Diagnostics for Logistic-Type Regression Models Using Stochastic Processes

• Activity Number: 31
• Type: Contributed
• Date/Time: Sunday, July 29, 2012 : 2:00 PM to 3:50 PM
• Sponsor: Biometrics Section
• Author(s): Ivy Liu*+ and Estate Khmaladze and Haizhen Wu
• Companies: Victoria University of Wellington and Victoria University of Wellington and Massey University
• Keywords: Brownian motion ; Goodness-of-fit ; Logistic regression ; Proportional odds models ; Stochastic process

Abstract

Traditional methods to detect lack of fit for logistic-type regression models (such as logistic regression models for binary responses and proportional odds models for ordinal responses) use either likelihood-ratio or Pearson chi-squared tests. The test statistics follow asymptotically a chi-squared distribution when data are not sparse. It applies when the explanatory variable is categorical. When the model contains a continuous explanatory variable, these goodness-of-fit tests are not valid. Furthermore, we can partition observed and fitted values according to the predicted probabilities of success using the original data, and then apply a Pearson statistic, which is known as the Hosmer-Lemeshow method. Unfortunately, methods based on the grouping strategy do not have good power. This talk provides an alternative diagnostic method using a process that converges in distribution to a Brownian motion. The Kolmogorov-Smirnov statistics are constructed to assess the adequacy of the model. For various cases, we will show that empirical distributions of statistics are very close to the limiting distribution under the null.