Note: This example was done using Mplus version 6.12.

Logistic regression, also called a logit model, is used to model dichotomous
outcome variables. In the logit model the log odds of the outcome is modeled as a linear
combination of the predictor variables.

Please note: The purpose of this page is to show how to use various data analysis commands.
It does not cover all aspects of the research process which researchers are expected to do. In
particular, it does not cover data cleaning and checking, verification of assumptions, model
diagnostics and potential follow-up analyses.

Examples

Example 1:  Suppose that we are interested in the factors

that influence whether a political candidate wins an election.  The

outcome (response) variable is binary (0/1);  win or lose. 

The predictor variables of interest are the amount of money spent on the campaign, the

amount of time spent campaigning negatively, and whether the candidate is an

incumbent.

Example 2:  A researcher is interested in how variables, such as GRE (Graduate Record Exam scores),

GPA (grade

point average) and prestige of the undergraduate institution, effect admission into graduate

school. The outcome variable, admit/don’t admit, is binary.

Description of the data

For our data analysis below, we are going to expand on Example 2 about
getting into graduate school.  We have generatedhypotheticaldata,
which can be obtained by clicking on https://stats.idre.ucla.edu/wp-content/uploads/2016/02/binary.dat. You can store this anywhere you like, but our examples will
assume it has been stored in c:data.  (Note that the names of
variables should NOT be included at the top of the data file.  Instead, the
variables are named as part of the variable command.)  You may want to do your
descriptive statistics in a general use statistics package, such as SAS, Stata
or SPSS, because the options for obtaining descriptive statistics are limited in
Mplus. Even if you chose to run descriptive statistics in another package, it is
a good idea to run a model with type=basic before you do anything else,
just to make sure the dataset is being read correctly.

This dataset has data on 400 cases. There is a binary response (outcome, dependent) variable called
admit and there are three predictor variables: gre, gpa, and
rank. We will treat the
variables gre and gpa as continuous. The variable rank takes on the values 1
through 4. Institutions with a rank of 1 have the highest prestige, while those
with a rank of 4 have the lowest. The dataset also contains four dummy
variables, one for each level of rank, named rank1 to rank4,
for example, rank1 is equal to 1 when rank=1, and 0 otherwise.
Lets start by running a model with type=basic.

Data:
  File is c:datahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/binary.dat ;
Variable:
  Names are 
     admit gre gpa rank rank1 rank2 rank3 rank4;
Analysis: 
  Type = basic ;

As we mentioned above, you will want to look at this carefully to be sure that
the dataset was read into Mplus correctly.  You will want to make sure that
you have the correct number of observations, and that the variables all have
means that are close to those from the descriptive statistics generated in a
general purpose statistical package. If there are missing values for some or all
of the variables, the descriptive statistics generated by Mplus will not match
those from a general purpose statistical package exactly, because by default, Mplus versions
5.0 and later use maximum likelihood based procedures for handling missing
values. The main point of running this model is to make sure that the data is
being read correct by Mplus, if the number of cases and variables is correct,
and the means are reasonable, then it is probably safe to proceed.

<output omitted>

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         400

<output omitted>


     SAMPLE STATISTICS


           Means
              ADMIT         GRE           GPA           RANK          RANK1
              ________      ________      ________      ________      ________
      1         0.318       587.700         3.390         2.485         0.152


           Means
              RANK2         RANK3         RANK4
              ________      ________      ________
      1         0.378         0.302         0.168

Analysis methods you might consider

Below is a list of some analysis methods you may have encountered.
Some of the methods listed are quite reasonable while others have either
fallen out of favor or have limitations.

Using the logit model

The Mplus input file for a logistic regression model is shown below. Because
the data file contains variables that are not used in the model, the usevariables subcommand
is used to list the variables that appear in the model (i.e., admit,
gre, gpa, rank1, rank2, and rank3). Note that because Mplus uses the names subcommand to determine the order of variables
in the data file, the number and order of variables in the names subcommand should not be changed unless the
data file is also changed. The
categorical subcommand is used to identify binary and ordinal outcome
variables. Only the categorical outcome
variable (i.e., admit) is included in the categorical subcommand. Categorical predictor
variables should be included as a series of dummy variables (e.g., rank1,
rank2, and rank3). Under analysis we have specified estimator=ml,
this requests a logit model, rather than the default probit model. Finally,
in the model command we specify that the outcome (i.e., admit) should be regressed
on the predictor variables (i.e., gre, gpa, rank1, rank2,

and rank3).

