臺灣博碩士論文加值系統

對於彼此獨立且具有相同多項式分配的類別資料，一般常使用具卡方分配的皮爾森統計量來進行適合度檢定。該統計量乃是由各類別觀測總數減去期望總數之差所構成的二次型。關於一般多類別迴歸模型的適合度檢定，文獻上已有一些藉由拓展皮爾森卡方檢定法所提出的統計量，包括統計量$Q$, Hosmer及Lemeshow的$hat{C}$,及Pigeon的$J^2$。我們將報告這些統計量的缺點，並提出一個利用更合適共變異矩陣所建造的二次型統計量$W$。根據模擬試驗，除了使用自然聯結函數的模型外，我們所提出的統計量$W$在虛無假設下近似於卡方分配，其自由度為反應變數類別數減去一。經由模擬研究，我們檢視以上這些統計量的表現，並與近來利用殘差所建造的$SD_1$統計量作比較。結果如同我們所臆測， $Q$和$J^2$不適於實際使用， $W$和$SD_1$有令人滿意的『型一錯誤』模擬值。此外，對於順序型資料，當聯結函數為累積probit函數時， $W$檢定出模型為錯誤的能力表現優於$SD_1$，尤其是樣本數小的時候。在名目型資料的自然聯結迴歸模型下， $SD_1$檢定出模型為錯誤的能力不穩定。

Goodness-of-fit tests for i.i.d. multinomials are usually based on Pearson''s $chi^2$ statistic, which is a quadractic form in the observed totals minus expected totals. For general polytomous regression models with independent multinomials, several
statistics extending Pearson''s $chi^2$ statistic have been proposed in the literature for goodness-of-fit assessment, including $Q$, Hosmer and Lemeshow''s $hat{C}$, Pegion''s $J^2$. We present the drawback of these approaches and propose a new quadratic form statistic $W$, which uses a more appropriate covariance matrix to form the quadratic form. We present a simulation study to show that for models without using the natural link the statistic $W$ has an approximate $chi_{k-1}^2$ distribution under the null hypothesis, where $k$ denotes the number of categories of the response variable. We compare by simulation the performance, in terms of both size and power, of these statistics to the recent residual based Pearson''s $SD_1$, a member in the family of power-divergence statistics $SD_{lambda}$. These simulations confirm our conjecture that $Q$ and $J^2$ are not appropriate for practical use and demonstrate that both $W$ and $SD_1$ have satisfactory size performance. In addition, it was found that for ordinal data the power performance of the simulated $W$ is better than that of $SD_1$ in the cumulative probit regression model, especially when the sample size is small, and that for nominal data the power performance of the simulated $SD_1$ is not stable in the multinomial logit regression model.

Introduction (1)
Regression Models for Categorical Data (4)
Existing Omnibus Goodness-of-Fit Statistics (15)
The Proposed Goodness of Fit Statistic (22)
Simulation Studies (32)
Concluding Remarks (52)
References (54)

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