Issues in the Analysis of Failure Time Data

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Issues in the Analysis of Failure Time Data

• Rebecca Betensky
• Harvard School of Public Health
• FDA/Industry Statistical Workshop
• September 19, 2003

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Current Issues in Failure Time Studies

• Unrecognized heterogeneity
• Bivariate estimation of safety and efficacy
• Dependent interval censoring
• Dependent truncation

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Unrecognized Heterogeneity

• Highly relevant as genetic discoveries are and
  are not being made
• Known to undermine the power of clinical trials
• Calls for adaptive designs

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Background

• Histologic diagnosis is principal eligibility
  criterion for clinical trials in oncology. Based
  on assumption that cancers that seem
  histologically identical constitute a single
  disease.
• This notion is false cancers of single
  histologic type often respond differently to
  treatment, likely due to biologic differences.

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• In neuro-oncology, at least 3 genetic subtypes of
  anaplastic oligodendroglioma have been identified
  (1pLOH, TP53 mutation).
• These clinically distinct subtypes are
  indistinguishable microscopically.
• Identification of molecular subtypes and the use
  of molecular signatures to direct treatment is
  becoming part of clinical practice in
  neuro-oncology.

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• If molecular heterogeneity implies that only some
  patients in a clinical trial will respond to
  treatment, important implications for the design
  and interpretations of clinical trials.

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• North American and European Intergroup trials
  comparing chemoRT versus RT for patients with AO
  nearing completion.
• Trials were designed prior to discovery of 3
  clinically distinct genetic subtypes
• 1pLOH good response and survival
• 1p intact, no TP53 mutation poor response and
  survival
• 1p intact, TP53 mutation intermediate response
  and survival

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• Proportions of these genetic subtypes differ
  across cohorts in uncontrolled Phase II studies
  proportion of 1pLOH is 90, versus 60 among all
  newly diagnosed cases.
• Leads to unrealistic design assumptions for Phase
  III trials.

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• Investigators now question whether these trials
  might have been under-powered due to this
  unrecognized heterogeneity.
• Both consortia collected tumor tissue on all
  randomized cases post hoc clinical molecular
  correlative studies will be feasible and may
  enhance interpretation.

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In Statistical Terms

• The investigators assume that the data follow a
  simple proportional hazards model
• However, the data really follow a different
  proportional hazards model

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Resulting Hazard Ratios

• Under the simple model, the hazard ratio for
  treatment is exp(?1).
• Under the true model, the hazard ratio is a
  complicated function of t, and thus the derived
  model for treatment is not a proportional hazards
  model.

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Illustrative Scenarios

• In all cases, assume true effect of treatment is
  to increase survival of patients with genetic
  subtype 1.
• Treatment is equally effective for genetic
  subtypes 1 and 2.
• Treatment is ineffective for genetic subtype 2.
• Treatment is detrimental.

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Sample Size Calculation

• In pilot data, genetic subtype 1 comprises 90 of
  the population.
• Under simple model, 5 years of accrual, 3 years
  of follow-up, median survival of 4 years not on
  treatment, require 286 subjects to detect an
  increase of 50 in median survival with 80 power.

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Power Calculations

• For a sample size of 286 cases, we evaluate power
  of the simple logrank test to detect survival
  differences between the two groups, given the
  true statistical model.
• Hazard functions are no longer propotional
  simple formulas no longer apply.

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Frailty Model

• Unrecognized heterogeneity can undermine the
  power of a randomized trial to detect a
  beneficial treatment.
• There will frequently be an interaction between
  treatment and the omitted covariate when it
  describes a genetic subtype.

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• Realistically, heterogeneity is likely to confer
  a continuum of relative risk.
• Assume each individual has a distinct realization
  of a frailty that modifies his/her hazard.
• Allow frailty to interact with treatment.

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Statistical Model

• Now, the true model is expressed as
• where, bF(b?).

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Numerical Studies

• Accrual period a1
• Follow-up period f0.5,1,1.5,2
• ?0(t)1
• ?0,0.1,0.2,,1.0
• bN(0,?) and exp(b)gamma(?-1,?-1)
• ?0.3, ?0.3

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Results

• With interaction, ARE decreases with increasing
  variance of b, and for fixed ?, ARE decreases
  with mean time to censoring (with less censoring,
  more of an opportunity for heterogeneity to have
  an impact).
• Without interaction, ARE remains close to 1 over
  ? and mean times to censoring.

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• Power decreases with ?, and for fixed ?, with
  mean time to censoring moreso with interaction.
• This said, frailty may increase the power of the
  log rank test, due to its dual action
• The frailty tends to attenuate the treatment
  effect ? decreasing the power
• The frailty tends to increase the baseline hazard
  ? increasing the power relative to the sample
  size based on the non-frailty model, especially
  when using the weighted log rank test.

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Bivariate Estimation of Safety and Efficacy

• Often both endpoints of equal importance
• Often highly dependent
• Desirable to jointly estimate
• Especially true within the context of
  sequentially designed clinical trials

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Efficacy and Safety Data

• Efficacy
• Continuous time to event subject to right
  censoring
• Survival time
• Safety
• Discrete measure also subject to right censoring
• rounds of chemotherapy tolerable to cancer
  patients
• Tumor grade when stereotactic biopsy is used

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Data Structure
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Estimator
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Properties of the Estimator

• NPML by invariance property
• Completely non-parametric
• Results in very little negative mass
• Computationally simple
• Closed-form covariance estimator

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Simulation Study

• Campbell Földes (1982) estimator
• Simulated (X,Y) and (C,D) times using Gumbel
  (1960) bivariate exponential distribution
• Tried both continuous and discretized
  times

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Simulation Results
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Simulation Results
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E2290 Data
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E2290 Data
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Sequential Analysis
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Sequential Estimator
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Properties of Sequential Estimator

• Covariance estimate with respect to calendar time
  t
• First bivariate survival estimator with
  sequential application
• Other estimators for sequential analysis of
  safety and efficacy data rely on summary
  statistics or assumptions that are inappropriate
  for bivariate right-censored data.

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Conclusions

• A number of issues arise in bivariate survival
  estimation we chose to focus on computational
  simplicity and closed-form covariance estimation.
• The form of our estimator and its
  interpretability allows for practical application
  in sequential clinical trials.

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Dependent Interval Censoring

• In certain situations can test for dependent
  censoring (not true for right censored or current
  status data).
• Use frailty models to handle dependence.

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Dependent Truncation

• With truncated data, view only part of the
  population of interest.
• Important assumption is quasi-independence, i.e.
  independence of failure and truncation over the
  observable region.
• Can test for quasi-independence!