HSRP 734: Advanced Statistical Methods July 31, 2008
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Transcript of HSRP 734: Advanced Statistical Methods July 31, 2008
HSRP 734: HSRP 734: Advanced Statistical Advanced Statistical
MethodsMethodsJuly 31, 2008July 31, 2008
ObjectivesObjectives
Describe the general form of the Cox Describe the general form of the Cox proportional hazards model proportional hazards model extended for time-dependent extended for time-dependent variablesvariables
Describe the analysis for staggered Describe the analysis for staggered entryentry
Review for Final examReview for Final exam
Time-Dependent Time-Dependent VariablesVariables
Time-dependent variable: covariate Time-dependent variable: covariate whose value may vary over time.whose value may vary over time.
Two options if the proportional Two options if the proportional hazards assumption is not satisfied hazards assumption is not satisfied for one or more of the predictors in for one or more of the predictors in the model.the model. Use a stratified Cox modelUse a stratified Cox model Use time-dependent variablesUse time-dependent variables
Time-Dependent Time-Dependent VariablesVariables
Time-dependent variables may be:Time-dependent variables may be: Inherently time-dependentInherently time-dependent
Internal – only have meaning when subject is aliveInternal – only have meaning when subject is alive smoking status at time tsmoking status at time t white blood count at time twhite blood count at time t
External – can be obtained whether or not subject is External – can be obtained whether or not subject is alivealive
Air pollution index at time tAir pollution index at time t Part internal and part ancillaryPart internal and part ancillary
E.g., heart transplant status at time tE.g., heart transplant status at time t Defined to analyze a time-independent Defined to analyze a time-independent
predictor not satisfying the PH assumptionpredictor not satisfying the PH assumption E.g., RACE x time; RACE x log(time+1)E.g., RACE x time; RACE x log(time+1)
Internal Time-Dependent Internal Time-Dependent VariableVariable
Internal time-dependent variables Internal time-dependent variables are particularly susceptible to be are particularly susceptible to be inappropriately controlled. inappropriately controlled.
They often lie in the causal pathway They often lie in the causal pathway about which we want to make about which we want to make inferences.inferences.
Internal Time-Dependent Internal Time-Dependent VariableVariableexampleexample
Clinical trial for treatment of Clinical trial for treatment of metastatic colorectal cancer – do we metastatic colorectal cancer – do we adjust for most recent WBC?adjust for most recent WBC?
Internal Time-Dependent Internal Time-Dependent VariableVariableexampleexample
Clinical trial for treatment of metastatic Clinical trial for treatment of metastatic colorectal cancer – do we adjust for most colorectal cancer – do we adjust for most recent WBC?recent WBC? Treatment comparison among subjects with Treatment comparison among subjects with
like prognosis at each timelike prognosis at each time But treatment might improve prognosis by But treatment might improve prognosis by
improving depressed WBC over timeimproving depressed WBC over time Adjustment for WBC over time might remove Adjustment for WBC over time might remove
the apparent effect of treatment, since the apparent effect of treatment, since patients with the same WBC in either patients with the same WBC in either treatment group might have similar prognosistreatment group might have similar prognosis
Extended Cox Model for Extended Cox Model for Time-Dependent VariablesTime-Dependent Variables
ModelModel
Assumption:Assumption: The effect of a time-dependent variable on the The effect of a time-dependent variable on the
survival probability at time t depends on the survival probability at time t depends on the value of this variable at that same time t.value of this variable at that same time t.
Statistical inferences:Statistical inferences: Wald, Score, Likelihood ratio testsWald, Score, Likelihood ratio tests
)(exp)())(,( 22110 tXXthtXth
Extended Cox Model for Extended Cox Model for Time-Dependent VariablesTime-Dependent Variables
Even though the values of the time-Even though the values of the time-dependent variable may change over dependent variable may change over time, the hazard model provides only time, the hazard model provides only one coefficient for each time-one coefficient for each time-dependent variable in the model.dependent variable in the model.
