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.
Examples
Example 1: Suppose that we are interested in the factors that influence whether a political candidate wins an election. The outcome 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.
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 binary.sav. You can store this anywhere you like, but our examples will assume it has been stored in c:data. First, we read the data file into SPSS.
get file = "c:\data\probit.sav".
This data set has a binary response (outcome, dependent) variable called admit. There are three predictor variables: gre, gpa and rank. We will treat the variables gre and gpa as continuous. The variable rank is ordinal, it 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. We will treat rank as categorical. Lets start by looking at descriptive statistics.
descriptives /variables=gre gpa.
Descriptive Statistics
N Minimum Maximum Mean Std. Deviation
gre 400 220 800 587.70 115.517
gpa 400 2.26 4.00 3.3899 .38057
Valid N (listwise) 400
frequencies /variables = rank admit.
Statistics
rank admit
N Valid 400 400
Missing 0 0
Frequency Table
rank
Frequency Percent Valid Percent Cumulative Percent
Valid 1 61 15.3 15.3 15.3
2 151 37.8 37.8 53.0
3 121 30.3 30.3 83.3
4 67 16.8 16.8 100.0
Total 400 100.0 100.0
admit
Frequency Percent Valid Percent Cumulative Percent
Valid 0 273 68.3 68.3 68.3
1 127 31.8 31.8 100.0
Total 400 100.0 100.0
crosstabs /tables = admit by rank.
Case Processing Summary
Cases
Valid Missing Total
N Percent N Percent N Percent
admit * rank 400 100.0% 0 .0% 400 100.0%
admit * rank Crosstabulation
Count
rank Total
1 2 3 4
admit 0 28 97 93 55 273
1 33 54 28 12 127
Total 61 151 121 67 400
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.
- Probit regression, the focus of this page.
- Logistic regression. A logit model will produce results similarprobit regression. The choice of probit versus logit depends largely on
individual preferences.
- OLS regression. When used with a binary response variable, this model is knownas 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 T2. 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.”
Probit regression
Below we use the plum command with the subcommand /link=probit to run a probit regression model. After the command name (plum), the outcome variable (admit) is followed with by rank which indicates that rank is a categorical predictor, followed by with gre gpa, indicating that the predictors gre and gpa should be treated as continuous.
plum admit BY rank WITH gre gpa /link=probit /print= parameter summary.
The output from the plum command is broken into several sections, each of which is discussed below
Case Processing Summary
N Marginal Percentage
admit 0 273 68.3%
1 127 31.8%
rank 1 61 15.3%
2 151 37.8%
3 121 30.3%
4 67 16.8%
Valid 400 100.0%
Missing 0
Total 400
- The plum output is labeled as an ordinal regression, however, we can confirm below (see the note in the next set of tables) that the probit link function was used. Note that a model with a binary outcome can be view as a special case of an ordinal model, where there are only two categories.
- The table above includes frequencies for the two categorical variables admit (the outcome) and rank (one of the predictors).
- We can see that all 400 observations have been used. Fewer observations would have been used if any of our variables had missing values.
Model Fitting Information
Model -2 Log Likelihood Chi-Square df Sig.
Intercept Only 493.620
Final 452.057 41.563 5 .000
Link function: Probit.
Pseudo R-Square
Cox and Snell .099
Nagelkerke .138
McFadden .083
Link function: Probit.
- The table labeled Model Fitting Information includes two rows, one for the model we requested (labeled Final) and one for a so called null model (Intercept Only). The -2 log likelihoods can be used to compare model fit of the two models. The final -2 log likelihood for our model is 452.057. The intercept-only model has a -2 log likelihood of 493.620.
- The chi-square test statistic of 41.563 is the difference between the two -2 log likelihoods, this test statistic, with 5 degrees of freedom and an associated p-value of less than 0.0004 tells us that the current model fits better than a model with just an intercept.
- Pseudo-R-squared values are another way to assess model fit. Three different pseudo-R-squared are given in the output, but many different measures of pseudo-R-squareds 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?
Parameter Estimates
Estimate Std. Error Wald df Sig. 95% Confidence Interval
Lower Bound Upper Bound
Threshold [admit = 0] 3.323 .663 25.090 1 .000 2.023 4.623
Location gre .001 .001 4.478 1 .034 .000 .003
gpa .478 .197 5.869 1 .015 .091 .864
[rank=1] .936 .245 14.560 1 .000 .455 1.417
[rank=2] .520 .211 6.091 1 .014 .107 .934
[rank=3] .124 .224 .305 1 .581 -.315 .563
[rank=4] 0a . . 0 . . .
Link function: Probit.
a. This parameter is set to zero because it is redundant.
