本文在病例對照研究之下，探求相對於一種既存標準的醫學診斷方法，
另一種嫌究中的醫學診斷方法是否具備非劣性。此一研究中的每一個受試
者皆接受兩種不同的診斷方法，所得到的是具有相關性的成對診斷值。本
文考慮將成對資料利用冪轉換，轉換成近似二元常態分布的資料，然後進
行現有文獻中的有母數非劣性檢定。另一方面，本文應用不同的廣義伽瑪
分布描述右偏分布的二個診斷資料，並且使用適當的關聯結構函數聯結上
述的兩個邊際分布，用以描述成對資料的聯合分布。然後，在此一聯合分
布之下，建構兩條受試者操作特徵曲線，並且根據二條估計的曲線下的部
分面積之差異進行非劣性檢定。本文進一步在不同的關聯結構函數、廣義
伽瑪分邊際分布、相關係數等條件下，藉由模擬研究探討本文所提檢定方
法的型I 誤差率和檢定力表現。最後藉由分析一個實例說明上述檢定方法的
應用。

In this paper, we consider testing the non-inferiority of two medical
diagnostic methods in a case-control study where each subject receiving the two
different diagnostics produces correlated paired measurements. Note that it
occurs often in practice that the marginal distributions of the measurements are
right-skewed. Therefore, we first apply the power transformation to the
paired data so that they would behave like the bivariate normal data. One
parametric non-inferiority test is then implemented based on the transformed
data. On the other hand, we suggest and employ appropriate copula function
which links two generalized gamma distributions to describe the joint
distribution of the paired measurements. Under the joint distribution, an
approximate test based on the difference between the partial areas under the two
estimated Receiver Operating Characteristic (ROC) curves is then constructed.
In this paper, we would like to test if the difference between the true areas is
within an allowable region. The results of a simulation investigation of the
level and power performances of the approximate test for different degrees of
correlation in several possible copula functions with a variety of marginal
distributions are reported. Finally, a real data set is illustrated by using the
approximate test.

1. Li, C.R., Liao, C.T. and Liu, J.P.(2006) A non-inferiority test for diagnostic accuracy based on the paired partial areas under ROC curves. Computational Statistics & Data Analysis. 50: 1855 – 1876.
2. Box, G.E.P. and Cox, D.R. (1964) An analysis of transformations. Journal of the Royal Statistical Society, Series B. 26:211-252.
3. Cox, C., Chu, H., Schneider, M.F. and Munoz, A .(2007)Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine. 26:4352-4374.
4. DeLong, E., DeLong, D. and Clarke-Pearson, D.(1988) Comparing the areas under two or more correlated receiver operation characteristic curves: a non-parametric approach. Biometrics.44:837-845.
5. Faraggi, D. and Reiser, B.(2002) Estimation of the area under the ROC curve. Statistics in Medicine. 21:3093-3106.
6. Greiner, M., Pfeiffer, D. and Smith, R.(2005) The partial area under the summary ROC curve. Statistics in Medicine. 24: 2025–2040.
7. Liu, J.P., Ma, M.C., Wu, C.Y. and Tai, J.Y.(2006) Tests of equivalence and non-inferiority for diagnostic accuracy based on the paired areas under ROC curves. Statistics in Medicine. 25:1219-1238.
8. Molodianovitch, K., Faraggi, D. and Reiser, B. (2006) Comparing the areas under two correlated ROC curves: parametric and non-parametric approaches. Biometrical Journal. 48: 745–757
9. Nelsen, R.B.( 2006) An Introduction to Copulas. Second Edition. Springer: New York.
10. Stacy, E.(1962) A generalization of the gamma distribution. Annals of Mathematical Statistics. 33:1187-1192.
11. McClish, D.K.(1989) Analyzing a portion of the ROC curve. Medical Decision Making.9:190–195.
12. Wieand, S., Gail, M.H., James, B.R. and James, K.L.(1989) A family of non-parametric statistics for comparing diagnostic markers with paired or unpaired data. Biometrika. 76:585-592.
13. Zhou, X.H., Obuchowski, N.A., and McClish, D.K.(2002) Statistical Methods in Diagnostic Medicine. Wiley: New York.
14. Pepe, M.S.(2003) The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford University Press:New York.
15. 陳秀琴 (2011). 針對右偏分布資料進行兩個醫學診斷方法之相等性檢定。國立中央大學統計研究所碩士論文。
16. 李念純 (2011). 一維及二維右設限存活資料的適合度檢定。國立中央大學統計研究所碩士論文。