A comparison of residual diagnosis tools for diagnosing regression models for count data

Longhai Li,Cindy Feng,Alireza Sadeghpour

doi:10.1186/s12874-020-01055-2

Longhai Li, Cindy Feng + Show 1 more

Open Access

https://doi.org/10.1186/s12874-020-01055-2

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Abstract

BackgroundExamining residuals is a crucial step in statistical analysis to identify the discrepancies between models and data, and assess the overall model goodness-of-fit. In diagnosing normal linear regression models, both Pearson and deviance residuals are often used, which are equivalently and approximately standard normally distributed when the model fits the data adequately. However, when the response vari*able is discrete, these residuals are distributed far from normality and have nearly parallel curves according to the distinct discrete response values, imposing great challenges for visual inspection.MethodsRandomized quantile residuals (RQRs) were proposed in the literature by Dunn and Smyth (1996) to circumvent the problems in traditional residuals. However, this approach has not gained popularity partly due to the lack of investigation of its performance for count regression including zero-inflated models through simulation studies. Therefore, we assessed the normality of the RQRs and compared their performance with traditional residuals for diagnosing count regression models through a series of simulation studies. A real data analysis in health care utilization study for modeling the number of repeated emergency department visits was also presented.ResultsOur results of the simulation studies demonstrated that RQRs have low type I error and great statistical power in comparisons to other residuals for detecting many forms of model misspecification for count regression models (non-linearity in covariate effect, over-dispersion, and zero inflation). Our real data analysis also showed that RQRs are effective in detecting misspecified distributional assumptions for count regression models.ConclusionsOur results for evaluating RQRs in comparison with traditional residuals provide further evidence on its advantages for diagnosing count regression models.

Highlights

Examining residuals is a crucial step in statistical analysis to identify the discrepancies between models and data, and assess the overall model goodness-of-fit
For the remaining of this article, in “Methods” section, we review the commonly used count regression models and discuss the inadequacies of the Pearson and deviance residuals for detecting model misspecification followed by a brief review of randomized quantile residual (RQR)
For testing the normality of the residuals based on the SW normality test, the results clearly showed that both true and wrong models were rejected according to the Pearson residual, deviance residual, and Middle-point quantile residuals (MQR)

Summary

Introduction

Examining residuals is a crucial step in statistical analysis to identify the discrepancies between models and data, and assess the overall model goodness-of-fit. In diagnosing normal linear regression models, both Pearson and deviance residuals are often used, which are equivalently and approximately standard normally distributed when the model fits the data adequately. In non-normal regression, in data with the response variable being distributed on a small number of distinct values, the Pearson and deviance residuals do not typically follow (marginally and conditionally) a normal distribution. The plots of their residuals contain nearly parallel curves, making their assessment difficult to interpret due to the lack of a unified reference distribution for comparison. The χ 2 tests with χn2−p as the null distribution are widely used for quantitatively measuring the overall GOF in non-normal regression, the χn2−p distribution is often very poor for approximating the true null sampling distribution of χ 2 statistic [8, 9] except for situations where the response variable is approximately normally distributed (e.g., Poisson with large means)

Methods

Results

Conclusion