### Example 83.1 Modeling Salaries of Major League Baseball Players

This example features the use of ODS Graphics in the process of building models by using the REG procedure and highlights the use of fit and influence diagnostics.

The `Sashelp.Baseball` 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). The following step displays in Output 83.1.1 the variables in the data set:

```proc contents varnum data=sashelp.baseball;
ods select position;
run;
```

Output 83.1.1: `Sashelp.Baseball` Data Set

The CONTENTS Procedure

Variables in Creation Order
# Variable Type Len Label
1 Name Char 18 Player's Name
2 Team Char 14 Team at the End of 1986
3 nAtBat Num 8 Times at Bat in 1986
4 nHits Num 8 Hits in 1986
5 nHome Num 8 Home Runs in 1986
6 nRuns Num 8 Runs in 1986
7 nRBI Num 8 RBIs in 1986
8 nBB Num 8 Walks in 1986
9 YrMajor Num 8 Years in the Major Leagues
10 CrAtBat Num 8 Career Times at Bat
11 CrHits Num 8 Career Hits
12 CrHome Num 8 Career Home Runs
13 CrRuns Num 8 Career Runs
14 CrRbi Num 8 Career RBIs
15 CrBB Num 8 Career Walks
16 League Char 8 League at the End of 1986
17 Division Char 8 Division at the End of 1986
18 Position Char 8 Position(s) in 1986
19 nOuts Num 8 Put Outs in 1986
20 nAssts Num 8 Assists in 1986
21 nError Num 8 Errors in 1986
22 Salary Num 8 1987 Salary in \$ Thousands
23 Div Char 16 League and Division
24 logSalary Num 8 Log Salary

Suppose you want to investigate whether you can model the players’ salaries for the 1987 season based on batting statistics for the previous season and lifetime batting performance. Since the variation in salaries is much greater for higher salaries, it is appropriate to apply a log transformation for this analysis. The following statements begin the analysis:

```ods graphics on;

proc reg data=sashelp.baseball;
id name team league;
model logSalary = nhits nruns nrbi nbb yrmajor crhits;
run;
```

Output 83.1.2 shows the default output produced by PROC REG. The number of observations table shows that 59 observations are excluded because they have missing values for at least one of the variables used in the analysis. The analysis of variance and parameter estimates tables provide details about the fitted model.

Output 83.1.2: Default Output from PROC REG

The REG Procedure
Model: MODEL1
Dependent Variable: logSalary Log Salary

 Number of Observations Read 322 263 59

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 6 121.53052 20.25509 60.56 <.0001
Error 256 85.62322 0.33447
Corrected Total 262 207.15373

 Root MSE R-Square 0.57833 0.5867 5.92722 0.5770 9.75719

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept Intercept 1 4.14614 0.13612 30.46 <.0001
nHits Hits in 1986 1 0.00663 0.00210 3.15 0.0018
nRuns Runs in 1986 1 0.00019890 0.00398 0.05 0.9602
nRBI RBIs in 1986 1 0.00125 0.00235 0.53 0.5947
nBB Walks in 1986 1 0.00672 0.00239 2.81 0.0054
YrMajor Years in the Major Leagues 1 0.07108 0.01925 3.69 0.0003
CrHits Career Hits 1 0.00023910 0.00014571 1.64 0.1020

Before you accept a regression model, it is important to examine influence and fit diagnostics to see whether the model might be unduly influenced by a few observations and whether the data support the assumptions that underlie the linear regression. To facilitate such investigations, you can obtain diagnostic plots by enabling ODS Graphics.

Output 83.1.3: Fit Diagnostics

Output 83.1.3 shows a panel of diagnostic plots. The plot of externally studentized residuals (RStudent) by leverage values reveals that there is one observation with very high leverage that might be overly influencing the fit produced. The plot of Cook’s D by observation also indicates two highly influential observations. To investigate further, you can use the PLOTS= option in the PROC REG statement as follows to produce labeled versions of these plots:

```proc reg data=sashelp.baseball
plots(only label)=(RStudentByLeverage CooksD);
id name team league;
model logSalary = nhits nruns nrbi nbb yrmajor crhits;
run;
```

Output 83.1.4 and Output 83.1.5 reveal that Pete Rose is the highly influential observation. You might obtain a better fit to the remaining data if you omit his statistics when building the model.

