One of the beautiful things about Mplus is that there are only three rudimentary model commands. One of these is “WITH” which asks Mplus to correlate/covariate variables that fall on either side of it.
Here is an generic syntax applying the WITH model command:
TITLE: Simple correlation analysis; DATA: File is FILENAME.dat; VARIABLE: Names are VARx VARy; Missing are all(-999); Usevariables = VARx VARy; MODEL: VARx with VARy; OUTPUT: Standardized Sampstat;
Visually the above is asking, what is the relationship between VARx and VARy (i.e., no causation is inferred):
Imagine you have a bunch of variables you want to correlate, how would you write the syntax so that you can create a correlation matrix? Below is an applied example using real data to answer this question.
Here we are looking at the correlations between political knowledge (i.e., an employee’s collection of strategic and potentially sensitive information about his or her supervisor), political will (i.e., an individual’s motivation to engage in political behaviour), political skill (i.e., an individual’s interpersonal effectiveness), and change-oriented organizational citizenship behaviour (i.e., an individual’s extra-role behaviour enacted to bring around change in the workplace).
The above syntax produces the output below. There are actually two places where standardized correlations are provided because I also asked for the sample statistics (sampstat) under the output command: one under SAMPLE STATISTICS and one under STANDARDIZED MODEL RESULTS (see highlighted areas):
Mplus VERSION 7.4 (Mac) MUTHEN & MUTHEN 04/29/2017 12:32 AM INPUT INSTRUCTIONS TITLE: Simple Correlation Analysis; DATA: File is PK4correlations.dat; VARIABLE: Names are PK PW PS PT CHOCB LMX; Missing are all(-999); Usevariables = PK PW PS CHOCB; ANALYSIS: Estimator = ML; MODEL: PK PW PS CHOCB with PK PW PS CHOCB; OUTPUT: Standardized sampstat; *** WARNING Data set contains cases with missing on all variables. These cases were not included in the analysis. Number of cases with missing on all variables: 1 1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS Simple Correlation Analysis; SUMMARY OF ANALYSIS Number of groups 1 Number of observations 494 Number of dependent variables 4 Number of independent variables 0 Number of continuous latent variables 0 Observed dependent variables Continuous PK PW PS CHOCB Estimator ML Information matrix OBSERVED Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Maximum number of iterations for H1 2000 Convergence criterion for H1 0.100D-03 Input data file(s) PK4correlations.dat Input data format FREE SUMMARY OF DATA Number of missing data patterns 3 COVARIANCE COVERAGE OF DATA Minimum covariance coverage value 0.100 PROPORTION OF DATA PRESENT Covariance Coverage PK PW PS CHOCB ________ ________ ________ ________ PK 0.998 PW 0.996 0.996 PS 0.996 0.996 0.996 CHOCB 0.998 0.996 0.996 1.000 SAMPLE STATISTICS ESTIMATED SAMPLE STATISTICS Means PK PW PS CHOCB ________ ________ ________ ________ 1 3.459 4.130 5.100 3.642 Covariances PK PW PS CHOCB ________ ________ ________ ________ PK 0.547 PW 0.215 1.640 PS 0.360 0.347 1.042 CHOCB 0.238 0.232 0.384 0.586 Correlations PK PW PS CHOCB ________ ________ ________ ________ PK 1.000 PW 0.227 1.000 PS 0.476 0.265 1.000 CHOCB 0.421 0.237 0.492 1.000 MAXIMUM LOG-LIKELIHOOD VALUE FOR THE UNRESTRICTED (H1) MODEL IS -2477.074 UNIVARIATE SAMPLE STATISTICS UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS Variable/ Mean/ Skewness/ Minimum/ % with Percentiles Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median PK 3.458 -0.433 1.040 0.20% 2.870 3.350 3.520 493.000 0.547 0.200 5.000 1.01% 3.700 4.090 PW 4.130 -0.403 1.000 2.24% 3.130 3.880 4.250 492.000 1.640 -0.338 7.000 0.20% 4.500 5.250 PS 5.100 -0.581 1.220 0.20% 4.280 4.940 5.220 492.000 1.043 0.542 7.000 2.03% 5.440 5.940 CHOCB 3.642 -0.583 1.000 0.40% 3.000 3.500 