Effects of violating the assumptions of equal variance and independent and identically distributed random variables: A case using {B}ayesian statistical modeling

Abstract

All statistical methods involve assumptions about the data and the output of the methods can be biased when the assumptions are not supported by the data. One of the common assumptions is equal variance across the conditions. Another common assumption is that variables are independently sampled from identically distributed populations (i.i.d.). The present study describes an example of such a violation of these assumptions and its effect on the results of Bayesian statistical analyses. Yu et al. (2021) developed a Bayesian statistical model that can analyze the same type of data as the one-way repeated-measure ANOVA. Their model assumed equal variance and i.i.d. Unfortunately, these assumptions were not satisfied by their data. In the present study, their model was revised to allow variance to vary with the conditions, and their data was reanalyzed. The results of the analyses using these models were compared with the psychophysical results of Yu et al. (2021). This comparison showed that the violated assumptions biased the results of the analysis. This bias made the results of the analysis appear more supportive of Yu et al.’s (2021) conclusion, but the validity of the analysis’s results needs to be re-considered. Note that it is important that one carefully scrutinizes the data and understands the statistical method used to discuss the results of the analysis.

Keywords

• bayesian statistics
• bayesian modeling
• type-1 error
• unequal variance
• statistical literacy
• r