Cox Regression II

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Cox Regression II
Kristin Sainani Ph.D.http//www.stanford.edu/kco
bbStanford UniversityDepartment of Health
Research and Policy
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Topics

• Stratification
• Age as time scale
• Residuals
• Repeated events
• Intention-to-treat analysis for RCTs

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1. Stratification

• Violations of PH assumption can be resolved by
• Adding timecovariate interaction
• Adding other time-dependent version of the
  covariate
• Stratification

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Stratification

• Different stratum are allowed to have different
  baseline hazard functions.
• Hazard functions do not need to be parallel
  between different stratum.
• Essentially results in a weighted hazard ratio
  being estimated weighted over the different
  strata.
• Useful for nuisance confounders (where you do
  not care to estimate the effect).
• Does not allow you to evaluate interaction or
  confounding of stratification variable (will miss
  possible interactions).

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Example stratify on gender

• Males 1, 3, 4, 10, 12, 18 (subjects 1-6)
• Females 1, 4, 5, 9 (subjects 7-10)

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The PL
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2. Using age as the time-scale in Cox Regression

• Age is a common confounder in Cox Regression,
  since age is strongly related to death and
  disease.
• You may control for age by adding baseline age as
  a covariate to the Cox model.
• A better strategy for large-scale longitudinal
  surveys, such as NHANES, is to use age as your
  time-scale (rather than time-in-study).
• You may additionally stratify on birth cohort to
  control for cohort effects.

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Age as time-scale

• The risk set becomes everyone who was at risk at
  a certain age rather than at a certain event
  time.
• The risk set contains everyone who was still
  event-free at the age of the person who had the
  event.
• Requires enough people at risk at all ages (such
  as in a large-scale, longitudinal survey).

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The likelihood with age as time
Event times 3, 5, 7, 12, 13 (years-in-study) Ba
seline ages 28, 25, 40, 29, 30 (years) Age at
event or censoring 31, 30, 47, 41, 43
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3. Residuals

• Residuals are used to investigate the lack of fit
  of a model to a given subject.
• For Cox regression, theres no easy analog to the
  usual observed minus predicted residual of
  linear regression

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Martingale residual

• ci (1 if event, 0 if censored) minus the
  estimated cumulative hazard to ti (as a function
  of fitted model) for individual i
• ci-H(ti,Xi,?ßi)
• E.g., for a subject who was censored at 2 months,
  and whose predicted cumulative hazard to 2 months
  was 20
• Martingale0-.20 -.20
• E.g., for a subject who had an event at 13
  months, and whose predicted cumulative hazard to
  13 months was 50
• Martingale1-.50 .50
• Gives excess failures.
• Martingale residuals are not symmetrically
  distributed, even when the fitted model is
  correctly, so transform to deviance residuals...

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Deviance Residuals

• The deviance residual is a normalized transform
  of the martingale residual. These residuals are
  much more symmetrically distributed about zero.
• Observations with large deviance residuals are
  poorly predicted by the model.

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Deviance Residuals

• Behave like residuals from ordinary linear
  regression
• Should be symmetrically distributed around 0 and
  have standard deviation of 1.0.
• Negative for observations with longer than
  expected observed survival times.
• Plot deviance residuals against covariates to
  look for unusual patterns.

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Deviance Residuals

• In SAS, option on the output statement
• Output outoutdata resdevVarname
• Cannot get diagnostics in SAS if time-dependent
  covariate in the model

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Example uis data
Pattern looks fairly symmetric around 0.
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Example uis data
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Example censored only
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Example had event only
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Schoenfeld residuals

• Schoenfeld (1982) proposed the first set of
  residuals for use with Cox regression packages
• Schoenfeld D. Residuals for the proportional
  hazards regresssion model. Biometrika, 1982,
  69(1)239-241.
• Instead of a single residual for each individual,
  there is a separate residual for each individual
  for each covariate
• Note Schoenfeld residuals are not defined for
  censored individuals.

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Schoenfeld residuals

• The Schoenfeld residual is defined as the
  covariate value for the individual that failed
  minus its expected value. (Yields residuals for
  each individual who failed, for each covariate).
• Expected value of the covariate at time ti a
  weighted-average of the covariate, weighted by
  the likelihood of failure for each individual in
  the risk set at ti.

