Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
09/08/2025 7:02 AM
INPUT INSTRUCTIONS
TITLE: DCSM, MSQ sum scores from EPESE
Input from LDSM.EXE
y1, y2, and y3 are MSQTOT at wave 1, 4, and 7
DATA: FILE = ex0201.dat;
VARIABLE: NAMES = y1 y2 y3 ; ! msqtot1 msqtot4 msqtot7 age black ;
MISSING = ALL (-9999) ;
OUTPUT: TECH1 ;
TECH4 ;
SAVEDATA: SAVE = FSCORES ;
FILE = ex0399.dat ;
MODEL:
!For the variable y
!Observed variables and latent level
ly1 by y1 @1;
ly2 by y2 @1;
ly3 by y3 @1;
!Autoregressive part
ly2 on ly1 @1;
ly3 on ly2 @1;
!Difference score on latent level
dy1 by ly2 @1;
dy2 by ly3 @1;
!Auto-proportion of difference score on level
dy1 on ly1 * (1);
dy2 on ly2 * (1);
!Model relationship between slope and ds
sy by dy1 @1;
sy by dy2 @1;
y0 by ly1 @1;
! Set the means and variance to be 0
[y1@0]; [ly1@0]; [dy1@0]; ly1@0; dy1@0;
[y2@0]; [ly2@0]; [dy2@0]; ly2@0; dy2@0;
[y3@0]; [ly3@0];ly3@0;
!Estimate means and variances for intercept and slope
[y0 sy];
y0 sy;
!Set all residuals to be equal
y1* (2);
y2* (2);
y3* (2);
*** 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: 397
1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS
DCSM, MSQ sum scores from EPESE
Input from LDSM.EXE
y1, y2, and y3 are MSQTOT at wave 1, 4, and 7
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 14059
Number of dependent variables 3
Number of independent variables 0
Number of continuous latent variables 7
Observed dependent variables
Continuous
Y1 Y2 Y3
Continuous latent variables
LY1 LY2 LY3 DY1 DY2 SY
Y0
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)
ex0201.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 7
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y1 Y2 Y3
________ ________ ________
Y1 0.974
Y2 0.752 0.775
Y3 0.565 0.561 0.581
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
Y1 4.768 -1.152 0.000 0.70% 4.000 5.000 5.000
13698.000 1.621 1.156 6.000 34.84% 5.000 6.000
Y2 4.732 -1.233 0.000 1.17% 4.000 5.000 5.000
10899.000 1.822 1.268 6.000 35.05% 5.000 6.000
Y3 4.569 -1.160 0.000 2.00% 4.000 5.000 5.000
8169.000 2.033 1.060 6.000 30.31% 5.000 6.000
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 7
Loglikelihood
H0 Value -52686.644
H1 Value -52661.224
Information Criteria
Akaike (AIC) 105387.289
Bayesian (BIC) 105440.146
Sample-Size Adjusted BIC 105417.901
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 50.840
Degrees of Freedom 2
P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.042
90 Percent C.I. 0.032 0.052
Probability RMSEA <= .05 0.906
CFI/TLI
CFI 0.993
TLI 0.989
Chi-Square Test of Model Fit for the Baseline Model
Value 6609.379
Degrees of Freedom 3
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.034
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
LY1 BY
Y1 1.000 0.000 999.000 999.000
LY2 BY
Y2 1.000 0.000 999.000 999.000
LY3 BY
Y3 1.000 0.000 999.000 999.000
DY1 BY
LY2 1.000 0.000 999.000 999.000
DY2 BY
LY3 1.000 0.000 999.000 999.000
SY BY
DY1 1.000 0.000 999.000 999.000
DY2 1.000 0.000 999.000 999.000
Y0 BY
LY1 1.000 0.000 999.000 999.000
LY2 ON
LY1 1.000 0.000 999.000 999.000
LY3 ON
LY2 1.000 0.000 999.000 999.000
DY1 ON
LY1 0.617 0.144 4.274 0.000
