Learnings from Study Group on Interval Censored Data

DSBS Nytrskur 14 Jan 2010Learnings from Study Group on Interval Censored DataStudy group and litteratureWhat is interval censoringAnalysis MethodsGraphical presentationSoftwareExamplesStudy GroupGenmab: Kristian Winfeld, Birgitte Rnn, Wan Hui Ong Clausen, Kim Knudsen, Kamilla BegtrupNovo Nordisk: Mette Suntum(Lundbeck: Ole Lemmecke, Nycomed: Henrik Andersen)

Spring 2008-Authumn 2008

LitteratureBook by Jianguo Sun: The analysis of Interval-censored Failure Time DataDraft book manuscript by Philip Hougaard: Analysis of interval-Censored Survival DataHalina Frydman and Michael Szarek, Non-parametric estimation in a Markov illness-death process from interval censored observations with missing intermediate transition status, Biometrics, 2008, 65, 143-151 and Biostat Research ReportInterval censored dataT failure timeT interval censored if T (L,R]R= corresponds to right-censored data

Types of Interval CensoringCase I interval-censored failure time data/current status data Observations either right or left censoredT (L,R] where either L=0 or R=

Case II interval-censored failure time dataT (L,R] with at least some L and R belonging to(0,)

Doubly censored failure time data.T=S-X where X and S two related events, XS and both are interval censoredMost common type of doubly censored data is when S is only right censoredExample: AIDS data

Panel count data/multivariate dataThe event can happen multiple times Only knows the number of occurrences between two time pointsExample: recurrence rate of tumor at multiple locationsThe situation in oncology trialsWish to evaluate Progression-free Survival (PFS), i.e., time to death or progression of disease

Progression of disease evaluated by physical examination and CT-scan

Example: Effect of Ofatumumab in patients with Follicular Lymphoma08369121824345500 mgN=311000 mgN=86TreatmentPeriod8 infusionsFollow-upPeriodExtended Follow-upPeriodScrRandomizationWeeksMonthsYearsVisits300 mg InfusionResponse EvaluationFollow-up ContactInfusionsFDA cancer endpoints guidelineWe recommend assigning the progression date to the earliest time when any progression is observed without prior missing assessments and censoring at the date when the last radiological assessment determined a lack of progression.

FDA guideline

Interval censoringThe values of the survivor function can only be estimated at the endpoints of the intervals generated by (Li, Ri].

Sufficient only to consider intervals with positive probablity mass the Peto-Turnbull intervals

Peto-Turnbull estimates

ExampleIntervals : (1,2], (1,4], (3,6], and (5,6]

Corresponding Peto-Turnbull estimates:(1,2], (3,4] and (5,6]

Intervals with probability mass zero:(2,3] and (4,5] are not (L,R] intervalsMethodsParametric modelsRelative easy. The likelihood can be written explicitely Semi-Paramteric modelEstimation can be done via the Newton-Raphson algorithm but unstable estimation problems can occur in situations with large number of different observation time points. When no exact failure times are observed, Hypothesis testing can be performed via the Wald test or a Score test. For the general case, the asymptotic properties of the estimates are not fully explored. Ordering of the events is unknown i.e., impossible to separate the effect of covariates from the effect of time.

NPMLE for Type II censored dataStudy contains of n independent subjects from survival function S(t). gg Likelihood:

Maximise Ls(p) with respect to p, with pj=1

are the ordered distinct time points of {Li,Ri}ij = indicator for time point sj in (Li,Ri] NPMLE for type II censored dataEstimation done via iteration several approaches available (Turnbull, ICM, EMICM, etc). convergence can be difficultglobal maximum not guaranteed.

Estimation problem can be simplified by finding the intervals with positive probability mass. (Peto-Turnbull intervals).

If enough exact failure times are available or if the inspection process is non-random the NPMLE can have n1/2-convergence rate and as. normal dist. Otherwise, order is only n1/3 and asymptotic distribution is not normal.

Generalized Log-rank Test for type II censored data

Notation:= ML estimator of common Surv funct under H0are the ordered distinct time points of {Li,Ri}i = indicator for Ti not right censored, I(Rism)ij = indicator for Ti right censored and subject i still at riskat sj-, I(i = 0, Li sj)ij = indicator for time point sj in (Li,Ri] Generalized log-rank test for case II censored data

Under H0: overall estimate of numbers failing at sj is:The number at risk at sj is estimated as:

= number of failures at sj or later + number right censoredand at risk at time sjCase II interval censored dataGeneralized log-rank test

Covariance matrix, Vr for Ur estimated by multiple imputation. Then

Principle: Log-rank test = sum of observed minus expected no. of eventsHow to draw survival curve when S(t) only estimated at endpoints of the intervals (Li, Ri] ?

