### Annotated Mplus Output Logit Regression

This page shows an example of 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 3.5) 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 Stata

Here is a 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/ex3.5.dat, clear
tabulate u1

u1 |      Freq.     Percent        Cum.
------------+-----------------------------------
0 |        327       65.40A       65.40
1 |        173       34.60A      100.00
------------+-----------------------------------
Total |        500      100.00


A. These are the percent of cases with 0 and 1 on the variable u1

logit u1 x1 x3

Iteration 0:   log likelihood = -322.46763
Iteration 1:   log likelihood = -216.57883
Iteration 2:   log likelihood = -203.79479
Iteration 3:   log likelihood = -202.63515
Iteration 4:   log likelihood = -202.61995
Iteration 5:   log likelihood = -202.61995

Logistic regression                               Number of obs   =        500
LR chi2(2)      =     239.70
Prob > chi2     =     0.0000
Log likelihood = -202.61995                       Pseudo R2       =     0.3717

------------------------------------------------------------------------------
u1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 |   1.071767E  .1428573     7.50   0.000      .791772    1.351762
x3 |   1.838588E  .1794923    10.24   0.000     1.486789    2.190386
_cons |  -1.025842D  .1369173    -7.49   0.000    -1.294195   -.7574886
------------------------------------------------------------------------------
logit , or

Logistic regression                               Number of obs   =        500
LR chi2(2)      =     239.70
Prob > chi2     =     0.0000
Log likelihood = -202.61995                       Pseudo R2       =     0.3717

------------------------------------------------------------------------------
u1 | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 |   2.920536F   .4172198     7.50   0.000     2.207304    3.864229
x3 |   6.287652F   1.128585    10.24   0.000     4.422872    8.938663
------------------------------------------------------------------------------
estat ic

------------------------------------------------------------------------------
Model |    Obs    ll(null)   ll(model)B    df         AICC        BICC
-------------+----------------------------------------------------------------
. |    500   -322.4676   -202.6199      3     411.2399    423.8837
------------------------------------------------------------------------------

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 x2, are significantly related to the outcome variable, u1. The coefficients from the logit output can be exponentiated to obtain odds ratios, as shown in the output from the logit, or command. For a one unit increase in x1, the odds of u1 equaling 1 (as compared to u1 equaling 0) increases by a factor of 2.92. 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 #1

Here is the same example illustrated in Mplus based on the ex3.5.dat data file.

TITLE:
this is an example of a logistic
regression for a categorical observed
dependent variable with two covariates
DATA:
FILE = ex3.5.dat;
VARIABLE:
NAMES = u1 x1 x3;
CATEGORICAL = u1;
ANALYSIS:
ESTIMATOR = ML;
! need to use estimator = ml to make this a logistic model;
MODEL:
u1 ON x1 x3;

SUMMARY OF ANALYSIS
Number of observations                                         500
Estimator                                                      MLR

<some output was omitted to save space>

SUMMARY OF CATEGORICAL DATA PROPORTIONS

U1
Category 1    0.654A
Category 2    0.346A

TESTS OF MODEL FIT

Loglikelihood

H0 Value                        -202.620B

Information Criteria

Number of Free Parameters              3
Akaike (AIC)                     411.240C
Bayesian (BIC)                   423.884C
(n* = (n + 2) / 24)
MODEL RESULTS

Estimates     S.E.  Est./S.E.

U1         ON
X1                 1.072D    0.143      7.502
X3                 1.839D    0.179     10.243

Thresholds
U1$1 1.026E 0.137 7.492 LOGISTIC REGRESSION ODDS RATIO RESULTS U1 ON X1 2.921F X3 6.288F  1. These are the percent of cases with 0 and 1 on the variable u1, see output of tabulate command from Stata 2. This is the log likelihood value associated with the model (see the ll(model) from the estat ic command in Stata. 3. These are the AIC and BIC values, see the AIC and BIC values from the estat ic command in Stata. 4. These are the logit coefficients expressing the relationship between x1 x2 and u1 in the logit scale, corresponding to the results of the Stata logit command. This is followed by the S.E. column (standard error) and the estimate divided by the standard error (Est./S.E). This final column is used for assessing significance by treating this like a Z test. 5. A logit model can incorporate either an intercept or a threshold in the model. Mplus uses a threshold (which Stata uses for ordinal models and calls them cutpoints). If you replaced the Stata logit command with ologit you would get the same threshold value as shown in Mplus. 6. These are the exponentiated logit coefficients, aka the odds ratios. These correspond to the values shown by the logit, or command in Stata. Mplus Example #2 Here is another version of this example in Mplus. Note that by using estimator=ml; (maximum likelihood) the results are shown in a logit metric. Had we specified something like estimator=wls; (weighted least squares) then the results would be shown in a probit scale. Because this analysis does not use the type=logistic option (unlike example #1), the format of the output is somewhat different (notably omitting odds ratios from the output). TITLE: this is an example of a logistic regression for a categorical observed dependent variable with two covariates. DATA: FILE = ex3.5.dat; VARIABLE: NAMES = u1 x1 x3; CATEGORICAL = u1; ! note using Maximum Likelihood produces results in Logit scale ! using GLS produces results in Probit scale analysis: estimator=ml; MODEL: u1 ON x1 x3; SUMMARY OF ANALYSIS Number of observations 500 Estimator ML <some output omitted to save space> SUMMARY OF CATEGORICAL DATA PROPORTIONS U1 Category 1 0.654A Category 2 0.346A THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Loglikelihood H0 Value -202.620B Information Criteria Number of Free Parameters 3 Akaike (AIC) 411.240C Bayesian (BIC) 423.884C Sample-Size Adjusted BIC 414.362 (n* = (n + 2) / 24) MODEL RESULTS Estimates S.E. Est./S.E. U1 ON X1 1.072E 0.143 7.503 X3 1.839E 0.179 10.245 Thresholds U1$1               1.026D    0.137      7.493
1. These are the percent of cases with 0 and 1 on the variable u1, see output of tabulate command from Stata
2. This is the log likelihood value associated with the model (see the ll(model) from the estat ic command in Stata).
3. These are the AIC and BIC values, see the AIC and BIC values from the estat ic command in Stata.
4. A logit model can incorporate either an intercept or a threshold in the model. Mplus uses a threshold (which Stata uses for ordinal models and calls them cutpoints). If you replaced the Stata logit command with ologit you would get the same threshold value as shown in Mplus.
5. These are the logit coefficients expressing the relationship between x1 x2 and u1 in the logit scale, corresponding to the results of the Stata logit command. You would need to exponentiate these values to convert them into odds ratios. This is followed by the S.E. column (standard error) and the estimate divided by the standard error (Est./S.E). This final column is used for assessing significance by treating this like a Z test.

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