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Annotated Mplus Output
Latent Growth Curve Model, Example 2

This page shows an example of a latent growth curve model (LGCM) with footnotes explaining the output. A LGCM can be similar to a multilevel model (a model many people have seen). To help you understand the LGCM and its output, first a multilevel model is shown using HLM and then using Stata, and then the same data are analyzed using Mplus using a LGCM. The Mplus output is related to the multilevel model results. We suggest that you view this page using two web browsers so you can show the page side by side showing the Stata output in one browser and the corresponding Mplus output in the other browser. 

This example is drawn from the Mplus User's Guide (example 6.10) and we suggest that you see the Mplus User's Guide for more details about this example. We thank the kind people at Muthén & Muthén for permission to use examples from their manual.

Example using HLM

Each subject is observed on the variable Y at four different times. A covariate called a is measured at each of the four time points. Also, a variable x1 and x2 are measured for each person. Conceptualized as a multilevel model, the variable time and a are level 1 variables. (Note that time is coded 0, 1, 2, and 3). The variables x1 and x2 are level two variables. The model uses time and a to predict the values of y at level 1, and uses x1 and x2 to predict the intercept and slope of time at level 2. We can write this model using multiple equations as shown below. This uses the ex610.mdm file.

Level-1 Model
	Y = B0 + B1*(A) + B2*(TIME) + R
Level-2 Model
	B0 = G00 + G01*(X1) + G02*(X2) + U0
	B1 = G10 
	B2 = G20 + G21*(X1) + G	B2 = G20 + G21*(X1) + G22*(X2) + U2

Here is the output from HLM, condensed to save space. Footnotes are included for relating the output to Mplus.

 Sigma_squared =      0.54200I

 Tau
 INTRCPT1,B0      1.08757F       0.05079 
     TIME,B2      0.05079H       0.20495G 

Tau (as correlations)
 INTRCPT1,B0  1.000  0.108
     TIME,B2  0.108  1.000

Final estimation of fixed effects:
----------------------------------------------------------------------------
                                      Standard             Approx.
   Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
For       INTRCPT1, B0
   INTRCPT2, G00           0.570413A   0.054807    10.408       497    0.000
         X1, G01           0.560548B   0.054574    10.271       497    0.000
         X2, G02           0.716557B   0.055865    12.827       497    0.000
For        A slope, B1
   INTRCPT2, G10           0.296872E   0.021381    13.885      1993    0.000
For     TIME slope, B2
   INTRCPT2, G20           1.010207C   0.025332    39.879       497    0.000
         X1, G21           0.263030D   0.025223    10.428       497    0.000
         X2, G22           0.473419D   0.025819    18.336       497    0.000
----------------------------------------------------------------------------

Example using Stata

Combining the two equations into one by substituting the level 2 equation into the level 1 equation, we have the equation below, with the random effects identified by placing them in square brackets.

Composite model
	Y = G00 + G01*(X1) + G02*(X2) + G10*A + G20*TIME + G21*X1*TIME + G22*X2*TIME + [ U0 + U2*TIME + r ]

Based on the composite model, this is the same example using Stata.  Please note that this is Stata 12 code.

infile y1-y4 x1 x2 a1-a4 using http://www.ats.ucla.edu/stat/mplus/output/ex6.10.dat
generate id = _n
reshape long y a, i(id) j(time)
generate t1 = time - 1
xtmixed y a c.t1##c.x1 c.t1##c.x2 || id: t1, cov(un) var mle

Mixed-effects ML regression                     Number of obs      =      2000
Group variable: id                              Number of groups   =       500

                                                Obs per group: min =         4
                                                               avg =       4.0
                                                               max =         4

