Stepwise Regression
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The PHREG Procedure

 

Example 66.1 Stepwise Regression

Krall, Uthoff, and Harley (1975) analyzed data from a study on multiple myeloma in which researchers treated 65 patients with alkylating agents. Of those patients, 48 died during the study and 17 survived. The following DATA step creates the data set Myeloma. The variable Time represents the survival time in months from diagnosis. The variable VStatus consists of two values, 0 and 1, indicating whether the patient was alive or dead, respectively, at the end of the study. If the value of VStatus is 0, the corresponding value of Time is censored. The variables thought to be related to survival are LogBUN (log(BUN) at diagnosis), HGB (hemoglobin at diagnosis), Platelet (platelets at diagnosis: 0=abnormal, 1=normal), Age (age at diagnosis, in years), LogWBC (log(WBC) at diagnosis), Frac (fractures at diagnosis: 0=none, 1=present), LogPBM (log percentage of plasma cells in bone marrow), Protein (proteinuria at diagnosis), and SCalc (serum calcium at diagnosis). Interest lies in identifying important prognostic factors from these nine explanatory variables. 

data Myeloma;                                                       
   input Time VStatus LogBUN HGB Platelet Age LogWBC Frac
         LogPBM Protein SCalc;
   label Time='Survival Time'
         VStatus='0=Alive 1=Dead';
   datalines;
 1.25  1  2.2175   9.4  1  67  3.6628  1  1.9542  12  10
 1.25  1  1.9395  12.0  1  38  3.9868  1  1.9542  20  18
 2.00  1  1.5185   9.8  1  81  3.8751  1  2.0000   2  15
 2.00  1  1.7482  11.3  0  75  3.8062  1  1.2553   0  12
 2.00  1  1.3010   5.1  0  57  3.7243  1  2.0000   3   9
 3.00  1  1.5441   6.7  1  46  4.4757  0  1.9345  12  10
 5.00  1  2.2355  10.1  1  50  4.9542  1  1.6628   4   9
 5.00  1  1.6812   6.5  1  74  3.7324  0  1.7324   5   9
 6.00  1  1.3617   9.0  1  77  3.5441  0  1.4624   1   8
 6.00  1  2.1139  10.2  0  70  3.5441  1  1.3617   1   8
 6.00  1  1.1139   9.7  1  60  3.5185  1  1.3979   0  10
 6.00  1  1.4150  10.4  1  67  3.9294  1  1.6902   0   8
 7.00  1  1.9777   9.5  1  48  3.3617  1  1.5682   5  10
 7.00  1  1.0414   5.1  0  61  3.7324  1  2.0000   1  10
 7.00  1  1.1761  11.4  1  53  3.7243  1  1.5185   1  13
 9.00  1  1.7243   8.2  1  55  3.7993  1  1.7404   0  12
11.00  1  1.1139  14.0  1  61  3.8808  1  1.2788   0  10
11.00  1  1.2304  12.0  1  43  3.7709  1  1.1761   1   9
11.00  1  1.3010  13.2  1  65  3.7993  1  1.8195   1  10
11.00  1  1.5682   7.5  1  70  3.8865  0  1.6721   0  12
11.00  1  1.0792   9.6  1  51  3.5051  1  1.9031   0   9
13.00  1  0.7782   5.5  0  60  3.5798  1  1.3979   2  10
14.00  1  1.3979  14.6  1  66  3.7243  1  1.2553   2  10
15.00  1  1.6021  10.6  1  70  3.6902  1  1.4314   0  11
16.00  1  1.3424   9.0  1  48  3.9345  1  2.0000   0  10
16.00  1  1.3222   8.8  1  62  3.6990  1  0.6990  17  10
17.00  1  1.2304  10.0  1  53  3.8808  1  1.4472   4   9
17.00  1  1.5911  11.2  1  68  3.4314  0  1.6128   1  10
18.00  1  1.4472   7.5  1  65  3.5682  0  0.9031   7   8
19.00  1  1.0792  14.4  1  51  3.9191  1  2.0000   6  15
19.00  1  1.2553   7.5  0  60  3.7924  1  1.9294   5   9
24.00  1  1.3010  14.6  1  56  4.0899  1  0.4771   0   9
25.00  1  1.0000  12.4  1  67  3.8195  1  1.6435   0  10
26.00  1  1.2304  11.2  1  49  3.6021  1  2.0000  27  11
32.00  1  1.3222  10.6  1  46  3.6990  1  1.6335   1   9
35.00  1  1.1139   7.0  0  48  3.6532  1  1.1761   4  10
37.00  1  1.6021  11.0  1  63  3.9542  0  1.2041   7   9
41.00  1  1.0000  10.2  1  69  3.4771  1  1.4771   6  10
41.00  1  1.1461   5.0  1  70  3.5185  1  1.3424   0   9
51.00  1  1.5682   7.7  0  74  3.4150  1  1.0414   4  13
52.00  1  1.0000  10.1  1  60  3.8573  1  1.6532   4  10
54.00  1  1.2553   9.0  1  49  3.7243  1  1.6990   2  10
58.00  1  1.2041  12.1  1  42  3.6990  1  1.5798  22  10
66.00  1  1.4472   6.6  1  59  3.7853  1  1.8195   0   9
67.00  1  1.3222  12.8  1  52  3.6435  1  1.0414   1  10
88.00  1  1.1761  10.6  1  47  3.5563  0  1.7559  21   9
89.00  1  1.3222  14.0  1  63  3.6532  1  1.6232   1   9
92.00  1  1.4314  11.0  1  58  4.0755  1  1.4150   4  11
 4.00  0  1.9542  10.2  1  59  4.0453  0  0.7782  12  10
 4.00  0  1.9243  10.0  1  49  3.9590  0  1.6232   0  13
 7.00  0  1.1139  12.4  1  48  3.7993  1  1.8573   0  10
 7.00  0  1.5315  10.2  1  81  3.5911  0  1.8808   0  11
 8.00  0  1.0792   9.9  1  57  3.8325  1  1.6532   0   8
12.00  0  1.1461  11.6  1  46  3.6435  0  1.1461   0   7
11.00  0  1.6128  14.0  1  60  3.7324  1  1.8451   3   9
12.00  0  1.3979   8.8  1  66  3.8388  1  1.3617   0   9
13.00  0  1.6628   4.9  0  71  3.6435  0  1.7924   0   9
16.00  0  1.1461  13.0  1  55  3.8573  0  0.9031   0   9
19.00  0  1.3222  13.0  1  59  3.7709  1  2.0000   1  10
19.00  0  1.3222  10.8  1  69  3.8808  1  1.5185   0  10
28.00  0  1.2304   7.3  1  82  3.7482  1  1.6721   0   9
41.00  0  1.7559  12.8  1  72  3.7243  1  1.4472   1   9
53.00  0  1.1139  12.0  1  66  3.6128  1  2.0000   1  11
57.00  0  1.2553  12.5  1  66  3.9685  0  1.9542   0  11
77.00  0  1.0792  14.0  1  60  3.6812  0  0.9542   0  12
;