DY2 ON
LY2 0.617 0.144 4.274 0.000
Y0 WITH
SY -0.497 0.134 -3.716 0.000
Means
SY -3.103 0.682 -4.552 0.000
Y0 4.782 0.011 448.388 0.000
Intercepts
Y1 0.000 0.000 999.000 999.000
Y2 0.000 0.000 999.000 999.000
Y3 0.000 0.000 999.000 999.000
LY1 0.000 0.000 999.000 999.000
LY2 0.000 0.000 999.000 999.000
LY3 0.000 0.000 999.000 999.000
DY1 0.000 0.000 999.000 999.000
DY2 0.000 0.000 999.000 999.000
Variances
SY 0.328 0.139 2.353 0.019
Y0 0.902 0.021 43.107 0.000
Residual Variances
Y1 0.759 0.011 66.171 0.000
Y2 0.759 0.011 66.171 0.000
Y3 0.759 0.011 66.171 0.000
LY1 0.000 0.000 999.000 999.000
LY2 0.000 0.000 999.000 999.000
LY3 0.000 0.000 999.000 999.000
DY1 0.000 0.000 999.000 999.000
DY2 0.000 0.000 999.000 999.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.370E-05
(ratio of smallest to largest eigenvalue)
TECHNICAL 1 OUTPUT
PARAMETER SPECIFICATION
NU
Y1 Y2 Y3
________ ________ ________
0 0 0
LAMBDA
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
Y1 0 0 0 0 0
Y2 0 0 0 0 0
Y3 0 0 0 0 0
LAMBDA
SY Y0
________ ________
Y1 0 0
Y2 0 0
Y3 0 0
THETA
Y1 Y2 Y3
________ ________ ________
Y1 1
Y2 0 1
Y3 0 0 1
ALPHA
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
0 0 0 0 0
ALPHA
SY Y0
________ ________
2 3
BETA
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0 0 0 0 0
LY2 0 0 0 0 0
LY3 0 0 0 0 0
DY1 4 0 0 0 0
DY2 0 4 0 0 0
SY 0 0 0 0 0
Y0 0 0 0 0 0
BETA
SY Y0
________ ________
LY1 0 0
LY2 0 0
LY3 0 0
DY1 0 0
DY2 0 0
SY 0 0
Y0 0 0
PSI
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0
LY2 0 0
LY3 0 0 0
DY1 0 0 0 0
DY2 0 0 0 0 0
SY 0 0 0 0 0
Y0 0 0 0 0 0
PSI
SY Y0
________ ________
SY 5
Y0 6 7
STARTING VALUES
NU
Y1 Y2 Y3
________ ________ ________
0.000 0.000 0.000
LAMBDA
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
Y1 1.000 0.000 0.000 0.000 0.000
Y2 0.000 1.000 0.000 0.000 0.000
Y3 0.000 0.000 1.000 0.000 0.000
LAMBDA
SY Y0
________ ________
Y1 0.000 0.000
Y2 0.000 0.000
Y3 0.000 0.000
THETA
Y1 Y2 Y3
________ ________ ________
Y1 0.810
Y2 0.000 0.911
Y3 0.000 0.000 1.016
ALPHA
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
ALPHA
SY Y0
________ ________
0.000 0.000
BETA
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0.000 0.000 0.000 0.000 0.000
LY2 1.000 0.000 0.000 1.000 0.000
LY3 0.000 1.000 0.000 0.000 1.000
DY1 0.000 0.000 0.000 0.000 0.000
DY2 0.000 0.000 0.000 0.000 0.000
SY 0.000 0.000 0.000 0.000 0.000
Y0 0.000 0.000 0.000 0.000 0.000
BETA
SY Y0
________ ________
LY1 0.000 1.000
LY2 0.000 0.000
LY3 0.000 0.000
DY1 1.000 0.000
DY2 1.000 0.000
SY 0.000 0.000
Y0 0.000 0.000
PSI
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0.000
LY2 0.000 0.000
LY3 0.000 0.000 0.000
DY1 0.000 0.000 0.000 0.000
DY2 0.000 0.000 0.000 0.000 0.000
SY 0.000 0.000 0.000 0.000 0.000
Y0 0.000 0.000 0.000 0.000 0.000
PSI
SY Y0
________ ________
SY 0.050
Y0 0.000 0.050
TECHNICAL 4 OUTPUT
ESTIMATES DERIVED FROM THE MODEL
ESTIMATED MEANS FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
4.782 4.631 4.386 -0.151 -0.245
ESTIMATED MEANS FOR THE LATENT VARIABLES
SY Y0
________ ________
-3.103 4.782
S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