                                                Wald chi2(6)       =   2871.89
Log likelihood = -3075.8519                     Prob > chi2        =    0.0000
------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           a |   .2967777E   .0213597    13.89   0.000     .2549135    .3386419
          t1 |   1.010207C   .0252504    40.01   0.000     .9607168    1.059696
          x1 |    .560547B   .0544035    10.30   0.000     .4539181    .6671759
             |
   c.t1#c.x1 |   .2630303D   .0251425    10.46   0.000     .2137519    .3123087
             |
          t1 |          0   (omitted)
          x2 |    .716562B   .0556897    12.87   0.000     .6074121    .8257118
             |
   c.t1#c.x2 |   .4734171D   .0257359    18.40   0.000     .4229756    .5238586
             |
       _cons |    .570413A   .0546356    10.44   0.000     .4633292    .6774968
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured             |
                   var(time) |     .20304G   .0203019      .1669054    .2469978
                  var(_cons) |   1.178701F   .1311813      .9476997    1.466008
             cov(time,_cons) |  -.1515308H    .041814     -.2334848   -.0695768
-----------------------------+------------------------------------------------
               var(Residual) |   .5416011I   .0242353      .4961241    .5912467
------------------------------------------------------------------------------
LR test vs. linear regression:       chi2(3) =  1344.80   Prob > chi2 = 0.0000

Mplus example #1

Here is the same example analyzed as a Latent Growth Curve Model using Mplus based on the ex6.10.dat data file. We should reiterate that the multilevel model is not identical to the LGCM model, but only similar, so the results are analogous, not identical, but we use this as a means of helping you understand a technique and output below that might be new to you. 

TITLE:
  this is an example of a linear growth
  model for a continuous outcome with time-
  invariant and time-varying covariates
DATA:
  FILE IS ex6.10.dat;
VARIABLE:
  NAMES ARE y11-y14 x1 x2 a31-a34;
MODEL:
  i s | y11@0 y12@1 y13@2 y14@3;
  i s ON x1 x2;
  y11 ON a31 ;
  y12 ON a32 ;
  y13 ON a33 ;
  y14 ON a34 ;

SUMMARY OF ANALYSIS
Number of observations                                         500

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

          Value                             25.786
          Degrees of Freedom                    21
          P-Value                           0.2147

Chi-Square Test of Model Fit for the Baseline Model

          Value                           2862.582
          Degrees of Freedom                    30
          P-Value                           0.0000

CFI/TLI

          CFI                                0.998
          TLI                                0.998

Loglikelihood

          H0 Value                       -7255.873
          H1 Value                       -7242.980

Information Criteria

          Number of Free Parameters             17
          Akaike (AIC)                   14545.745
          Bayesian (BIC)                 14617.393
          Sample-Size Adjusted BIC       14563.434
            (n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

          Estimate                           0.021
          90 Percent C.I.                    0.000  0.046
          Probability RMSEA <= .05           0.978

SRMR (Standardized Root Mean Square Residual)
          Value                              0.014

MODEL RESULTS
                   Estimates     S.E.  Est./S.E.

 I        |
    Y11                1.000    0.000      0.000
    Y12                1.000    0.000      0.000
    Y13                1.000    0.000      0.000
    Y14                1.000    0.000      0.000

 S        |
    Y11                0.000    0.000      0.000
    Y12                1.000    0.000      0.000
    Y13                2.000    0.000      0.000
    Y14                3.000    0.000      0.000

 I        ON
    X1                 0.557B   0.054     10.286
    X2                 0.718B   0.055     12.953

 S        ON
    X1                 0.264D   0.025     10.549
    X2                 0.473D   0.026     18.438

 Y11      ON
    A31                0.190E   0.044      4.302

 Y12      ON
    A32                0.323E   0.038      8.433

 Y13      ON
    A33                0.344E   0.038      9.016

 Y14      ON
    A34                0.303E   0.050      6.004

 S        WITH
    I                  0.055H   0.035      1.588

 Intercepts
    Y11                0.000    0.000      0.000
    Y12                0.000    0.000      0.000
    Y13                0.000    0.000      0.000
    Y14                0.000    0.000      0.000
    I                  0.570A    0.054     10.477
    S                  1.010C    0.025     40.112

