Logit Regression


Description Usage Output Method See Also

Description

Logit Regression attempts to fit the logit distribution to the specified data. Logit is most often used in fitting binomial distributions where the data of interest can have two states. The logit function is defined as:

logit(p) = log(p) - log(1-p)
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 Logit regression can be started by clicking Logit... in the Data Window's Regression menu. The Logit regression dialog appears below:
Logit Regression Window
To perform a logit 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 Logit regression should be binomial, and must lie between 0 and 1.
Logit Regression Menu
The Draco Logit regression is performed using an iterative technique. Prior to executing the Logit 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 Logit 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 Logit regression.

After performing the regression, the independent and dependent check boxes in the Logit 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 Logit 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 Logit regression is shown below:

Results of the Logit MLE Regression Model

Regression Variable: Married_Or_Not

Sum Squared of the Residuals:
1.0482E03
Standard Error of the Fit:
0.39149
R-Squared Value:
0.80507
Adjusted R-Squared Value:
0.80504

Coefficient Value Std. Err. t-Score
Constant
0.59012
0.05367
10.99615
Family_Income
.9299E-05
.3187E-06
14.63489


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 Logit 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 logit regression is defined as:

Logit Likelihood

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
Probit Regression

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