Satorra-Bentler量尺化檢定統計量於模型錯誤界定之適用性

Abstract

Satorra-Bentler量尺化檢定統計量（Satorra-Bentler scaled test statistic, TSB, Satorra & Bentler, 1994）為檢定結構方程模式於資料違反常態之統計量。學者發現當模型錯誤界定時，TSB的平均數與決策力會因偏離常態程度提高而下降，可能不適用於高峰度等嚴重偏離常態情況（Curran, West, & Finch, 1996; Foss, Joreskog, & Olsson, 2011）。基於在何種偏離常態假設時不宜使用TSB依然未解，本模擬研究檢視模型錯誤界定下TSB可能適用之變項偏態與峰度範圍，操弄的因子包括三種不同大小的模型、模型錯誤界定的程度 (定義為root mean square error of approximation [RMSEA] 的數值大小 = .025, .05, .08, .1) 、樣本數與模型參數數目的比值 (5, 10, 15, 20, 50) 、變項的偏態 (0~3) 與峰度 (-1~21) 。結果顯示隨著RMSEA提高，TSB過度校正的偏誤會加劇。當RMSEA為.025時，偏離常態的對於TSB的影響較不明顯，該結果亦支持Satorra與Bentler (1994) 認為TSB校正應適用於輕微模型界定錯誤的觀點。當RMSEA數值更高時，除少數情境外，變項偏態小於1且峰度介於-1和4會使得TSB的相對平均偏誤絕對值均小於20%，而偏態介於0到2峰度介於-1至7則使得TSB的實證決策力偏誤絕對值小於.1。本研究提供TSB於模型錯誤界定情境下可能適用之變項偏態與峰度範圍參考，以便協助實證研究者使用Satorra-Bentler量尺化檢定統計量。

As the violation of normality impacts statistical inferences in structural equation modeling, Satorra and Bentler (1988, 1994) proposed the Satorra-Bentler scaled test statistic (TSB) to adjust the normal theory test statistic for non-normal data. Scholars found this test statistic tended to decrease as non-normality increased with model misspecification, and indicated it should not be used with extreme kurtosis (Curran, West, & Finch, 1996; Foldnes, Olsson, & Foss, 2012; Foss, Joreskog, & Olsson, 2011). However, when non-normality would be problematic for TSB usage remains unknown. The present simulation study investigated the applicable distributional situation of skewness and kurtosis for TSB. The manipulated conditions included modelling model sizes, degrees of model misspecification (defined by root mean square error of approximation [RMSEA] = .025, .05, .08, .1), sample size to number of parameters ratios (5, 10, 15, 20, 50), marginal skewness (0~3) and marginal kurtosis (-1~21) of indicators. The results suggested that over-correction of TSB was severer as RMSEA increased. When RMSEA = .025, the effect of non-normality was minor, and it supported the proposition by Satorra and Bentler (1994) that the correction was applicable for minimal model misspecification. For higher RMSEAs, the skewness being 0 or 1 and kurtosis between -1 and 4 yielded the absolute values of relative bias of mean lower than 20% in most cases. For the empirical power loss of TSB to be less than .1 as caused by non-normality, the skewness would need to range from 0 to 2 with kurtosis between -1 and 7. This study provided the references for possible performance of Satorra-Bentler scaled test statistic at various distributional situations to assist researchers’ use of this test statistic in structural equation modeling.