復發事件 (recurrent event) 在長期追蹤以及臨床實驗中是相當常見的資料型態。對於復發型資料分析而言，實驗者通常對共變數因子 (covariates) 對復發事件的頻率函數(rate function)的影響感到興趣。文獻中有許多統計方法可用於估計共變數因子對頻率函數的影響 (effect)，但大多需要設限時間獨立(independent censoring)以及共變數測量值準確等假設。然而，在真實資料分析中，復發事件可能被其他事件 (例如:死亡) 中止而違反設限時間獨立之假設。此種情況，我們稱之為訊息設限 (informative censoring)。此外，共變數因子的測量值可能受限於測量誤差 (measurement errors) 而需要被校正。本篇論文主要提出半母數估計方法，在訊息設限和共變數因子有測量誤差的情況下，對復發型資料的共變數因子進行迴歸分析。本論文總共分為兩部分: 第一部分探討單一復發事件 (univariate recurrent event) 的估計方法。我們利用共享脆弱模型 (shared frailty model)來解釋訊息設限和復發事件之間的關聯性以及發生於同一人之事件的相關性。詳細而言，假設當脆弱變數 (frailty variable) 給定之後，復發事件服從一個普瓦松過程，其強度函數為一共享脆弱模型，且不假定脆弱變數的分配。在共變數和測量誤差通同時服從常態分配的假設下，我們提出迴歸校正法(regression calibration approach)和動差校正法(moment corrected approach) 去修正測量誤差在迴歸參數估計中造成的偏誤。此兩種方法皆屬於有母數校正方法且需要重複測量資料 (replicated data) 去估計測量誤差的變異數 (variance)。在第二部分，我們將第一部份的方法延伸到多變量復發事件 (multivariate recurrent event data) 分析。在此類資料中，研究者會對兩種類型以上的復發事件同時感到興趣。另外，我們考慮的情況為:每個樣本都有一個不偏測量值(surrogate)，但只有一部分的樣本有工具變數 (instrumental variable)。重複測量資料和驗證資料 (validation data)皆不可得。我們假設不同類型復發事件的頻率函數服從不同的共享脆弱模型，其中脆弱變數用來描述訊息設限和復發事件之間的關聯以及不同復發事件之間的相關性。為修正測量誤差，我們提出兩個無母數校正方法(non-parametric correction approaches)去估計迴歸參數。第一個無母數校正方法只用工具變數可得之部份樣本來進行估計。為增進估計效率，我們提出第二個校正方法將其餘的樣本也納入估計。不同於第一部分，第二部分之方法不需要普瓦松過程的假設以及共變數和測量誤差的分配假設 (distributional assumption)。在估計過程中，我們亦不假定脆弱函數之分配。在兩個部分中，我們分別對本文提出之估計統計量建立大樣本理論，且利用模擬實驗來檢查估計量的表現。最後，我們將本文提出之估計方法套用到硒與癌症預防之雙盲實驗資料 (the Nutritional Prevention of Cancer trial)，估計硒的補充對預防鱗狀細胞癌 (squamous cell carcinoma) 和 基底細胞癌 (basal cell carcinoma)的復發之效用。

Recurrent event data are frequently observed in many longitudinal and clinical studies. In the literature, various methods have been proposed to analyze covariate effects on the occurrence rate of a recurrent event, yet these methods usually require the assumption of independent censoring and accurately measured covariates. However, in many real data applications, informative censoring occurs when the recurrent event process is stopped by some terminal events that are related to the recurrent event (e.g. death). Additionally, the covariates could be measured with errors and need to be corrected. In this doctoral dissertation, we develop semi-parametric estimation to deal with informative censoring and measurement errors for recurrent event data. This dissertation contains two works. In the first work, we propose two approaches to estimate regression parameters for univariate recurrent event data in the presence of informative censoring and measurement errors. Explicitly, we impose a shared frailty model on the intensity function of a Poisson process to characterize the informative censoring and the dependence of the events within a subject without specifying the frailty distribution. To estimate the regression parameters, a regression calibration method and a moment corrected method are proposed for adjusting measurement errors. Both methods are referred to as the parametric correction because they assume that the underlying covariates and error terms are normally distributed. Moreover, the replicated data is needed to estimate the measurement error variance. In the second work, we extend the first work to accommodate informative censoring and measurement errors in multivariate recurrent event data, in which more than one type of events is of interest. Also, we consider a situation that a surrogate is available for all subjects but an instrumental variable is obtained only for a fraction of subjects. No replicated data or a validation set is available. To formulate the dependence of the informative censoring on the recurrent event processes, a shared frailty model is imposed on the rate function for each type of recurrent event, where the frailty distribution is unspecified. The shared frailty model also characterizes the association among different types of recurrent events. For regression parameter estimation, we first construct a simple correction approach, in which only subjects with an observed instrumental variable are involved in the estimation. To gain the efficiency of the simple correction estimator, we further develop a new correction approach to incorporate the information from the whole cohort. Distinct from the approaches in our first work, the approaches in the second work require neither the assumption of a Poisson process nor the distributional assumption of the underlying covariates and measurement errors. The asymptotic properties of the four proposed estimators are established. The performance of all proposed methods is investigated through simulation studies. We illustrate the proposed methods with the Nutritional Prevention of Cancer data, which aims to assess the effect of plasma selenium supplement on recurrences of squamous cell carcinoma and basal cell carcinoma.

1 Introduction 1
2 Literature reviews 5
2.1 Modelling and statistical methods for recurrent event data . . . . 5
2.1.1 Conditional model . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Marginal model . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Frailty model . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Methods for measurement errors in various models . . . . . . . . 10
2.2.1 Generalized linear model . . . . . . . . . . . . . . . . . . . 11
2.2.2 Cox model . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Recurrent event model . . . . . . . . . . . . . . . . . . . . 14
3 Parametric corrections for univariate recurrent event model
with informative censoring and measurement error 16
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Model illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Recurrent event model . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Measurement model . . . . . . . . . . . . . . . . . . . . . 19
3.3 Estimating methods . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 Regression Calibration . . . . . . . . . . . . . . . . . . . . 22
3.3.2 Moment corrected approach . . . . . . . . . . . . . . . . . 24
3.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Real data application . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Non-Parametric corrections for multivariate recurrent event
model with informative censoring and measurement error 34
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Model illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 Multivariate recurrent event model . . . . . . . . . . . . . 36
4.2.2 Measurement error model . . . . . . . . . . . . . . . . . . 37
4.3 Estimating methods . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.1 Simple non-parametric correction method . . . . . . . . . 41
4.3.2 GMM non-parametric correction method . . . . . . . . . 43
4.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Real data application . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Conclusions and future works 53
Appendix A Asymptotic properties of the RC and MC estimators 76
Appendix B Proof of RC = MC for regression parameters 80
Appendix C Asymptotic properties of the SNC and GNC estimators
81

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