# Mixed Factorial ANOVA in R

library(afex)
## Loading required package: lme4
## Loading required package: Matrix
## ************
## Welcome to afex. For support visit: http://afex.singmann.science/
## - Functions for ANOVAs: aov_car(), aov_ez(), and aov_4()
## - Methods for calculating p-values with mixed(): 'S', 'KR', 'LRT', and 'PB'
## - 'afex_aov' and 'mixed' objects can be passed to emmeans() for follow-up tests
## - NEWS: emmeans() for ANOVA models now uses model = 'multivariate' as default.
## - Get and set global package options with: afex_options()
## - Set orthogonal sum-to-zero contrasts globally: set_sum_contrasts()
## - For example analyses see: browseVignettes("afex")
## ************
##
## Attaching package: 'afex'
## The following object is masked from 'package:lme4':
##
##     lmer
library(tidyr)
##
## Attaching package: 'tidyr'
## The following objects are masked from 'package:Matrix':
##
##     expand, pack, unpack

# Data Description

The fictional data set is part of the data library of the JASP software, and is from the book by Field (2017)1. Here it only contains a subset of the data, where 10 men and 10 women for three speed-dating partners. All three partners were low in attractiveness (however defined), but varied in charisma: high, some, or none.

## Data Import

charisma_data <- read.csv("charisma_ugly.csv")
# Show the first six rows
head(charisma_data)

## Mixed ANOVA

In R, one needs to first convert the data to a long format, so that each row represents an observation (instead of a person). In this data, there were 60 observations.

# First, convert to long format data
long_data <- tidyr::pivot_longer(
charisma_data,
cols = ug_high:ug_none,
# name of IV
names_to = "charisma",
# remove the "ug_" prefix
names_prefix = "ug_",
# name of DV
values_to = "rating"
)
long_data
mixed_anova <- aov_ez(
id = "participant",
dv = "rating",
data = long_data,
between = "gender",
within = "charisma"
)
## Converting to factor: gender
## Contrasts set to contr.sum for the following variables: gender
test_sphericity(mixed_anova)
## Warning: Functionality has moved to the 'performance' package.
## Calling 'performance::check_sphericity()'.
## Warning in summary.Anova.mlm(object\$Anova, multivariate = FALSE): HF eps > 1
## treated as 1
## OK: Data seems to be spherical (p > 0.612).

When one or more independent variable is a within-subject variable, mixed or repeated-measure ANOVA requires testing the assumption of sphericity. Here, the Mauchly’s test of sphericity provided insufficient evidence that the assumption of sphericity was violated, $$p = .612$$.

### ANOVA Table

mixed_anova
## Anova Table (Type 3 tests)
##
## Response: rating
##            Effect          df   MSE          F  ges p.value
## 1          gender       1, 18 25.42  71.82 *** .586   <.001
## 2        charisma 1.89, 34.09 24.42 167.84 *** .857   <.001
## 3 gender:charisma 1.89, 34.09 24.42  58.10 *** .676   <.001
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1
##
## Sphericity correction method: GG

Note that the result by default applies the Greenhousse-Geisser correction for violations of the sphericity assumption, but since the Mauchly’s test was not statistically significant, the results would be similar with or without the correction. Also note that the result here reports the generalized $$\eta^2$$ (the ges column), which is different from some other software. See the documentation of the ?afex::nice page.

### Plot

afex_plot(mixed_anova, x = "charisma", trace = "gender")
## Warning: Panel(s) show a mixed within-between-design.
## Error bars do not allow comparisons across all means.
## Suppress error bars with: error = "none"

## Sample Result Reporting

Table 1:
Two-Way ANOVA Statistics for Study Variables
Effect $$\hat{\eta}^2_G$$ 90% CI $$F$$ $$\mathit{df}^{\mathrm{GG}}$$ $$\mathit{df}_{\mathrm{res}}^{\mathrm{GG}}$$ $$p$$
Gender .586 [.311, .739] 71.82 1 18 < .001
Charisma .857 [.781, .899] 167.84 1.89 34.09 < .001
Gender $$\times$$ Charisma .676 [.516, .768] 58.10 1.89 34.09 < .001

A two-way mixed ANOVA was conducted, with gender (female vs. male) as the between-subject independent variable and charisma (high, some, none) as the within-subject independent variable. There was a significant main effect of charisma, $$F(1.89, 34.09) = 167.84$$, $$p < .001$$, $$\hat{\eta}^2_G = .857$$, 90% CI $$[.781, .899]$$. As shown in the figure, overall female participants provided a higer rating than male participants. There was also a significant main effect of gender, $$F(1, 18) = 71.82$$, $$p < .001$$, $$\hat{\eta}^2_G = .586$$, 90% CI $$[.311, .739]$$. As shown in the figure, ratings were higher for partners who had high charisma. Finally, there was a significant charisma $$\times$$ gender interaction, $$F(1.89, 34.09) = 58.10$$, $$p < .001$$, $$\hat{\eta}^2_G = .676$$, 90% CI $$[.516, .768]$$. As shown in the figure, while the ratings were similar between female and male participants for partners who had some or no charisma, female participants gave higher ratings for the partner with high charisma, as compared to male participants.

1. Field, A. P. (2017). Discovering Statistics Using IBM SPSS Statistics (5th ed.). Sage.↩︎

##### Hok Chio (Mark) Lai 黎學昭
###### Assistant Professor of Psychology (Quantitative Methods)

My research interests include statistics, multilevel and latent variable models, and psychometrics.