Hazard Ratio for the Hazard Ratio for the Extended Cox ModelExtended Cox Model
Let XLet X11 = smoking yes/no; = smoking yes/no; XX22(t) = X(t) = X11 x tx t The hazard ratio (or RR) is a function of The hazard ratio (or RR) is a function of
time.time.
PH assumption is not satisfied for the PH assumption is not satisfied for the extended Cox modelextended Cox model
ttRH
ttRH
tXth
XthtRH
as
as
)(ˆ0ˆ
)(ˆ0ˆ
ˆˆexp)0,(ˆ)1,(ˆ
)(ˆ
2
2
21
1
1
Hazard Ratio for the Hazard Ratio for the Extended Cox ModelExtended Cox Model
Coefficient represents the “overall” Coefficient represents the “overall” effect of the corresponding time-dependent effect of the corresponding time-dependent variable, considering all times at which this variable, considering all times at which this variable has been measured in the study.variable has been measured in the study.
Another model with a time-dependent Another model with a time-dependent variablevariable
compares an exposed person to an compares an exposed person to an unexposed person at time t.unexposed person at time t.
2
01ˆexp)0)(,(ˆ)1)(,(ˆ
)(ˆ
tEth
tEthtRH
e
Time-Dependent VariablesTime-Dependent Variablesin SASin SAS
Do not define the time-dependent Do not define the time-dependent variable in a data stepvariable in a data step The variable will be time-independentThe variable will be time-independent
Use the programming statements in Use the programming statements in proc tphregproc tphreg
time depend example.sastime depend example.sas
Left Truncation of Left Truncation of Failure TimesFailure Times
Also know as staggered entryAlso know as staggered entry
Left truncation arises when Left truncation arises when individuals only come under individuals only come under observation some known time after observation some known time after the natural time origin of the the natural time origin of the phenomenon under study. phenomenon under study.
Left Truncation Left Truncation ExamplesExamples
Ex 1 Atomic bomb survivors studyEx 1 Atomic bomb survivors study Time zero is August 1945 – time is time Time zero is August 1945 – time is time
since radiation exposuresince radiation exposure Observation of subjects begins with the Observation of subjects begins with the
1950 census1950 census People who died before 1950 are not in People who died before 1950 are not in
the sample - survival times are left the sample - survival times are left truncated at 5 yearstruncated at 5 years
Left Truncation Left Truncation ExamplesExamples
Ex 2 Welsh nickel refinersEx 2 Welsh nickel refiners Time zero is employee’s start date - all Time zero is employee’s start date - all
were before 1925were before 1925 Observation of most subjects begin in Observation of most subjects begin in
1934, some in 1939, 1944, or 19491934, some in 1939, 1944, or 1949 In contrast to example 1, each subject In contrast to example 1, each subject
has his own truncation time i.e. has his own truncation time i.e. staggered entrystaggered entry
Left Truncation Example 2Left Truncation Example 2cont.cont.
calender year
12
34
56
1900 1925 1939 1944 1949 1979
C
C
C
C
D
D
time since employment
12
34
56
0 20 40 60
C
C
C
C
D
D
Left TruncationLeft Truncation
The risk set just prior to an event The risk set just prior to an event time does not include individuals time does not include individuals whose left truncation times exceed whose left truncation times exceed the given event time. Thus, any the given event time. Thus, any contribution to the likelihood must contribution to the likelihood must be conditional on the truncation be conditional on the truncation limit having been exceeded. limit having been exceeded.
Left TruncationLeft Truncation Please do not confuse this with left censoringPlease do not confuse this with left censoring
Recall – left censoring occurs when the true Recall – left censoring occurs when the true survival time is less than what we observedsurvival time is less than what we observed
We may not know a left censored participant's We may not know a left censored participant's exact survival time, but at least we know exact survival time, but at least we know he/she existed; i.e. he/she did get observedhe/she existed; i.e. he/she did get observed
In a staggered entry situation, we may not In a staggered entry situation, we may not know how many participants we missed.know how many participants we missed.
