Chapter 11 Survival Analysis Part 3. 2 Considering Interactions Adapted from "Anderson" leukemia data as presented in Survival Analysis: A Self-Learning.

Chapter 11Chapter 11Survival AnalysisSurvival Analysis

Part 3Part 3

2

Considering InteractionsConsidering Interactions

Adapted from "Anderson" leukemia data as Adapted from "Anderson" leukemia data as presented in presented in Survival Analysis: A Self-Learning Survival Analysis: A Self-Learning TextText. .

Survival time in weeks in remission for 42 Survival time in weeks in remission for 42 leukemia patients. leukemia patients.

This was a randomized clinical trial where 21 This was a randomized clinical trial where 21 patients were randomized to a treatment group patients were randomized to a treatment group and 21 to a placebo group (standard of care). and 21 to a placebo group (standard of care).

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The DataThe Data

OutcomeOutcome status=1 for event, status=0 for status=1 for event, status=0 for

censoredcensored Time in remissionTime in remission

time=time until eventtime=time until event White Blood Cell Count(WBC)White Blood Cell Count(WBC)

logwbc=Log(WBC)logwbc=Log(WBC) TreatmentTreatment

trt=0 for new treatmenttrt=0 for new treatment trt=1 of standard of caretrt=1 of standard of care

Survival Plot by TreatmentSurvival Plot by Treatment

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SAS coding - LifetestSAS coding - Lifetest

proc lifetest data=leuk plots=(s);proc lifetest data=leuk plots=(s);

time time*status(0);time time*status(0);

strata trt;strata trt;

symbol1 v=none color=blue line=1;symbol1 v=none color=blue line=1;

symbol2 v=none color=red line=2;symbol2 v=none color=red line=2;

run;run;

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SAS coding - PHREGSAS coding - PHREG

proc phreg data=leuk;proc phreg data=leuk;

model time*status(0)=trt/risklimits;model time*status(0)=trt/risklimits;

run;run;


model time*status(0)=logwbc/risklimits;model time*status(0)=logwbc/risklimits;

run;run;


model time*status(0)=trt logwbc/risklimits;model time*status(0)=trt logwbc/risklimits;

run;run;


model time*status(0)=trt logwbc trt_logwbc/risklimits;model time*status(0)=trt logwbc trt_logwbc/risklimits;

run;run;

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Testing Equality over Testing Equality over StrataStrata

Test Chi-Square DF Pr Chi-SquareTest Chi-Square DF Pr Chi-Square

Log-Rank 16.7929 1 <.0001Log-Rank 16.7929 1 <.0001

Wilcoxon 13.4579 1 0.0002Wilcoxon 13.4579 1 0.0002

Conclude there is a difference in the survival curves for the two Conclude there is a difference in the survival curves for the two treatments.treatments.

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OUTPUT from PHREGOUTPUT from PHREGThe PHREG ProcedureThe PHREG Procedure

Analysis of Maximum Likelihood EstimatesAnalysis of Maximum Likelihood Estimates

Parameter Standard Hazard 95% Hazard RatioParameter Standard Hazard 95% Hazard Ratio

Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence LimitsVariable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits

trt 1 1.50919 0.40956 13.5783 0.0002 4.523 2.027 10.094trt 1 1.50919 0.40956 13.5783 0.0002 4.523 2.027 10.094

The risk of remission for standard of care is 4.523 time the risk for The risk of remission for standard of care is 4.523 time the risk for the new treatment.the new treatment.

There is a 77.8% reduction in risk of remission for the new treatment There is a 77.8% reduction in risk of remission for the new treatment compared to the standard of care.compared to the standard of care.

How did I get the second sentence?How did I get the second sentence?

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logwbc 1 1.59375 0.29673 28.8480 <.0001 4.922 2.752 8.805logwbc 1 1.59375 0.29673 28.8480 <.0001 4.922 2.752 8.805

The log of the white blood cell count is significant in predicting time The log of the white blood cell count is significant in predicting time until event. We want to include this in our analysis.until event. We want to include this in our analysis.