Output 83.1.4: Outlier and Leverage Diagnostics

Output 83.1.5: Cook’s D

The following statements use a WHERE statement to omit Pete Rose’s statistics when building the model. An alternative way to do this within PROC REG is to use a REWEIGHT statement. See Reweighting Observations in an Analysis for details about reweighting.

```proc reg data=sashelp.baseball
plots=(RStudentByLeverage(label) residuals(smooth));
where name^="Rose, Pete";
id name team league;
model logSalary = nhits nruns nrbi nbb yrmajor crhits;
run;
```

Output 83.1.6 shows the new fit diagnostics panel. You can see that there are still several influential and outlying observations. One possible reason for observing outliers is that the linear model specified is not appropriate to capture the variation in this data. You can often see evidence of an inappropriate model by observing patterns in plots of residuals.

Output 83.1.6: Fit Diagnostics

Output 83.1.7 shows plots of the residuals by the regressors in the model. When you specify the RESIDUALS(SMOOTH) suboption of the PLOTS option in the PROC REG statement, a loess fit is overlaid on each of these plots. You can see the same clear pattern in the residual plots for `YrMajor` and `CrHits`. Players near the start of their careers and players near the end of their careers get paid less than the model predicts.

Output 83.1.7: Residuals by Regressors

You can address this lack of fit by using polynomials of degree 2 for these two variables as shown in the following statements:

```data baseball;
set sashelp.baseball(where=(name^="Rose, Pete"));
YrMajor2 = yrmajor*yrmajor;
CrHits2  = crhits*crhits;
run;

proc reg data=baseball
plots=(diagnostics(stats=none) RStudentByLeverage(label)
CooksD(label) Residuals(smooth)
DFFITS(label) DFBETAS ObservedByPredicted(label));
id name team league;
model logSalary = nhits nruns nrbi nbb yrmajor crhits
yrmajor2 crhits2;
run;
ods graphics off;
```

Output 83.1.8 shows the analysis of variance and parameter estimates for this model. Note that the R-square value of 0.787 for this model is considerably larger than the R-square value of 0.587 for the initial model shown in Output 83.1.2.

Output 83.1.8: Output from PROC REG

The REG Procedure
Model: MODEL1
Dependent Variable: logSalary Log Salary

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 8 162.73473 20.34184 117.13 <.0001
Error 253 43.93712 0.17366
Corrected Total 261 206.67186

 Root MSE R-Square 0.41673 0.7874 5.92458 0.7807 7.03393

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept Intercept 1 3.78922 0.11581 32.72 <.0001
nHits Hits in 1986 1 -0.00012753 0.00159 -0.08 0.9363
nRuns Runs in 1986 1 0.00215 0.00288 0.75 0.4549
nRBI RBIs in 1986 1 0.00431 0.00172 2.51 0.0127
nBB Walks in 1986 1 0.00501 0.00173 2.90 0.0040
YrMajor Years in the Major Leagues 1 0.23908 0.03443 6.94 <.0001
CrHits Career Hits 1 0.00170 0.00027562 6.18 <.0001
YrMajor2   1 -0.01440 0.00165 -8.73 <.0001
CrHits2   1 -3.31739E-7 1.001272E-7 -3.31 0.0011

The plots of residuals by regressors in Output 83.1.9 and Output 83.1.10 show that the strong pattern in the plots for `CrMajors` and `CrHits` has been reduced, although there is still some indication of a pattern remaining in these residuals. This suggests that a quadratic function might be insufficient to capture dependence of salary on these regressors.

Output 83.1.9: Residuals by Regressors

Output 83.1.10: Residuals by Regressors

Output 83.1.11 show the diagnostics plots; three of the plots, with points of interest labeled, are shown individually in Output 83.1.12, Output 83.1.13, and Output 83.1.14. The STATS=NONE suboption specified in the PLOTS=DIAGNOSTICS option replaces the inset of statistics with a box plot of the residuals in the fit diagnostics panel. The observed by predicted value plot reveals a reasonably successful model for explaining the variation in salary for most of the players. However, the model tends to overpredict the salaries of several players near the lower end of the salary range. This bias can also be seen in the distribution of the residuals that you can see in the histogram, Q-Q plot, and box plot in Output 83.1.11.

Output 83.1.11: Fit Diagnostics

Output 83.1.12: Outlier and Leverage Diagnostics

Output 83.1.13: Observed by Predicted Values

Output 83.1.14: Cook’s D

The RStudent by leverage plot in Output 83.1.12 and the Cook’s D plot in Output 83.1.14 show that there are still a number of influential observations. By specifying the DFFITS and DFBETAS suboptions of the PLOTS= option, you obtain additional influence diagnostics plots shown in Output 83.1.15 and Output 83.1.16. See Influence Statistics for details about the interpretation DFFITS and DFBETAS statistics.

Output 83.1.15: DFFITS

Output 83.1.16: DFBETAS

You can continue this analysis by investigating how the influential observations identified in the various influence plots affect the fit. You can also use PROC ROBUSTREG to obtain a fit that is resistant to the presence of high leverage points and outliers.