0.011 0.012 0.016 0.010 0.013
S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
SY Y0
________ ________
0.682 0.011
EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
448.388 394.210 275.395 -15.793 -18.578
EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
SY Y0
________ ________
-4.552 448.388
TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
SY Y0
________ ________
0.000 0.000
ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0.902
LY2 0.962 1.080
LY3 1.059 1.271 1.615
DY1 0.060 0.118 0.213 0.058
DY2 0.097 0.191 0.344 0.094 0.153
SY -0.497 -0.475 -0.441 0.021 0.035
Y0 0.902 0.962 1.059 0.060 0.097
ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 0.328
Y0 -0.497 0.902
S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0.021
LY2 0.018 0.022
LY3 0.023 0.027 0.041
DY1 0.009 0.012 0.018 0.010
DY2 0.016 0.012 0.022 0.010 0.017
SY 0.134 0.148 0.171 0.018 0.027
Y0 0.021 0.018 0.023 0.009 0.016
S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 0.139
Y0 0.134 0.021
EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 43.107
LY2 53.199 50.236
LY3 46.331 47.044 39.626
DY1 6.657 10.152 11.839 6.089
DY2 6.153 15.811 15.494 9.333 9.154
SY -3.716 -3.203 -2.583 1.166 1.275
Y0 43.107 53.199 46.331 6.657 6.153
EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 2.353
Y0 -3.716 43.107
TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0.000
LY2 0.000 0.000
LY3 0.000 0.000 0.000
DY1 0.000 0.000 0.000 0.000
DY2 0.000 0.000 0.000 0.000 0.000
SY 0.000 0.001 0.010 0.244 0.202
Y0 0.000 0.000 0.000 0.000 0.000
TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 0.019
Y0 0.000 0.000
ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 1.000
LY2 0.974 1.000
LY3 0.877 0.963 1.000
DY1 0.261 0.471 0.693 1.000
DY2 0.261 0.471 0.693 1.000 1.000
SY -0.913 -0.799 -0.605 0.155 0.155
Y0 1.000 0.974 0.877 0.261 0.261
ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 1.000
Y0 -0.913 1.000
S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0.000
LY2 0.005 0.000
LY3 0.015 0.005 0.000
DY1 0.050 0.035 0.019 0.000
DY2 0.050 0.035 0.019 0.000 0.000
SY 0.051 0.086 0.112 0.147 0.147
Y0 0.000 0.005 0.015 0.050 0.050
S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 0.000
Y0 0.051 0.000
EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 999.000
LY2 214.409 999.000
LY3 57.189 189.614 999.000
DY1 5.209 13.417 36.841 999.000
DY2 5.209 13.417 36.841 *********** 999.000
SY -17.849 -9.308 -5.423 1.052 1.052
Y0 *********** 214.409 57.189 5.209 5.209
EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 999.000
Y0 -17.849 999.000
TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
LY1 LY2 LY3 DY1 DY2
________ ________ ________ ________ ________
LY1 0.000
LY2 0.000 0.000
LY3 0.000 0.000 0.000
DY1 0.000 0.000 0.000 0.000
DY2 0.000 0.000 0.000 0.000 0.000
SY 0.000 0.000 0.000 0.293 0.293
Y0 0.000 0.000 0.000 0.000 0.000
TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
SY Y0
________ ________
SY 0.000
Y0 0.000 0.000
SAMPLE STATISTICS FOR ESTIMATED FACTOR SCORES
SAMPLE STATISTICS
Means
LY1 LY1_SE LY2 LY2_SE LY3
________ ________ ________ ________ ________
4.782 0.513 4.631 0.530 4.386
Means
