, 56, 62, 67)))

# use contr.sum for correct sum of squares type 3
dfMD$IV1s <- C(dfMD$IV1, "contr.sum")
dfMD$IV2s <- C(dfMD$IV2, "contr.sum")
dfMD$IV1t <- C(dfMD$IV1, "contr.treatment")
dfMD$IV2t <- C(dfMD$IV2, "contr.treatment")

EtaSq(aov(DV ~ IV1s*IV2s, data=dfMD), type=3)
EtaSq(aov(DV ~ IV1t*IV2t, data=dfMD), type=1)

#### Example 4: two-way split-plot ANOVA -> EtaSq.aovlist ####

DV_t1 <- round(rnorm(3*10, -0.5, 1), 2)
DV_t2 <- round(rnorm(3*10,  0,   1), 2)
DV_t3 <- round(rnorm(3*10,  0.5, 1), 2)
dfSPF <- data.frame(id=factor(rep(1:(3*10), times=3)),
                    IVbtw=factor(rep(LETTERS[1:3], times=3*10)),
					IVwth=factor(rep(1:3, each=3*10)),
					DV=c(DV_t1, DV_t2, DV_t3))
spf <- aov(DV ~ IVbtw*IVwth + Error(id/IVwth), data=dfSPF)
EtaSq(spf, type=1, anova=TRUE)

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