計數型及二元型資料的非傳統迴歸模型｜國立清華大學博碩士論文庫

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研究生：	蘇尼爾 Chatla, Suneel Babu
論文名稱：	計數型及二元型資料的非傳統迴歸模型 Unconventional Regression Models for Count and Binary Data
指導教授：	徐茉莉 Shmueli, Galit
口試委員:	雷松亞 Ray, Soumya 盧鴻興 Lu, Horng-Shing 陳君厚 Chen, Chun-Houh 黃禮珊 Huang, Li-Shan
學位類別：	博士 Doctor
系所名稱：	科技管理學院 - 服務科學研究所 Institute of Service Science
論文出版年：	2018
畢業學年度：	106
語文別：	英文
論文頁數：	128
中文關鍵詞：	Probit 模型、疊代重新加權最小平方法、懲罰樣條、過度分散與分散不足、時間序列、轉折點、廣義概似比、選擇性偏差、Logit 模型
外文關鍵詞：	Probit, IRLS, Penalized Splines, Over and under dispersion, Time series, Change point, Generalized Likelihood Ratio, Selection bias, Logit
相關次數：	點閱：3 下載：0
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由於近期資料的多元化，計數以及二元反應變數已成為各領域中熱門的資料型態，特別是在人類社會行為學中。傳統在處理計數型反應變數的統計模型包含了卜瓦松迴歸以及負二項迴歸，而羅吉斯迴歸以及Probit迴歸則是使用於二元反應變數。如同大多數的迴歸模型，上述的傳統方法皆存在限制條件而無法應付於各類情況。為了因應這些傳統模型不能解決的問題，許多替代的模型孕育而生。我們討論了其中兩個的非傳統模型:針對計數資料我們研究了Conway-Maxwell Poisson (CMP) 迴歸而二元變數則採用線性機率模型 (LPM)。對於CMP迴歸，我們發展新的模型推廣，包含generalized additive model以及tree-based 變化係數模型，也同時提出包含IRLS在內的估計方法。對於LPM，我們評估其估計以及預測的性質。

(a) CMP迴歸模型能有效的掌握資料分散不足或者過度分散的現象，因此實務上CMP迴歸模型被廣泛使用於處理計數資料。然而，我們已知CMP迴歸模型使用於複雜非線性關係時是有其限制的。隨著資料多元性的增加，對於處理複雜非線性關係的計數模型的需求也隨之上升。在此篇論文中，我們首先提出在CMP分布下的變化係數模型，其迴歸係數被建構為調節變數的低維度函數，而後將此模型推廣至可容許高維度係數函數。後者我們採用新的tree-based方法。整體而言，我們在CMP迴歸研究上的貢獻如下:
(i) 我們提出一個在CMP迴歸上基於iterative reweighted least square (IRLS) 的較彈性估計過程，接著使用penalized spline估計法將其模型延伸至允許additive components (GAMs) 和 varying coefficient models (VCMs)。在一般條件下，CMP分布屬於指數族的特性可確保IRLS對於此模型的收斂性。我們透過大量的模擬研究與華盛頓特區單車分享系統的實際資料來展現此方法的可用性。
(ii) 我們提出的tree-based 變化係數模型是較有彈性的。在CMP分布的條件下，我們考慮基於模型的遞歸分割算法來有效率的改良tree-based 變化係數模型。除此之外，此篇文章也提供了另外的方法去估計切割點，進而取代預設的窮舉法，使得CMP分佈的模型配適更容易。同樣使用大量的模擬研究與華盛頓特區單車分享系統的實際資料來展現該方法的可用性。

