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 is 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, the variables x1 and x2 are measured for each person. Conceptualized as a multilevel model, the variables time and a are level 1 variables. (Note that time is coded 0, 1, 2, and 3.) The variables x1 and x2 are level 2 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
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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.
infile y1 y2 y3 y4 x1 x2 a1 a2 a3 a4 using https://stats.idre.ucla.edu/stat/mplus/output/ex6.10.dat
generate id = _n
reshape long y a, i(id) j(time)
replace time = time-1
generate timeBYx1 = time*x1
generate timeBYx2 = time*x2
mixed y x1 x2 a time timeBYx1 timeBYx2 || 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(6) = 2871.89
Log likelihood = -3075.8518 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .560547B .0544035 10.30 0.000 .4539181 .6671759
x2 | .716562B .0556897 12.87 0.000 .6074121 .8257118
a | .2967777E .0213597 13.89 0.000 .2549134 .3386419
time | 1.010207C .0252504 40.01 0.000 .9607168 1.059696
timeBYx1 | .2630303D .0251425 10.46 0.000 .2137518 .3123087
timeBYx2 | .4734171D .0257359 18.40 0.000 .4229756 .5238586
_cons | .5704131A .0546356 10.44 0.000 .4633293 .6774969
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(time) | .2030399G .0203019 .1669053 .2469977
var(_cons) | 1.078679F .0937659 .9097041 1.27904
cov(time,_cons) | .0515093H .0314154 -.0100638 .1130825
-----------------------------+------------------------------------------------
var(Residual) | .5416011I .0242353 .496124 .5912467
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LR test vs. linear regression: chi2(3) = 1344.80 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Mplus Example
Here is the same example analyzed as a multilevel using Mplus based on the ex6.10.dat data file.
Title:
Two level multilevel model in Mplus
Data:
File is ex6.10.dat ;
Variable:
Names are id time y x1 x2 a;
WITHIN = time a;
BETWEEN = x1 x2;
CLUSTER = id;
ANALYSIS:
TYPE = TWOLEVEL RANDOM;
MODEL:
%WITHIN%
s | y ON time;
y on a;
%BETWEEN%
y on x1 x2;
s on x1 x2;
y with s;
SUMMARY OF ANALYSIS
Number of observations 2000
SUMMARY OF DATA
Number of clusters 500
Average cluster size 4.000
Estimated Intraclass Correlations for the Y Variables
Intraclass Intraclass
Variable Correlation Variable Correlation
Y 0.615
TESTS OF MODEL FIT
Loglikelihood
H0 Value -3075.853
Information Criteria
Number of Free Parameters 11
Akaike (AIC) 6173.706
Bayesian (BIC) 6235.316
Sample-Size Adjusted BIC 6200.369
(n* = (n + 2) / 24)
MODEL RESULTS
Estimates S.E. Est./S.E.
Within Level
Y ON
A 0.297E 0.022 13.300
Residual Variances
Y 0.541I 0.024 22.178
Between Level
S ON
X1 0.263D 0.027 9.801
X2 0.473D 0.025 18.909
Y ON
X1 0.561B 0.054 10.296
X2 0.717B 0.054 13.264
Y WITH
S 0.050H 0.033 1.529
Intercepts
Y 0.570A 0.055 10.400
S 1.010C 0.025 39.763
Residual Variances
Y 1.081F 0.093 11.617
S 0.204G 0.020 10.236
- This is G00 in the multilevel model. It is the predicted value of y when time and a are both 0.
- This is 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.
- This is G20 in the multilevel model. It is the slope for time when x1 and x2 are held constant at 0.
- This is 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.
- This is G10 from the multilevel model, representing the regression of y on a.
- This is the variance of the intercept, the variance component for the intercept in the multilevel model.
- This is the variance of the slope for time, the variance component for the time slope in the multilevel model.
- This is the covariance of the intercept and slope, the covariance of B0 and B1 from the multilevel model.
- This is the residual variance.