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
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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.
