Getting Started: GLMSELECT Procedure

The following data set contains salary and performance information for Major League Baseball players who played at least one game in both the 1986 and 1987 seasons, excluding pitchers. The salaries (Sports Illustrated, April 20, 1987) are for the 1987 season and the performance measures are from 1986 (Collier Books, The 1987 Baseball Encyclopedia Update).

 data baseball;
   length name $ 18;
   length team $ 12;
   input name $ 1-18 nAtBat nHits nHome nRuns nRBI nBB
         yrMajor crAtBat crHits crHome crRuns crRbi crBB
         league $ division $ team $ position $ nOuts nAssts
         nError salary;
   label name="Player's Name"
      nAtBat="Times at Bat in 1986"
      nHits="Hits in 1986"
      nHome="Home Runs in 1986"
      nRuns="Runs in 1986"
      nRBI="RBIs in 1986"
      nBB="Walks in 1986"
      yrMajor="Years in the Major Leagues"
      crAtBat="Career times at bat"
      crHits="Career Hits"
      crHome="Career Home Runs"
      crRuns="Career Runs"
      crRbi="Career RBIs"
      crBB="Career Walks"
      league="League at the end of 1986"
      division="Division at the end of 1986"
      team="Team at the end of 1986"
      position="Position(s) in 1986"
      nOuts="Put Outs in 1986"
      nAssts="Assists in 1986"
      nError="Errors in 1986"
      salary="1987 Salary in $ Thousands";
      logSalary = log(Salary);
Allanson, Andy       293    66     1    30    29    14
                       1   293    66     1    30    29    14
                    American East Cleveland C 446 33 20 .
Ashby, Alan          315    81     7    24    38    39
                      14  3449   835    69   321   414   375
                    National West Houston C 632 43 10 475

   ... more lines ...   

Wilson, Willie       631   170     9    77    44    31
                      11  4908  1457    30   775   357   249
                    American West KansasCity CF 408 4 3 1000

Suppose you want to investigate whether you can model the players’ salaries for the 1987 season based on performance measures for the previous season. The aim is to obtain a parsimonious model that does not overfit this particular data, making it useful for prediction. This example shows how you can use PROC GLMSELECT as a starting point for such an analysis. Since the variation of salaries is much greater for the higher salaries, it is appropriate to apply a log transformation to the salaries before doing the model selection.

The following code selects a model with the default settings:

ods graphics on;
proc glmselect data=baseball plots=all;
  class league division; 
  model logSalary = nAtBat nHits nHome nRuns nRBI nBB
                    yrMajor crAtBat crHits crHome crRuns crRbi
                    crBB league division nOuts nAssts nError  
                  / details=all stats=all;
ods graphics off;

PROC GLMSELECT performs effect selection where effects can contain classification variables that you specify in a CLASS statement. The "Class Level Information" table shown in Figure 44.1 lists the levels of the classification variables "division" and "league."

Figure 44.1 Class Level Information
The GLMSELECT Procedure

Class Level Information
Class Levels Values
league 2 American National
division 2 East West

When you specify effects that contain classification variables, the number of parameters is usually larger than the number of effects. The "Dimensions" table in Figure 44.2 shows the number of effects and the number of parameters considered.

Figure 44.2 Dimensions
Number of Effects 19
Number of Parameters 21

Figure 44.3 Model Information
The GLMSELECT Procedure

Dependent Variable logSalary
Selection Method Stepwise
Select Criterion SBC
Stop Criterion SBC
Effect Hierarchy Enforced None

You find details of the default search settings in the "Model Information" table shown in Figure 44.3. The default selection method is a variant of the traditional stepwise selection where the decisions about what effects to add or drop at any step and when to terminate the selection are both based on the Schwarz Bayesian information criterion (SBC). The effect in the current model whose removal yields the maximal decrease in the SBC statistic is dropped provided this lowers the SBC value. Once no decrease in the SBC value can be obtained by dropping an effect in the model, the effect whose addition to the model yields the lowest SBC statistic is added and the whole process is repeated. The method terminates when dropping or adding any effect increases the SBC statistic.

Figure 44.4 Candidates for Entry at Step Two
Best 10 Entry Candidates
Rank Effect SBC
1 nHits -252.5794
2 nAtBat -241.5789
3 nRuns -240.1010
4 nRBI -232.2880
5 nBB -223.3741
6 nHome -208.0565
7 nOuts -205.8107
8 division -194.4688
9 crBB -191.5141
10 nAssts -190.9425

The DETAILS=ALL option requests details of each step of the selection process. The "Best 10 Entry Candidates" table at each step shows the candidates for inclusion or removal at that step ranked from best to worst in terms of the selection criterion, which in this example is the SBC statistic. By default only the 10 best candidates are shown. Figure 44.4 shows the candidate table at step two.

