Effect Size Calculations for ANOVAs

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

Usage

etaSq(fit, type = 2, anova = FALSE)

# S3 method for class 'lm'
etaSq(fit, type = 2, anova = FALSE)

# S3 method for class 'aovlist'
etaSq(fit, type = 2, anova = FALSE)

aovlDetails(fit)

aovlErrorTerms(fit)

Arguments

fit: an analysis of variance object of class "aov" or "aovlist"
type: type of sums of squares to calculate. etaSq.aovlist() requires type = 1.
anova: logical; whether to include the full ANOVA table with the effect sizes

Value

for etaSq.lm(), a numeric matrix with one row per model term and columns eta.sq and eta.sq.part; for etaSq.aovlist(), a numeric matrix that additionally contains eta.sq.gen. If anova = TRUE, ANOVA statistics are included in additional columns. aovlDetails() returns a data frame of effect terms and their ANOVA statistics, while aovlErrorTerms() returns a list with components:

SS: error sums of squares
MS: error mean squares
DF: error degrees of freedom

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 fit 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.

Note

Based on code by Danielle Navarro, and Daniel Wollschlaeger.

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.

Examples


#### 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
#>             Df Sum Sq Mean Sq F value Pr(>F)  
#> treatment1   2  7.936   3.968   3.663 0.0913 .
#> Residuals    6  6.500   1.083                 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
etaSq(anova1)                                        # effect size
#>               eta.sq eta.sq.part
#> treatment1 0.5497229   0.5497229

#### 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
#>             Df Sum Sq Mean Sq F value  Pr(>F)   
#> treatment1   2  7.936   3.968    55.8 0.00120 **
#> treatment2   2  6.216   3.108    43.7 0.00191 **
#> Residuals    4  0.284   0.071                   
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
etaSq(anova2)                                    # effect size
#>               eta.sq eta.sq.part
#> treatment1 0.5497229   0.9653961
#> treatment2 0.4305727   0.9562393

#### 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)
#>                eta.sq eta.sq.part
#> IV1s      0.086255016  0.16919863
#> IV2s      0.497535493  0.54017354
#> IV1s:IV2s 0.005976273  0.01391427
etaSq(aov(DV ~ IV1t*IV2t, data=dfMD), type=1)
#>                eta.sq eta.sq.part
#> IV1t      0.042592627  0.09137634
#> IV2t      0.527900579  0.55484898
#> IV1t:IV2t 0.005976273  0.01391427

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

set.seed(1)
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)
#>                 eta.sq eta.sq.part eta.sq.gen         SS df        MS      SSE
#> IVbtw       0.01359217  0.03775582 0.01693141  1.1645622  2 0.5822811 29.68002
#> IVwth       0.18569613  0.29547219 0.19048057 15.9102422  2 7.9551211 37.93659
#> IVbtw:IVwth 0.01152556  0.02536984 0.01439415  0.9874978  4 0.2468744 37.93659
#>             dfE          F            p
#> IVbtw        27  0.5297029 5.947733e-01
#> IVwth        54 11.3235402 7.820776e-05
#> IVbtw:IVwth  54  0.3514079 8.419225e-01