COX REGRESSION ANALYSIS

 

 

COX PROPORTIONAL HAZARD REGRESSION:

 

            model:

                                    h(t,z) = ho(t) exp(b'z)

 

            where h(t,z) is the hazard function for an individual with covariates z,

            b is the vector of unknown regression coefficients,

            ho(t) is the unknown hazard function  for an individual with covariates z = 0, (completely arbitrary no parametric function),

           

            assumptions:

            1.  multiplicative relationship between underlyling hazard function and the log-linear function of the covariates (proportionality)

 

            2.  effect of covariates upon the hazard function is log-linear.

 

            problem:  from covariates (z), t (time to death) and censoring variable, estimate  b's and their standard errors, test hypotheses about their magnitude, singly and in groups.  Just like multiple linear regression.

 

            estimation: by Cox's partial likelihood method - regression coefficients

 

SAS APPROACH

EXAMPLE PROBLEM:

            PROGRAM:

 PROGRAM INSTRUCTIONS

1.      Stage and Trt

 

 data oroca;

  infile 'c:/my documents/courses/bios740/session 19/oroca.txt' delimiter=',';

  input id surv cens inst sex trt grade age cond site stage node;

  newsex = sex - 1;

  newtrt = trt - 1;

  stg3 = 0;  stg4 = 0;

   If(stage = 3) then stg3 = 1;

   If(stage = 4) then stg4 = 1;

 Data Stage12; Set oroca; if stg3 = 0 and stg4 = 0;

 Data Stage3; set oroca; if stg3 = 1;

 Data Stage4; set oroca; if stg4 = 1;

proc phreg data = oroca;

  model surv*cens(0) = trt stage; run;

 

                                      The PHREG Procedure

 

                                       Model Information

 

                              Data Set                 WORK.OROCA

                              Dependent Variable       surv

                              Censoring Variable       cens

                              Censoring Value(s)       0

                              Ties Handling            BRESLOW

                       Summary of the Number of Event and Censored Values

                                                              Percent

                             Total       Event    Censored    Censored

                               195         142          53       27.18

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

                                     Model Fit Statistics

                                             Without           With

                            Criterion     Covariates     Covariates

                            -2 LOG L        1324.530       1315.756

                            AIC             1324.530       1319.756

                            SBC             1324.530       1325.668

                           Testing Global Null Hypothesis: BETA=0

                    Test                 Chi-Square       DF     Pr > ChiSq

                    Likelihood Ratio         8.7743        2         0.0124

                    Score                    8.5160        2         0.0142

                    Wald                     8.4629        2         0.0145

                           Analysis of Maximum Likelihood Estimates

                          Parameter      Standard                                  Hazard

       Variable    DF      Estimate         Error    Chi-Square    Pr > ChiSq       Ratio

       trt          1       0.16154       0.16853        0.9188        0.3378       1.175

       stage        1       0.31660       0.11459        7.6338        0.0057       1.372

 

                                      COX REGRESSION

2.  STAGE AND TRT - DUMMY VARIABLES FOR STAGE

  . . .

  proc phreg data = oroca;

  model surv*cens(0) = trt stg3 stg4; run;

 

                                    The PHREG Procedure

                                       Model Information

                              Data Set                 WORK.OROCA

                              Dependent Variable       surv

                              Censoring Variable       cens

                              Censoring Value(s)       0

                              Ties Handling            BRESLOW

                       Summary of the Number of Event and Censored Values

                                                               Percent

                             Total       Event    Censored    Censored

                               195         142          53       27.18

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

                                     Model Fit Statistics

                                             Without           With

                            Criterion     Covariates     Covariates

                            -2 LOG L        1324.530       1313.867

                            AIC             1324.530       1319.867

                            SBC             1324.530       1328.734

                            Testing Global Null Hypothesis: BETA=0

                    Test                 Chi-Square       DF     Pr > ChiSq

                    Likelihood Ratio        10.6637        3         0.0137

                    Score                   11.2132        3         0.0106

                    Wald                    10.9521        3         0.0120

                            Analysis of Maximum Likelihood Estimates

                          Parameter      Standard                                  Hazard

       Variable    DF      Estimate         Error    Chi-Square    Pr > ChiSq       Ratio

       trt          1       0.14207       0.16892        0.7073        0.4003       1.153

       stg3         1       0.23231       0.24788        0.8783        0.3487       1.262

       stg4         1       0.70369       0.25398        7.6767        0.0056       2.021

3.  TRT - STRATIFY by STAGE

Stratification:   The name of the variable will group the data and the model will be

 

            hj(t,z) = hoj(t)exp{bz}

 

where hj(t,z) is the hazard for an individual with covariates z at time t in the jth strata and hoj(t) is the underlying hazard for the jth strata.  Note that in this model the underlying hazards can change from strata to strata, however, the effect of the covariates is assumed to be the same in all strata.  If you want strata specific effects, you must model each strata separately.  This is a good check on the assumption. If the strata specific b's are  close to the b's obtained from the stratified model then the assumption is OK.  Problem- seldom have sufficient data to get good estimates of strata specific effects.

