1. Does smiling increase leniency? Are different types of smiles differentially effective? Subjects in this experiment judged cases of possible academic misconduct. They read descriptions of an alleged infraction while viewing a picture of the accused person. The severity (leniency) of the penalty/punishment was the dependent variable. Actors used the Ekman facial expression system to generate faces with neutral (no smile) expressions, contempt smiles, miserable smiles, and real smiles. Each subject was tested with all four types of expression. The data can be loaded with the following commands:

options(contrasts=c("contr.sum","contr.poly"))
load(url("http://pnb.mcmaster.ca/bennett/psy710/datasets/smile-2023.rda") )
sapply(smile.df,class)
##     smile  leniency    subjID 
##  "factor" "numeric"  "factor"
sapply(smile.wide,class)
##   neutral  contempt miserable      real 
## "numeric" "numeric" "numeric" "numeric"
class(smile.mat)
## [1] "matrix" "array"

The command loads the long-format data frame smile.df, the wide-format data frame smile.wide, and the wide-format matrix smile.mat.

1. Use a within-subject ANOVA to evaluate the statistical significance of the fixed effect of smile. Adjust the \(p\) value with the Geisser-Greenhouse or Huynh-Feldt adjustment to correct for deviations from sphericity.

2. Use the lmer command in the lmerTest package to evaluate the effect of smile with a mixed model. Evaluate the effect with an \(F\) test that assumes sphericity and with a chi-square test that does not assume sphereicity.

3. Use the lme command in the nlme package to evaluate the effect of smile with two mixed models: one that assumes the variance-covariance matrix of the residuals is compound symmetric (i.e., the correlations between conditions is constant), and another that assumes the variance-covariance matrix is symmetric (i.e., the correlations between conditions are allowed to differ). Do you prefer one model over the other? Why or why not?

4. Calculate the ANOVA estimates of the variance component and intra-class correlation for subjID.

5. Use a linear contrast to evaluate the null hypothesis that leniency in the real smile condition did not differ from average leniency in the other three conditions.