This page shows an example of a two level multilevel model. First a multilevel model is shown using HLM and then using
Stata, and then the same data are analyzed using Mplus using a multilevel model.
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 multilevel model using Mplus based on the
ex61l.dat data file.
Title:
Multilevel model
Data:
File is ex61l.dat ;
Variable:
Names are
id time y;
WITHIN = time ;
CLUSTER = id;
ANALYSIS:
TYPE = TWOLEVEL RANDOM;
MODEL:
%WITHIN%
s | y ON time;
%BETWEEN%
y s ;
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 0.452
TESTS OF MODEL FIT
Loglikelihood
H0 Value -3016.699
Information Criteria
Number of Free Parameters 6
Akaike (AIC) 6045.397
Bayesian (BIC) 6079.003
Sample-Size Adjusted BIC 6059.940
(n* = (n + 2) / 24)
MODEL RESULTS
Estimates S.E. Est./S.E.
Within Level
Residual Variances
Y 0.487F 0.024 20.455
Between Level
Y WITH
S 0.131E 0.032 4.078
Means
Y 0.523A 0.051 10.154
S 1.026B 0.025 40.291
Variances
Y 0.986C 0.081 12.130
S 0.227D 0.020 11.262
- This is γ00 in the multilevel model. It is the predicted value of y when time is 0.
- This is γ10 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, the variance of u0j in the multilevel model.
- This is the variance of the slope, the variance of u1j in the multilevel model.
- This is the covariance of the intercept and slope, the covariance β0j and β1j from the multilevel model.
- This is the residual variance at level 1.