A Mathematical Tool for Inference in Logistic Regression with Small-Sized Data Sets: A Practical Application on ISW-Ridge Relationships

Abstract

The general approach to modeling binary data for the purpose of estimating the propagation of an internal solitary wave (ISW) is based on the maximum likelihood estimate (MLE) method. In cases where the number of observations in the data is small, any inferences made based on the asymptotic distribution of changes in the deviance may be unreliable for binary data (the model's lack of fit is described in terms of a quantity known as the deviance). The deviance for the binary data is given by D. Collett (2003). may be unreliable for binary data. Logistic regression shows that theP-values for the likelihood ratio test and the score test are both<0.05. However, the null hypothesis is not rejected in the Wald test. The seeming discrepancies inP-values obtained between the Wald test and the other two tests are a sign that the large-sample approximation is not stable. We find that the parameters and the odds ratio estimates obtained via conditional exact logistic regression are different from those obtained via unconditional asymptotic logistic regression. Using exact results is a good idea when the sample size is small and the approximateP-values are<0.10. Thus in this study exact analysis is more appropriate.