3.750 494.000 0.586 0.441 5.000 5.47% 4.000 4.250 THE MODEL ESTIMATION TERMINATED NORMALLY MODEL FIT INFORMATION Number of Free Parameters 14 Loglikelihood H0 Value -2477.074 H1 Value -2477.074 Information Criteria Akaike (AIC) 4982.147 Bayesian (BIC) 5040.983 Sample-Size Adjusted BIC 4996.546 (n* = (n + 2) / 24) Chi-Square Test of Model Fit Value 0.000 Degrees of Freedom 0 P-Value 0.0000 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <= .05 0.000 CFI/TLI CFI 1.000 TLI 1.000 Chi-Square Test of Model Fit for the Baseline Model Value 341.302 Degrees of Freedom 6 P-Value 0.0000 SRMR (Standardized Root Mean Square Residual) Value 0.000 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value PK WITH PW 0.215 0.044 4.910 0.000 PS 0.360 0.038 9.548 0.000 CHOCB 0.238 0.028 8.612 0.000 PW WITH PS 0.347 0.061 5.687 0.000 CHOCB 0.232 0.045 5.111 0.000 PS WITH CHOCB 0.384 0.039 9.796 0.000 Means PK 3.459 0.033 103.886 0.000 PW 4.130 0.058 71.548 0.000 PS 5.100 0.046 110.878 0.000 CHOCB 3.642 0.034 105.760 0.000 Variances PK 0.547 0.035 15.700 0.000 PW 1.640 0.105 15.686 0.000 PS 1.042 0.066 15.692 0.000 CHOCB 0.586 0.037 15.716 0.000 STANDARDIZED MODEL RESULTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value PK WITH PW 0.227 0.043 5.307 0.000 PS 0.476 0.035 13.677 0.000 CHOCB 0.421 0.037 11.350 0.000 PW WITH PS 0.265 0.042 6.328 0.000 CHOCB 0.237 0.043 5.563 0.000 PS WITH CHOCB 0.492 0.034 14.385 0.000 Means PK 4.678 0.156 30.066 0.000 PW 3.225 0.112 28.738 0.000 PS 4.996 0.165 30.206 0.000 CHOCB 4.758 0.158 30.130 0.000 Variances PK 1.000 0.000 999.000 999.000 PW 1.000 0.000 999.000 999.000 PS 1.000 0.000 999.000 999.000 CHOCB 1.000 0.000 999.000 999.000 STDY Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value PK WITH PW 0.227 0.043 5.307 0.000 PS 0.476 0.035 13.677 0.000 CHOCB 0.421 0.037 11.350 0.000 PW WITH PS 0.265 0.042 6.328 0.000 CHOCB 0.237 0.043 5.563 0.000 PS WITH CHOCB 0.492 0.034 14.385 0.000 Means PK 4.678 0.156 30.066 0.000 PW 3.225 0.112 28.738 0.000 PS 4.996 0.165 30.206 0.000 CHOCB 4.758 0.158 30.130 0.000 Variances PK 1.000 0.000 999.000 999.000 PW 1.000 0.000 999.000 999.000 PS 1.000 0.000 999.000 999.000 CHOCB 1.000 0.000 999.000 999.000 STD Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value PK WITH PW 0.215 0.044 4.910 0.000 PS 0.360 0.038 9.548 0.000 CHOCB 0.238 0.028 8.612 0.000 PW WITH PS 0.347 0.061 5.687 0.000 CHOCB 0.232 0.045 5.111 0.000 PS WITH CHOCB 0.384 0.039 9.796 0.000 Means PK 3.459 0.033 103.886 0.000 PW 4.130 0.058 71.548 0.000 PS 5.100 0.046 110.878 0.000 CHOCB 3.642 0.034 105.760 0.000 Variances PK 0.547 0.035 15.700 0.000 PW 1.640 0.105 15.686 0.000 PS 1.042 0.066 15.692 0.000 CHOCB 0.586 0.037 15.716 0.000 R-SQUARE QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.130E-01 (ratio of smallest to largest eigenvalue) Beginning Time: 00:32:46 Ending Time: 00:32:46 Elapsed Time: 00:00:00 MUTHEN & MUTHEN 3463 Stoner Ave. Los Angeles, CA 90066 Tel: (310) 391-9971 Fax: (310) 391-8971 Web: www.StatModel.com Support: Support@StatModel.com Copyright (c) 1998-2015 Muthen & Muthen
We can conclude that all of the variables are correlated significantly (ps < .001) but that there are stronger correlations between political knowledge, political skill, and change-oriented organizational citizenship behaviour. So individuals who have a deep understanding of their supervisor are also more socially astute and also try to bring around more change in the workplace. However, as any lesson on correlation goes, causation cannot be inferred! All we can tell from this analysis is that these variables go hand-in-hand in the same direction (i.e., as one goes up, so does the other and vice versa).
Finally, you can take the correlations in the output and create a beautiful table:
Okay, maybe not beautiful, but informative at least! And that’s about sums up basic correlation analysis.
Steve Granger 