LY3_SE DY1 DY1_SE DY2 DY2_SE
________ ________ ________ ________ ________
0.704 -0.151 0.210 -0.245 0.340
Means
SY SY_SE Y0 Y0_SE
________ ________ ________ ________
-3.103 0.398 4.782 0.513
Covariances
LY1 LY1_SE LY2 LY2_SE LY3
________ ________ ________ ________ ________
LY1 0.634
LY1_SE -0.011 0.005
LY2 0.704 -0.012 0.788
LY2_SE -0.019 0.008 -0.021 0.012
LY3 0.817 -0.014 0.923 -0.023 1.095
LY3_SE -0.028 0.011 -0.031 0.017 -0.035
DY1 0.070 -0.001 0.084 -0.002 0.106
DY1_SE -0.003 0.001 -0.004 0.002 -0.004
DY2 0.113 -0.002 0.135 -0.003 0.172
DY2_SE -0.005 0.002 -0.006 0.003 -0.007
SY -0.321 0.006 -0.351 0.010 -0.398
SY_SE -0.002 0.001 -0.002 0.001 -0.002
Y0 0.634 -0.011 0.704 -0.019 0.817
Y0_SE -0.011 0.005 -0.012 0.008 -0.014
Covariances
LY3_SE DY1 DY1_SE DY2 DY2_SE
________ ________ ________ ________ ________
LY3_SE 0.024
DY1 -0.002 0.014
DY1_SE 0.003 0.000 0.000
DY2 -0.004 0.022 -0.001 0.036
DY2_SE 0.004 -0.001 0.001 -0.001 0.001
SY 0.015 -0.029 0.002 -0.047 0.003
SY_SE 0.002 0.000 0.000 0.000 0.000
Y0 -0.028 0.070 -0.003 0.113 -0.005
Y0_SE 0.011 -0.001 0.001 -0.002 0.002
Covariances
SY SY_SE Y0 Y0_SE
________ ________ ________ ________
SY 0.169
SY_SE 0.001 0.000
Y0 -0.321 -0.002 0.634
Y0_SE 0.006 0.001 -0.011 0.005
Correlations
LY1 LY1_SE LY2 LY2_SE LY3
________ ________ ________ ________ ________
LY1 1.000
LY1_SE -0.202 1.000
LY2 0.996 -0.196 1.000
LY2_SE -0.218 0.987 -0.212 1.000
LY3 0.981 -0.187 0.994 -0.202 1.000
LY3_SE -0.226 0.958 -0.220 0.992 -0.211
DY1 0.746 -0.111 0.802 -0.125 0.862
DY1_SE -0.233 0.884 -0.228 0.946 -0.220
DY2 0.746 -0.111 0.802 -0.125 0.862
DY2_SE -0.233 0.884 -0.228 0.946 -0.220
SY -0.982 0.210 -0.961 0.224 -0.925
SY_SE -0.141 0.888 -0.135 0.808 -0.127
Y0 1.000 -0.202 0.996 -0.218 0.981
Y0_SE -0.202 1.000 -0.196 0.987 -0.187
Correlations
LY3_SE DY1 DY1_SE DY2 DY2_SE
________ ________ ________ ________ ________
LY3_SE 1.000
DY1 -0.134 1.000
DY1_SE 0.978 -0.145 1.000
DY2 -0.134 1.000 -0.145 1.000
DY2_SE 0.978 -0.145 1.000 -0.145 1.000
SY 0.232 -0.605 0.236 -0.605 0.236
SY_SE 0.731 -0.069 0.612 -0.069 0.612
Y0 -0.226 0.746 -0.233 0.746 -0.233
Y0_SE 0.958 -0.111 0.884 -0.111 0.884
Correlations
SY SY_SE Y0 Y0_SE
________ ________ ________ ________
SY 1.000
SY_SE 0.148 1.000
Y0 -0.982 -0.141 1.000
Y0_SE 0.210 0.888 -0.202 1.000
SAVEDATA INFORMATION
Save file
ex0399.dat
Order and format of variables
Y1 F10.3
Y2 F10.3
Y3 F10.3
LY1 F10.3
LY1_SE F10.3
LY2 F10.3
LY2_SE F10.3
LY3 F10.3
LY3_SE F10.3
DY1 F10.3
DY1_SE F10.3
DY2 F10.3
DY2_SE F10.3
SY F10.3
SY_SE F10.3
Y0 F10.3
Y0_SE F10.3
Save file format
17F10.3
Save file record length 10000
Save missing symbol *
Beginning Time: 07:02:05
Ending Time: 07:02:05
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) 1998-2024 Muthen & MuthenDual Change Score Model
Rich Jones
2025-09-04
Where to learn more
McArdle, J. (2009). Latent variable modeling of differences and changes with longitudinal data. Annual Review of Psychology, 60, 577-605.
Serang, S., Grimm, K. J., & Zhang, Z. (2019). On the correspondence between the latent growth curve and latent change score models. Structural Equation Modeling: A Multidisciplinary Journal, 26(4), 623-635.
