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