區間設限資料經常出現於臨床實驗中，此時觀察對象是被週期性追蹤，而且只知道其失敗時間落在某一區間中。在這篇論文中，我們考慮兩組或多組區間設限樣本的比較問題。對於離散型的區間設限資料，我們提出一個多重插補法，這個方法可從一組區間設限數據中抽出多組相對應的完整數據，然後將現存用於處理完整數據的檢定法，例如對數秩檢定法(log-rank test)，應用於被抽出的完整數據。檢定統計量及共變異數矩陣是藉由我們所提出的多重插補法進行計算。其中共變異數矩陣的公式與Follmann，Proschan 及Leifer (2003)提出應用於群集資料的變異數估計量是相似的。經由模擬研究，我們發現藉由多重插補法提出的對數秩型態檢定法，其表現可比得上由Finkelstein (1986)所提出的檢定法，而且優於另外兩個分別由Sun (2001)及Zhao與Sun (2004)所提出的對數秩型態檢定法。我們的對數秩型態檢定法與其他兩個對數秩型態檢定法的差異，在於使用不同的多重插補法與共變異矩陣估計量。我們將以一個乳癌病患的實例作說明。我們也研究如何將所提出多重插補法，應用於其他針對完整數據處理兩組樣本比較問題的檢定法，例如Mantel檢定法及累積加權差異(integrated weighted difference)檢定法。

Interval-censored failure time data often occur in follow-up studies where subjects can only be followed periodically and the failure time can only be known to lie in an interval. In this paper we consider the problem of comparing two or more interval-censored samples. We propose a multiple imputation method for discrete interval-censored data to impute exact failure times from interval-censored observations and then apply existing test for exact data, such as the log-rank test, to imputed exact data. The test statistic and covariance matrix are calculated by our proposed multiple imputation technique. The formula of covariance matrix estimator is similar to the estimator used by Follmann, Proschan and Leifer (2003) for clustered data. Through simulation studies we find that the performance of the proposed log-rank type test is comparable to that of the test proposed by Finkelstein (1986), and is better than that of the two existing log-rank type tests proposed by Sun (2001) and Zhao and Sun (2004) due to the differences in the method of multiple imputation and the covariance matrix estimation. The proposed method is illustrated by means of an example involving patients with breast cancer. We also investigate applying our method to the other two-sample comparison tests for exact data, such as Mantel's test (1967) and the integrated weighted difference test.

1 Introduction
2 A Generalized Log-Rank Test for Interval- Censored Data
2.1 Assumptions and notation
2.2 Review of the log-rank test for exact data
2.3 A test procedure based on multiple imputation for interval-censored data
2.4 Construction of the covariance matrix estimator
2.5 Simulation studies
2.5.1 Simulation procedure
2.5.2 Reviews of Finkelstein's , Sun's and Zhao and Sun's tests
2.5.3 Simulation results
2.6 An example
2.7 Remarks
3 Application of Multiple Imputation Approach to the IWD Test
3.1 Review
3.2 IWD test via multiple imputation for interval-censored data
3.3 Comparison of the IWD test and the IWD-MI test
4 Application of Multiple Imputation Approach to Mantel's Test
4.1 Review
4.2 Mantel's test via multiple imputation for interval-censored data
4.3 Comparison of Mantel's test and Mantel-MI test
5 Comparisons of Tests under the proportional hazards and non-proportional hazards models
5.1 Proportional hazards model
5.2 Non-proportional hazards model
6 Discussions

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