 

Example:

 proc phreg data = oroca;

  model surv*cens(0) = trt; strata stg3 stg4;run;

 

                                      The PHREG Procedure

                                       Model Information

                              Data Set                 WORK.OROCA

                              Dependent Variable       surv

                              Censoring Variable       cens

                              Censoring Value(s)       0

                              Ties Handling            BRESLOW

                       Summary of the Number of Event and Censored Values

                                                                                Percent

        Stratum    stg3        stg4           Total       Event    Censored    Censored

              1    0           0                 35          22          13       37.14

              2    0           1                 67          55          12       17.91

              3    1           0                 93          65          28       30.11

        -------------------------------------------------------------------------------

          Total                                 195         142          53       27.18

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

                                     Model Fit Statistics

                                             Without           With

                            Criterion     Covariates     Covariates

                            -2 LOG L        1035.340       1034.265

                            AIC             1035.340       1036.265

                            SBC             1035.340       1039.221

                            Testing Global Null Hypothesis: BETA=0

                    Test                 Chi-Square       DF     Pr > ChiSq

                    Likelihood Ratio         1.0742        1         0.3000

                    Score                    1.0804        1         0.2986

                    Wald                     1.0778        1         0.2992

                                The PHREG Procedure

                            Analysis of Maximum Likelihood Estimates

                          Parameter      Standard                                  Hazard

       Variable    DF      Estimate         Error    Chi-Square    Pr > ChiSq       Ratio

       trt          1       0.17664       0.17015        1.0778        0.2992       1.193

 

4,  STRATA SPECIFIC ESTIMATES OF THE TREATMENT MODEL

proc phreg data = Stage12;

  model surv*cens(0) = trt; run;

 proc phreg data = Stage3;

  model surv*cens(0) = trt; run;

 proc phreg data = Stage4;

  model surv*cens(0) = trt; run;

                              The PHREG Procedure

                                       Model Information

                             Data Set                 WORK.STAGE12

                             Dependent Variable       surv

                             Censoring Variable       cens

                             Censoring Value(s)       0

                             Ties Handling            BRESLOW

                       Summary of the Number of Event and Censored Values

                                                               Percent

                             Total       Event    Censored    Censored

                                35          22          13       37.14

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

                                     Model Fit Statistics

                                             Without           With

                            Criterion     Covariates     Covariates

                            -2 LOG L         130.448        130.314

                            AIC              130.448        132.314

                            SBC              130.448        133.406

                            Testing Global Null Hypothesis: BETA=0

                    Test                 Chi-Square       DF     Pr > ChiSq

                    Likelihood Ratio         0.1338        1         0.7145

                    Score                    0.1339        1         0.7144

                    Wald                     0.1337        1         0.7146

                            Analysis of Maximum Likelihood Estimates

                          Parameter      Standard                                  Hazard

       Variable    DF      Estimate         Error    Chi-Square    Pr > ChiSq       Ratio

       trt          1       0.16132       0.44113        0.1337        0.7146       1.175

 

                                      The PHREG Procedure

                                       Model Information

                              Data Set                 WORK.STAGE3

                              Dependent Variable       surv

                              Censoring Variable       cens

                              Censoring Value(s)       0

                              Ties Handling            BRESLOW

                       Summary of the Number of Event and Censored Values

                                                               Percent

                             Total       Event    Censored    Censored

                                93          65          28       30.11

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

                                     Model Fit Statistics

                                             Without           With

                            Criterion     Covariates     Covariates

                            -2 LOG L         515.747        514.854

                            AIC              515.747        516.854

                            SBC              515.747        519.028

                            Testing Global Null Hypothesis: BETA=0

                    Test                 Chi-Square       DF     Pr > ChiSq

                    Likelihood Ratio         0.8932        1         0.3446

                    Score                    0.9044        1         0.3416

                    Wald                     0.9004        1         0.3427

                            Analysis of Maximum Likelihood Estimates

                          Parameter      Standard                                  Hazard

       Variable    DF      Estimate         Error    Chi-Square    Pr > ChiSq       Ratio

       trt          1       0.23806       0.25089        0.9004        0.3427       1.269

