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.
$$
\begin{eqnarray}
\mbox{Level 1:} \quad Y_{ij} & = & \beta_{0j} + \beta_{1j} Time + r_{ij} \\
\mbox{Level 2:} \quad \beta_{0j} & = & \gamma_{00} + u_{0j} \\
\beta_{1j} & = & \gamma_{10} + u_{1j}
\end{eqnarray}
$$
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.
$$
MathAch_{ij} = \gamma_{00} + \gamma_{10}(MeanSES) + [ u_{0j} + r_{ij}]
$$
-
The term u0j is
a random effect at level 2, representing random variation in the average
math achievement among (between) schools. -
The term rij 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 https://stats.idre.ucla.edu/stat/mplus/output/ex6.1.dat, clear generate id = _n reshape long y, i(id) j(time) mixed y time || id: time, cov(un)
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 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
- A. This analogous to
γ00 in the multilevel model. It is the predicted value of
y when time is 0. - B. This analogous to
γ01 in the multilevel model. It is the predicted value
increase in y for a one unit increase in time. - C. This is the variance of the intercept, analogous to the
variance of
u0j in the multilevel model. - D. This is the variance of the slope, analogous to the
variance of
u1j in the multilevel model. - E. This is the covariance of the intercept and slope, analogous to the
covariance β0j
and β1j from the multilevel model. - F. 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.