#### Date of Award

1983

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Abstract

A large sample relative efficiency of estimation for multinomial logistic regression compared to multiple group discriminant analysis has been derived and evaluated for parameter values relevant to epidemiological research. The large sample distributions of the two procedures are based on the assumptions of multivariate normality and common covariance structure among groups. Matrix calculus methods were found to be valuable in obtaining concise expressions for the large sample variances.;Relative efficiency does not decrease as the number of response categories increases, although increases tend to be small. The number of explanatory variables and the magnitude of the odds ratios associated with them are the main factors determining relative efficiency, with the correlation among the explanatory variables and the distribution of the response frequencies being secondary factors. Values of odds ratios typical in practice can give relative efficiencies greater than two-thirds for a small number of variables.;An approximation to the large sample distribution of logistic regression has been extended and used to develop methods for sample size estimation in the multinomial case. Matrix calculus was involved in developing a matrix Taylor expansion for the Fisher information matrix. Sample size was approximated by the first term in this expansion. The approximation was evaluated for two particular distributions of a single explanatory variable by comparing it to the precise sample size based on the full expansion. It was found to be inaccurate for more than two response groups except in limited circumstances. However, a correction factor was derived that, when applied, gave sample size estimates of reasonable accuracy for several response groups. We also found that the sample size required to look at the risk of response in one particular category is greater than that for all responses combined when alternative hypotheses are the same.

#### Recommended Citation

Bull, Shelley Brenda, "Theoretical Properties Of Multinomial Logistic Regression" (1983). *Digitized Theses*. 1291.

https://ir.lib.uwo.ca/digitizedtheses/1291