- In the table labeled Parameter Estimates, we see the coefficients, their standard errors, Wald test statistic
with associated df and p-values,
and the 95% confidence interval of the coefficients. The variables
gre, gpa, and the terms for
rank=1 and
rank=2 are statistically significant. The probit regression coefficients give the change in the z-score
(also called the probit index) for a one unit change in the predictor.
- For a one unit increase in gre, the z-score increases by 0.001.
- For each one unit increase in gpa, the z-score increases by 0.478.
- The terms for 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 reference group), increases the z-score by 0.936.
We may also want to test the overall effect of rank, we can do this using the test subcommand. The test subcommand is followed by the name of the variable we wish to test (i.e., rank), and then one value for each level of that variable (including the omitted category). The first line of the test subcommand rank 1 0 0 0 indicates that we want to test that the coefficient for rank=1 is 0. To perform a multiple degree of freedom test, we include multiple lines in the test subcommand, all but the last line is separated by a semicolon. The second and third rows indicate that we wish to test that the coefficients for rank=2 and rank=3 are equal to 0. Note that there is no need to include a row for the fourth category of rank.
plum admit by rank with gre gpa /link=probit /print= parameter summary /test rank 1 0 0 0; rank 0 1 0 0; rank 0 0 1 0.
Because the models are the same, most of the output produced by the above plum command is the same as before. The only difference is the additional output produced by the test subcommand, only this portion of the output is shown below.
Custom Hypothesis Tests 1
Contrast Coefficients
C1 C2 C3
Threshold [admit = 0] 0 0 0
Location gre 0 0 0
gpa 0 0 0
[rank=1] 1 0 0
[rank=2] 0 1 0
[rank=3] 0 0 1
[rank=4] 0 0 0
Contrast Results
Contrasts Estimate Std. Error Test value Wald df Sig. 95% Confidence Interval
Lower Bound Upper Bound
C1 .936 .245 0 14.560 1 .000 .455 1.417
C2 .520 .211 0 6.091 1 .014 .107 .934
C3 .124 .224 0 .305 1 .581 -.315 .563
Link function: Probit.
Test Results
Wald df Sig.
21.361 3 .000
Link function: Probit.
- The first table above, labeled Contrast Coefficients, shows the hypotheses we are testing.
- The second table gives the contrast results, because each row in the test subcommand tests that a coefficient in the model is equal to 0, these estimates, standard errors, etc. are equal to those from the table labeled Parameter Estimates in the main part of the output. The one difference in this table is that the column labeled Test which explicitly gives the null hypothesis, in our case, that each of the terms is equal to 0. (Note that other null hypotheses can be specified.)
- The final table produced by the test subcommand, labeled Test Results, gives the multiple degree of freedom test we are interested in, the Wald test statistic of 21.361, with 3 degrees of freedom, and an associated p-value of less than 0.001, tells us that the overall effect of rank is statistically significant.
The table labeled Parameter Estimates gives hypothesis tests for differences between each level of rank and the reference category. We can use the test subcommand to test for differences between the other levels of rank. For example, we might want to test for a difference in coefficients for rank=2 and rank=3. In the syntax below we have added a second test subcommand. This time, the values given are 0 1 -1 0 this indicates that we want to calculate the difference between the coefficients for rank=2 and rank=3 (i.e., rank=2 – rank=3).
plum admit by rank with gre gpa /link=probit /print= parameter summary /test rank 1 0 0 0; rank 0 1 0 0; rank 0 0 1 0 /test rank 0 1 -1 0.
Again the output from the model, as well as the output associated with the first test subcommand are identical to those shown above, so they are omitted.
Custom Hypothesis Tests 2
Contrast Coefficients
C1
Threshold [admit = 0] 0
Location gre 0
gpa 0
[rank=1] 0
[rank=2] 1
[rank=3] -1
[rank=4] 0
Contrast Results
Contrasts Estimate Std. Error Test value Wald df Sig. 95% Confidence Interval
Lower Bound Upper Bound
C1 .397 .168 0 5.573 1 .018 .067 .726
Link function: Probit.
In the table labeled Contrast Results we see the difference in the coefficients (i.e., 0.397). The Wald test statistic of 5.573, with one degree of freedom, and associated p-value of less than 0.02, indicates that the difference between the coefficients for rank=2 and rank=3 is statistically significant. Because only one estimate was specified in the test subcommand, the multiple degree of freedom test (i.e. the Test Results table) is not printed.
Things to consider
- Empty cells or small cells: You should check for empty or smallcells 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 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.
See also
- Annotated Output for Probit Regression
- Stat Books for Loan, Logistic Regression and Limited Dependent Variables
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.