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Example

• 5 people left in our risk set at event time7
  months
• Female 55-year old smoker
• Male 45-year old non-smoker
• Female 67-year old smoker
• Male 58-year old smoker
• Male 70-year old non-smoker
• The 55-year old female smoker is the one who has
  the event

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Example

• Based on our model, we can calculate a predicted
  probability of death by time 7 for each person
  (call it p-hat)
• Female 55-year old smoker p-hat.10
• Male 45-year old non-smoker p-hat.05
• Female 67-year old smoker p-hat.30
• Male 58-year old smoker p-hat.20
• Male 70-year old non-smoker p-hat.30
• Thus, the expected value for the AGE of the
  person who failed is
• 55(.10) 45 (.05) 67(.30) 58 (.20) 70
  (.30) 60
• And, the Schoenfeld residual is 55-60 -5

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Example

• Based on our model, we can calculate a predicted
  probability of death by time 7 for each person
  (call it p-hat)
• Female 55-year old smoker p-hat.10
• Male 45-year old non-smoker p-hat.05
• Female 67-year old smoker p-hat.30
• Male 58-year old smoker p-hat.20
• Male 70-year old non-smoker p-hat.30
• The expected value for the GENDER of the person
  who failed is
• 0(.10) 1(.05) 0(.30) 1 (.20) 1 (.30) .55
• And, the Schoenfeld residual is 0-.55 -.55

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Schoenfeld residuals

• Since the Schoenfeld residuals are, in principle,
  independent of time, a plot that shows a
  non-random pattern against time is evidence of
  violation of the PH assumption.
• Plot Schoenfeld residuals against time to
  evaluate PH assumption
• Regress Schoenfeld residuals against time to test
  for independence between residuals and time.

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Example no pattern with time
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Example violation of PH
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Schoenfeld residuals

• In SAS
• option on the output statement
• Output outoutdata ressch Covariate1 Covariate2
  Covariate3

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Summary of the many ways to evaluate PH
assumption

• 1. Examine log(-log(S(t)) plots
• PH assumption is supported by parallel lines and
  refuted by lines that cross or nearly cross
• Must use categorical predictors or categories of
  a continuous predictor
• 2. Include interaction with time in the model
• PH assumption is supported by non-significant
  interaction coefficient and refuted by
  significant interaction coefficient
• Retaining the interaction term in the model
  corrects for the violation of PH
• Dont complicate your model in this way unless
  its absolutely necessary!
• 3. Plot Schoenfeld residuals
• PH assumption is supported by a random pattern
  with time and refuted by a non-random pattern
• 4. Regress Schoenfeld residuals against time to
  test for independence between residuals and time.
• PH assumption is supported by a non-significant
  relationship between residuals and time, and
  refuted by a significant relationship

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4. Repeated events

• Death (presumably) can only happen once, but many
  outcomes could happen twice
• Fractures
• Heart attacks
• Pregnancy
• Etc

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Repeated events 1

• Strategy 1 run a second Cox regression (among
  those who had a first event) starting with first
  event time as the origin
• Repeat for third, fourth, fifth, events, etc.
• Problems increasingly smaller and smaller sample
  sizes.

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Repeated events Strategy 2

• Treat each interval as a distinct observation,
  such that someone who had 3 events, for example,
  gives 3 observations to the dataset
• Major problem dependence between the same
  individual

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Strategy 3

• Stratify by individual (fixed effects partial
  likelihood)
• In PROC PHREG strata id
• Problems
• does not work well with RCT data
• requires that most individuals have at least 2
  events
• Can only estimate coefficients for those
  covariates that vary across successive spells for
  each individual this excludes constant personal
  characteristics such as age, education, gender,
  ethnicity, genotype

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5. Considerations when analyzing data from an RCT
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Intention-to-Treat Analysis

• Intention-to-treat analysis compare outcomes
  according to the groups to which subjects were
  initially assigned, regardless of which
  intervention they actually received.
• Evaluates treatment effectiveness rather than
  treatment efficacy

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Why intention to treat?

• Non-intention-to-treat analyses lose the benefits
  of randomization, as the groups may no longer be
  balanced with regards to factors that influence
  the outcome.
• Intention-to-treat analysis simulates real
  life, where patients often dont adhere
  perfectly to treatment or may discontinue
  treatment altogether.

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Drop-ins and Drop-outs example, WHI
Womens Health Initiative Writing Group.
JAMA. 2002288321-333.
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Effect of Intention to treat on the statistical
analysis

• Intention-to-treat analyses tend to underestimate
  treatment effects increased variability waters
  down results.

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Example

• Take the following hypothetical RCT
• Treated subjects have a 25 chance of dying
  during the 2-year study vs. placebo subjects have
  a 50 chance of dying.
• TRUE RR 25/50 .50 (treated have 50 less
  chance of dying)
• You do a 2-yr RCT of 100 treated and 100 placebo
  subjects.
• If nobody switched, you would see about 25 deaths
  in the treated group and about 50 deaths in the
  placebo group (give or take a few due to random
  chance).
• ?Observed RR? .50

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Example, continued

• BUT, if early in the study, 25 treated subjects
  switch to placebo and 25 placebo subjects switch
  to control.
• You would see about
• 25.25 75.50 43-44 deaths in the placebo
  group
• And about
• 25.50 75.25 31 deaths in the treated group
• Observed RR 31/44 ? .70
• Diluted effect!

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References

• Paul Allison. Survival Analysis Using SAS. SAS
  Institute Inc., Cary, NC 2003.