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;