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SPSS Data Analysis Examples
Probit Regression

Examples

Example 2:  Some people have heart attacks and others don't.  We would like to see if exercise, age and gender influences whether or not someone has a heart attack.  Again, we have a binary outcome:  have heart attack or not. 

Example 3:  Many undergraduates wish to continue their education in graduate school.  In their application to any given graduate program, they include their GRE scores and their GPA from their undergraduate institution.  Some students are graduating from very prestigious institutions, while others are graduating from not-so-prestigious institutions.  Many months after sending in their applications, students receive either a thick or a thin envelope from the graduate program to which they applied:  some were admitted and others were not.

Description of the Data

get file = "c:\data\probit.sav".

This hypothetical data set has a 0/1 variable called admit that we will use as our response (i.e., outcome, dependent) variable.  We also have three variables that we will use as predictors:  gre, which is the student's Graduate Record Exam score; gpa, which is the student's grade point average; and topnotch, which is a 0/1 variable where 1 indicates that the undergraduate institution was "top notch" and 0 indicates that it is not. 

descriptives /variables=gre gpa.

frequencies /variables=topnotch.

Some Strategies You Might Try

• OLS regression:  This analysis is problematic because the assumptions of OLS are violated when it is used with a non-interval outcome variable.
• t-test:  If you use only one continuous predictor, you could "flip" the model around so that, say, gpa was the outcome variable and admit was the predictor variable.  Then you could run a t-test.  This isn't a bad thing to do if you only have one predictor variable and it is continuous.
• Logit regression:  This is very, very similar to doing probit regression.  The similarities and differences between these two analyses will be discussed shortly.

Using the Probit Model

Before we run our probit model, we will see if any cells (created by the crosstab of our categorical and response variables) are empty or particularly small.  If any are, we may have difficulty running our model. 

crosstabs /tables=admit by topnotch.

None of the cells is too small or empty (has no cases), so we will run our model.

PLUM

admit WITH gre topnotch gpa

/LINK = PROBIT

/PRINT = PARAMETER SUMMARY .

In the output the first thing we see is a warning that about 48% of cells have zero frequencies. There are empty cells because there are continuous variables (gre and gpa) in our model. However, we checked to be sure there were no empty cells when we did the crosstab of the categorical predictor variable (topnotch) and our outcome variable (admit). Because we did this, and everything looked okay, we will continue with the analysis.

 This shows the number of observations and the coding for the outcome variable, admit. We see that all 400 observations have been used. Fewer observations would have been used if any of our variables had missing values.  By default, SPSS does a listwise deletion of cases with missing values. 

The Model Fitting Information table contains information on a model containing an intercept only (labeled "Intercept Only" in the left column) and the model we asked SPSS to estimate (labeled "Final").  Looking at the column labeled "-2 Log Likelihood" we can compare the values for the intercept only model and the final model, to determine if including gre, topnotch, and gpa in the model improves the fit of the model. To do this, we could subtract the -2 log likelihood for the final model from the same for the intercept only model, but SPSS has done this for us and placed it in the next column over, labeled "Chi-Square" because this value is distributed chi-square with degrees of freedom equal to the number of independent variables in the model (in this case, 3). The likelihood ratio chi-square with a p-value of less than 0.0004 tells us that our model as a whole fits significantly better than the model with no predictors (the "intercept only" model).

Three different pseudo-R-squared are given, there are many different types of pseudo-R-squareds.  It is a pseudo-R-squared because there is no direct equivalent of an R-squared (from OLS regression) in non-linear models.  Because it is an approximation to the R-squared values you see in OLS, each type of pseudo-R-squared will give a different estimate.

 In the table we see the coefficients, their standard errors, the Wald tests, df (degrees of freedom), the associated p-values (labeled "Sig."), and the 95% confidence interval of the coefficients.  Both gre and gpa are statistically significant; topnotch is not.  A discussion of the interpretation of the coefficients can be found in the sample write up section below.

Sample Write-up of the Analysis

Below is one way of describing the results.  Please note that the coefficients can be discussed in terms of either Z-scores or probit index.  These are equivalent terms. Note that they can not be interpreted the way OLS coefficients are interpreted.

The Z-score of a person with a zero GRE score and zero GPA at a non-topnotch school is about -2.8.  For each point of increase in GRE score, the Z-score is increased by .0015244; for each point of increase in GPA, the probit index increases by .4.

Similarities and differences between logit and probit models

Neither the logit model nor the probit model are linear, which makes things difficult.  To make the model linear, a transformation is done on the dependent variable.  In logit regression, the transformation is the logit function which is the natural log of the odds.  In probit models, the function used is the inverse of the standard normal cumulative distribution (a.k.a. a z-score).  In reality, this difference isn't too important:  both transformations are equally good at linearizing the model; which one you use is a matter of personal preference.  Both models need to have diagnostics done afterwards to check that the assumptions of the model have not been violated.  Both methods use maximum likelihood, and so require more cases than a similar OLS model.  Unlike logit models, you don't get odds ratios with probit models.  In general, the logit coefficients are larger than the probit coefficients by a factor of 1.7.  However, this rule often does not apply when an independent variable has a high standard error (lots of variability).

Cautions, Flies in the Ointment

• Perfect prediction:  Perfect prediction means that only one value of a predictor variable is associated with only one value of the response variable.  If this happens, Stata will usually issue a note at the top of the output and will drop the cases so that the model can run.
• Sample size:  Both logit and probit models require more cases than OLS regression because they are maximum likelihood techniques.
• 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.
• Pseudo-R-squared:  There is no analog of the R-squared found in OLS.  There are many versions of pseudo-R-squares, but the emphasis should be put on "pseudo."
• Diagnostics:  Diagnostics still have to be done to check that the assumptions of the probit analysis have not been violated, but these are different from the diagnostics done in OLS regression and they are often more tricky.  They are the same as those done for logit regression.

See Also

• Regression Models for Categorical and Limited Dependent Variables, Revised Edition by J. Scott Long (2 copies)

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