This example demonstrates two algorithms of automatic variable selection in the TCOUNTREG procedure. This method is most effective
when the number of possible candidates for explaining the variation of some variable is large. For clarity of exposition,
this example uses only a small number of variables. The data set ARTICLE published by Long (1997) contains six variables.
This data set was already used in ZIP and ZINB Models for Data Exhibiting Extra Zeros. The dependent variable called art
records the number of articles published by a graduate student in the last three years of their program. Explanatory factors
include sex of a student (fem
), his or her marital status (mar
), number of children (kid5
), prestige of the program (phd
), and publishing activity of the academic adviser (ment
). All these variables intuitively suggest their affect on students’ primary academic output.
First, for comparison purposes, estimate the simple Poisson model. The choice of model is specified by DIST= option in the MODEL statement.
proc tcountreg data = long97data; model art = fem mar kid5 phd ment / dist = poisson; run;
The output of these statements is shown in Output 30.3.1.
Output 30.3.1: Poisson Model for the Number of Published Articles
Model Fit Summary | |
---|---|
Dependent Variable | art |
Number of Observations | 915 |
Data Set | WORK.LONG97DATA |
Model | Poisson |
Optimization Method | Newton-Raphson |
Log Likelihood | -1651 |
Maximum Absolute Gradient | 3.5741E-9 |
Number of Iterations | 5 |
AIC | 3314 |
SBC | 3343 |
Algorithm converged. |
Parameter Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | t Value | Approx Pr > |t| |
Intercept | 1 | 0.304617 | 0.102982 | 2.96 | 0.0031 |
fem | 1 | -0.224594 | 0.054614 | -4.11 | <.0001 |
mar | 1 | 0.155243 | 0.061375 | 2.53 | 0.0114 |
kid5 | 1 | -0.184883 | 0.040127 | -4.61 | <.0001 |
phd | 1 | 0.012823 | 0.026397 | 0.49 | 0.6271 |
ment | 1 | 0.025543 | 0.002006 | 12.73 | <.0001 |
Note that the Newton-Raphson optimization algorithm took five steps to converge. All parameters, except for one, are significant
at a 1% or 5% level, while phd
is not significant even at the 10% level.
In this case, it might be easy to identify variables with limited explanatory power. However, if the number of variables were large, the manual selection could be time demanding and inaccurate. For a large number of variables, you would be better off by applying one of the automatic algorithms of variable selection. The following statements use the penalized likelihood method, which is indicated by SELECT=PEN option in the MODEL statement:
proc tcountreg data = long97data method = qn; model art = fem mar kid5 phd ment / dist = poisson select = PEN; run;
The output of these statements is shown in Output 30.3.2.
Output 30.3.2: Poisson Model for the Number of Published Articles with Penalized Likelihood Method
Model Fit Summary | |
---|---|
Dependent Variable | art |
Number of Observations | 915 |
Data Set | WORK.LONG97DATA |
Model | Poisson |
Optimization Method | Quasi-Newton |
Log Likelihood | -1651 |
Maximum Absolute Gradient | 4.20414E-6 |
Number of Iterations | 7 |
AIC | 3312 |
SBC | 3336 |
Algorithm converged. |
Parameter Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | t Value | Approx Pr > |t| |
Intercept | 1 | 0.345174 | 0.060125 | 5.74 | <.0001 |
fem | 1 | -0.225303 | 0.054615 | -4.13 | <.0001 |
mar | 1 | 0.152175 | 0.061067 | 2.49 | 0.0127 |
kid5 | 1 | -0.184993 | 0.040139 | -4.61 | <.0001 |
ment | 1 | 0.025761 | 0.001950 | 13.21 | <.0001 |
The “Parameter Estimates” table shows that the variable was dropped from the model.
The next statements use the information criterion by specifying the SELECT=INFO option.The direction of search is chosen to be FORWARD, and the information criterion is AIC. In order to achieve the same selection of variables as for the penalized likelihood method, 0.001 is specified for the percentage of decrease in the information criterion necessary for the algorithm to stop.
proc tcountreg data = long97data; model art = fem mar kid5 phd ment / dist = poisson select = INFO ( direction = forward criter = AIC lstop = 0.001 ); run;
The output of these statements is shown in Output 30.3.3.
Output 30.3.3: Poisson Model for the Number of Published Articles with Search Method Using Information Criterion
Variable Selection Information | ||||
---|---|---|---|---|
Step | Effect Entered | Effect Removed | AIC | SBC |
0 | Base Model | 3487.146950 | 3491.965874 | |
1 | ment | 3341.286487 | 3350.924335 | |
2 | fem | 3330.744604 | 3345.201376 | |
3 | kid5 | 3316.593036 | 3335.868733 | |
4 | mar | 3312.348824 | 3336.443445 |
Model Fit Summary | |
---|---|
Dependent Variable | art |
Number of Observations | 915 |
Data Set | WORK.LONG97DATA |
Model | Poisson |
Optimization Method | Newton-Raphson |
Log Likelihood | -1651 |
Maximum Absolute Gradient | 1.28434E-9 |
Number of Iterations | 0 |
AIC | 3312 |
SBC | 3336 |
Algorithm converged. |
Parameter Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | t Value | Approx Pr > |t| |
Intercept | 1 | 0.345174 | 0.060125 | 5.74 | <.0001 |
fem | 1 | -0.225303 | 0.054615 | -4.13 | <.0001 |
mar | 1 | 0.152175 | 0.061067 | 2.49 | 0.0127 |
kid5 | 1 | -0.184993 | 0.040139 | -4.61 | <.0001 |
ment | 1 | 0.025761 | 0.001950 | 13.21 | <.0001 |
From the output, it is clear that the same set of variables was chosen as the result of information criterion algorithm. Note that the forward optimization algorithm starts with the constant as the only explanatory variable.