臺灣博碩士論文加值系統

因素分析（factor analysis, FA）已被廣泛地使用在心理學、精神醫學、及其他社會科學的研究中，用以探索或驗證一群可觀測到的指示變數之間所存在的潛在測度結構。該方法通常假設指示變數與潛在變數均為連續性隨機變數，且至少為對稱分佈。在過去三十多年中，有些研究者嚐試將因素分析方法推廣至類別形態的指示變數與（或）類別形態的潛在變數上，如：潛在架構分析（latent structure analysis）、潛在輪廓分析（latent profile analysis）、潛在種類分析（latent class analysis）、潛在特性分析（latent trait analysis）、及類別觀測資料的因素分析（見Bartholomew 1987）等。
本研究的目的在於針對連續性、間斷性（discrete）、或混合形態的指示變數－只要其母群體的分佈是屬於指數族（包括常態、二項、與波瓦松等分佈），發展一整合性的因素分析方法，稱為「廣義因素分析」（generalized factor analysis, GFA）。就像變異數分析、線性迴歸、邏輯斯迴歸、與波瓦松迴歸等模式可視為廣義線性模式（generalized linear models, GLMs）的特例一樣，廣義因素分析運用統一的方式，將傳統的因素分析方法推廣至探索或驗證一群連續性、二分、順序性、計數、或混合形態的指示變數之間所存在的連續性潛在變數之潛在測度結構。
我們先應用廣義線性模式中的「疊代重新加權最小平方法」（iterative reweighted least squares, IRLS）程序來線性化（linearize）廣義因素模式；然後，才利用一般因素分析的估計方法來估計各因素負荷量（factor loading）。我們所獨立發展統一的三步驟估計程序相似於Bartholomew（1987, Sec. 6.1, Pp. 107-115）書中所討論的E-M algorithm。另一方面，我們將廣義因素模式中各因素負荷量的估計視同在廣義線性模式中的變數有估計誤差存在的問題，然後以經濟學家在聯立方程模式 (simultaneous equations model, SiEM) 中的輔助變數法 (instrumental variable approach, IV)來估計因素負荷量。我們將討論統計模擬分析的結果，並且以數據比較不同的估計方法的表現。

Factor analysis (FA) has been used widely in various areas of sciences to explore or examine the latent measurement structure from a set of observed indicator variables. Both the observed and the latent variables are usually assumed to be continuous and, at least, symmetrically distributed. In the past 30 years or so, several methods had been proposed to extend the FA method for categorical observed indicator variables and/or latent variables, which include latent structure analysis, latent profile analysis, latent class analysis, latent trait analysis, and factor analysis of categorical data. See, for example, the books written by Bartholomew (1987) and Basilevsky (1994) and the references therein. We are interested in developing a general framework for FA, called the " generalized factor analysis" (GFA), for continuous, discrete, or mixed observed indicator variables, as long as they belong to the exponential family of distributions such as Normal, Binomial, and Poisson distributions. Just like the generalized linear models (GLMs), which include analysis of variances (ANOVA), linear regression, logistic regression, and Poisson regression as the special cases, we hope that the GFA method extends the standard FA method to build a measurement structure of continuous latent variable(s) from observed continuous, binary, ordinal, count, or mixed indicator variables in a unified way. Yet, before doing that, we investigate the equivalence between exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). To estimate the factor loadings in a GFA model, we apply the iterative reweighted least squares (IRLS) algorithm of GLMs to "linearize" the generalized factor model first, and then use the usual estimation methods of factor analysis to obtain the estimates of the factor loadings. Specifically, we develop independently a unified three-step estimation procedure for GFA, which is similar to the E-M algorithm discussed in Bartholomew (1987, Sec. 6.1, Pp. 107-115). On the other hand, we treat the estimation of factor loadings in GFA models as an error-in-variable problem of GLMs, and then take an econometricians'' instrumental variable (IV) approach for simultaneous equations model (SiEM) to estimating factor loadings. We shall discuss the results of our simulation study and compare the performances of different estimators numerically.

Cover
Contents
1 Introduction
1.1 Development
1.2 Motivation
1.3 Focus
1.3.1 Discrete Indicator Variables
1.3.2 Mixed Indicator Variables
2 Review #1: Exploratory and Confirmatory Factor Analyses
2.1 Factor Analysis of Continuous Data
2.1.1 Exploratory Factor Analysis (EFA)
2.1.2 Confirmatory Factor Analysis (CFA)
2.2 Factor Analysis of Discrete Data
3 Review #2: Generalized Linear Models (GLMs)
3.1 Model Specification and Interpretation
3.2 Estimation: The Iteratively Reweighted Least Squares (IRLS) Algorithm
3.3 Statistical Inference
3.4 Model-Fitting Techniques
4 Project #1: Exploratory vs Confirmatory Factor Analysis - Are There Two Kinds of Factor Analyses?
4.1 Introduction
4.1.1 Modes of Statistical Modeling
4.1.2 Motivation
4.2 Comparison between EFA and CFA
4.3 Working Example (A)
4.3.1 Analysis I
4.3.2 Analysis II
4.4 Working Example (B)
4.4.1 CFA Solution
4.4.2 EFA Solution
4.5 Issues
5 Project #2: Can the Factor Analysis Be Generalized?
5.1 Introduction
5.1.1 Motivation
5.1.2 Outline
5.2 Model Specification and Interpretation
5.2.1 Factor Analysis (FA) Model
5.2.2 Generalized Factor Analysis (GFA) Model
5.3 Estimation
5.3.1 Tools
5.3.2 Methods
5.4 An Illustration: A One-Factor Three-Indicator GFA Model
6 Simulations
6.1 Design
6.2 Results
7 Discussions
7.1 Summary
7.2 Future Work
8 Appendices
8.1 Appendix I: EFA vs CFA - Working Example (A)
8.1.1 Appendix 1.1: The Summarized Results for Various Values of λi1 and λi2
8.1.2 Appendix 1.2: The Detailed Result for λ12 = λ41 =1.0
8.2 Appendix 2: EFA vs CFA - Working Example (B)
8.2.1 Appendix 2.1: The Results for the CFA analysis
8.2.2 Appendix 2.2: The Results for the EFA analysis
8.2.3 Appendix 2.3: S-PLUS Plot Program and Output for Factor Rotation
8.3 Appendix 3: S-PLUS Simulation Program for Examining the Correlation be-tween the Derived Residuals δz1,δz2 and δz3
8.4 Appendix 4: S-PLUS Simulation Program for GFA
9 References
10 Tables
11 Figures

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