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Probit Regression

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 outcome variable, admit/don't admit, is binary.

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 from our website by clicking on binary.sas7bdat.
You can store this anywhere you like, but our the example syntax assumes it has been
stored in **c:\data**.
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** 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 start out by looking at the data.

proc means data="c:\data\binary"; var gre gpa; run;The MEANS Procedure Variable N Mean Std Dev Minimum Maximum ------------------------------------------------------------------------------- GRE 400 587.7000000 115.5165364 220.0000000 800.0000000 GPA 400 3.3899000 0.3805668 2.2600000 4.0000000 -------------------------------------------------------------------------------proc freq data="c:\data\binary"; tables rank admit admit*rank; run;The FREQ Procedure Cumulative Cumulative RANK Frequency Percent Frequency Percent ---------------------------------------------------------- 1 61 15.25 61 15.25 2 151 37.75 212 53.00 3 121 30.25 333 83.25 4 67 16.75 400 100.00 Cumulative Cumulative ADMIT Frequency Percent Frequency Percent ---------------------------------------------------------- 0 273 68.25 273 68.25 1 127 31.75 400 100.00 Table of ADMIT by RANK ADMIT RANK Frequency| Percent | Row Pct | Col Pct | 1| 2| 3| 4| Total ---------+--------+--------+--------+--------+- 0 | 28 | 97 | 93 | 55 | 273 | 7.00 | 24.25 | 23.25 | 13.75 | 68.25 | 10.26 | 35.53 | 34.07 | 20.15 | | 45.90 | 64.24 | 76.86 | 82.09 | ---------+--------+--------+--------+--------+- 1 | 33 | 54 | 28 | 12 | 127 | 8.25 | 13.50 | 7.00 | 3.00 | 31.75 | 25.98 | 42.52 | 22.05 | 9.45 | | 54.10 | 35.76 | 23.14 | 17.91 | ---------+--------+--------+--------+--------+- Total 61 151 121 67 400 15.25 37.75 30.25 16.75 100.00

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. Logistic regression will produce results similar to probit 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 the grouping 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."

There are multiple ways to run a probit model in SAS, this page uses **proc logistic**
with **link=probit** on the **model**
statement. Alternative methods not shown on this page include using **proc probit**,
or **proc genmod**. The advantage of running the model using **proc logistic **is that it is
easier to specify the ordering of the categories than it is in **proc probit**.
One possible advantage of using **proc probit** is that it will produce graphs that
may help you
interpret and explain the model.

Below we run the probit regression model using **proc logistic**. To model 1s rather than 0s, we
use the **descending** option. We do this because by default, **proc logistic** models
0s rather than 1s, in this case that would mean predicting the probability of
not getting into graduate school (**admit**=0) versus getting in (**admit**=1).
Mathematically, the models are equivalent, but conceptually, it probably makes
more sense to model the probability of getting into graduate school versus not getting in. The **class** statement tells SAS that **rank** is a
categorical variable. The **parm=ref** option after the slash requests dummy
coding, rather than the default effects coding, for the levels of **rank**.
For more information on dummy versus effects coding in **proc logistic**, see
our FAQ page: In PROC LOGISTIC why aren't the coefficients consistent with the odds ratios?.
The **model** statement specifies that we are modeling the outcome **admit** as a
function of the predictor variables **gre**, **gpa**, and **rank**. The
**link=probit** option fits a probit model rather than the default logit
model.

proc logistic data=data.binary descending; class rank / param=ref ; model admit = gre gpa rank /link=probit; run;

The output from **proc logistic** is broken into several sections each of which is discussed below.

The LOGISTIC Procedure Model Information Data Set DATA.BINARY Written by SAS Response Variable ADMIT Number of Response Levels 2 Model binary probit Optimization Technique Fisher's scoring Number of Observations Read 400 Number of Observations Used 400 Response Profile Ordered Total Value ADMIT Frequency 1 1 127 2 0 273 Probability modeled is ADMIT=1. Class Level Information Class Value Design Variables RANK 1 1 0 0 2 0 1 0 3 0 0 1 4 0 0 0 Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied.

