本篇文章將介紹針對結構方程常用的配適度指標作介紹,除了整理各指標的判斷標準及參考文獻之外,亦針對一些特殊的情況進行說明。

一、卡方檢定(Chi square test

卡方值是SEM最原始的指標,因為它直接從ML估計法的函數【(N-1FML】計算而得。卡方值是愈小愈好,但也沒有一定的標準,因為卡方值不但會受到樣本數的影響,也會受到模型複雜度的影響,幾乎所有的模式都可能被拒絕(Bnetler & Bonett, 1980; Marsh & Hocevar, 1985; Marsh, Balla, & McDonald., 1988),算不上是實用的指標,因此顯少採用,但它是許多配適度指標的計算基礎,所以在SEM分析中需要呈現。

Bentler, P. M. & Bonett, D. G. (1980). Significance tests and goodness-of –fit in the analysis of covariance structures. Psychological Bulletin, 88, 588-606.

Marsh, H. W., & Hocevar, D. (1985). Application of confirmatory factor analysis to the study of self-concept: First- and higher order factor models and their invariance across groups. Psychological Bulletin, 97, 562-582.

Marsh, H. W., Balla, J. R., & McDonald, R. P. (1988). Goodness-of-fit indexes in confirmatory factor analysis: The effect of sample size. Psychological Bulletin, 103(3), 391-410.

二、配適度指標(goodness of fit index, GFI

GFI值越接近1,表示模式配適度越高;反之,則表示模式配適度越低。通常學者建議GFI值大於0.9時表示模式有良好的適配(Bentler, 1983; Hu & Bentler, 1999; 黃芳銘,2007)。樣本數越大時,GFI也會愈大,但此時若剩下的自由度也大時,GFI則會產生向下偏誤(低估),除非估計的參數非常多,在此一情形下建議採用AGFI(不過Bollen1990)提出,樣本數小的時候AGFI會低估),Doll, Xia, and Torkzadeh1994)認為,當模型所估計的參數變多時,要達到0.9的標準就會有困難,建議可酌量放寬到0.8之標準

Bentler, P. M. (1983). Comfirmatory factor analysis via noniterative estimation: A fast, inexpensive method. Journal of Marketing Research, 19, 417-424.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1-55.

黃芳銘(2007)。結構方程模式理論與應用(五版)。台北:五南。

Bollen, K. A. (1990). Overall fit in covariance structure models: Two types of sample size effects. Psychological Bulletin, 107, 256-259.

Doll, W. J., Xia, W., & Torkzadeh, G. (1994). A Confirmatory Factor Analysis of the End-User Computing Satisfaction Instrument. MIS Quarterly, 12(2), 259-274.

 

三、調整之配適度指標(adjusted goodness of fit index, AGFI

在計算GFI時,將自由度納入考慮之後所設計出來的模型配適度指數,當參數越多時,AGFI指數數值將會越大,越有利得到理想之配適度,當模式處於恰好辨識時,AGFI值可能會超過1。通常採用AGFI值大於0.9為適配度門檻(Bentler, 1983; Hu & Bentler, 1999; 黃芳銘,2007),表示有良好的配適度。但模型一旦估計的參數變多,有時要達到0.9就會有困難,而Bollen1990)、Hu and Bentler1995)也提當,樣本數小的時候AGFI會低估,因此MacCallum and Hong1997)建議可酌量放寬到0.8

Bentler, P. M. (1983). Comfirmatory factor analysis via noniterative estimation: A fast, inexpensive method. Journal of Marketing Research, 19, 417-424.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1-55.

黃芳銘(2007)。結構方程模式理論與應用(五版)。台北:五南。

Bollen, K. A. (1990). Overall fit in covariance structure models: Two types of sample size effects. Psychological Bulletin, 107, 256-259.

Hu, L., & Bentler, P. M. (1995). Evaluating model fit. In R. H. Hoyle (Ed.), Structural equation modeling: Concepts, issues and applications (pp. 76-99). Thousand Oaks, CA: Sage.

