Empirical Investigation of Randomized Quantile Residuals for Diagnosis of Non-Normal Regression Models

Traditional tools for model diagnosis for Generalized Linear Model (GLM), such as deviance and Pearson residuals, have been often utilized to examine goodness of fit of GLMs. In normal linear regression, both of these residuals coincide and are normally distributed; however in non-normal regression models, such as Logistic or Poisson regressions, the residuals are far from normality, with residuals aligning nearly parallel curves according to distinct response values, which imposes great challenges for visual inspection. As such, the residual plots for modeling discrete outcome variables convey very limited meaningful information, which render it of limited practical use. Randomized quantile residuals was proposed in literature to circumvent the above-mentioned problems in the traditional residuals in modeling discrete outcomes. However, this approach has not gained deserved awareness and attention. Therefore, in this thesis, we theoretically justify the normality of the randomized quantile residuals and compare their performance with the traditional ones, Pearson and deviance residuals, through a set of simulation studies. Our simulation studies demonstrate the normality of randomized quantile residuals when the fitted model is true. Further, we show that randomized quantile residual is able to detect many kinds of model inadequacies. For instance, the linearity assumption of the covariate effect in GLM can be examined by visually checking the plots of randomized quantile residuals against the predicted values or the covariates. Randomized quantile residuals can be also used to detect overdispersion and zero-inflation, two commonly occurred cases associated with count data. We advocate examining normality of the randomized quantile residuals as a unifying way for examining the goodness of fit for regression model, especially for modeling the discrete outcomes. We also demonstrate this approach in a real application studying the independent association between air pollution and daily influenza incidence in Beijing, China. ii Acknowledgements I would like to first and foremost express my sincere gratitude to my supervisors, Dr. Longhai Li and Dr. Cindy Feng, for their academic and financial support as well as their encouragements and patience throughout the period of my MSc. education. It was an honour to work with two of the most dedicated academics I have ever met. I like to thank Dr. Chris Soteros for agreeing to serve as my thesis supervisor when I applied to study for my MSc. in Canada. It was her willingness to trust in my academic abilities which allowed me to have the opportunity to study at the University of Saskatchewan in Canada. I like to thank Dr. Shahedul Khan for serving as a member of my thesis committee as well as teaching me various statistical skills which will be useful in my career as a statistician. My special thanks goes to my dear friend Mehdi Rostami for all the wonderful friendship and companionship we have had in the last nine years. His advice and guidance in Mathematics, Statistics, and computer programming have always been very insightful in all aspects of my academic life. I would like to thank the Department of Mathematics and Statistics at the University of Saskatchewan for academic and financial support as well as opportunities given to me. It has been a wonderful two years. Finally, I would specially like to express my very profound gratitude to my family for providing me with continuous love and support during my whole life. Their love, enthusiasm, and encouragements have always inspired me to strive towards my goals. My accomplishments would not have been possible without them. Thank you for everything.