(b)線性迴歸在社會科學研究中是最為常見的方法。而針對二元型態的反應變數
的線性機率模型(LPM)同樣也被廣泛的使用在不同學科中。在此篇論文中，我們首先使用在二元反應變數模型中三種常見的指標來衡量LPM，包含了模型推論與估計、預測與分類以及選擇性偏差。我們在不同的樣本數、誤差分布等的條件下，比較該模型與羅吉斯迴歸以及Probit迴歸的優缺點。其次，我們在LPM中放寬了有母數的參數假設，同時也使用類似的比較方法來測試無母數LPM、羅吉斯迴歸與Probit迴歸的特點。我們研究發現在係數的正負號、統計顯著性以及邊際效性中，LPM的結果與羅吉斯和Probit模型相似。除此之外，LPM的估計量也收斂到真實參數的倍數。在分類以及選擇性偏差上，LPM同樣不亞於羅吉斯和Probit模型分析的結果。然而當有興趣的分析目的是在預測機率上時，LPM的實用性可能就相對不足，主因是LPM的機率預測值有機會高於一。最後，我們將呈現該模型在線上拍賣網(eBay)資料的價格分析結果。

針對每個研究主題，我們除了數學推導也都提供了充分的模擬數據分析以及真實數據來驗證對於所有提出模型的可用性。

Count and binary data have become popular dependent variables in studies in various areas, especially due to the growing availability of data on human and social behavior. While the standard models for count data include Poisson regression and Negative Binomial regression, the standard models for binary data include Logistic regression and Probit regression. Like any other regression models, these standard models have limitations in some cases and there may be other methods that are good in those situations. Among those alternatives we choose two unconventional models: Conway-Maxwell-Poisson (CMP) regression for count data and Linear Probability Model (LPM) for binary data. For CMP, we develop new model extensions and estimation frameworks including an IRLS algorithm, a generalized additive model as well as a tree-based varying coefficient model. For LPM, we critically evaluate its properties in terms of estimation and prediction.

(a) The Conway-Maxwell-Poisson (CMP) regression is a popular model for count data due to its ability to capture both under dispersion and over dispersion. However, CMP regression is limited when dealing with complex nonlinear relationships. With today's wide availability of count data, there is need for count data models that can capture complex nonlinear relationships. In this dissertation, we first present a varying coefficient model for the CMP distribution in which the regression coefficient is modeled as a low dimensional function of moderator variables and then we extend the model formulation in such a way that it allows high dimensional coefficient functions. The latter is fitted using a new tree-based method. Our contributions for the CMP regression include:

1. We propose a flexible estimation framework for CMP regression based on iterative reweighed least squares (IRLS) and then extend this model to allow for additive components (GAMs) as well as varying coefficient models (VCMs), using penalized splines estimation method. Because CMP distribution belongs to the exponential family, convergence is guaranteed for IRLS under some regularity conditions. We illustrate the usefulness of this method through extensive simulation studies and using real data from a bike sharing system in Washington, DC.

2. Our proposed tree-based varying coefficient model offers further flexibility. We consider a model based (MOB) recursive partitioning framework to implement a tree-based varying coefficient model for CMP distribution, as it is computationally less intensive. We also provide an alternative method to estimate the split point than the default exhaustive search which in turn eases model fitting in general, and for CMP distribution in particular. We illustrate the usefulness of our method by extensive simulation and a real application from a bike sharing system in Washington, DC.

(b) Linear regression is among the most popular statistical models in social sciences research. Linear probability models (LPMs) - linear regression models applied to a binary outcome - are used in various disciplines. In this dissertation, first, we evaluate LPM for three common uses of binary outcome models: inference and estimation, prediction and classification, and selection bias. We compare its performance to Logit and Probit regression models under different sample sizes, error distributions, and more. Second, we relax the parametric assumption and perform a similar type of comparison study for the nonparametric extensions of LPM, logistic and probit models. We find that coefficient directions, statistical significance, and marginal effects from LPM yield results similar to logit and probit. In addition, LPM estimators are consistent for the true parameters up to a multiplicative scalar. For classification and selection bias, LPM is on par with logit and probit in terms of class separation and ranking, and is a viable alternative in selection models. LPM is lacking when the predicted probabilities are of interest, because predicted probabilities can exceed the unit interval. We illustrate some of these results by modeling price in online auctions, using data from eBay.

