**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 generated** **hypothetical** **data,
which can be obtained by clicking on http://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:datahttp://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.

- Logistic regression, the focus of this page.
- Probit regression. Probit analysis will produce results similar
logistic regression. The choice of probit versus logit depends largely on

individual preferences.

- OLS regression. When used with a binary response variable, this model is known
as a linear probability model and can be used as a way to

describe conditional probabilities. However, the errors (i.e., residuals) from the linear probability model violate the homoskedasticity and

normality of errors assumptions of OLS

regression, resulting in invalid standard errors and hypothesis tests. For

a more thorough discussion of these and other problems with the linear

probability model, see Long (1997, p. 38-40).

- Two-group discriminant function analysis. A multivariate method for dichotomous outcome variables.
- Hotelling’s T
^{2}. The 0/1 outcome is turned into thegrouping variable, and the former predictors are turned into outcome

variables. This will produce an overall test of significance but will not

give individual coefficients for each variable, and it is unclear the extent

to which each "predictor" is adjusted for the impact of the other

"predictors."

## 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:datahttp://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

- At the top of the output we see that 400 observations were used.
- From the output we see that the model includes one binary dependent (i.e., outcome) variable and 5 independent (predictor) variables.
- The analysis summary is followed by a block of technical information about the model, we
won’t discuss most of this information, but we will note two things:
- The estimator, given on the first line of the block is listed as ML, which is what we intended.
- The link function, given on the second to the last line of the block, is listed as logit, which is also what we intended.

Input data file(s) C:datahttp://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)

- Several measures of model fit are included in the output. The log likelihood (-229.259) can be used in comparisons of nested models, but we won’t show an example of that here.
- The Akaike information criterion (AIC) and the Bayesian information criterion (BIC, sometimes also called the Schwarz criterion), can also be used to compare models.

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

- The section titled MODEL RESULTS includes the coefficients (labeled Estimate),
their standard errors, the ratio of each estimate to its standard error (i.e., the z-score,
labeled Est./S.E.), and the associated p-values. Both
**gre**and**gpa**are statistically significant, as are the terms for**rank**=1 and**rank**=2 (versus the omitted category**rank**=4). The logistic regression coefficients give the change in the log odds of the outcome for a one unit increase in the predictor variable.- For every one unit change in
**gre**, the log odds of admission (versus non-admission) increases by 0.002. - For a one unit increase in
**gpa**, the log odds of being admitted to graduate school increases by 0.804. - The coefficients for the categories of
**rank**have a slightly different interpretation. For example, having attended an undergraduate institution with a**rank**of 1, versus an institution with a**rank**of 4 (the omitted category), increases the log odds of admission by 1.551.

- For every one unit change in
- Below the coefficients for each of the predictor variables, under the heading Thresholds, is the threshold for the model (sometimes also called a cutpoint). Mplus reports a threshold in place of the intercept, the two are the same except that they have opposite signs (so the intercept for this model would be -5.541). For more information on the differences between intercepts and thresholds, please see http://www.stata.com/support/faqs/stat/oprobit.html.

LOGISTIC REGRESSION ODDS RATIO RESULTS ADMIT ON GRE 1.002 GPA 2.235 RANK1 4.718 RANK2 2.401 RANK3 1.235

- Mplus also gives the model results as odds ratios. An odds
ratio is the exponentiated coefficient, and can be interpreted as the multiplicative
change in the odds for a one unit change in the predictor variable. For example,
for a one unit
increase in
**gpa**, the odds of being admitted to graduate school (versus not being admitted) increase by a factor of 2.24. For more information on interpreting odds ratios see our FAQ page: How do I interpret odds ratios in logistic regression?

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:datahttp://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:datahttp://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

- Empty cells or small cells: You should check for empty or small
cells by doing a crosstab between categorical predictors and the outcome variable. If a cell has very few cases (a small cell), the model may become unstable or it might not run at all.

- Separation or quasi-separation (also called perfect prediction), a condition in which the outcome does not vary at some levels of the independent variables. See our page FAQ: What is complete or quasi-complete separation in logistic/probit regression and how do we deal with them? for information on models with perfect prediction.
- Sample size: Both logit and probit models require more cases than OLS
regression because they use maximum likelihood estimation techniques. It is
sometimes possible to estimate models for binary outcomes in datasets with
only a small number of cases using exact logistic regression (using the
**exlogistic**command). For more information see our data analysis example for exact logistic regression. It is also important to keep in mind that when the outcome is rare, even if the overall dataset is large, it can be difficult to estimate a logit model. - Pseudo-R-squared: Many different measures of pseudo-R-squared exist. They all attempt to provide information similar to that provided by R-squared in OLS regression; however, none of them can be interpreted exactly as R-squared in OLS regression is interpreted. For a discussion of various pseudo-R-squareds see Long and Freese (2006) or our FAQ page What are pseudo R-squareds?
- Diagnostics: The diagnostics for logistic regression are different from those for OLS regression. For a discussion of model diagnostics for logistic regression, see Hosmer and Lemeshow (2000, Chapter 5). Note that diagnostics done for logistic regression are similar to those done for probit regression.
- Clustered data: Sometimes observations are clustered into groups (e.g.,
people within families, students within classrooms). In such cases, you may
want to consider using either a multilevel model or the
**cluster**option of the**variable**command.

## 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

- Mplus Annotated Output: Logit Regression
- References
- Long, J. S. 1997.
*Regression Models for Categorical and Limited Dependent Variables.*Thousand Oaks, CA: Sage Publications.