 Residual Variances
    Y11                0.509I   0.068      7.512
    Y12                0.597I   0.048     12.348
    Y13                0.481I   0.049      9.858
    Y14                0.579I   0.088      6.607
    I                  1.074F   0.098     10.922
    S                  0.201G   0.022      9.092
  1. This analogous to G00 in the multilevel model. It is the predicted value of y when time and a are both  0.
  2. This analogous to G01 and G02 in the multilevel model. It is the predicted increase in the intercept for a one unit increase in x1 and x2, respectively.
  3. This is analogous to G20 in the multilevel model. It is the slope for time when  x1 and x2 are held constant at 0.
  4. This analogous to G21 and G22 in the multilevel model. It is the predicted increase in the time slope for a one unit increase in x1 and x2, respectively.
  5. These are the four slopes representing the regression of y1 on a1, y2 on a2, y3 on a3, and y4 on a4. Note that in the multilevel model there is only one such relationship, whereas in this model there is a separate coefficient for each time point.
  6. This is the variance of the intercept, analogous to the variance component for the intercept in the multilevel model.
  7. This is the variance of the slope for time, analogous to the variance component for the intercept for time in the multilevel model.
  8. This is the covariance of the intercept and slope, analogous to the covariance of B0 and B1 from the multilevel model.
  9. This is the residual variance for each time point. Note that in the LGCM there is a separate residual variance at each time point. This is analogous to the rij value from the multilevel model. Note that in the multilevel model there is a single residual value.

Mplus example #2

Here is a second example which is a variation that uses constraints to make the assumptions more similar to the assumptions of the multilevel model. We should reiterate that the multilevel model is not identical to the LGCM model, but only similar, so the results are analogous, not identical, but we use this as a means of helping you understand a technique and output below that might be new to you. 

TITLE:
  this is an example of a linear growth
  model for a continuous outcome with time-
  invariant and time-varying covariates
DATA:
  FILE IS ex6.10.dat;
VARIABLE:
  NAMES ARE y11-y14 x1 x2 a31-a34;
MODEL:
  i s | y11@0 y12@1 y13@2 y14@3;
  i s ON x1 x2;
  y11 ON a31 (1);
  y12 ON a32 (1);
  y13 ON a33 (1);
  y14 ON a34 (1);
  y11 y12 y13 y14 (2);

SUMMARY OF ANALYSIS
Number of observations                                         500

TESTS OF MODEL FIT

Loglikelihood
          H0 Value                       -7261.105
          H1 Value                       -7242.980

Information Criteria
          Number of Free Parameters             11
          Akaike (AIC)                   14544.210
          Bayesian (BIC)                 14590.571
          Sample-Size Adjusted BIC       14555.656
            (n* = (n + 2) / 24)

MODEL RESULTS
                   Estimates     S.E.  Est./S.E.
 I        |
    Y11                1.000    0.000      0.000
    Y12                1.000    0.000      0.000
    Y13                1.000    0.000      0.000
    Y14                1.000    0.000      0.000

 S        |
    Y11                0.000    0.000      0.000
    Y12                1.000    0.000      0.000
    Y13                2.000    0.000      0.000
    Y14                3.000    0.000      0.000

 I        ON
    X1                 0.561B   0.054     10.303
    X2                 0.717B   0.056     12.867

 S        ON
    X1                 0.263D   0.025     10.462
    X2                 0.473D   0.026     18.395

 Y11      ON
    A31                0.297E   0.021     13.894

 Y12      ON
    A32                0.297E   0.021     13.894

 Y13      ON
    A33                0.297E   0.021     13.894

 Y14      ON
    A34                0.297E   0.021     13.894

 S        WITH
    I                  0.052I   0.031      1.641

 Intercepts
    Y11                0.000    0.000      0.000
    Y12                0.000    0.000      0.000
    Y13                0.000    0.000      0.000
    Y14                0.000    0.000      0.000
    I                  0.570A   0.055     10.440
    S                  1.010C   0.025     40.008

 Residual Variances
    Y11                0.542I   0.024     22.361
    Y12                0.542I   0.024     22.361
    Y13                0.542I   0.024     22.361
    Y14                0.542I   0.024     22.361
    I                  1.079F   0.094     11.506
    S                  0.203G   0.020     10.012

See the footnotes above for descriptions of the results. Exceptions are noted below.

E. Note how the coefficients predicting y from a are all the same. Now they are closer to the multilevel model because the LGCM has been constrained to be more similar to the multilevel model.

F. Note how the residual errors are the same. They are closer to (but not the same as) the multilevel model.

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