- The first part of the output tells us the file being analyzed (c:\data\binary) and the number of observations used. We see that all 400 observations in our data set were used in the analysis (fewer observations would have been used if any of our variables had missing values).
- We also see that SAS is modeling
**admit**using a binary probit model and that the probability that of**admit**= 1 is being modeled. (If we omitted the**descending**option, SAS would model**admit**being 0 and our coefficients would be reversed.)

Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 501.977 470.413 SC 505.968 494.362 -2 Log L 499.977 458.413 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 41.5633 5 <.0001 Score 40.1603 5 <.0001 Wald 38.6596 5 <.0001 Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq GRE 1 4.4767 0.0344 GPA 1 5.8685 0.0154 RANK 3 21.3611 <.0001

- The portion of the output labeled Model Fit Statistics describes and tests the overall fit of the model. The -2 Log L (499.977) can be used in comparisons of nested models, but we won't show an example of that here.
- In the next section of output, the likelihood ratio chi-square of 41.5633 with a p-value of 0.0001 tells us that our model as a whole fits significantly better than an empty model. The Score and Wald tests are asymptotically equivalent tests of the same hypothesis tested by the likelihood ratio test, not surprisingly, these tests also indicate that the model is statistically significant.
- The section labeled Type 3 Analysis of Effects, shows the hypothesis
tests for each of the variables in the model individually. The chi-square
test statistics and associated p-values shown in the table indicate that
each of the three variables in the model significantly improve the fit of
the model. For continuous variables, such as
**gre**, and**gpa**, this test duplicates the test of the coefficients shown below. However, for class variables, such as**rank**, this table gives the multiple degree of freedom test for the overall effect of the variable.

Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -3.3225 0.6633 25.0872 <.0001 GRE 1 0.00138 0.000650 4.4767 0.0344 GPA 1 0.4777 0.1972 5.8685 0.0154 RANK 1 1 0.9359 0.2453 14.5606 0.0001 RANK 2 1 0.5205 0.2109 6.0904 0.0136 RANK 3 1 0.1237 0.2240 0.3053 0.5806

- The above table shows the coefficients (labeled Estimate), their
standard errors (error), the Wald Chi-Square statistic, and associated
p-values. The coefficients for
**gre**, and**gpa**are statistically significant, as are the terms for**rank**=1 and**rank**=2 (versus the omitted category**rank**=4). The probit regression coefficients give the change in the probit index, also called a z-score, for a one unit increase in the predictor variable.- For every one unit change in
**gre**, the z-score increases by 0.001. - For a one unit increase in gpa, the z-score increases by 0.478.
- 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, increases the z-score by 0.936.

- For every one unit change in

Association of Predicted Probabilities and Observed Responses Percent Concordant 69.1 Somers' D 0.385 Percent Discordant 30.6 Gamma 0.387 Percent Tied 0.4 Tau-a 0.167 Pairs 34671 c 0.693

The table above gives information about the relationship between the predicted probabilities from our model, and the actual outcomes in our data.

The output shown above gives a test for the overall effect of **rank** as well as coefficients
that describe the difference between the reference group (**rank**=4) and each of the other
three groups. We can also 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. We can test this type of hypothesis by adding a **contrast**
statement to the code for **proc logistic**. The syntax shown below is the same
as that shown above, except that it uses the **contrast** statement.
Following the word contrast, is the label that will appear in the output,
enclosed in single quotes (i.e. **'rank 2 vs. rank 3'**). This is followed by
the name of the variable we wish to test hypotheses about (i.e. **rank**),
and a vector (i.e. **0 1 -1**) that describes the desired comparison. In this
case the value computed is the difference between the coefficients for **rank**=2 and **rank**=3.
After the slash (i.e. **/ **) we use the **estimate = parm** option to
request that the estimate be the difference in coefficients. For
more information on the contrast statement, see our
FAQ page
How can I create contrasts with proc logistic?.

