在獨立及成對樣本下探討以參數方式建構之兩種診斷試劑準確度之估計方法及檢定

Evaluation of the performance of the estimations and tests for comparing accuracy between two parametric diagnostic tests under the independent and paired samples.

巫品秀(Ping-Hsiu Wu)

指導教授 ： 黃怡婷

國立臺北大學/商學院/統計學系/碩士(2013年)

摘要

預防醫學常會使用診斷試劑來了解受測者患病狀況,良好的診斷試劑能及早發現疾病,讓病人獲得即時治療。一般透過比較受測者實際患病狀況及診斷試劑判定結果的一致性來評估試劑的準確度。當判定結果為二元時,敏感度和專一度是衡量準確度的基本準則。當判定結果為連續或順序型時,則透過選擇臨界值來設定患病的標準;隨著臨界值不同,檢驗試劑的標準也會不同,此時,接收操作者特徵曲線(Receiver Operating Characteristic Curve,簡稱ROC 曲線) 便是常用於評估準確度的工具。
假設患病和未患病受測者的判定結果均服從某一母數分配族,則可建構出參數 ROC 曲線,該曲線可由 a 及 b 兩參數描述,其中 a 及 b 為位置及尺度參數的函數。對此兩個參數的估計,當兩種診斷試劑受測者為獨立樣本時,本論文透過 Green 和 Swet (1966) 所建立的信號診斷理論之架構來建構資料的概似函數,求得最大概似估計式,更進一步利用費雪資訊矩陣(Fisher Information) 來求得估計式的估計共變異數矩陣; 而 Zhou 等人 (2002) 利用試驗結果的分配來建構概似函數,並提出估計式及估計變異數和共變異數之公式,本論文更進一步求出在 Binormal 分配之下, Zhou 等人所提出之估計式變異數和共變異數的精確公式。再者,當兩種診斷試劑受測者為成對樣本時,本論文建議利用 Biswas 和 Hwang (2002) 所提出之新型態的二元二項分配 (Bivariate Binomial Distribution) 來建構概似函數,用以得到ROC 曲線參數估計值,及其大樣本分配。本論文將就不同資料型態,設定比較兩種診斷試劑的準確度的檢定與其表現。最後,利用統計模擬來說明推導公式及檢定的表現,及二元二項分配假設的可行性。

關鍵字

ROC 曲線 ； 診斷試劑 ； binormal 分配 ； 敏感度 ； 專一度

並列摘要

In preventive medical science, diagnostic tests have been developed to detect the diseases. A diagnostic test that can detect the disease early and correctly is preferred. In general, the consistency of true disease status and test results is used to assessed the accuracy of the diagnostic test. For binary outcomes, the accuracy of the diagnostic test can be summarized by two basic indices, sensitivity and specificity. For continuous or ordinal outcomes, choose the threshold to set up the criteria of tests. However, as the threshold changes, the criteria will also be different. The receiver operating characteristic (ROC) curve is used to assess the performance of the diagnostic test when the test results are continuous or ordinal. This paper use parametric approaches to construct the ROC curve. We assume that the distributions of the test results are from a parametric family. Under this assumption,the ROC curve can be described by two parameters, a and b, which are the functions of location and scale parameters. When the participants of two diagnostic tests are independent, we use the signal detection theory proposed by Green and Swet (1966) to construct the likelihood function and obtain the maximum likelihood estimators of a and b. Furthermore, we can use the distribution of test results to construct the likelihood function. Zhou et al (2002) propose the corresponding formula of estimators of a and b. When two test results are paired, we use the bivariate binomial distribution proposed by Biswas and Hwang (2002) to construct the likelihood function. In this paper, we derive the estimators of a and b and their asymptotic distributions. When there are two diagnostic tests can be used to detect the disease, comparing their accuracies can help us to figure out which test is better. This paper compare the accuracy of two diagnostic tests and test if the parameters of two ROC curves are equal or not.

並列關鍵字

ROC curve ； diagnostic test ； binormal distribution ； sensitivity ； specificity

參考文獻

Agresti, A. (2002). Categorical Data Analysis. John Wiley and Sons, Inc., New York.

Biswasa, A. and Hwang, J. S. (2002). A new bivariate binomial distribution. Statistics and Probability Letters, 60, 231-240.

Dorfman, D. D. and Alf, Jr. E. (1968). Maximum likelihood estimation of parameters of signal-detection theory-A direct solution. Psychometrika, 33, 117-124.

Dorfman, D. D. and Alf, Jr. E. (1969). Maximum likelihood estimation of parameters of signal-detection theory and determination of confidence intervals-Rating method data. Journal of Mathematical Psychology, 6, 487-496.

Green, D. M. and Swets, J. A. (1966). Signal detection theory and psychophysics. John Wiley and Sons, Inc., New York.