Note: This example was done using Mplus version 5.2. The syntax may not work, or may function differently, with other versions of Mplus.
Probit regression, also called a probit model, is used to model dichotomous or binary outcome variables. In the probit model, the inverse standard normal distribution of the probability is modeled as a linear combination of the predictors.
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.
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 response variable, admit/don't admit, is a binary variable.
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 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 in 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.
Data: File is C:\data\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.
<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
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.
The Mplus input file for a probit 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 are used 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. Categorical predictor variables should be included as a series of dummy variables (e.g., rank1, rank2, and rank3). We do not need to specify that we wish to run a probit model because probit models are the default for binary outcome variables. Finally, under model 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 D:\documents\dae_updating\binary.dat ; Variable: names = admit gre gpa rank rank1 rank2 rank3 rank4; categorical = admit; usevariables = admit gre gpa rank1 rank2 rank3; 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 WLSMV Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Parameterization DELTA
Input data file(s) C:\data\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 Chi-Square Test of Model Fit Value 0.000* Degrees of Freedom 0** P-Value 0.0000 * The chi-square value for MLM, MLMV, MLR, ULSMV, WLSM and WLSMV cannot be used for chi-square difference tests. MLM, MLR and WLSM chi-square difference testing is described in the Mplus Technical Appendices at www.statmodel.com. See chi-square difference testing in the index of the Mplus User's Guide. ** The degrees of freedom for MLMV, ULSMV and WLSMV are estimated according to a formula given in the Mplus Technical Appendices at www.statmodel.com. See degrees of freedom in the index of the Mplus User's Guide. Chi-Square Test of Model Fit for the Baseline Model Value 33.821 Degrees of Freedom 5 P-Value 0.0000 CFI/TLI CFI 1.000 TLI 1.000 Number of Free Parameters 6 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 WRMR (Weighted Root Mean Square Residual) Value 0.005
MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value ADMIT ON GRE 0.001 0.001 2.122 0.034 GPA 0.478 0.189 2.529 0.011 RANK1 0.936 0.248 3.781 0.000 RANK2 0.520 0.211 2.464 0.014 RANK3 0.124 0.224 0.553 0.580 Thresholds ADMIT$1 3.315 0.670 4.950 0.000 R-SQUARE Observed Residual Variable Estimate Variance ADMIT 0.165 1.000
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 could 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:\data\binary.dat ; Variable: names = admit gre gpa rank rank1 rank2 rank3 rank4; categorical = admit; usevariables = admit gre gpa rank1 rank2 rank3; 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 the part of the output that is associated with the model test command.
Wald Test of Parameter Constraints Value 21.132 Degrees of Freedom 3 P-Value 0.0001
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. The test statistic is 21.132, 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:\data\binary.dat; Variable: names = admit gre gpa rank rank1 rank2 rank3 rank4; categorical = admit; usevariables = admit gre gpa rank1 rank2 rank3; 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).
Wald Test of Parameter Constraints Value 5.682 Degrees of Freedom 1 P-Value 0.0171
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.
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