For each of the studies, in addition to the methodological derivations, we use both extensive simulation studies and real data applications to illustrate the usefulness of the proposed methodologies.

Introduction 1
1 Count Data and the CMP Distribution................. 1
1.1 Varying Coefficient Model .................... 7
1.2 Tree-based Varying Coefficient Model . . . . . . . . . . . . . . 8
1.3 Model Based Recursive Partitioning (MOB) . . . . . . . . . . 10
1.4 PartReg .............................. 12
2 Binary Data................................ 12
3 Applications................................ 14 1.3.1 Bike Sharing............................ 14 1.3.2 Online Auctions.......................... 16
4 Outline................................... 17
An Additive Model for the COM-Poisson Distribution 19
1 Introduction................................ 19
2 Challenges of IRLS Estimation with CMP Regression . . . . . . . . . 20
3 IRLS Framework for CMP Regression.................. 23
3.1 Calculation of Cumulants .................... 23
3.2 Two Step Method......................... 26
3.3 Proof of Convergence of the Two Step Method. . . . . . . . . 28
3.4 Practical Issues .......................... 29
3.5 Inference.............................. 31
4 Simulation Study for the CMP Regression . . . . . . . . . . . . . . . 32
5 A CMP Generalized AdditiveModel .................. 37 2.5.1 Background ............................ 37
5.2 Implementation of the CMP Generalized Additive Model . . . 38
6 Simulation Study for the CMP Generalized Additive Model . . . . . . 42 2.6.1 Example1............................. 42 2.6.2 Example2............................. 44
7 Example: Modeling Data from Bike Sharing Systems . . . . . . . . . 47
8 Conclusions and Future Directions ................... 52
Tree-based Varying Coefficient Model for COM-Poisson Distribution
53
1 Introduction................................ 53 3.1.1 Contributions ........................... 54
2 Methodology ............................... 55
2.1 Split Variable Selection...................... 56
2.2 Split Point Estimation using Exhaustive Search . . . . . . . . 58
2.3 Split Point Estimation using Change Point Theory . . . . . . 60
3 Simulation Study ............................. 62
4 Modeling Data from Bike Sharing Systems . . . . . . . . . . . . . . . 66 3.4.1 Registered Users ......................... 67
4.2 Casual Users ........................... 72
5 Summary and Conclusions........................ 75
An Extensive Evaluation of Linear Probability Models 76
1 Introduction................................ 76
2 Background ................................ 78
2.1 Criticisms ............................. 78
2.2 Justifications ........................... 79
2.3 Connection between LPM and Logistic Regression . . . . . . . 81
3 Frame work for Inference ......................... 83
3.1 Methodology ........................... 83
3.2 Simulation Design......................... 87
3.3 Results and Analyses....................... 87
4 Framework for Selection Bias and Prediction . . . . . . . . . . . . . . 91
4.1 Methodology ........................... 91
4.2 Simulation Design......................... 92
4.3 Results and Analyses....................... 93
5 Extensions to Nonparametric Models .................. 94 4.5.1 Simulation Study ......................... 97
6 An Application to Online Auctions ................... 101
7 Summary and Conclusions........................ 105
Future Directions         108
1 CMB Distribution............................. 108
1.1 CMB Regression ......................... 110
2 Regularization............................... 111
2.1 Regularized CMP and CMB Regressions . . . . . . . . . . . . 112
Bibliography                112
Appendices 122
A     122
A.1 R Package................................. 122
A.2 Bike sharing Data Description and Results . . . . . . . . . . . . . . . 123
A.3 Model Summary:Registered Users ................... 126
A.4 Model Summary:Casual Users ..................... 127
B 128
B.1 LPMs in the Information Systems Literature . . . . . . . . . . . . . . 128
                                

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