Abstract
Many statistical tests rely on the assumption that the residuals of a model are normally distributed. Rank-based inverse normal transformation (INT) of the dependent variable is one of the most popular approaches to satisfy the normality assumption. When covariates are included in the analysis, a common approach is to first adjust for the covariates and then normalize the residuals. This study investigated the effect of regressing covariates against the dependent variable and then applying rank-based INT to the residuals. The correlation between the dependent variable and covariates at each stage of processing was assessed. An alternative approach was tested in which rank-based INT was applied to the dependent variable before regressing covariates. Analyses based on both simulated and real data examples demonstrated that applying rank-based INT to the dependent variable residuals after regressing out covariates re-introduces a linear correlation between the dependent variable and covariates, increasing type-I errors and reducing power. On the other hand, when rank-based INT was applied prior to controlling for covariate effects, residuals were normally distributed and linearly uncorrelated with covariates. This latter approach is therefore recommended in situations were normality of the dependent variable is required.
Highlights
Many statistical tests rely on the assumption that the residuals of a model are normally distributed [1]
We demonstrate that regressing covariate effects from the dependent variable creates a covariate-based rank, which is subsequently distorted by rank-based inverse normal transformation (INT), leading to increased type-I errors and reduced power
This study has demonstrated that regressing covariates against the dependent variable and using rank-based INT to transform the residuals to normality re-introduces a correlation between the covariates and the normalized dependent variable
Summary
Many statistical tests rely on the assumption that the residuals of a model are normally distributed [1]. In genetic analyses of complex traits, the normality of residuals is largely determined by the normality of the dependent variable (phenotype) due to the very small effect size of individual genetic variants [2]. Many traits do not follow a normal distribution. In behavioral research in particular, questionnaire data often exhibit marked skew as well as a large number of ties between individuals. Normality of residuals can lead to heteroskedasticity (comparison of variables with unequal variance) potentially resulting in increased type-I error rates and reduced power [3]. Most genome-wide analysis software currently implements only linear and logistic models, precluding the use of more general parametric or nonparametric analyses
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