Methods of finding the initial values of parameters in the maximum likelihood estimating equations for a logistic regression model and comparison of their final solutions using different criterion functions

Abstract

We present two methods of finding the initial values of parameters of the maximum likelihood estimating equations (MLEE) for a logistic regression model using two criterion functions. We then use the initial values and the corresponding criterion functions to obtain the final solutions of MLEE. Most experiments include more than two doses for determining a lethal dose like E D 50 . With more than two doses, we do not have an exact analytical expression for the solution of estimating equations. However, for two doses, we do have an exact analytic expression for the solution of estimating equations. The iterative methods make use of the initial values of the parameters. We have used the search algorithm for performing the optimization to find the final solutions of MLEE. The proposed approach starts with all possible pairs of doses from the doses considered in the experiment. It then chooses the pair giving the optimum value of a criterion function and the corresponding exact solutions for the parameters based on two observations in the pair as the initial values of parameters for solving MLEE for all observations. The proposed methods are transparent in the selection of the initial values of parameters. The proposed methods are computer intensive like bootstrap and jackknife methods popular among statisticians. We illustrate our two methods of finding the initial values with an observed beetle mortality data. We then apply them to obtain the final solutions using two criterion functions. We observe that the numerical values of E D 50 for the initial values of the parameters obtained by our approach are almost the same as the numerical values of E D 50 for the final solutions. This closeness of the estimated E D 50 values from our initial parameter values to the estimated E D 50 values from the final solutions is a strong feature of our proposed methods. Moreover, the proposed methods compare favorably with SAS and R in terms of CPU time values.