Probit Regression


Description Usage Output Method See Also

Description

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:
Probit Window
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.
Probit Menu
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:

Maximum Likelihood Estimator Convergence

The likelihood function for the probit regression is defined as:

Probit Likelihood Equation

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.

See Also

Least Squares Fit
Iteration Controls
Logit Regression

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