| Summary: | 碩士 === 國立東華大學 === 應用數學系 === 101 === In order to investigate the relationship between nominal multicategorical response variable and some covariates, it is customary to fit multinomial logit models to the data at hand. To assess the goodness of fit of the multinomial logit model, several omnibus statistics have been proposed in the literature, including Ĉ of Fagerland, Hosmer and Bofin (2008), J^2 of Pigeon and Heyse (1999), and the family of power-divergence statistics SD_λ of Osius and Rojek (1992). Recently, Hsu (2003) proposed the W statistic for assessing the goodness of fit of polytomous regression models with nonnatural link functions. The W statistic is a quadratic form in the differences between the observed and fitted totals over response categories. Because the multinomial logit link is the natural link function for nominal response, these differences are identical to zero when fitting multinomial logit model. Thus, the W statistic can't be used for assessing the goodness of fit of multinomial logit model. To overcome this difficulty in the presence of continuous covariates, Hsu (2003) considered partitioning the covariate space into groups, calculating the W statistic for every group, and summing up these W values to form the W_G statistic. Since the summands of W_G statistic are dependent, he was not able to derive the asymptotic null distribution of W_G statistic. In this thesis, we propose a new W_G statistic which also involves partitioning the covariate space into groups. The main distinction is that we calculate the aforementioned differences for each group and then compute a quadratic form in all these differences to form the new W_G statistic. We compare the performance of these aforementioned statistics by simulation studies, in terms of both size and power, for assessing the goodness of fit of multinomial logit model in the presence of continuous covariates.
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