Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 186372, 12 pages
Research Article

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

Tsung-Hao Chen,1 Chen-Yuan Chen,2 Hsien-Chueh Peter Yang,3 and Cheng-Wu Chen4

1Department of Business Administration, Shu-Te University, Yen Chau, Kaohsiung, Taiwan 82445, Taiwan
2Department of Management Information System, Yung-Ta Institute of Technology and Commerce, Pingtung, Taiwan 90941, Taiwan
3Department of Risk Management and Insurance, Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan 811, Taiwan
4Department of Logistics Management, Shu-Te University, Yen Chau, Kaohsiung, Taiwan 82445, Taiwan

Received 6 October 2007; Revised 3 March 2008; Accepted 26 August 2008

Academic Editor: Irina Trendafilova

Copyright © 2008 Tsung-Hao Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 the P-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 in P-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 approximate P-values are <0.10. Thus in this study exact analysis is more appropriate.