Using time-dependent covariates in the Cox model

THIS MATERIAL IS NOT REQUIRED FOR YOUR METHODS II EXAM

With some examples taken from Fisher and Lin (1999) American Review of Public Health.

Examples of time-dependent covariates

1) Discrete events (recurrent MIs)

2) Exposure Histories (smoking history)

3) History of vital measures (heart-rate every 4 months)

Time

CovariateValue

t

How to model the effect of covariate history on the hazard?

h(t)

-Instantaneous effect

-Lagged effect

-Cumulative effect

- Attenuation after a spike in exposure

How to model the effect of covariate history on the hazard?

Time

CurrentSmoker

death

yes

no

Example: Using current smoking status to model the hazard of death

Example: Cholesterol-lowering drugs

CovariateValue

High blood lipid levels lipid build-up in vascular lesionsLow blood lipid levels leaching from established build-up

Lipid build-up hazard of death

“Simplest” case involves a single instantaneous change inan otherwise constant value.

Time TreatmentInitiated

Models for treatment effects:

Constant effect of treatment

0

1

V(t)

Time TreatmentInitiated

Models for treatment effects:

Time TreatmentInitiated

Temporary Benefit

Increasing Effect

0

0

Time

Long-term

Risks associated with interventions:

Time

Year 1

Time

Interventionperiod

0

1

0

1

0

1

Time-dependent covariates lead to complex modeling decisions:

- Many more modeling options compared to fixed covariates

- Models can be very interesting, or very misleading

- The choice of a model and the interpretation of its fit to data can depend heavily on background knowledge.

- Danger of overfitting is increased

Time-dependent covariates create more opportunities for confounding:

Are marker and event both associated with some underlying temporal process?

Example: Circadian Rhythms:

Time

UnderlyingCircadian Rhythm

Drug plasma level

Hazard of Death

Time-dependent covariates create more opportunities for confounding:

Treatment assignment and health status: Consider a bad health outcome that increases the hazard of death and also triggers a particular treatment assignment.

Age: Things that accumulate over time are associated with age, and thereforewith the hazard of death. Important to appropriately control for age.

With time-dependent covariates, standard methods may not be Appropriate for predicting survival

With fixed covariates, we can estimate

Pr( survive beyond time t given baseline measures)

The corresponding quantity for a model with time dependent covariates is

Pr( survive beyond time t given covariate history )

The properties of this quantity depend on the kinds of covariatesIn our model.

“External” covariates

- External to the failure process.

- Future covariate values don’t depend on whether or not the patient dies today.

-Equivalently, a patient’s prob. of instant death depends only the covariate history up to the present, not on future covariate values.

- Examples: Air Pollution, pre-defined treatment regimes, fixed covariates.

If all of our covariates are external then we can estimate

Pr( survive beyond time t given covariate history )

as long as we plug in a covariate history (but where do we get that?).

“External” covariates

- External to the failure process.

- Future covariate values don’t depend on whether or not the patient dies today.

-Equivalently, a patient’s prob. of instant death depends only the covariate history up to the present, not on future covariate values.

- Examples: Air Pollution, pre-defined treatment regimes, fixed covariates.

If all of our covariates are external then we can estimate

Pr( survive beyond time t given covariate history )

as long as we plug in a covariate history (but where do we get that?).

-For external covariates: A patient’s prob. of instant death depends only the covariate history up to the present, not on future covariate values.

-This is a special property—not true for all covariates—even though we model the hazard only as a function of past history.

Consider:

Asthma exacerbation / air pollutionstroke / ICU visit

Does knowing the covariate value on Thursday tell you anything moreabout the probability of event on Wednesday given that you know allcovariate values up to and including Wednesday’s?

“Internal” covariates

- Internal to the failure process.

- Formally, not external

- Can require survival of patient for measurement

-Examples: blood pressure, or any other vital measure, number of recurrent events, etc.

If our model contains internal covariates then we often know

Pr( survive beyond time t given covariate history )

with certainty—it is either 0 or 1 depending on whether we havemeasurements at time t.

Of course one can always condition on covariate history and survival up to acertain point in time, say t0, then stratify patients and compute empiricalsurvival curves for times beyond t0 for each group.

This doesn’t use a model for the hazard to predict survival for individualpatients. It just gives an aggregate prediction for each group.

Suppose our time-varying covariate is time of recurrent MI. Can we justdivide the population into two groups, those with and without recurrent MI,and estimate survival curves in each group?

Implementation issues for time-dependent covariates

Follow-up time

Pat

ient

His

torie

s

Censored survival data

Follow-up time

Pat

ient

His

torie

s

… with time-dependent covariates

Follow-up time

Pat

ient

His

torie

s

Follow-up time

Pat

ient

His

torie

s

Follow-up time

Pat

ient

His

torie

s

Cox model’s view of survival data with time-dependent covariates

Follow-up time

Pat

ient

His

torie

s

a

ba

b

1) Split each subject’s history into little intervals between successive failure times

2) Assign each interval the covariate Values at its endpoint.

3) Unless the interval ends in an observed failure, code it as censored.

dc

Note that this process will generate a table with approximately

(num. patients) x (num unique failure times) / 2 rows.

In a large trial with ~14,000 patients and 900 unique failure time This would give > 6 million rows!

Follow-up time

Pat

ient

His

torie

s

When the covariate only changes value a few times, it may be easier toSplit the data only at these change points:

This will lead to roughly

(num. patients) + (num change-points)

rows in the table.

- It’s best to chose the splitting strategy that results in the fewest intervals

- Of course if the time-dependent covariate is continuous, we must split by every failure time.

- Once the data have been appropriately split, they can be fed into any program that fits Cox models for fixed covariates.

Example

Basline(index MI) Recurrent MI t

M E

For patients with recurrent MI:

Basline(index MI) Recurrent MI t

R E

Hypothesis is that durations R and E should influence the hazard ofdeath at time t when there is a recurrent MI.

How should we model the effect of R and E ?

Is it better to have an early or later recurrent MI?