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trt 1 1.29405 0.42210 9.3987 0.0022 3.648 1.595 8.342trt 1 1.29405 0.42210 9.3987 0.0022 3.648 1.595 8.342

logwbc 1 1.60432 0.32933 23.7318 <.0001 4.974 2.609 9.486 logwbc 1 1.60432 0.32933 23.7318 <.0001 4.974 2.609 9.486

The risk of remission for standard of care is 3.648 time the risk for The risk of remission for standard of care is 3.648 time the risk for the new treatment adjusted for log(wbc).the new treatment adjusted for log(wbc).

There is a 72.6% reduction in risk of remission for the new treatment There is a 72.6% reduction in risk of remission for the new treatment compared to the standard of care adjusted for log(wbc).compared to the standard of care adjusted for log(wbc).

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Interactions in proportional hazards Interactions in proportional hazards regressionregression

Effect modification leads to complications in Effect modification leads to complications in interpreting parameter estimates (the Binterpreting parameter estimates (the Biis)s)

Example;Example;

(t)(t) = = 00(t)(t) exp(B exp(B11trt + Btrt + B2 2 log(wbc)+ Blog(wbc)+ B33trt*log(wbc))trt*log(wbc))

What is relative risk for those on the new treatment vs. What is relative risk for those on the new treatment vs. the standard treatment?the standard treatment?

How does log(wbc) effect this RR?How does log(wbc) effect this RR?

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trt 1 2.35494 1.68102 1.9625 0.1612 10.537 0.391 284.201trt 1 2.35494 1.68102 1.9625 0.1612 10.537 0.391 284.201

logwbc 1 1.80279 0.44672 16.2864 <.0001 6.067 2.528 14.561logwbc 1 1.80279 0.44672 16.2864 <.0001 6.067 2.528 14.561

trt_logwbc 1 -0.34220 0.51974 0.4335 0.5103 0.710 0.256 1.967trt_logwbc 1 -0.34220 0.51974 0.4335 0.5103 0.710 0.256 1.967

There does not appear to be an interaction between treatment and There does not appear to be an interaction between treatment and the log(WBC), we can leave this out of the model.the log(WBC), we can leave this out of the model.

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Interactions in proportional hazards regressionInteractions in proportional hazards regressionfor two binary variables.for two binary variables.

OutcomeOutcome

status=1 for event, status=0 for censoredstatus=1 for event, status=0 for censored

Time in remissionTime in remission

time=time until eventtime=time until event

White Blood Cell Count(WBC)White Blood Cell Count(WBC)

logwbc=Log(WBC)logwbc=Log(WBC)

Prior treatmentPrior treatment

Prior=1 if they have had prior treatment; 0 otherwisePrior=1 if they have had prior treatment; 0 otherwise

TreatmentTreatment

trt=0 for new treatmenttrt=0 for new treatment

trt=1 of standard of caretrt=1 of standard of care

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(t)(t) = = 00(t)(t) exp(B exp(B11trt + Btrt + B22prior + Bprior + B33trt*prior)trt*prior)

For those with For those with no priorno prior treatment (prior = 0); treatment (prior = 0);

trt)exp()(

0)*trt 0* trt exp()( )(

10

3210

t

tt

Relative risk for standard of care vs. new treatment for no prior trt group;

)exp( 0)*exp()(

1)*exp()(

newfor hazard

standardfor hazard1

10

10

t

t

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(t)(t) = = 00(t)(t) exp(B exp(B11trt + Btrt + B22prior + Bprior + B33trt*prior)trt*prior)

For those with For those with priorprior treatment (prior = 0); treatment (prior = 0);

trt) trt exp()(

trt) 1* trt exp()( )(

3210

3210

t

tt

Relative risk for standard of care vs. new treatment for prior trt group;

)exp( )exp()(

)exp()(

newfor hazard

standardfor hazard31

22

3210

t

t

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Interactions in proportional hazards regressionInteractions in proportional hazards regression

)exp()exp(

)exp(3

1

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Question: Is the treatment effect the same for those who Question: Is the treatment effect the same for those who have had prior treatment vs. those who have not had prior have had prior treatment vs. those who have not had prior treatment?treatment?

What do I look for?What do I look for?

