Probit Regression attempts to fit the cumultive distribution function
of the standard normal distribuion to the
specified data. Probit is most often used in fitting binomial
distributions where the data of interest can have two states.
The regression is performed using the Maximum Likelihood Estimator
technique, as described in the Method
section. The coefficients resulting from the regression show the
influence of variables on the binomial probabilities.
Usage
The Probit regression can be started by clicking Probit...
in the Data
Window's Regression menu. The Probit regression dialog
appears
below:
To perform a proit regression, the appropriate data must be
chosen. To select a variable as either dependent or independent, click
the check box associated with each. Only one variable may be chosen as
the dependent variable. Note that the dependent variable in a Probit
regression should be binomial.
The Draco Probit regression is performed using an iterative technique.
Prior to executing the Probit regression, the convergence
criteria can
be modified from the Options menu. Select Iterations and
Convergence... from the Options menu. The resulting
dialog
will allow the setting of the maximum number of iterations and the
convergence criteria; the defaults should be sufficient for most
well-formed problems.
From the Probit regression's menu, choose Compute
Regression...
from the Perform Fit menu. This menu item launches an iterative
progress dialog to provide updates on the fitting procedure. When the
dialog signals that computations are complete, press the Close
button to view the results of the Probit regression.
After performing the regression, the independent and dependent check
boxes in the Probit regression window will no longer be
changeable.
Also, the "Options" menu associated with this regression will
be deactivated. The results window can be closed at any time;
to reopen the results, simply select the "Compute Regression..." menu
item again.
Output
The Probit regression will generate a table of coefficient
values as
well as the errors asociated with each. The results of the
fit, including the R2 value are also output.
An example from a Probit regression is shown below:
Results of the Probit MLE Regression Model
Regression Variable:
postsec
Sum Squared of the Residuals:
1.3919E03
Standard Error of the Fit:
0.43562
R-Squared Value:
0.58916
Adjusted R-Squared Value:
0.5891
Coefficient
Value
Std. Err.
t-Score
Constant
-0.96955
0.03615
-26.82159
asvab_percentile
0.02309
0.00079
29.24396
Further information can be generated after the regression. By
selecting Generate Column Data from the Perform Fit
menu, the
estimated dependent variable data will be output as a new variable in
the Data window. The resulting covariance matrix can be
viewed by selecting View Covariance Matrix... from
the Supporting
Data menu. The residuals from the regression can also be
output to the Data window as a new variable by selecting Output
Residuals to Column from the Supporting Data menu.
Method
The Probit Regression is performed using a Maximum Likelihood Estimator
technique. Initial estimates of the parameters are computed
from
a simple Least Squares Fit of the data. All subsequent
iterations
then perform updates to these parameters in an attempt to drive the
partial derivatives of the likelihood function with respect to the
parameters to zero. Therefore, the estimates of the
parameters
are converged when:
The likelihood function for the probit regression is defined as:
The likelihood function in Draco is maximized using a Nelder-Mead optimization algorithm built into the Apache Commons Math library. The derivatives of the likelihood function are numerically computed within the algortihm.
The covariance matrix as computed in the logit regression is
represented by the inverse of the Hessian matrix associated with the
logit function. The Hessian matrix is numerically computed within Draco.