The stepwise selection process consists of a series of alternating forward selection and backward elimination steps. The former adds variables to the model, while the latter removes variables from the model.

The following statements use PROC PHREG to produce a stepwise regression analyis. Stepwise selection is requested by specifying the SELECTION=STEPWISE option in the MODEL statement. The option SLENTRY=0.25 specifies that a variable has to be significant at the 0.25 level before it can be entered into the model, while the option SLSTAY=0.15 specifies that a variable in the model has to be significant at the 0.15 level for it to remain in the model. The DETAILS option requests detailed results for the variable selection process. 

proc phreg data=Myeloma;
   model Time*VStatus(0)=LogBUN HGB Platelet Age LogWBC
                         Frac LogPBM Protein SCalc
                         / selection=stepwise slentry=0.25
                           slstay=0.15 details;
run;

Results of the stepwise regression analysis are displayed in Output 66.1.1 through Output 66.1.7.

Individual score tests are used to determine which of the nine explanatory variables is first selected into the model. In this case, the score test for each variable is the global score test for the model containing that variable as the only explanatory variable. Output 66.1.1 displays the chi-square statistics and the corresponding p-values. The variable LogBUN has the largest chi-square value (8.5164), and it is significant (p=0.0035) at the SLENTRY=0.25 level. The variable LogBUN is thus entered into the model.

Output 66.1.1 Individual Score Test Results for All Variables
The PHREG Procedure

Model Information
Data Set WORK.MYELOMA  
Dependent Variable Time Survival Time
Censoring Variable VStatus 0=Alive 1=Dead
Censoring Value(s) 0  
Ties Handling BRESLOW  

Summary of the Number of Event and Censored
Values
Total Event Censored Percent
Censored
65 48 17 26.15

Analysis of Effects Eligible for
Entry
Effect DF Score
Chi-Square		 Pr > ChiSq
LogBUN 1 8.5164 0.0035
HGB 1 5.0664 0.0244
Platelet 1 3.1816 0.0745
Age 1 0.0183 0.8924
LogWBC 1 0.5658 0.4519
Frac 1 0.9151 0.3388
LogPBM 1 0.5846 0.4445
Protein 1 0.1466 0.7018
SCalc 1 1.1109 0.2919

Residual Chi-Square Test
Chi-Square DF Pr > ChiSq
18.4550 9 0.0302

Output 66.1.2 displays the results of the first model. Since the Wald chi-square statistic is significant () at the SLSTAY=0.15 level, LogBUN stays in the model.

