Residuals and diagnostic methods in models for polytomous data - Araripe, Patricia Peres (2022)
Experiments and observational studies that result in polytomous data, nominal or ordinal, are frequently conducted in different areas of knowledge, especially in the agricultural or biological sciences. The generalized logit model is the alternative used for the analysis of this type of data and based on it, conclusions and decision-making are obtained. In statistical inference, it is very important to validate a model that has been fitted to the data using diagnostic methods based on appropriate residuals. However, residual analysis and diagnostics for models associated with polytomous response are still emerging in scientific research, constituting an object of research in the area of Statistics. As the polytomous categorical variable is multivariate, Pearsons ordinary residuals and deviance are vectors per individual with unknown distribution, which creates challenges in graphical visualization and interpretation. Randomized quantile residuals can be used to circumvent problems. However, it is observed that there is a lack of an investigation of its performance for the polytomous regression through simulation studies. As an alternative to reduce the dimension of the residuals and study outliers, this work proposes to use Euclidean and Mahalanobis distance measures, since there are no records of their use for the multinomial case. In this context, the methodological contributions of this work are: review of existing residuals for the class of models associated with polytomous data; study of the normality of randomized quantile residuals; proposition of using Euclidean and Mahalanobis distances to reduce the dimension of ordinary residuals, thus constituting a procedure for the diagnosis of generalized logit models, allowing the identification of the presence of outliers. Two applications illustrate the utility of the randomized quantile residuals and distance measurements. The performance of the proposed methods was done through simulation studies. In these studies, we evaluated the performance of randomized quantile residuals for individual nominal data as well as the use of Euclidean and Mahalanobis distances for grouped data. Graphic techniques such as the half-normal plot were used to assess the model and the Shapiro-Wilk test were used to verify normality of residuals. Under different scenarios, simulation studies have shown that the approaches are relevant to assess the goodness of fit of the generalized logits model to the data. Additionally, it is noted that such studies are just the beginning of a research area with many gaps to be filled.