Implications of left Implications of left truncation Ex. 1truncation Ex. 1
We have no way of making We have no way of making inferences about risk of death before inferences about risk of death before 5 years5 years
In a Cox model, if there are different In a Cox model, if there are different relationships between the covariates relationships between the covariates and and λλ((tt) when t<5 and when t>5, we ) when t<5 and when t>5, we have no way to detect this.have no way to detect this.
Implications of staggered Implications of staggered entry Ex. 2entry Ex. 2
Any subject in the cohort had to Any subject in the cohort had to survive from initial employment to survive from initial employment to beginning of observationbeginning of observation
If we ignore this in a Cox model, we If we ignore this in a Cox model, we will compare the covariates of subject will compare the covariates of subject 2 (for example) to all other subjects2 (for example) to all other subjects This is not fair. There would be subjects This is not fair. There would be subjects
in the denominator who could not in the denominator who could not possible be in the numerator possible be in the numerator
i
i
D
N
SolutionSolution
At each event time, include in the risk At each event time, include in the risk set only those subjects who have not set only those subjects who have not yet died yet died and who are under and who are under observationobservation
Risk sets are not necessarily nested Risk sets are not necessarily nested and can get bigger as time progressesand can get bigger as time progresses
Every inferential statement we make Every inferential statement we make must be made conditional on must be made conditional on surviving to beginning of observationsurviving to beginning of observation
Solution – main Solution – main assumptionassumption
The sampling process leading to late The sampling process leading to late entry into the sample does not entry into the sample does not preferentially select subjects with preferentially select subjects with unusual risks or covariate valuesunusual risks or covariate values
??
How are the coefficient estimates How are the coefficient estimates from a Cox model for example 1 from a Cox model for example 1 ((atomic bomb survivors studyatomic bomb survivors study) different ) different if we correct for left truncation from if we correct for left truncation from those from an uncorrected model?those from an uncorrected model?
??
How are the coefficient estimates How are the coefficient estimates from a Cox model for example 1 from a Cox model for example 1 ((atomic bomb survivors studyatomic bomb survivors study) different ) different if we correct for left truncation from if we correct for left truncation from those from an uncorrected model?those from an uncorrected model?
Answer:Answer:
They do not change. No one fails They do not change. No one fails until after everyone has entered. The until after everyone has entered. The risk sets do not change.risk sets do not change.
? 2? 2
What changes in example 2 (What changes in example 2 (Welsh Welsh nickel refinersnickel refiners) if instead of correcting ) if instead of correcting for left truncation we change the for left truncation we change the time scale to be time since each time scale to be time since each subject’s entry into observation?subject’s entry into observation?
This is not the same as accounting This is not the same as accounting for left truncationfor left truncation
New Time ScaleNew Time Scale
calender year
12
34
56
1900 1925 1939 1944 1949 1979
C
C
C
C
D
D
time since observation
12
34
56
0 10 20 30 40
C
C
C
C
D
D
? 2? 2
What changes in example 2 (What changes in example 2 (Welsh Welsh nickel refinersnickel refiners) if instead of correcting ) if instead of correcting for left truncation we change the for left truncation we change the time scale to be time since each time scale to be time since each subject’s entry into observation?subject’s entry into observation?
AnswerAnswer
The risk set compositions change. The risk set compositions change. Thus, the coefficient estimates and Thus, the coefficient estimates and hazard function changes.hazard function changes.
Left TruncationLeft Truncation
Coding example from SAS manualCoding example from SAS manual
proc tphreg data=one;proc tphreg data=one;
model t2*dead(0)=x1-x10/entry=t1;model t2*dead(0)=x1-x10/entry=t1;
baseline out=out1 survival=s;baseline out=out1 survival=s;
run; run;