Output 66.1.2 First Model in the Stepwise Selection Process

Step 1. Effect LogBUN is entered. The model contains the following effects:


LogBUN

Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics
Criterion Without
Covariates		 With
Covariates
-2 LOG L 309.716 301.959
AIC 309.716 303.959
SBC 309.716 305.830

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 7.7572 1 0.0053
Score 8.5164 1 0.0035
Wald 8.3392 1 0.0039

Analysis of Maximum Likelihood Estimates
Parameter DF Parameter
Estimate		 Standard
Error		 Chi-Square Pr > ChiSq Hazard
Ratio
LogBUN 1 1.74595 0.60460 8.3392 0.0039 5.731

The next step consists of selecting another variable to add to the model. Output 66.1.3 displays the chi-square statistics and p-values of individual score tests (adjusted for LogBUN) for the remaining eight variables. The score chi-square for a given variable is the value of the likelihood score test for testing the significance of the variable in the presence of LogBUN. The variable HGB is selected because it has the highest chi-square value (4.3468), and it is significant () at the SLENTRY=0.25 level.

Output 66.1.3 Score Tests Adjusted for the Variable LogBUN
Analysis of Effects Eligible for
Entry
Effect DF Score
Chi-Square		 Pr > ChiSq
HGB 1 4.3468 0.0371
Platelet 1 2.0183 0.1554
Age 1 0.7159 0.3975
LogWBC 1 0.0704 0.7908
Frac 1 1.0354 0.3089
LogPBM 1 1.0334 0.3094
Protein 1 0.5214 0.4703
SCalc 1 1.4150 0.2342

Residual Chi-Square Test
Chi-Square DF Pr > ChiSq
9.3164 8 0.3163

Output 66.1.4 displays the fitted model containing both LogBUN and HGB. Based on the Wald statistics, neither LogBUN nor HGB is removed from the model.

Output 66.1.4 Second Model in the Stepwise Selection Process

Step 2. Effect HGB is entered. The model contains the following effects:


LogBUN HGB

Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics
Criterion Without
Covariates		 With
Covariates
-2 LOG L 309.716 297.767
AIC 309.716 301.767
SBC 309.716 305.509

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 11.9493 2 0.0025
Score 12.7252 2 0.0017
Wald 12.1900 2 0.0023

Analysis of Maximum Likelihood Estimates
Parameter DF Parameter
Estimate		 Standard
Error		 Chi-Square Pr > ChiSq Hazard
Ratio
LogBUN 1 1.67440 0.61209 7.4833 0.0062 5.336
HGB 1 -0.11899 0.05751 4.2811 0.0385 0.888

Output 66.1.5 shows Step 3 of the selection process, in which the variable SCalc is added, resulting in the model with LogBUN, HGB, and SCalc as the explanatory variables. Note that SCalc has the smallest Wald chi-square statistic, and it is not significant () at the SLSTAY=0.15 level.

Output 66.1.5 Third Model in the Stepwise Regression

Step 3. Effect SCalc is entered. The model contains the following effects:


LogBUN HGB SCalc

Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics
Criterion Without
Covariates		 With
Covariates
-2 LOG L 309.716 296.078
AIC 309.716 302.078
SBC 309.716 307.692

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 13.6377 3 0.0034
Score 15.3053 3 0.0016
Wald 14.4542 3 0.0023

Analysis of Maximum Likelihood Estimates
Parameter DF Parameter
Estimate		 Standard
Error		 Chi-Square Pr > ChiSq Hazard
Ratio
LogBUN 1 1.63593 0.62359 6.8822 0.0087 5.134
HGB 1 -0.12643 0.05868 4.6419 0.0312 0.881
SCalc 1 0.13286 0.09868 1.8127 0.1782 1.142

The variable SCalc is then removed from the model in a step-down phase in Step 4 (Output 66.1.6). The removal of SCalc brings the stepwise selection process to a stop in order to avoid repeatedly entering and removing the same variable.

Output 66.1.6 Final Model in the Stepwise Regression

Step 4. Effect SCalc is removed. The model contains the following effects:


LogBUN HGB

Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics
Criterion Without
Covariates		 With
Covariates
-2 LOG L 309.716 297.767
AIC 309.716 301.767
SBC 309.716 305.509

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 11.9493 2 0.0025
Score 12.7252 2 0.0017
Wald 12.1900 2 0.0023

Analysis of Maximum Likelihood Estimates
Parameter DF Parameter
Estimate		 Standard
Error		 Chi-Square Pr > ChiSq Hazard
Ratio
LogBUN 1 1.67440 0.61209 7.4833 0.0062 5.336
HGB 1 -0.11899 0.05751 4.2811 0.0385 0.888


Note: Model building terminates because the effect to be entered is the effect that was removed in the last step.

The procedure also displays a summary table of the steps in the stepwise selection process, as shown in Output 66.1.7.

Output 66.1.7 Model Selection Summary
Summary of Stepwise Selection
Step Effect DF Number
In		 Score
Chi-Square		 Wald
Chi-Square		 Pr > ChiSq
Entered Removed
1 LogBUN   1 1 8.5164   0.0035
2 HGB   1 2 4.3468   0.0371
3 SCalc   1 3 1.8225   0.1770
4   SCalc 1 2   1.8127 0.1782

The stepwise selection process results in a model with two explanatory variables, LogBUN and HGB.

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