A CLASS OF OPTIMAL EXPERIMENTAL DESIGNS FOR SOME POISSON REGRESSION MODELS WITH APPLICATION TO FERTILITY STUDIES

Abstract

Optimal experimental designs are a set of array of design points that are most efficient with respect to some statistical criteria. Russell et al. (2009) found locally D-optimal designs for onevariable first order model and additive two linear predictors on standardized design space [-1, 1], through the use of canonical form. The possibility of having negative criteria values ensues. This study investigated the Fisher information matrix-based optimal experimental design criteria for some Poisson regression models. The study specifically constructed D-, A-, and E-optimal experimental designs for two-variable, three-variable, quadratic and cubic Poisson regression models. The study determined the criterion values of the constructed optimal designs and also evaluated the efficiencies of the generated optimal designs. The imperialist competitive algorithmic search procedure in the construction of optimal designs using restricted design space [0, 1] was employed in order to ensure that the criteria values are nonnegative. In generating the optimal designs, model parameter estimates were assumed for each Poisson regression model considered and the experiment was conducted with 1000 search iterations. D-optimal designs were constructed for the two-variable, three-variable, quadratic and cubic Poisson regression models considered in the study and were found to be 75.00%, 71.43%, 100.00% and 99.99% Defficient respectively. The D-optimal design for the quadratic Poisson regression model was found to be the most D-efficient. A-optimal designs were generated for the two-variable, threevariable, quadratic and cubic Poisson regression models and were found to be 100.00%, 88.00%, 99.99% and 99.96% A-efficient respectively. The A-optimal design for the two-variable Poisson regression model was found to be the most A-efficient. Also, E-optimal designs were generated for the Poisson regression models with two and three predictor variables. The E-optimal designs were found to be optimal at 4-design points. All the constructed optimal designs were verified through the general equivalence theorem using the maximum sensitivity functions and were found to be indeed optimal. The powers of the designs were examined for the two-variable Poisson regression model. The generated powers are 0.9300, 0.9930 and 0.9890; which imply that the two-variable Poisson regression model has at most 99.30% probability of ending up with a P-value that is less than 5% in a statistical test and at least 93% to achieve this. Generally, the high power implies the significant relevance of the study. The concept of optimal experimental designs for Poisson regression models was applied in fertility studies. Approximate A-optimal design weights of educational levels of women were obtained for each marriage duration period vii with respect to their places of residence. The study revealed that women with secondary education and above were found to be consistently more than the number of women who had no education, lower primary education and upper primary education respectively for all the marriage duration periods considered and at each place of residence except for marriage duration of 0 – 4 years at Suva where the percentage of women with no education was more than the percentage of women with lower primary education, upper primary education and secondary education and above. Application of weighted A-optimal designs was also considered for Nigerian women of child bearing age. It was shown that the number of women with no education, primary education and secondary education were consistently higher in the rural areas than urban areas while women with higher education were more in the urban areas than rural areas.