Dual Change Score Model
Shape of the curve driven by \(\beta\)
| \(\beta\) | Equilibrium | Implication | Typical trajectory |
|---|---|---|---|
| \(\beta>0\) | Exists but unstable | Proportional change, exponential growth or decay away from a equilibrium point (\(-s/b\)) | Moves away from equilibrium, monotone, no oscillation |
| \(\beta=0\) | None (unless \(s=0\)) | Constant change, linear growth or decay | Straight line |
| \(-1<\beta<0\) | Stable \(y^* = -s/b\) | Proportional change, exponential monotone decay toward an equilibrium point | Monotone decay toward equilibrium |
| \(-2<\beta<-1\) | Stable | Converges to equilibrium, oscillates around (\(-s/b\)) | Damped oscillation toward equilibrium |
| \(\beta<-2\) | Unstable | Diverges with oscillation | Growing flips around equilibrium |
Equilibrium value for y is \(y^* = -s/\beta\) and if \(y\) ever reached this value the change would be 0 and \(y\) would remain at this value. Related to an asymptote. Unstable equilibrium means that if \(y\) is perturbed away from this value it will move further away. Stable means that if \(y\) is perturbed away from this value it will move back toward it.
Baseline–Change Association
In a Dual Chnge Score Model (DCSM), the baseline-change association has a dual nature: a between-person component (\(\text{Cov}(y_0, s)\)) and a within-person component (\(\beta\)). The random effects covariance \(\text{Cov}(y_0, s)\) parameter captures the correlation between the baseline level (\(y_0\)) and constant change component (\(s\)). Although they are both random effects, this effect is at the “between-person” level in a multilevel framework. The covariance \(\text{Cov}(y_0, s)\) tells us whether individuals who start higher also tend to have faster/slower long-run slopes. This is the between-person piece, describing population heterogeneity.
The dynamic feedback parameter (\(\beta\)) captures the structural regression of a person’s current level on their own subsequent change. It is interpreted as a within-person dynamic: does being higher now predict more or less change next time? Unlike the LGCM, where such dynamics are absorbed into the intercept–slope covariance, the DCSM models them explicitly. The recursion multiplier means that even though \(\beta\) is fixed across individuals, the effect of \(\beta\) on change varies as a function of the individual’s prior level of y.
In contrast, unconditional linear growth curve model (LGCM), the correlation between baseline and change is captured in one model parameter: the covariance between intercept and slope (\(\text{Cov}(i,s)\)). This is a between-persons effect.
However in practice, the constant change component and the proportional change parameter in the DCSM are often highly correlated, sometimes nearly collinear. This means the dual nature of baseline–change correlation is not always empirically separable. More time points, smaller proportional change values, and constant change means near zero reduce the problem; few time points and extreme constant change means exacerbate it (Jacobucci R et al. Structural Equation Modeling: A Multidisciplinary Journal. 2019:26(6);924-930).
LDSM.EXE
LDSM.EXE is a Windows program that generates Mplus code for latent difference score models (LDSM), including dual change score models (DCSM).
!*********************************************!
!This M+ latent difference score model scripts!
!are generated by LDSM generator. !
!*********************************************!
MODEL:
!For the variable y
!Observed variables and latent level
ly1 by y1 @1;
ly2 by y2 @1;
ly3 by y3 @1;
!Autoregressive part
ly2 on ly1 @1;
ly3 on ly2 @1;
!Difference score on latent level
dy1 by ly2 @1;
dy2 by ly3 @1;
!Auto-proportion of difference score on level
dy1 on ly1 * (1);
dy2 on ly2 * (1);
!Model relationship between slope and ds
sy by dy1 @1;
sy by dy2 @1;
y0 by ly1 @1;
!Set the means and variance to be 0
[y1@0]; [ly1@0]; [dy1@0]; ly1@0; dy1@0;
[y2@0]; [ly2@0]; [dy2@0]; ly2@0; dy2@0;
[y3@0]; [ly3@0];ly3@0;
!Estimate means and variances for intercept and slope
[y0 sy];
y0 sy;
!Set all item-level residuals to be equal
y1* (2);
y2* (2);
y3* (2);
DCSM of MSQTOT in EPESE
- MSQTOT assessed at baseline, +3 years, and +6 years
- Scores on 0-6 scale, higher score is more correct answers
Scroll down to see the entire Mplus output
Using R to explore the output and results
- MplusAutomation package commands to extract information about the model and results
- Load the savedata file and plot the estimated latent trajectories
Load the data in the SAVEDATA file
I have a program to do this on my GitHub (rnj0nes/RNJmisc::runmplus_load_savedata.R)
This program will read the OUT file, find the name and contents of the SAVEDATA file, and read that file into R.