To help in the interpretation of the selection process, you can use graphics supported by PROC GLMSELECT. ODS Graphics must be enabled before requesting plots. For general information about ODS Graphics, see Chapter 21, Statistical Graphics Using ODS. With ODS Graphics enabled, the PLOTS=ALL option together with the DETAILS=STEPS option in the MODEL statement produces a needle plot view of the "Candidates" tables. The plot corresponding to the "Candidates" table at step two is shown in Figure 44.5. You can see that adding the effect "nHits" yields the smallest SBC value, and so this effect is added at step two.

Figure 44.5 Needle Plot of Entry Candidates at Step Two
Needle Plot of Entry Candidates at Step Two

The "Stepwise Selection Summary" table in Figure 44.6 shows the effect that was added or dropped at each step of the selection process together with fit statistics for the model at each step. The STATS=ALL option in the MODEL statement requests that all the available fit statistics are displayed. See the section Criteria Used in Model Selection Methods for descriptions and formulas. The criterion panel in Figure 44.7 provides a graphical view of the progression of these fit criteria as the selection process evolves. Note that none of these criteria has a local optimum before step five.

Figure 44.6 Selection Summary Table
The GLMSELECT Procedure

Stepwise Selection Summary
Step Effect
Effects In
Parms In
0 Intercept   1 1 0.0000 0.0000 204.2238 204.2699 -60.6397 375.9275 -57.2041 208.7381 0.7877 0.00 1.0000
1 crRuns   2 2 0.4187 0.4165 63.5391 63.6318 -200.7872 111.2315 -194.3166 123.9195 0.4578 188.01 <.0001
2 nHits   3 3 0.5440 0.5405 1.7041 1.8592 -261.8807 33.4438 -252.5794 97.6368 0.3592 71.42 <.0001
3 yrMajor   4 4 0.5705 0.5655 -12.0208 -11.7873 -275.3333 18.5870 -262.7322 92.2998 0.3383 15.96 <.0001
4   crRuns 3 3 0.5614 0.5581 -8.5517 -8.3967 -271.9095 22.3357 -262.8353 93.1482 0.3454 5.44 0.0204
5 nBB   4 4 0.5818 0.5770* -19.0690* -18.8356* -282.1700* 11.3524* -269.7804* 89.5434* 0.3294 12.62 0.0005
* Optimal Value Of Criterion

Figure 44.7 Criterion Panel
Criterion Panel

The stop reason and stop details tables in Figure 44.8 gives details of why the selection process terminated. This table shows that at step five the best add candidate, "division," and the best drop candidate, "nBB," yield models with SBC values of and , respectively. Both of these values are larger than the current SBC value of , and so the selection process stops at the model at step five.

Figure 44.8 Stopping Details
Selection stopped at a local minimum of the SBC criterion.

Stop Details
Effect Candidate
Entry division -268.6094 > -269.7804
Removal nBB -262.8353 > -269.7804

The coefficient panel in Figure 44.9 enables you to visualize the selection process. In this plot, standardized coefficients of all the effects selected at some step of the stepwise method are plotted as a function of the step number. This enables you to assess the relative importance of the effects selected at any step of the selection process as well as providing information as to when effects entered the model. The lower plot in the panel shows how the criterion used to choose the selected model changes as effects enter or leave the model.

Figure 44.9 Coefficient Progression
Coefficient Progression

The selected effects, analysis of variance, fit statistics, and parameter estimates tables shown in Figure 44.10 give details of the selected model.

Figure 44.10 Details of the Selected Model
The GLMSELECT Procedure
Selected Model

The selected model is the model at the last step (Step 5).

Effects: Intercept nHits nBB yrMajor

Analysis of Variance
Source DF Sum of
F Value
Model 3 120.52553 40.17518 120.12
Error 259 86.62820 0.33447  
Corrected Total 262 207.15373    

Root MSE 0.57834
Dependent Mean 5.92722
R-Square 0.5818
Adj R-Sq 0.5770
AIC -19.06903
AICC -18.83557
BIC -282.17004
C(p) 11.35235
PRESS 89.54336
SBC -269.78041
ASE 0.32938

Parameter Estimates
Parameter DF Estimate Standard Error t Value
Intercept 1 4.013911 0.111290 36.07
nHits 1 0.007929 0.000994 7.98
nBB 1 0.007280 0.002049 3.55
yrMajor 1 0.100663 0.007551 13.33

PROC GLMSELECT provides you with the flexibility to use several selection methods and many fit criteria for selecting effects that enter or leave the model. You can also specify criteria to determine when to stop the selection process and to choose among the models at each step of the selection process. You can find continued exploration of the baseball data that uses a variety of these methods in Example 44.1.