Goodness‐of‐fit tests for ordinal response regression models

Abstract

It is well documented that the commonly used Pearson chi-square and deviance statistics are not adequate for assessing goodness-of-fit in logistic regression models when continuous covariates are modelled. In recent years, several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently. However, these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case. We present the modified Pearson chi-square and deviance tests that are appropriate for assessing goodness-of-fit in ordinal response models when both categorical and continuous covariates are present. The methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate. These tests also provide immediate information as to where a model may not fit well. In addition, the methods are simple to understand and implement, and are non-specific. That is, they do not require prespecification of a type of lack-of-fit to detect.

Regressão logística ordinal em estudos epidemiológicos

“…These tests involve the constitution of a contingency table in which the lines comprise all the possible confi gurations of the covariables of the model and the columns are the categories of the ordinal response. 14 The expected counts of this table are expressed as , where N l is the total number of individuals classifi ed in line l and represents the probability of an individual in line l having the response j calculated from the model adopted. 14 The Pearson test for evaluating the suitability of the adjustment compares these expected counts with those actually observed, using the formula:…”

“…A signifi cant p-value leads to the conclusion of a lack of adjustment of the model to the data being studied. 14 Pulkstenis & Robinson 14 (2004) report that statistics (1) and (2) do not provide a good approximation of the chi-squared distribution when continuous covariables are adjusted. They suggest small modifi cations in this case.…”

“…1 There are several ordinal models, such as the proportional odds, partial proportional odds, continuation-ratio and stereotype logistic models. Despite this diversity and the great variety of studies [1][2][3][4][7][8][9][10][11][12][13][14][15] on the subject their use in the public health area is still rare. 1 This may be attributed not only to their complexity, but specially to the diffi culty encountered when it comes to validating their assumptions.…”

Notes on the Goodness-of-Fit Tests for the Ordinal Response Model

“…The ordinal response data has become increasingly common in many areas such as biomedical and health sciences. Pulkstenis and Robinson (2004) already proposed a chi-squared type statistic by forming a table using the ordinal scores and the patterns of categorical covariates. However, the requirement of both types of categorical and continuous covariates in the model is a weakness of the test by Pulkstenis and Robinson (2004) even though it can simply be approximated by a chi-square distribution with appropriate degrees of freedom.…”

“…Pulkstenis and Robinson (2004) already proposed a chi-squared type statistic by forming a table using the ordinal scores and the patterns of categorical covariates. However, the requirement of both types of categorical and continuous covariates in the model is a weakness of the test by Pulkstenis and Robinson (2004) even though it can simply be approximated by a chi-square distribution with appropriate degrees of freedom. Because the ordinal response model has r response categories we consider a linear combination of the response probabilities which is the so called ordinal scores by Lipsitz et al (1996).…”