Transcript Regression Analysis

Chapter 11
Survival Analysis
Part 3
Considering Interactions
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Adapted from "Anderson" leukemia data as
presented in Survival Analysis: A Self-Learning
Text.
Survival time in weeks in remission for 42
leukemia patients.
This was a randomized clinical trial where 21
patients were randomized to a treatment group
and 21 to a placebo group (standard of care).
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The Data
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Outcome
 status=1 for event, status=0 for censored
Time in remission
 time=time until event
White Blood Cell Count(WBC)
 logwbc=Log(WBC)
Treatment
 trt=0 for new treatment
 trt=1 of standard of care
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Survival Plot by Treatment
SAS coding - Lifetest
proc lifetest data=leuk plots=(s);
time time*status(0);
strata trt;
symbol1 v=none color=blue line=1;
symbol2 v=none color=red line=2;
run;
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SAS coding - PHREG
proc phreg data=leuk;
model time*status(0)=trt/risklimits;
run;
proc phreg data=leuk;
model time*status(0)=logwbc/risklimits;
run;
proc phreg data=leuk;
model time*status(0)=trt logwbc/risklimits;
run;
proc phreg data=leuk;
model time*status(0)=trt logwbc trt_logwbc/risklimits;
run;
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Testing Equality over Strata
Test
Log-Rank
Wilcoxon
Chi-Square
DF
16.7929
13.4579
1
1
Pr Chi-Square
<.0001
0.0002
Conclude there is a difference in the survival curves for the two
treatments.
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OUTPUT from PHREG
The PHREG Procedure
Analysis of Maximum Likelihood Estimates
Variable
trt
DF
Parameter
Estimate
Standard
Error
Chi-Square
Pr > ChiSq
Hazard
Ratio
1
1.50919
0.40956
13.5783
0.0002
4.523
95% Hazard Ratio
Confidence Limits
2.027
10.094
The risk of remission for standard of care is 4.523 time the risk for the new
treatment.
There is a 77.8% reduction in risk of remission for the new treatment
compared to the standard of care.
How did I get the second sentence?
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OUTPUT from PHREG
The PHREG Procedure
Analysis of Maximum Likelihood Estimates
Variable
logwbc
DF
Parameter
Estimate
Standard
Error
Chi-Square
Pr > ChiSq
Hazard
Ratio
1
1.59375
0.29673
28.8480
<.0001
4.922
95% Hazard Ratio
Confidence Limits
2.752
8.805
The log of the white blood cell count is significant in predicting time until
event. We want to include this in our analysis.
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OUTPUT from PHREG
The PHREG Procedure
Analysis of Maximum Likelihood Estimates
Variable
trt
logwbc
DF
Parameter
Estimate
Standard
Error
Chi-Square
Pr > ChiSq
Hazard
Ratio
1
1
1.29405
1.60432
0.42210
0.32933
9.3987
23.7318
0.0022
<.0001
3.648
4.974
95% Hazard Ratio
Confidence Limits
1.595
2.609
8.342
9.486
The risk of remission for standard of care is 3.648 time the risk for the new
treatment adjusted for log(wbc).
There is a 72.6% reduction in risk of remission for the new treatment
compared to the standard of care adjusted for log(wbc).
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Interactions in proportional hazards
regression
Effect modification leads to complications in
interpreting parameter estimates (the Bis)
Example;
(t) = 0(t) exp(B1trt + B2 log(wbc)+ B3trt*log(wbc))
What is relative risk for those on the new treatment vs.
the standard treatment?
How does log(wbc) effect this RR?
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OUTPUT from PHREG
The PHREG Procedure
Analysis of Maximum Likelihood Estimates
Variable
trt
logwbc
trt_logwbc
DF
Parameter
Estimate
Standard
Error
Chi-Square
Pr > ChiSq
Hazard
Ratio
1
1
1
2.35494
1.80279
-0.34220
1.68102
0.44672
0.51974
1.9625
16.2864
0.4335
0.1612
<.0001
0.5103
10.537
6.067
0.710
95% Hazard Ratio
Confidence Limits
0.391
2.528
0.256
284.201
14.561
1.967
There does not appear to be an interaction between treatment and the
log(WBC), we can leave this out of the model.
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Interactions in proportional hazards regression
for two binary variables.
Outcome
status=1
Time
for event, status=0 for censored
in remission
time=time
White
until event
Blood Cell Count(WBC)
logwbc=Log(WBC)
Prior
treatment
Prior=1
if they have had prior treatment; 0 otherwise
Treatment
trt=0
for new treatment
trt=1
of standard of care
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Interactions in proportional hazards regression
for two binary variables.
(t) = 0(t) exp(B1trt + B2prior + B3trt*prior)
For those with no prior treatment (prior = 0);
 (t )  0 (t )exp(1trt   2 * 0   3 trt* 0)
 0 (t )exp(1trt)
Relative risk for standard of care vs. new treatment for no prior trt group;
hazardfor standard 0 (t )exp(1 * 1)

