Annotated Mplus Output Multinomial Logit Regression

This page shows an example of multinomial logit regression with footnotes explaining the output. First an example is shown using Stata, and then an example is shown using Mplus, to help you relate the output you are likely to be familiar with (Stata) to output that may be new to you (Mplus). 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 from the Mplus User's Guide (example 3.6) 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.

Stata Example

Here is a multinomial logit regression example using Stata with two continuous predictors x1 and x2 used to predict a binary outcome variable, u1.

infile u1 x1 x3 using http://www.ats.ucla.edu/stat/mplus/output/ex3.6.dat, clear
mlogit u1 x1 x3

Iteration 0:   log likelihood =  -539.2303
Iteration 1:   log likelihood = -446.49742
Iteration 2:   log likelihood = -434.20483
Iteration 3:   log likelihood =  -433.4331
Iteration 4:   log likelihood = -433.42628
Iteration 5:   log likelihood = -433.42628

Multinomial logistic regression                   Number of obs   =        500
LR chi2(4)      =     211.61
Prob > chi2     =     0.0000
Log likelihood = -433.42628                       Pseudo R2       =     0.1962

------------------------------------------------------------------------------
u1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
0            |
x1 |   .7686261C   .1567749     4.90   0.000      .461353    1.075899
x3 |   2.259422C   .2144306    10.54   0.000     1.839146    2.679699
_cons |  -.7488877E   .1702198    -4.40   0.000    -1.082512   -.4152631
-------------+----------------------------------------------------------------
1            |
x1 |   .2798667D   .1131474     2.47   0.013     .0581018    .5016316
x3 |    .885101D   .1402897     6.31   0.000     .6101382    1.160064
_cons |   .2622508E   .1198104     2.19   0.029     .0274268    .4970748
------------------------------------------------------------------------------
(u1==2 is the base outcome)

estat ic

------------------------------------------------------------------------------
Model |    Obs    ll(null)   ll(model)     df          AIC         BIC
-------------+----------------------------------------------------------------
. |    500   -539.2303   -433.4263A      6     878.8526B    904.1402B
------------------------------------------------------------------------------

The output is labeled with superscripts to help you relate the later Mplus output to this Stata output. To summarize the output, both predictors in this model, x1 and x3, are significantly related to predicting the comparison of level 0 to level 2 of the outcome variable, u1. Likewise, x1 and x3, are significantly related to predicting the comparison of level 1 to level 2 of the outcome variable, u1. The estat ic command produces fit indices for the model including the log likelihood for the empty (null) model, the log likelihood for the model, as well as the AIC and BIC fit indices.

Mplus Example

Here is the same example illustrated in Mplus based on the ex3.6.dat data file.

TITLE:
this is an example of a multinomial
logistic regression for an unordered
categorical (nominal) dependent variable
with two covariates
DATA:
FILE IS ex3.6.dat;
VARIABLE:
NAMES ARE u1 x1 x3;
NOMINAL IS u1;
MODEL:
u1#1 u1#2 ON x1 x3;
Number of observations                                         500
Estimator                                                      MLR

THE MODEL ESTIMATION TERMINATED NORMALLY

TESTS OF MODEL FIT

Loglikelihood

H0 Value                        -433.426A

Information Criteria

Number of Free Parameters              6
Akaike (AIC)                     878.853B
Bayesian (BIC)                   904.140B
(n* = (n + 2) / 24)

MODEL RESULTS
Estimates     S.E.  Est./S.E.

U1#1       ON
X1                 0.769C    0.165      4.670
X3                 2.259C    0.203     11.148

U1#2       ON
X1                 0.280D    0.114      2.444
X3                 0.885D    0.143      6.200

Intercepts
U1#1              -0.749E    0.158     -4.728
U1#2               0.262E    0.120      2.192

1. This is the log likelihood value associated with the model (see the ll(model) from the estat ic command in Stata.
2. These are the AIC and BIC values, see the AIC and BIC values from the estat ic command in Stata.
3. These are the logit coefficients expressing the relationship between x1 x3 and u1 in the logit scale. Note that u1 is coded 0, 1, 2 and Mplus converts those values into groups 1, 2 and 3 respectively. So these coefficients represent the comparison of u1 for the first level of u1 (i.e., 0) to the omitted level of u1 (i.e., 2). These results parallel the results of the Stata mlogit command.
4. These are the logit coefficients expressing the relationship between x1 x3 and u1 in the logit scale. These coefficients represent the comparison of u1 for the second level of u1 (i.e., 1) to the omitted level of u1 (i.e., 2). These results parallel the results of the Stata mlogit command.
5. These are the intercepts for the model, representing the intercepts for the first and second levels of  u1 (i.e., levels 0 and 1).

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