MacCallum, R. C., & Hong, S. (1997). Power analysis in covariance structure modeling using GFI and AGFI. Multivariate Behavioral Research, 32, 193-210.

 

四、標準化均方根殘差值(standardized root mean square residual, SRMR

RMR只有下限為0,沒有上限的標準。RMR愈接近0則配適度愈好,一般建議RMR<0.05表示模型配適度佳。但由於RMR沒有上限,是一非標準化的值,即使高於一般認定的門檻,也不必然代表模型不佳。由於RMR較難以解釋,SRMR被建議用來取代RMRSRMR愈小,表示模型配適度愈好。SRMR=0表示完美配適,小於0.05一般稱為良好配適(Jöreskog & Sörbom, 1989),小於0.08一般稱為可接受配適(邱皓政,2011;張偉豪,2011),不過也有學者認為數值低於0.08就算是模式配適度佳(Hu & Bentler, 1999SRMR也會受到樣本數影響,樣本數愈大或估計的參數愈多,SRMR愈小。

Jöreskog, K. G., & Sörbom, D. (1989). LISREL 7: A guide to the program and applications. Chicago: SPSS Inc.

邱皓政(2011)。結構方程模式:LISREL的理論、技術與應用(二版)。臺北市:雙葉書廊。

張偉豪(2011)。論文寫作-SEM不求人。台北:鼎茂。

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1-55.

 

五、近似均方根誤差(root mean square error of approximation, RMSEA

RMSEA也是一種缺適度指標,值越大表示假設模型與資料愈不配適,是近年來相當重視的一個模式配適指標,且許多研究顯示此指標的表現比許多其他指標更為理想(Browne & Arminger, 1995; Browne & Cudeck, 1993; Marsh & Balla, 1994; Steiger, 1990; Sugawara & MaCallum, 1983)。假如RMSEA小於0.05,表示有好的模型配適(Browen & Mels, 1990; McDonald & Ho, 2002; Schumacker & Lomax, 2004; Steiger, 1989),Hu and Bentler1999)建議RMSEA要小於等於0.06,如果介於0.05~0.08之間,則稱模型有不錯的配適度(fair fit)(McDonald & Ho, 2002; 黃芳銘,2007),若指標超過0.10則表示模型相當不理想(Browne & Cudeck, 1993)。RMSEA雖較不受樣本數的影響,但在很小的樣本時,RMSEA會被高估(Fan, Thompson, & Wang, 1999)。

Browne, M. W. & Arminger, G. (1995). Specification and estimation of mean- and covariance-structure models. In G. Arminger, C. C. Clogg, & M. E. Sobel (Eds.) , Handbook of statistical modeling for the social and behavioral sciences (pp.185-249). New York: Plenum Press.

Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136-162). Newbury Park, CA: Sage.

Marsh, H. W., & Balla, J. R. (1994). Goodness of fit in confirmatory factor analysis: The effect of sample size and model parsimony. Quality & Quality, 28, 185-217.

Steiger, J. H. (1990). Structural model evaluation and modification: An interval estimation approach. Multivariate Behavioral Research, 25, 173-180.

Sugawara, H. M., & MaCallum, R. C. (1983). Effect of estimation method on incremental fit indexes for covariance structure models. Applied Psychological Measurement, 17, 365-377.

Browne, M. W. & Mels, G. (1990). RAMONA user’s guide. Columbus: Department of Psychology, Ohio State University.

McDonald, R. P., & Ho, M. R. (2002). Principles and practice in reporting structural equation analysis. Psychological methods, 7, 64-82.

Schumacker, R. E., & Lomax, R. G. (2004). A beginner’s guide to structural equation modeling (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.

Steiger, J. H. (1989). EZPATH: A supplementary module for SYSTAT and SYSGRAPH. Evanston, IL: SYSTAT.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1-55.

黃芳銘(2007)。結構方程模式理論與應用(五版)。台北:五南。

Fan, X., Thompson, B., & Wang, L. (1999). Effects of sample size, estimation method, and model specification on structural equation modeling fit indexes. Structural Equation Modeling, 6, 56-83.

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