A Monte Carlo Comparison of Maximum Likelihood and Minimum Chi Square Sampling Distributions in Logit Analysis

Abstract

SUMMARY The finite-sample properties of maximum likelihood and minimum chi square estimators in a simple dichotomous logit regression model are studied by Monte Carlo methods. Convergence of the regression coefficient test statistics to normality is 'slow for designs in which doses are placed asymmetrically about the ED50; skewness and bias are problems even at sample sizes of 480. Designs with doses symmetric about the ED50 can be used with reasonable confidence at moderate sample sizes. There is some evidence that maximum likelihood is preferable to minimum chi square when statistical inferences are to be made with a logit model. Qualitative response models are commonly used to analyze the results of biological experiments involving a binary response variable and one or more explanatory variables. In insecticide research, probit or logit regression models are used to describe the relationship between dose of an insecticide and mortality of a target insect species. The toxicities of two or more candidate insecticides are typically compared by a statistical test of the equality of their regression coefficients. It is also standard practice to estimate the effective doses (ED) required to produce a certain probability of response for each of several insecticides, and to compare their toxicities on the basis of confidence intervals for the EDs. These examples from insectide research illustrate that confidence sets and hypothesis tests, as well as point estimates, may be used, in conjunction with a qualitative response model, to evaluate a biological phenomenon. Until now, the statistical literature has rarely addressed issues concerning the validity of inferences, typically based on asymptotic distributions, that are drawn from these models. Instead, attention has focused primarily on aspects of point estimation, particularly on the comparison between maximum likelihood (ML) and a minimum chi square (MCS) method for estimating the regression coefficients in logit and probit models. For certain simple logit (Berkson, 1955) and probit (Berkson, 1957) models, the MCS estimators of the regression coefficients had smaller mean square errors (MSE) than the ML estimators. Approximations to the MSEs of the two estimators to the order n-2 (where n is the 'average' number of observations per cell) were derived by Amemiya (1980) for the logit model; the second-order approximation to the MSE of the MCS estimator was smaller than the corresponding ML approximation in every example presented. Although these numerical examples are unanimously in support of the MCS estimator, there is no theoretical evidence that MCS always has smaller risk than ML in the estimation of logit or probit regression coefficients using a quadratic loss function.