結構方程模型兩階段漸進自由分配估計方法之評估

Abstract

非常態資料經常出現於實徵研究中。在共變結構分析（covariance structure analysis，簡稱CSA），亦即建基於樣本共變數抽樣分配的結構方程模型（structural equation modeling，簡稱SEM），Browne （1984）提出之漸進自由分配估計方法（asymptotically distribution-free，簡稱ADF）為處理非常態資料上之一重大貢獻。然而，ADF需要極大的樣本數方能得到可靠的分析結果。在立基於樣本相關係數抽樣分配的SEM，或稱相關結構分析（correlation structure analysis，簡稱RSA），Steiger與Hakstian（1982）提出兩階段ADF（two-stage ADF，簡稱TADF）估計法，以期改善ADF檢定統計量於有限樣本中之表現。TADF先以簡單的估計方法得到基於模型再製動差（model-reproduced moments）所估計的樣本動差之漸進共變數矩陣（asymptotic covariance matrix of sample moments，簡稱ACOV）。第二階段以此結構化之ACOV估計值進行ADF。目前僅有的少數RSA模擬研究（如Bentler & Savalei, 2010; Mels, 2000）顯示，TADF之表現優於ADF與再加權最小平方法結合強韌校正（reweighted least squares with robust corrections，簡稱RLS-C）。本研究將 TADF 延伸至CSA，系統性探討TADF在非常態資料與有限樣本下於CSA與RSA之表現。在ADF之外，本研究亦將TADF的表現與SEM中其他處理非常態資料的方法進行表現，包括CSA中的最大概似法結合強韌校正（maximum likelihood with robust corrections）與新發展的分配加權最小平方法（distributionally weighted least squares method; Du & Bentler, 2022a），以及RSA中的RLS-C。本論文分別針對CSA與RSA進行模擬研究，此二模擬研究皆考量兩母群模型，並系統性操弄樣本人數與變項分布。結果顯示，在CSA與RSA中，TADF能改善ADF的有限樣本表現，包括較少的不收斂與不適當解、更準確的參數與標準誤估計值，以及較佳的檢定統計量之型一錯誤控制（Type I error control）。與CSA和RSA中其他處理非常態資料的方法相比，TADF在參數估計上的表現相似，且其檢定統計量之實徵型一錯誤率表現優於或相當於其他方法。然而，當樣本數不足時，TADF 傾向低估參數估計值的實際波動情形。重抽樣方法（resampling methods）與Yuan-Bentler（1997b）校正可望改善TADF之標準誤估計以提升此方法的整體表現。結合改善後的標準誤估計，TADF期可為SEM應用中處理常見之非常態資料與有限樣本的推薦方法。

Deviation from normality is commonly observed in empirical data. In covariance structure analysis (CSA), the structural equation modeling (SEM) based on the asymptotic distribution of covariances, the asymptotically distribution-free (ADF) estimation method proposed by Browne (1984) stands as a landmark contribution for dealing with nonnormal data. Yet ADF requires extremely large samples to yield reliable parameter estimates, associated standard errors, and model test statistics. In correlation structure analysis (RSA), the SEM based on the asymptotic distribution of correlations, Steiger and Hakstian (1982) proposed the two-stage ADF (TADF) estimation method in hope of improving the finite sample behavior of ADF test statistics. TADF first employs a simple estimation method to obtain an estimate of the asymptotic covariance matrix of sample moments (ACOV) based on model-reproduced moments. In the second stage, TADF implements ADF with this structured ACOV estimate. TADF has been found to outperform ADF and reweighted least squares with robust corrections (RLS-C) in RSA under restricted conditions (e.g., Bentler & Savalei, 2010; Mels, 2000). In this study, TADF was extended to CSA and the behavior of TADF with nonnormal data under finite samples was systematically investigated in both CSA and RSA. In addition to ADF, the performance of TADF was also compared to other recommended methods for dealing with nonnormal data in SEM. These methods included maximum likelihood with robust corrections and the newly developed distributionally weighted least squares method (Du & Bentler, 2022a) in CSA and RLS-C in RSA. Two simulation studies were conducted, focusing on CSA and RSA, respectively. Both simulations considered two population models and systematically varied sample size and distribution of variables. TADF was found to improve the finite sample behavior of ADF in both CSA and RSA, showing fewer nonconvergence and improper solutions, more accurate parameter and standard error estimates, and better Type I error control of test statistics. Compared to other methods for dealing with nonnormal data in CSA and RSA, TADF behaved similarly in parameter estimation and yielded better or comparable empirical Type I error rates of test statistics. Yet TADF tended to underestimate the empirical fluctuation of parameter estimates when sample size was insufficient. Resampling methods and the Yuan-Bentler (1997b) correction were potential strategies to improve the estimation of standard errors in TADF to enhance the overall performance of this method. With the improved standard error estimates, TADF presents a promising approach for handling nonnormal data with finite sample sizes frequently encountered in SEM applications.