Also contents.r is a program to provide a summary of a data frame, and eme.r will be discussed in the Appendix.
library(dplyr)
library(tidyr)
library(ggplot2)
# RNJMisc programs
f1 <- "runmplus_load_savedata.R"
f2 <- "contents.r"
f3 <- "eme.r"
rnjmisc_url <- "https://raw.githubusercontent.com/rnj0nes/RNJmisc/master/"
devtools::source_url(paste0(rnjmisc_url, f1))
devtools::source_url(paste0(rnjmisc_url, f2))
devtools::source_url(paste0(rnjmisc_url, f3))contents of df (the SAVEDATA file)
| Variable | N | Mean | SD | Min | Max |
|---|---|---|---|---|---|
| y1 | 13698 | 4.768 | 1.273 | 0.000 | 6.000 |
| y2 | 10899 | 4.732 | 1.350 | 0.000 | 6.000 |
| y3 | 8169 | 4.569 | 1.426 | 0.000 | 6.000 |
| ly1 | 14059 | 4.782 | 0.796 | 1.474 | 5.765 |
| ly1_se | 14059 | 0.513 | 0.071 | 0.461 | 0.656 |
| ly2 | 14059 | 4.631 | 0.888 | 0.992 | 5.749 |
| ly2_se | 14059 | 0.529 | 0.109 | 0.445 | 0.723 |
| ly3 | 14059 | 4.386 | 1.046 | 0.213 | 5.723 |
| ly3_se | 14059 | 0.705 | 0.156 | 0.580 | 0.970 |
| dy1 | 14059 | -0.151 | 0.118 | -0.674 | 0.107 |
| dy1_se | 14059 | 0.210 | 0.018 | 0.195 | 0.237 |
| dy2 | 14059 | -0.245 | 0.190 | -1.090 | 0.173 |
| dy2_se | 14059 | 0.340 | 0.029 | 0.316 | 0.384 |
| sy | 14059 | -3.103 | 0.411 | -3.574 | -1.392 |
| sy_se | 14059 | 0.398 | 0.017 | 0.389 | 0.496 |
| y0 | 14059 | 4.782 | 0.796 | 1.474 | 5.765 |
| y0_se | 14059 | 0.513 | 0.071 | 0.461 | 0.656 |
y1-y3 are the observed MSQTOT scores at waves 1, 4, and 7. Everything else is a latent variable or factor score estimate, or a derivation of a factor score (the standard errors, XXX_se). Notice the sample sizes show missing data by death and attrition for the observed variables, but not for the latent variables (which are estimated for all persons).
The ly variables are the estimated true scores at each time point. Their _se variables are estimates of the standard errors of measurement for these factor scores, and are analogous to the standard deviations of the posterior distribution of the factor scores given the model and the data, where the factor score estimates are the means of these posterior distributions.
The dy variables are estimated latent difference scores, y0 is the estimated intercept, and sy is the estimated slope factor score.
Plots
We will make two plots. The first is a spaghetti plot of the estimated true scores (ly1, ly2, ly3) over time, with the mean curve overlaid. The second plot is a histogram of the person × occasion residuals (the differences between the observed scores and the estimated true scores).
Both of these will require data in “long” format, so we will reshape the data frame.
# Reshape wide to long
# df$ly1, ly2, and ly3 are the expected values
df_long <- df %>%
select(y1:y3, ly1, ly2, ly3) %>%
mutate(id = row_number()) %>% # step 1: row identifier
pivot_longer(
cols = c(y1:y3, ly1:ly3), # step 2: reshape
names_to = c(".value", "obs"), # split into 'y'/'ly' and 'obs'
names_pattern = "([a-z]+)([0-9]+)"
) %>%
arrange(id, obs)
head(df_long)With the reshaped data we can easily compute correlations and R-squared values.
In the spaghetti plot we will overlay the mean curve from the data (in red) and the expected mean curve implied by the estimated parameters of the model (in blue). We can compute the expected means from the estimated parameters in the Mplus output. This provides a check on my understanding of the model, as these two curves should overlap exactly. In this code I use model parameters copied from the Mplus output, but it would have been better to extract them using MplusAutomation functions. I will demonstrate that in an appendix.