  Data:
    File is c:datahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/binary.dat ;
  Variable:
    names = admit gre gpa rank rank1 rank2 rank3 rank4;
    usevariables = admit gre gpa rank1 rank2 rank3;
    categorical = admit;
  Analysis:
      estimator = ml;
  Model:
      admit on gre gpa rank1 rank2 rank3;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         400

Number of dependent variables                                    1
Number of independent variables                                  5
Number of continuous latent variables                            0

Observed dependent variables

  Binary and ordered categorical (ordinal)
   ADMIT

Observed independent variables
   GRE         GPA         RANK1       RANK2       RANK3


Estimator                                                       ML
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                 100
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-02
    Relative loglikelihood change                        0.100D-05
    Derivative                                           0.100D-02
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            15
  Minimum value for logit thresholds                           -15
  Minimum expected cell size for chi-square              0.100D-01
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            0
  Adaptive quadrature                                           ON
Link                                                         LOGIT
Cholesky                                                       OFF

Input data file(s)
  C:datahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/binary.dat
Input data format  FREE


SUMMARY OF CATEGORICAL DATA PROPORTIONS

    ADMIT
      Category 1    0.683
      Category 2    0.317



THE MODEL ESTIMATION TERMINATED NORMALLY


TESTS OF MODEL FIT

Loglikelihood

          H0 Value                        -229.259

Information Criteria

          Number of Free Parameters              6
          Akaike (AIC)                     470.517
          Bayesian (BIC)                   494.466
          Sample-Size Adjusted BIC         475.428
            (n* = (n + 2) / 24)

MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 ADMIT      ON
    GRE                0.002      0.001      2.070      0.038
    GPA                0.804      0.332      2.423      0.015
    RANK1              1.551      0.418      3.713      0.000
    RANK2              0.876      0.367      2.389      0.017
    RANK3              0.211      0.393      0.538      0.591

 Thresholds
    ADMIT$1            5.541      1.138      4.869      0.000

LOGISTIC REGRESSION ODDS RATIO RESULTS

 ADMIT      ON
    GRE                1.002
    GPA                2.235
    RANK1              4.718
    RANK2              2.401
    RANK3              1.235

We can also test that the coefficients for rank1, rank2, and rank3, are all
equal to zero using the model test command. This type of test can also be
described as an overall test for the effect of rank. There are
multiple ways to test this type of hypothesis, the model test command
requests a Wald test. The Mplus input file shown
below is similar to the first model, except that the coefficients for rank1, rank2, and rank3
are assigned the names r1, r2, and r3, respectively. In the
model test command,
these coefficient names (i.e., r1, r2 and r3) are used to test that each of the coefficients is equal to 0.

Data:
  File is C:datahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/binary.dat ;
Variable:
  names = admit gre gpa rank rank1 rank2 rank3 rank4;
  categorical = admit;
  usevariables = admit gre gpa rank1 rank2 rank3;
Analysis:
  estimator = ML;
Model:
  admit on gre gpa
    rank1 (r1)
    rank2 (r2)
    rank3 (r3);
Model test:
  r1 = 0;
  r2 = 0;
  r3 = 0;

The majority of the output from this model is the same as the first model, so we will only show part of
the output generated by the model test command.

TESTS OF MODEL FIT

Wald Test of Parameter Constraints

          Value                             20.895
          Degrees of Freedom                     3
          P-Value                           0.0001

Loglikelihood

          H0 Value                        -229.259

The portion of the output associated with the model test command is labeled “Wald Test of
Parameter Constraints” and appears under the heading TESTS OF MODEL FIT just before the likelihood
for the entire model is printed. The test statistic is 20.895, with three
degrees of freedom (one for each of the parameters tested), with an associated
p-value of 0.0001. This indicates that the overall effect of rank is
statistically significant.

We can also use the model test command to make pairwise comparisons among the terms for
rank. The Mplus input below tests the hypothesis that the coefficient for
rank2 (i.e., rank=2) is equal to the coefficient for rank3 (i.e.,
rank=3).

Data:
  File is C:datahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/binary.dat ;
Variable:
  names = admit gre gpa rank rank1 rank2 rank3 rank4;
  categorical = admit;
  usevariables = admit gre gpa rank1 rank2 rank3;
Analysis:
    estimator = ml;
Model:
  admit on gre gpa
    rank1 (r1)
    rank2 (r2)
    rank3 (r3);
Model test:
  r2 = r3;

Below is the output associated with the model test command (as before,
most of the model output is omitted).

MODEL FIT INFORMATION

Wald Test of Parameter Constraints

          Value                              5.505
          Degrees of Freedom                     1
          P-Value                           0.0190

The test statistic and associated p-value indicate that the coefficient for rank2 (i.e., rank=2)
is significantly different from the coefficient for rank3 (rank=3).

Things to consider

References

Hosmer, D. & Lemeshow, S. (2000). Applied Logistic Regression (Second Edition).
New York: John Wiley & Sons, Inc.

Long, J. Scott (1997). Regression Models for Categorical and Limited Dependent Variables.
Thousand Oaks, CA: Sage Publications.

See also