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 = 0.48774^{F}Tau INTRCPT1,B0 0.98667^{C}0.13242 TIME,B1 0.13242^{E}0.22750^{D }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 0.522793^{A}0.051538 10.144 499 0.000 For TIME slope, B1 INTRCPT2, G10 1.026268^{B}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 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 https://stats.idre.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 | 1.026268^{B}.0254715 40.29 0.000 .9763442 1.076191 _cons | .522793^{A}.0514865 10.15 0.000 .4218814 .6237047 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(time) | .2268527^{D}.0209755 .1892514 .2719247 var(_cons) | .9840141^{C}.0852067 .8304146 1.166025 cov(time,_cons) | .1324435^{E}.0300523 .0735421 .191345 -----------------------------+------------------------------------------------ var(Residual) | .4877361^{F}.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 0.133^{E}0.033 4.057 Means I 0.523^{A}0.051 10.153 S 1.026^{B}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 0.989^{C}0.089 11.097 S 0.224^{D}0.023 9.891 Residual Variances Y11 0.475^{F}0.059 7.989 Y12 0.482^{F}0.040 11.994 Y13 0.473^{F}0.047 10.007 Y14 0.545^{F}0.084 6.471

- This analogous to
γ
_{00}in the multilevel model. It is the predicted value of**y**when**time**is 0. - This analogous to
γ
_{0}_{1}in the multilevel model. It is the predicted value increase in**y**for a one unit increase in**time**. - This is the variance of the intercept, analogous to the
variance of
u
_{0j}in the multilevel model. - This is the variance of the slope, analogous to the
variance of
u
_{1j}in the multilevel model. - This is the covariance of the intercept and slope, analogous to the
covariance β
_{0j }and β_{1j}from the multilevel model. - 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.