Date of Award


Publication Type

Doctoral Thesis

Degree Name



Mathematics and Statistics




Paul, S. R.,




In this thesis we develop goodness of fit tests of the generalized linear model with non-canonical links for data that are extensive but sparse. We derive approximations to the first three moments of the deviance statistic. A supplementary estimating equation is proposed from which the modified deviance statistic is obtained. Applications of the modified deviance statistic to binomial and Poisson data are shown. A simulation study is conducted to compare the behavior, in terms of size and power, of the modified deviance statistic and the modified Pearson statistic developed earlier by Farrington (1996). Three sets of data with different degrees, of sparseness and different link functions are analyzed. The simulation results and examples indicate that both the modified Pearson statistic and the modified deviance statistic perform well in terms of holding nominal levels. However, the modified deviance statistic shows much better power properties for the range of parameters investigated under the alternative hypothesis. Theses results also answer a question posed by Farrington (1996) and extend results of McCullagh (1986) for Poisson log-linear models. In some instances a score or a C(alpha) statistic performs well. In this thesis we also develop a score test statistic to assess goodness of fit of the generalized linear model for data that are extensive but sparse. The performance of this statistic is then compared with the modified Pearson statistic. Results of simulation show that both the modified score test statistic developed in our paper and the modified Pearson statistic developed by Farrington (1996) maintain nominal levels. However the modified score test has some edge over the modified Pearson statistic in terms of power. In practice, sometimes, discrete data contain excess zeros that can not be explained by a simple model. In this thesis we develop score tests for testing zero-inflation in generalized linear models. These score tests are then applied to binomial models and Poisson models and their performances are evaluated. A limited simulation study shows that the score tests reasonably maintain the nominal levels. The power of the tests for detecting zero-inflation increases very slowly for Poisson mean mu or binomial parameter p. For large values of mu and p power increases very fast and approaches 1.0 even for moderate zero-inflation. A discrete generalized linear model (Poisson or binomial) may fall to fit a set of data having a lot of zeros either because of zero-inflation only, because of over-dispersion only, or because there is zero-inflation as well as over-dispersion in the data. In this thesis we obtain score tests (i) for zero-inflation in presence of over-dispersion, (ii) for over-dispersion in presence of zero-inflation, and (iii) simultaneously for testing for zero-inflation and over-dispersion. For Poisson and binomial data these score tests are compared with those obtained from the zero-inflated negative binomial model and the zero-inflated beta-binomial model. Some simulations are performed for Poisson data to study type I error properties of the tests. In general the score tests developed here hold nominal levels reasonably well. The data sets are analyzed to illustrate model section procedure by the score tests. (Abstract shortened by UMI.)Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2001 .D46. Source: Dissertation Abstracts International, Volume: 62-10, Section: B, page: 4616. Adviser: S. R. Paul. Thesis (Ph.D.)--University of Windsor (Canada), 2001.