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Annotated Mplus Output
Latent Growth Curve Model, Example 1

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). 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 6.1) 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. Conceptualized as a multilevel model, the variable time is a level 1 variable. Time is coded 0, 1, 2, and 3. The intercept is the predicted value when time is 0. Each subject has their own intercept and slope, expressed as random effects at level 2. We can write this model using multiple equations as shown below. This uses the ex61.mdm file.

Level 1:
  Y_ij =  β_0j + β_1jTime + r_ij  
Level 2:
  β_0j =  γ₀₀ + u_0j
   β_1j =  γ₁₀ + u_1j

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

Summary of the model specified (in equation format)
---------------------------------------------------

Level-1 Model

	Y = B0 + B1*(TIME) + R

Level-2 Model
	B0 = G00 + U0
	B1 = G10 + U1

Iterations stopped due to small change in likelihood function
******* ITERATION 2 *******

 Sigma_squared =      0.48774F

 Tau
 INTRCPT1,B0      0.98667C      0.13242 
     TIME,B1      0.13242E      0.22750D

Tau (as correlations)
 INTRCPT1,B0  1.000  0.279
     TIME,B1  0.279  1.000

Final estimation of fixed effects:
----------------------------------------------------------------------------
                                      Standard             Approx.
   Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
For       INTRCPT1, B0
   INTRCPT2, G00           0.522793A  0.051538    10.144       499    0.000
For     TIME slope, B1
   INTRCPT2, G10           1.026268B  0.025497    40.250       499    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.

MathAch_ij =  γ₀₀ + γ₀₁(time) + [ u_0j + r_ij ]

• The term u_0j is a random effect at level 2, representing random variation in the average math achievement among (between) schools.

• The term r_ij is a random effect at level 1, representing random variation in the math achievement of students within schools.

Here is an example using Stata.

infile y0 y1 y2 y3 using http://www.ats.ucla.edu/stat/mplus/output/ex6.1.dat, clear
generate id = _n
reshape long y, i(id) j(time)
xtmixed y time || id: time, 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(1)       =   1623.34
Log likelihood = -3016.6973                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        time |   1.026268B  .0254715    40.29   0.000     .9763442    1.076191
       _cons |    .522793A  .0514865    10.15   0.000     .4218814    .6237047
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured             |
                   var(time) |   .2268527D  .0209755      .1892514    .2719247
                  var(_cons) |   .9840141C  .0852067      .8304146    1.166025
             cov(time,_cons) |   .1324435E  .0300523      .0735421     .191345
-----------------------------+------------------------------------------------
               var(Residual) |   .4877361F  .0218122       .446805    .5324169
------------------------------------------------------------------------------
LR test vs. linear regression:       chi2(3) =  1601.32   Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference

Mplus Example

Here is the same example analyzed as a Latent Growth Curve Model using Mplus based on the ex6.1.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
DATA:
  FILE IS ex6.1.dat;
VARIABLE:
  NAMES ARE y11-y14;
MODEL:
  i s | y11@0 y12@1 y13@2 y14@3;

SUMMARY OF ANALYSIS
Number of observations                                         500

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                       -3016.386
          H1 Value                       -3014.089

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
 S        WITH
    I                  0.133E   0.033      4.057
 Means
    I                  0.523A   0.051     10.153
    S                  1.026B   0.025     40.268
 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
 Variances
    I                  0.989C   0.089     11.097
    S                  0.224D   0.023      9.891
 Residual Variances
    Y11                0.475F   0.059      7.989
    Y12                0.482F   0.040     11.994
    Y13                0.473F   0.047     10.007
    Y14                0.545F   0.084      6.471

1. This analogous to γ₀₀ in the multilevel model. It is the predicted value of y when time is 0.
2. This analogous to γ₀₁ in the multilevel model. It is the predicted value increase in y for a one unit increase in time.
3. This is the variance of the intercept, analogous to the variance of u_0j in the multilevel model.
4. This is the variance of the slope, analogous to the variance of u_1j in the multilevel model.
5. This is the covariance of the intercept and slope, analogous to the covariance β_0j and β_1j from the multilevel model.
6. 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 r_ij value from the multilevel model. Note that in the multilevel model there is a single residual value.

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