結構方程模型非中參數之估計：慣用法與拔靴法之比較

Abstract

非中參數估計在結構方程模型使用中扮演重要角色。然慣用的非中參數估計式存在偏誤，Raykov（2000）的推導指出非中參數慣用法之偏誤，及其大變異量和高均方誤差，並於2005年提出一非中參數拔靴偏誤校正估計式。非中參數奠基於卡方檢定量在對立假設下之大樣本分配，故其受樣本量與模型誤設程度之影響。為充分評估估計式之表現，亦將模型類型納入考量。因此，本研究採蒙地卡羅模擬方法，藉由操弄樣本量、模型誤設程度與模型類型，以評估比較非中參數慣用估計式（δ ̂）、基於拔靴偏誤校正概念之Raykov（2005）非中參數估計式（δ ̂_bc）與本研究提出之另一拔靴偏誤校正估計式（δ ̃_bc）三者的綜合表現。結果顯示，三非中參數估計式之偏誤、相對偏誤、標準差與均方根誤差主要受模型誤設程度影響，亦受樣本量影響。三估計式相對偏誤之絕對值均隨模型誤設程度和樣本量的增加而減小，標準差與均方根誤差則隨模型誤設程度和樣本量增加變大。δ ̂與δ ̃_bc表現相近，而δ ̂_bc出現過度校正的現象。模型正確時，δ ̂_bc的偏誤幾近於零，然於其他情境則易低估非中參數，其相對偏誤絕對值大都高於δ ̂與δ ̃_bc。δ ̂_bc之標準差基本上較小，然由於偏誤之影響，其均方根誤差於多數情況均較另兩估計式為大。綜整而言，於真實模型中，δ ̂_bc更接近真實值；然除真實模型和誤設程度很小之模型外，δ ̂與δ ̃_bc之偏誤與相對偏誤絕對值和均均方根誤差均較小。考量δ ̂於本研究展現之特性及其計算之簡便性，目前三估計式中仍推薦非中參數慣用估計法，然亦期待於未來能有更佳估計式之發展，以利結構方程模型之應用與分析。

Noncentrality parameter (NCP) plays an important role in evaluating structural equation models. The present simulation study compared the behaviors of three estimators of NCP. Raykov (2000) showed that the conventional noncentrality parameter estimator possessed asymptotically potentially large bias, variance, and mean squared error, and further developed a bias-corrected bootstrap estimator (Raykov, 2005) as a possible alternative for the conventional estimator. The noncentrality parameter is based on the asymptotic distribution of the chi-square test statistic under alternative hypothesis and is affected by sample size and degree of model misspecification. This Monte Carlo research therefore systematically manipulated degree of model misspecification, sample size and model type to evaluate the performance of the conventional noncentrality parameter estimator (δ ̂), Raykov’s bias-corrected bootstrap estimator (δ ̂_bc) and another bias-corrected bootstrap estimator proposed in this study (δ ̃_bc). The results showed that degree of model misspecification demonstrated the largest effect on the bias, relative bias, standard deviation and root mean squared error of these three noncentrality parameter estimators, followed by sample size. Absolute relative bias of these three estimators decreased with increasing sample size and degree of model misspecification, while their standard deviations and root mean squared errors increased with larger sample sizes and more severe model misspecifications. Raykov’s δ ̂_bc showed little bias under true models and tended to underestimate the true NCP value in other conditions, resulting in large absolute relative bias compared to the other two estimators. Although the standard deviation of δ ̂_bc was less than δ ̂ and δ ̃_bc, its root mean squared error was larger due to its bias. The other two estimators, δ ̂ and δ ̃_bc, performed similarly. Their absolute biases, absolute relative biases and mean squared errors were smaller than δ ̂_bc except for true models and very mild model misspecifications. The conventional NCP estimator is thus recommended considering its ease of computation and the behaviors shown in this simulation study. Further development of alternative estimators for this critical quantity in structural equation modeling is also encouraged.