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 = 0.54200^{I}Tau INTRCPT1,B0 1.08757^{F}0.05079 TIME,B2 0.05079^{H}0.20495^{G}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 0.570413^{A}0.054807 10.408 497 0.000 X1, G01 0.560548^{B}0.054574 10.271 497 0.000 X2, G02 0.716557^{B}0.055865 12.827 497 0.000 For A slope, B1 INTRCPT2, G10 0.296872^{E}0.021381 13.885 1993 0.000 For TIME slope, B2 INTRCPT2, G20 1.010207^{C}0.025332 39.879 497 0.000 X1, G21 0.263030^{D}0.025223 10.428 497 0.000 X2, G22 0.473419^{D}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.

infile y1 y2 y3 y4 x1 x2 a1 a2 a3 a4 using http://www.ats.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 xtmixed y x1 x2 a time timeBYx1 timeBYx2 || id: time, cov(un) var mleMixed-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 | .560547^{B}.0544035 10.30 0.000 .4539181 .6671759 x2 | .716562^{B}.0556897 12.87 0.000 .6074121 .8257118 a | .2967777^{E}.0213597 13.89 0.000 .2549134 .3386419 time | 1.010207^{C}.0252504 40.01 0.000 .9607168 1.059696 timeBYx1 | .2630303^{D}.0251425 10.46 0.000 .2137518 .3123087 timeBYx2 | .4734171^{D}.0257359 18.40 0.000 .4229756 .5238586 _cons | .5704131^{A}.0546356 10.44 0.000 .4633293 .6774969 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(time) | .2030399^{G}.0203019 .1669053 .2469977 var(_cons) | 1.078679^{F}.0937659 .9097041 1.27904 cov(time,_cons) | .0515093^{H}.0314154 -.0100638 .1130825 -----------------------------+------------------------------------------------ var(Residual) | .5416011^{I}.0242353 .496124 .5912467 ------------------------------------------------------------------------------ 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 0.297^{E}0.022 13.300 Residual Variances Y 0.541^{I}0.024 22.178 Between Level S ON X1 0.263^{D}0.027 9.801 X2 0.473^{D}0.025 18.909 Y ON X1 0.561^{B}0.054 10.296 X2 0.717^{B}0.054 13.264 Y WITH S 0.050^{H}0.033 1.529 Intercepts Y 0.570^{A}0.055 10.400 S 1.010^{C}0.025 39.763 Residual Variances Y 1.081^{F}0.093 11.617 S 0.204^{G}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.