# EtaSq: Effect Size Calculations for ANOVAs

### Description

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

### Usage

 ```1 2 3 4 5 6 7``` ```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.

`aov`, `anova`, `Anova`

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45``` ```#### 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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