 

                                      The PHREG Procedure

                                       Model Information

                              Data Set                 WORK.STAGE4

                              Dependent Variable       surv

                              Censoring Variable       cens

                              Censoring Value(s)       0

                              Ties Handling            BRESLOW

                       Summary of the Number of Event and Censored Values

                                                               Percent

                             Total       Event    Censored    Censored

                                67          55          12       17.91

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

                                     Model Fit Statistics

                                             Without           With

                            Criterion     Covariates     Covariates

                            -2 LOG L         389.144        388.977

                            AIC              389.144        390.977

                            SBC              389.144        392.985

                            Testing Global Null Hypothesis: BETA=0

                    Test                 Chi-Square       DF     Pr > ChiSq

                    Likelihood Ratio         0.1671        1         0.6827

                    Score                    0.1674        1         0.6824

                    Wald                     0.1684        1         0.6816

                           Analysis of Maximum Likelihood Estimates

                          Parameter      Standard                                  Hazard

       Variable    DF      Estimate         Error    Chi-Square    Pr > ChiSq       Ratio

       trt          1       0.11126       0.27117        0.1684        0.6816       1.118

 

 

SPSS APPROACH

 

1.  computed new dummy variables: stg3 and stg4 as above and stgcombo in which stages 1 and 2 are combined as 12 and stage 3 and 4 are the same as usual. 

Cox Regression - all cases trt and stage

 

Block 0: Beginning Block

 

Block 1: Method = Enter

 

Cox Regression - all cases - trt and stage as dummy variables stage 3 and 4 and stages 1 and 2 are background

Block 0: Beginning Block

 

Block 1: Method = Enter

 

Cox Regression -  all cases, trt and stratify by stage 3,4 and 1 and 2 combo

 

Block 0: Beginning Block

 

Block 1: Method = Enter

 

Cox Regression - trt specific for Stages 1 and 2

 

Block 0: Beginning Block

 

Block 1: Method = Enter

 

Cox Regression - trt specific for Stage 3

 

Block 0: Beginning Block

 

Block 1: Method = Enter

 

Cox Regression - trt specific for Stage 4

 

Block 0: Beginning Block

 

Block 1: Method = Enter

 

 

Cox Regression SPSS -Help

Click See Also above for Help on the Cox Regression dialog box, which provides access to the functionality of this command.

 

[TIME PROGRAM]

 [commands to compute time dependent covariates ]

 [CLEAR TIME PROGRAM]

COXREG [/VARIABLES=] depvar [ WITH indep_varlist]

   /STATUS=varname [EVENT](valuelist) [LOST(valuelist)]

 [/STRATA=varname]

 [/CATEGORICAL=varlist]

 [/CONTRASTS (indep_cat_var)={DEVIATION (refcat)}]

                             {SIMPLE (refcat)}

                             {DIFFERENCE}

                             {HELMERT}

                             {REPEATED}

                             {POLYNOMIAL(metric)}

                             {SPECIAL (matrix)}

                             {INDICATOR (refcat)}

 [/ [METHOD= {ENTER **}        [{mthd_varlist}]

             {BSTEP [{COND*}]  {ALL        }

                     {LR}

                     {WALD}

             {FSTEP [{COND*}

                     {LR}

                     {WALD}

 [/MISSING={EXCLUDE **}]

           {INCLUDE}

 [/PRINT=[{[ALL][SUMMARY][BASELINE][CORR][DEFAULT**]]

           [CI({95*})]

               {n}

 [/CRITERIA=[{BCON} ({1e-7**})]]

             {PCON}  {eps  }

            [ITERATE ({20**})]

                      {n  }

            [LCON ({1e-5**})]

                   {pct   }

            [PIN ({0.05 **})]

                 {eps    }

            [POUT ({0.1 **}) ]

                   {eps   }

 [/PLOT=[[NONE**] [SURVIVAL] [HAZARD] [LML] [OMS]]

 [/PATTERN=[varname (value) ...] [BY varname]]

 [/OUTFILE=[ COEFF (file)]  [ TABLE (file)]]

 [/SAVE=[tempvar [(newvarname)]], ...

 [/EXTERNAL]

 * Default if file is absent.

 ** Default if subcommand is omitted