Example 44.1: Stepwise Regression

The PHREG Procedure

Example 44.1: 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. In 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 step-up and step-down phases. The former adds variables to the model, while the latter removes variables from the model.

Stepwise regression analysis 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 44.1.1 through Output 44.1.7.

Output 44.1.1: 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 Variables Not in the Model
Variable	Score Chi-Square	Pr > ChiSq
LogBUN	8.5164	0.0035
HGB	5.0664	0.0244
Platelet	3.1816	0.0745
Age	0.0183	0.8924
LogWBC	0.5658	0.4519
Frac	0.9151	0.3388
LogPBM	0.5846	0.4445
Protein	0.1466	0.7018
SCalc	1.1109	0.2919

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

Output 44.1.2: First Model in the Stepwise Selection Process

The PHREG Procedure

Step 1: Variable LogBUN is entered. The model contains the following explanatory variables.

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
Variable	DF	Parameter Estimate	Standard Error	Wald Chi-Square	Pr > ChiSq	Hazard Ratio
LogBUN	1	1.74595	0.60460	8.3392	0.0039	5.731

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. The chi-squared statistic is compared to a chi-squared distribution with one degree of freedom. Output 44.1.1 displays the chi-squared statistics and the corresponding p-values. The variable LogBUN has the largest chi-squared 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 44.1.2 displays the model results. Since the Wald chi-squared statistic is significant (p=0.0039) at the SLSTAY=0.15 level, LogBUN stays in the model.

Output 44.1.3: Score Tests Adjusted for the Variable LogBUN

Analysis of Variables Not in
the Model

Variable Score
Chi-Square Pr > ChiSq

HGB 4.3468 0.0371

Platelet 2.0183 0.1554

Age 0.7159 0.3975

LogWBC 0.0704 0.7908

Frac 1.0354 0.3089

LogPBM 1.0334 0.3094

Protein 0.5214 0.4703

SCalc 1.4150 0.2342

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

Output 44.1.4: Second Model in the Stepwise Selection Process

Step 2: Variable HGB is entered. The model contains the following explanatory variables.

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
Variable	DF	Parameter Estimate	Standard Error	Wald 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

The next step consists of selecting another variable to add to the model. Output 44.1.3 displays the chi-squared 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-squared value (4.3468), and it is significant (p=0.0371) at the SLENTRY=0.25 level. Output 44.1.4 displays the fitted model containing both LogBUN and HGB. Based on the Wald statistics, neither LogBUN nor HGB is removed from the model.

Output 44.1.5: Third Model in the Stepwise Regression

Step 3: Variable SCalc is entered. The model contains the following explanatory variables.

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
Variable	DF	Parameter Estimate	Standard Error	Wald 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

Output 44.1.5 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-squared statistic, and it is not significant (p=0.1782) at the SLSTAY=0.15 level. The variable SCalc is then removed from the model in a step-down phase in Step 4 (Output 44.1.6). The removal of SCalc brings the stepwise selection process to a stop in order to avoid repeatedly entering and removing the same variable.

The procedure also displays a summary table of the steps in the stepwise selection process, as shown in Output 44.1.7. The stepwise selection process results in a model with two explanatory variables, LogBUN and HGB.

Output 44.1.6: Final Model in the Stepwise Regression

Step 4: Variable SCalc is removed. The model contains the following explanatory variables.

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
Variable	DF	Parameter Estimate	Standard Error	Wald 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 variable to be entered is the variable
that was removed in the last step.

Output 44.1.7: Model Selection Summary

Summary of Stepwise Selection

Step Variable Number
In Score
Chi-Square Wald
Chi-Square Pr > ChiSq

Entered Removed

1 LogBUN 1 8.5164 . 0.0035

2 HGB 2 4.3468 . 0.0371

3 SCalc 3 1.8225 . 0.1770

4 SCalc 2 . 1.8127 0.1782

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