# ensure obs is numeric (pivot_longer often makes it character)
df_long <- df_long %>% mutate(obs = as.integer(obs))
# Extract from Mplus output
mean_y0 <- 4.782 # mean of "y0" from Mplus output
beta <- 0.617 # beta from Mplus output (Dy on Ly)
mean_sy <- -3.103 # mean of "sy" from Mplus output
# Three different ways to compute the implied means all are equal
# implied_y1 <- mean_y0 # mean of "Y0" from Mplus output
# implied_y2 <- implied_y1 + beta*implied_y1 + mean_sy
# another way
# implied_y3 <- implied_y2 * (1 + beta) + mean_sy
# closed form showing proportional (~exponential) and linear change
implied_y1 <- (1 + beta)^(1-1) * (mean_y0 + mean_sy/beta) - (mean_sy/beta)
implied_y2 <- (1 + beta)^(2-1) * (mean_y0 + mean_sy/beta) - (mean_sy/beta)
implied_y3 <- (1 + beta)^(3-1) * (mean_y0 + mean_sy/beta) - (mean_sy/beta)
# expected means data
expected_df <- tibble::tibble(
obs = c(1L, 2L, 3L),
mu = c(implied_y1, implied_y2 , implied_y3)
)
# mean curve by obs
mean_curve <- df_long %>%
group_by(obs) %>%
summarize(ly_bar = mean(ly, na.rm = TRUE), .groups = "drop")Scroll down to see the entire code chunk
Code for the spaghetti plot using ggplot2.
spagplot <- ggplot(df_long, aes(x = obs, y = ly, group = id)) +
geom_line(alpha = 0.01, linewidth = 0.3) + # spaghetti
geom_point(alpha = 0.01, size = 0.6) + # optional points
# data mean curve (red)
geom_line(
data = mean_curve,
aes(x = obs, y = ly_bar, group = 1),
linewidth = 3.1, color = "red", inherit.aes = FALSE
) +
# expected mean curve (blue)
geom_line(
data = expected_df,
aes(x = obs, y = mu, group = 1),
linewidth = 1.1, color = "blue", inherit.aes = FALSE
) +
geom_point(
data = expected_df,
aes(x = obs, y = mu),
size = 0.6 , color = "blue", inherit.aes = FALSE
) +
scale_x_continuous(breaks = sort(unique(df_long$obs))) +
labs(x = "obs", y = "ly", title = "Spaghetti plot of ly by obs") +
theme_minimal()Scroll down to see the entire code chunk
Each line is one persons estimated trajectory of the “true score” (ly) over time. The thick red line is the mean of the estimated true scores at each time point. The blue line is the expected mean trajectory implied by the estimated parameters of the model. The fact that these overlap exactly provides a check on my understanding of the model. The observed and model-implied means are identical because there are no covariates and there is no missing data.
Code for the plot of person × occasion residuals.
Appendix: Extracting parameters from Mplus output
Here we show how to extract the estimated parameters from the Mplus output using MplusAutomation functions. This is better than copying and pasting numbers from the output, as it is less error-prone and can be automated.
Even better would be to run Mplus from R using MplusAutomation functions, and use the H5RESULTS to get parameter estimates, but that is for another time.
Assume we have already run the model and have the output file ex0399.out.
Let’s just look at the parameters that are really estimates, defined by having a standard error.
# let's at those entries in params with SE > 0
params |> dplyr::filter(se>0) |> head(n=sum(params$se > 0)) paramHeader param est se est_se pval
1 DY1.ON LY1 0.617 0.144 4.274 0.000
2 DY2.ON LY2 0.617 0.144 4.274 0.000
3 Y0.WITH SY -0.497 0.134 -3.716 0.000
4 Means SY -3.103 0.682 -4.552 0.000
5 Means Y0 4.782 0.011 448.388 0.000
6 Variances SY 0.328 0.139 2.353 0.019
7 Variances Y0 0.902 0.021 43.107 0.000
8 Residual.Variances Y1 0.759 0.011 66.171 0.000
9 Residual.Variances Y2 0.759 0.011 66.171 0.000
10 Residual.Variances Y3 0.759 0.011 66.171 0.000Pull the mean of y0 into an object in R:
That code is not hard but I do have a helper function to make it easier. This function is in my RNJmisc repo on GitHub, and it’s called eme.r.
(fin)
