病例對照研究中多元羅吉斯迴歸模型的適合度檢定

A Goodness-of-Fit Test of Multinomial Logistic Regression Model in Case-Control Studies

許育愷(Yu-kai Shiu)

指導教授 ： 陳麗菁

淡江大學/商管學院/統計學系碩士班/碩士(2014年)

https://doi.org/10.6846/TKU.2014.00291

摘要

在病例對照研究中，羅吉斯迴歸模型常用來推論疾病與風險因子之間的關係。當研究中的反應變數為多個類別時，需考慮使用多元羅吉斯迴歸模型。本文推廣Qin and Zhang (1997)的想法到多元羅吉斯迴歸模型，推導各個病例組與對照組之間的比例關係，並將此模型重參數化得到多組的半參數化模型，以此推論迴歸參數的半參數最大概似估計量。為了檢測模型的合適性，本文推廣Chen and Wang (2013)的方法，針對多元羅吉斯迴歸模型提出動差形式檢定統計量，並使用拔靴法求算檢定統計量的p值。經由模擬研究發現，即使在有限樣本下該檢定統計量亦表現良好。最後以兩組實際資料分析作為範例。

關鍵字

拔靴法 ； 病例對照研究 ； 適合度檢定 ； 多元羅吉斯迴歸 ； 半參數最大概似估計量

並列摘要

In case-control studies, the logistic regression model is used popularly for inferring the relationship of disease and risk factors. When a response has multiple categories, the multinomial logistic regression should be considered. By generalizing the concept of Qin and Zhang (1997), this study derives a proportional relationship between the control group and each case group. After reparameterisation, the logistic model is equivalent to several semiparametric models and then the semiparametric maximum likelihood estimator is derived based on this finding. This study generalizes the idea of Chen and Wang (2013) to propose a moment-type test statistic for the multinomial logistic regression. A bootstrap method is used to calculate the p-value of the proposed test. Simulation studies demonstrate that the proposed test performs well even in finite samples. An illustration with two real data sets is provided as well.

並列關鍵字

bootstrap method ； case-control studies ； goodness-of-fit test ； multinomial logistic regression ； semiparametric maximum likelihood estimator

參考文獻

Begg, C. B. and Gray, R. (1984). Calculation of polychotomous logistic regression parameters using individualized regressions, Biometrika, vol.71, pp.11-18.

Chen, L. C. and Wang, J. Y. (2013). Testing the fit of the logistic model for matched case-control studies, Computational Statistics and Data Analysis, vol.57, pp.309-319, 2013.

Cheng, K. F. and Chen, L. C. (2004). Testing goodness-of-fit of a logistic regression model with case-control data, Journal of Statistical Planning and Inference, vol.124, pp.409-422.

Cox, D. R. (1970). The Analysis of Binary Data. Methuen, London

Fagerland, M. W. and Hosmer, D. W. and Bofin, A. M. (2008). Multinomial goodness-of-fit test for logistic regression model, Statistics in medicine, vol. 27, pp.4238-4253.