 exp(1 )
hazardfor new
0 (t )exp(1 * 0)
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Interactions in proportional hazards regression
for two binary variables.
(t) = 0(t) exp(B1trt + B2prior + B3trt*prior)
For those with prior treatment (prior = 0);
 (t )  0 (t )exp(1trt   2 * 1   3 trt)
 0 (t )exp(1trt   2   3 trt)
Relative risk for standard of care vs. new treatment for prior trt group;
hazardfor standard 0 (t )exp(1   2   3 )

 exp(1   3 )
hazardfor new
2 (t )exp( 2 )
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Interactions in proportional hazards regression
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Question: Is the treatment effect the same for those who have had
prior treatment vs. those who have not had prior treatment?
exp(1   3 )
 exp( 3 )
exp(1 )
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What do I look for?
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Tied event times
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Observations with outcome on same time are
called “ties”
They complicate the calculations
Doesn’t happen too much in practice
Different methods for dealing with them
 Breslow, discrete, Efron, exact
Not many ties; all methods similar
 Exact best, but computer intensive
 Efron probably the next best
 Breslow default in SAS
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Methods for ties in PROC PHREG
PROC PHREG;
MODEL time*status(0) = trt / ties =
exact;
RUN;
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Use “exact” method for handling ties
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Other options “efron”, “breslow”, and “discrete”
Getting Confidence Intervals
PHREG
PROC PHREG DATA = vet;
MODEL SurvTime*death(0)
= treatment/risklimits;
RUN;
Provides 95% CI for HR
Testing for Proportionality
Assumption
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Main assumption in Cox Proportional Hazards
model is proportionality.
Look at Kaplan-Meier Survival Curves:
 This method does not work well for continuous
predictor
 or categorical predictors that have many levels
because the graph becomes to "cluttered".
 Univariate analysis: does not show whether a
predictor will be proportional with other predictors in
the model.
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Testing for Proportionality
Assumption
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Including time dependent covariates in the
Cox model is another way to test
proportionality
 Generate the time dependent covariates by
creating interactions of the predictors and a
function of survival time and include in the
model.
 If any of the time dependent covariates are
significant then those predictors are not
proportional.
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Testing Proportionality Assumption in SAS
PROC PHREG DATA = myel;
MODEL dur*status(0) = treat renal treatt renalt;
treatt=treat*log(time);
renalt=renal*log(time);
test_proportionality: test treatt, renalt;
RUN;
*Using log(time) because it is the most common
function of time used in time dependant covariates,
but any function of time could be used.
Test:
Ho: Proportionality
vs. Ha:
Not Proportional
Testing Proportionality Assumption in SAS
Output:
The SAS System
April 25, 2006
13:33 Tuesday,
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The PHREG Procedure
Analysis of Maximum Likelihood Estimates
Variabl DF
treat
renal
treatt
renalt
1
1
1
1
Parameter
Estimate
-1.30384
-64.44484
0.70127
31.45503
Standard
Error
1.96212
8356
0.52411
4019
Chi-Square
Pr > ChiSq
0.4416
0.0001
1.7903
0.0001
0.5064
0.9938
0.1809
0.9938
Independently the
time covariates are
not significant.
Hazard
Ratio
0.271
0.000
2.016
4.579E13
Testing Proportionality Assumption in SAS
Output:
Linear Hypotheses Testing Results
Label
test_proportionality
Chi-Square
DF
Wald
Pr > ChiSq
1.7903
2
0.4085
Collectively the
time covariates are
not significant.
If Proportionality Fails
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Choose a parametric regression model
OR
Include a time-dependent variable for nonproportional predictors in Cox PH
OR
Stratify on non-proportional predictors in Cox PH
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