外科手術時間對於醫院管理而言是一有用且重要的資訊。而醫院管理涵蓋了在不同診斷下病患手術時間估計、手術房排程與利用率等等。
在本論文中，我們探討婦產科十三種開刀手術的手術時間的分佈情形。由結果看來手術時間似乎呈現log-normal 分佈以及 mixture log-normal分佈。接著我們將手術時間依照小、中、中大、大手術來分群，並且針對手術時間為mixture log-normal分佈的手術，利用概似比檢定找出影響其手術時間分佈的因素。最後，我們試著說明導致某一些資料量多的手術時間呈現 mixture log-normal 分佈趨勢的原因以及未來研究方向。

Surgeon operation time is a useful and important information for hospital management, which involves operation time estimation for patients under different diagnoses, operation room scheduling, operating room utilization improvements and so on. In this work, we will focus on studying the operation time distributions of thirteen operations performed in the gynecology (GYN) department of one major teaching hospital in southern Taiwan. We firstly investigate what types of distributions are suitable in describing these operation times empirically, where log-normal and mixture log-normal distribution are identified to be acceptable statistically in describing these operation times. Then we compare and characterize the operations into different categories based on the operation time distribution estimates. Later we try to illustrate the possible reason why distributions
for some operations with large data set turn out to be mixture of certain log-normal distributions. Finally we end with discussions on possible future work.

Contents
List of Tables . . . . . . . . . . . . . . . . .. . . . 1
List of Figures . . . . . . . . . . . . . . . . . . . . 3
1. Introduction . . . . . . . . . . . . . . . . . . . . 4
2. Mixture distributions and method of estimation of parameters . . . 9
2.1 Expectation-maximization (EM)algorithm . . . . . . . 9
2.2 The standard errors of MLE . . . . . . . . . . .. . 10
2.3 Kolmogorov-Smirnov test . . . . . . . . . . . . . . 11
2.4 Tamhane T2 test . . . . . . . . . . . . . . . . . . 12
3. Efect testing . . . . . . . . . . . . . . . . . . . 12
3.1 Likelihood ratio test . . . . . . . . . . . . . . . 13
4. Statistical analysis results . . . . . . . . . . . . 15
4.1 Classi‾cation of operations based on operation time . . . . . 15
4.2 Mixture log-normal distribution . . . . . . . . . . 16
4.3 Efect of doctor for three operations . . . . . . . 20
4.4 E

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