在積體電路的製程中，進入機台的各種化學原料會黏附於機台內部，進而在生產時造成晶圓表面的殘留微粒，導致良率下降。因此不時清洗機台內部，以保持機台內部的清潔程度，是製程中的重要步驟，而此步驟被稱為清腔 (chamber purging)。在 Zhang (2015) 中，考慮了一清腔因子實驗，其利用廣義線性模型 (generalized linear model) 建立一清腔實驗模型，並對量測到的微粒顆數所可能反映的機台乾淨程度，提出兩種可能狀況。該論文針對第一種狀況，探討參數估計與檢定的方法。在本文中，首先根據 Zhang (2015) 建立的清腔實驗模型，探討第二種狀況下之模型的分析方法，包括參數估計與檢定。在判別數據較可能是由第一種或第二種模型所生成時，我們由最大概似估計的角度出發提出一個判別法，並利用電腦模擬來了解以錯誤模型分析數據會造成的影響。其次我們討論過度分散 (overdispersion) 的現象，藉由加入分散參數 (dispersion parameter) 以建構允許過度分散現象的模型，再利用準概似函數 (quasi-likelihood function)，調整參數估計及檢定方法，並將這些估計及檢定方法，應用在清腔實驗的真實數據分析上。最後，則探討樣本大小的設計問題，以在固定的顯著水準下，要求參數的檢定力達到標準之想法，來決定控片測試之重複次數。

In the manufacturing of semiconductor wafers, the chemicals released into chambers might remain on the surfaces of the chambers after the manufacturing operation. The remaining chemicals can cause the appearance of contaminant particles on the surfaces of the subsequently processed wafers, and consequently reduce the yields. An important step in the manufacturing of wafers is to regularly clean the chamber to remove residual chemicals and maintain chamber cleanliness at a desired level. It is thus critical to establish an effective clean recipe for the chamber clean process. Zhang (2015) studied a gas purge experiment, which was conducted to study the effects of some clean factors and to find an optimal clean recipe. In Zhang (2015), a statistical model based on generalized linear model was proposed for the data of the experiment. In the model, the responses are assumed to be random variables with Poisson distributions, of which the parameters must follow some recursive formula. According to the possible forms of the recursive formula, the model was further classified into type I and type II models. Zhang (2015) discussed and gave methods of parameter estimation and testing for the type I models. In the thesis, we identify and discuss several issues about the model. We first develop the parameter estimation and testing procedures for the type II model, and use computer simulations to examine the accuracy of the procedures. Second, a classification method based on the principle of maximum likelihood is proposed to determine whether the data is generated from type I or type II models. We also verify the effectiveness of the classification method by computer simulation, and briefly discuss the impact of fitting a model of wrong type. Third, we generalize the model by introducing a dispersion parameter for data exhibiting overdispersion relative to a Poisson model. Under the new model, we adopt the quasi-likelihood approach to resolve the problems of estimating parameters and testing hypotheses. The new model and the analysis methods are demonstrated
on a real data of purge experiment. Lastly, we discuss an issue of the design of the experiment, and obtain a criterion for determining the number of repeated measures of particle counts required to achieve a pre-specified level of testing power.

1 緒論 1
1.1 清腔因子實驗與其流程 1
1.2 清腔實驗模型 3
1.3 研究動機與目的 7
2 文獻探討 9
2.1 過度分散現象與分散參數 9
2.2 準概似函數 10
2.3 實驗重複次數的選擇準則 10
3 型 II 模型下之參數估計與檢定 12
3.1 參數估計 12
3.2 檢定 14
3.3 電腦模擬 14
4 型 I 與型 II 模型的判別 20
4.1 模型判別法 20
4.2 電腦模擬 21
5 過度分散現象下的模型與分析 26
5.1 模型調整 26
5.2 參數估計 27
5.3 檢定 32
5.4 電腦模擬 34
5.5 真實數據分析 36
6 控片測試重複次數的選擇 45
6.1 選擇準則 45
6.2 電腦模擬驗證 47
7 結論 51
參考文獻 53
附錄 A 型 II 模型的計分函數與費雪訊息矩陣之詳細推導 54
附錄 B 型 I 過散模型的計分函數與費雪訊息矩陣之詳細推導 59
附錄 C 型 II 過散模型的計分函數與費雪訊息矩陣之詳細推導 65

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