proc logistic data=data.binary descending; class rank / param=ref ; model admit = gre gpa rank /link=probit; contrast 'rank 2 vs. 3' rank 0 1 -1 / estimate=parm; run;Contrast Rows Estimation and Testing Results Standard Wald Contrast Type Row Estimate Error Alpha Confidence Limits Chi-Square Pr > ChiSq rank 2 vs. 3 PARM 1 0.3967 0.1681 0.05 0.0673 0.7261 5.5725 0.0182

Because the models are the same, most of the output produced by the above **proc logistic**
command is the same as before. The only difference is the additional output produced by the **contrast**
statement (shown above). Under the heading Contrast Test Results we see the label for the
contrast (rank 2 vs 3) along with its degrees of freedom, Wald chi-square
statistic, and p-value. Based on the p-value in this table we know that the coefficient for **
rank**=2 is significantly different from the coefficient for **rank**=3.
The second table, shows more detailed information, including the actual
estimate of the difference (under Estimate), it's standard error, confidence
limits, test statistic, and p-value. We can see that the estimated
difference was 0.3967, indicating that having attended an undergraduate
institution with a rank of 2,
versus an institution with a rank of 3, increases the z-score by 0.4.

You can also use predicted probabilities to help you understand the model.
The **contrast** statement can be used to estimate predicted probabilities by
specifying **estimate=prob**. In the syntax below we use multiple contrast
statements to estimate the predicted probability of admission as **gre**
changes from 200 to 800 (in increments of 100). When estimating the predicted
probabilities we hold **gpa** constant at
3.39 (its mean), and **rank** at 2. The word **intercept** followed by a **1** indicates that the intercept for
the model is to be included in estimate.

proc logistic data=data.binary descending; class rank / param=ref ; model admit = gre gpa rank /link=probit; contrast 'gre=200' intercept 1 gre 200 gpa 3.3899 rank 0 1 0 / estimate=prob; contrast 'gre=300' intercept 1 gre 300 gpa 3.3899 rank 0 1 0 / estimate=prob; contrast 'gre=400' intercept 1 gre 400 gpa 3.3899 rank 0 1 0 / estimate=prob; contrast 'gre=500' intercept 1 gre 500 gpa 3.3899 rank 0 1 0 / estimate=prob; contrast 'gre=600' intercept 1 gre 600 gpa 3.3899 rank 0 1 0 / estimate=prob; contrast 'gre=700' intercept 1 gre 700 gpa 3.3899 rank 0 1 0 / estimate=prob; contrast 'gre=800' intercept 1 gre 800 gpa 3.3899 rank 0 1 0 / estimate=prob; run;

Contrast Test Results Wald Contrast DF Chi-Square Pr > ChiSq gre=800 1 0.2452 0.6205 Contrast Rows Estimation and Testing Results Standard Wald Contrast Type Row Estimate Error Alpha Confidence Limits Chi-Square Pr > ChiSq gre=200 PROB 1 0.1821 0.0746 0.05 0.0720 0.3615 10.3375 0.0013 gre=300 PROB 1 0.2206 0.0662 0.05 0.1136 0.3698 11.8864 0.0006 gre=400 PROB 1 0.2635 0.0552 0.05 0.1676 0.3816 14.0168 0.0002 gre=500 PROB 1 0.3103 0.0442 0.05 0.2296 0.4014 15.6564 <.0001 gre=600 PROB 1 0.3604 0.0396 0.05 0.2861 0.4404 11.4014 0.0007 gre=700 PROB 1 0.4130 0.0480 0.05 0.3222 0.5087 3.1785 0.0746 gre=800 PROB 1 0.4672 0.0661 0.05 0.3415 0.5963 0.2452 0.6205

As with the previous example, we have omitted most of the **proc logistic** output, because it is the same as
before. The predicted probabilities are included in the column labeled Estimate in the second table
in the output. Looking at the estimates, we can see that the predicted
probability of being admitted is only 0.18 if one's **gre** score is 200,
but increases to 0.47 if one's gre score is 800, holding **gpa** at its mean
(3.39), and **rank** at 2.

- 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 (available
with the
**exact**option in**proc logistic**). 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 psuedo-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.
- By default,
**proc logistic**models the probability of the lower valued category (0 if your variable is coded 0/1), rather than the higher valued category.

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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