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 $ 118 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); datalines; 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; run; 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."
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.
Dimensions  

Number of Effects  19 
Number of Parameters  21 
Data Set  WORK.BASEBALL 

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.
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.
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.
Stepwise Selection Summary  

Step  Effect Entered 
Effect Removed 
Number Effects In 
Number Parms In 
Model RSquare 
Adjusted RSquare 
AIC  AICC  BIC  CP  SBC  PRESS  ASE  F Value  Pr > F 
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 
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.
Selection stopped at a local minimum of the SBC criterion. 
Stop Details  

Candidate For 
Effect  Candidate SBC 
Compare SBC 

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.
The selected effects, analysis of variance, fit statistics, and parameter estimates tables shown in Figure 44.10 give details of the selected model.
Effects:  Intercept nHits nBB yrMajor 

Analysis of Variance  

Source  DF  Sum of Squares 
Mean Square 
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 
RSquare  0.5818 
Adj RSq  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.