Only Q8 is shown in this paper. Other Q & A are now being translated into English.

Transcription

1 (c)ryo SASAKI, 8 Version. (Aug 8,8) Structural Equation Modeling (SEM) by AMOS (Last update:aug 8, 8) Q1 F-value Q Standardization Q3 Sample size Q Model fit Q5 Size of coefficients Q Equivalent model Q7 References Q8 Internal calculation in SEM Q9 Definition of SEM Q1 Report writing ( Only Q8 is shown in this paper. Other Q & A are now being translated into..

2 Q8 Internal Calculation in SEM 1

3 Q&A AMOS/LISREL で構造方程式構造方程式モデリング 1-8 What calculation is conducted inside SEM? First of all, SEM calculates (i) the fluctuation of the actual data and (ii) the fluctuation of the expected data using the obtained equations. Second, SEM tries the difference between (i) and (ii) the smallest. The equations we see in the output is the equations at the moment when the difference reaches the smallest.* * Formal name of fluctuation in the above sentences is variance and covariance. (Example) We have four observation variables.(,,,. (1) We calculate the fluctuation (= variance) of each variable.

4 We calculate the fluctuation between each variable ( = covariance). Since we have four variables, covariances are six. We make the summary table of variances and covariances. Also we show the graph. Actual Variance & Covariance

5 () By AMOS, five equations are calculated. Teaching Ability Student Satisfaction Also, the estimated data are calculated according to two factors ( Teaching Ability and Student Satisfaction ). We entered those data into the obtained equations. By doing so, we can get the table of Expected Variance & Covariance as follows. Expected Variance & Covariance

6 If we subtract the estimated table from the actual table, we can get the difference table as follows. Actual Variance & Covariance Expected Variance & Covariance Differences between "actual" and "expected" = We calculate the sum of squared differences. It is called as residual statistics ( Q ). AMOS will continue calculation until Q reaches the smallest. The equations we see in the AMOS output are the equations at the moment when the Q reaches the smallest. Also the estimated data of factors ( Teaching Ability and Student Satisfaction ) are the data at the moment when the Q reaches the smallest. Sum of squared difference Q This approach is called as the OLS methods (Ordinary Least Square method). 5

7 One caution is that the actual (default) method that AMOS (as well as LISREL) employ is the ML method (Maximum Likelihood method) instead of the OLS (Ordinary Least Square method).the ML method has some good features (e.g., The results of ML can be easily converted to several fit statistics such as Chi-square test) and thus, this method is accepted as the standard in SEM. However, the basic thought and calculation procedures do not change with the OLS.. (Reference)Wakui, R. & Wakui, S.(3)pp Kaplan,D. (). Pp.5-8. Wakui, R & Wakui, S.(3). Covariance Structure Analysis by Figures. Nipponjitsugyo Shuppansya Kaplan. ED (). Structural Equation Modeling: Foundation and Extensions. Thouand Oaks, CA: Sage Publication.

8 ( 参考 3) Calculation of Residual Statistics (Q) Actual Variance & Covariance Expected Variance & Covariance Differences between "actual" and "expected" = =>Finally, a sum of squared residual Q is calculated. 7

9 (Reference) Maximum Likelihood Method The ML method employs the following formula. f ML = tr( Σ S) logσ 1 1 S k Σ -----Actual Variance-Covariance Matrix S -----Expected Variance-Covariance Matrix k -----Number of observation variables tr Trace (the sum of diagonal elements) Sum of Matrix equation ( 行列式 ) Note1: Since the computer conducts the actual calculation, it is very rare (almost never) that an analyst should calculate this calculation by hand Note : If you want to see an example of actual hand calculation of ML methods, please access the following file.(excel file, some written in Japanese) 8