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Tied event timesTied event times

Observations with outcome on same time are Observations with outcome on same time are called “ties”called “ties”

They complicate the calculationsThey complicate the calculations Doesn’t happen too much in practiceDoesn’t happen too much in practice Different methods for dealing with themDifferent methods for dealing with them

Breslow, discrete, Efron, exactBreslow, discrete, Efron, exact

Not many ties; all methods similarNot many ties; all methods similar Exact best, but computer intensiveExact best, but computer intensive Efron probably the next bestEfron probably the next best Breslow default in SASBreslow default in SAS

Methods for ties in PROC PHREGMethods for ties in PROC PHREG

PROCPROC PHREGPHREG;; MODELMODEL time*status(0) = trt / ties = time*status(0) = trt / ties = exact; exact;

RUNRUN;;

Use “exact” method for handling tiesUse “exact” method for handling ties

Other options “efron”, “breslow”, and “discrete”Other options “efron”, “breslow”, and “discrete”

Getting Confidence IntervalsGetting Confidence IntervalsPHREGPHREG

PROCPROC PHREGPHREG DATA DATA = vet= vet;; MODELMODEL SurvTime*death(0) SurvTime*death(0) = treatment/= treatment/risklimits;risklimits; RUNRUN;;

Provides 95% CI for HR

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Testing for Proportionality Testing for Proportionality AssumptionAssumption

Main assumption in Cox Proportional Main assumption in Cox Proportional Hazards model is proportionalityHazards model is proportionality..

Look at Kaplan-Meier Survival Curves:Look at Kaplan-Meier Survival Curves: This method does not work well for continuous This method does not work well for continuous

predictor predictor or categorical predictors that have many levels or categorical predictors that have many levels

because the graph becomes to "cluttered". because the graph becomes to "cluttered". Univariate analysis: does not show whether a Univariate analysis: does not show whether a

predictor will be proportional with other predictor will be proportional with other predictors in the model.predictors in the model.

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Testing for Proportionality Testing for Proportionality AssumptionAssumption

Including time dependent covariates Including time dependent covariates in the Cox model is another way to in the Cox model is another way to test proportionalitytest proportionality Generate the time dependent covariates Generate the time dependent covariates

by creating interactions of the predictors by creating interactions of the predictors and a function of survival time and and a function of survival time and include in the model. include in the model.

If any of the time dependent covariates If any of the time dependent covariates are significant then those predictors are are significant then those predictors are not proportional. not proportional.

Testing Proportionality Assumption in Testing Proportionality Assumption in SASSAS

PROCPROC PHREGPHREG DATA DATA = myel= myel;; MODELMODEL dur*status(0) = treat renal treatt renalt; dur*status(0) = treat renal treatt renalt;

treatt=treat*log(time);treatt=treat*log(time);renalt=renal*log(time); renalt=renal*log(time); test_proportionality: test_proportionality: testtest treatt, renalt; treatt, renalt;

RUNRUN;;

*Using log(time) because it is the most common *Using log(time) because it is the most common function of time used in time dependant covariates, function of time used in time dependant covariates, but any function of time could be used.but any function of time could be used.

Test:Test:Ho: Proportionality vs. Ha: Not ProportionalHo: Proportionality vs. Ha: Not Proportional


Output:Output:

The SAS System 13:33 Tuesday, April 25, 2006 4 The PHREG Procedure Analysis of Maximum Likelihood Estimates Parameter Standard HazardVariabl DF Estimate Error Chi-Square Pr > ChiSq Ratio treat 1 -1.30384 1.96212 0.4416 0.5064 0.271renal 1 -64.44484 8356 0.0001 0.9938 0.000treatt 1 0.70127 0.52411 1.7903 0.1809 2.016renalt 1 31.45503 4019 0.0001 0.9938 4.579E13

Independently the Independently the time covariates are time covariates are not significant.not significant.


Output:Output:

Collectively the Collectively the time covariates are time covariates are not significant.not significant.

Linear Hypotheses Testing Results WaldLabel Chi-Square DF Pr > ChiSq test_proportionality 1.7903 2 0.4085

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If Proportionality FailsIf Proportionality Fails

Choose a parametric regression model Choose a parametric regression model

OROR Include a time-dependent variable for non-Include a time-dependent variable for non-

proportional predictors in Cox PHproportional predictors in Cox PH

OROR Stratify on non-proportional predictors in Stratify on non-proportional predictors in

Cox PHCox PH

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Chapter 11 Survival Analysis Part 3. 2 Considering Interactions Adapted from "Anderson" leukemia data as presented in Survival Analysis: A Self-Learning.

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