The estimated survival curve

SoftwareParametric models: R, SAS PROC LIFEREG or NLMIX

Semi-parametric models: SAS PROC NLMIX except when data includes exact observation times or many distinct timepoints. Software, NPMLEEstimationSplus: Estimates in KM procedure (Turnbulls self-consistency algorithm) Note!!! The plot interpolates over the holes in the curve by displaying the value from the right data point.SAS macro ICE from IML: NPMLE with CIs. 4 options for ML algorithms (Turnbulls self-consistency algorithm, Newton-Raphson Ridge, Quasi-Newton, Conjugate gradient). The produced plots have holes in the curve for all intervals.

Hypothesis testing: Must program explicitly

Example: Effect of Ofatumumab in patients with Follicular Lymphoma08369121824345500 mgN=311000 mgN=86TreatmentPeriod8 infusionsFollow-upPeriodExtended Follow-upPeriodScrRandomizationWeeksMonthsYearsVisits300 mg InfusionResponse EvaluationFollow-up ContactInfusionsConventional survival analysis based on FDA approachEstimated percentage alive and not progressed

Interval censoring approach with Turnbull estimator and generalized log-rank testEstimated percentage alive and not progressed

ComparisonEstimated percentage alive and not progressed

Results, various scenariosMedian durationP-valueConventional approachTurnbull estimateConventional approachGeneralized log-rankOriginal data 500 mg3.2[2.8;2.9]0.160.111000 mg6.0[3.8;4.4]4 patients changed in 500 mg group 500 mg3.2[0.6;1.3]0.0530.0371000 mg6 .0[3.8;4.4]500 mg patients systematicallyatt.visits 9 days earlier500 mg2.9[2.61;2.64]0.0450.0641000 mg6.0[3.8;4.4]Using scheduled visit 500 mg4.5[0;3.0]0.110.141000 mg6 .0[0;3.0]Patients dying without observed progr. have been assumed to have died without progressive disease: Progressed or DeadEntry*The real situation with PFSProgressed(Stage 2)Dead(Stage 3)Entry(Stage 1)Mortality introduces 2 problemsMay miss progressions if subjects die before being between examinations (interval censoring)Biased (upwards) estimation of TTPCant define marginal distribution of time to progressionMeaningless to consider risk of progression after deathUse (integrated) hazards instead of probabilities

Frydman & Szarek approachNon-parameteric Markov modelIntegrals simplify to finite sumsHazard contributions to progressions and deaths on non-overlapping setsDeath: at the observed death timesProgression: on intervals between the death timesFor progression hazard, the mass is concentrated on (P,Q] intervals generated from L,R,TFrydman & Szarek approach

Dont exit (1) before (Pj,Qj]Stay in (2) from Qj to TMake (1)-(2) transitionIn (Pj,Qj]Die or censorat tr=TSimilar expression for subjects where progression is not observed but with an additional term for dying directly from state 1 Contribution to likelihood from subject with observed progression:Progressed(Stage 2)Dead(Stage 3)Entry(Stage 1)Example: PFS for Cetuximab+RTC+RT vs RTPD assessed @ w4, w8, + every 4 months for 2 years, then every year during year 3-5Endpoint: PFSPD assigned to first time where PD observedIf no PD observed prior to death assume death occurred directly from state 1If censored at end of follow-up, censor at time of last PD assessmentIn accordance with FDA guidanceExample: PFS for Cetuximab+RTN=424 randomized207 had observed PD (of which 78.3% died prior to study end)214 had no observed PD prior to death or study end53 was assumed to have died directly from stage 1163 was right censored at last tumor assessment3 were known not to have PD and were alive at study endFour analysis methodsAnalysis I. Original analysis (the FDA method)

Observed PD times analyzed as interval censored:Analysis II: No other changesAnalysis III: Naive scenario (pts without observed PD or death are right censored for PFS at censoring time for death rather than at the last tumor ass.).Analysis IV: New method accounting for the observations with missing status of the intermediate transition.

ResultsAccounting for interval censoring decreases the estimated median PFS time for the RT groupThe largest difference in 2-year PFS proportion is seen in Analysis IV

Estimated proportion with eventF12F13F=F12+F13

No method for hypothesis testing is currently developed !!!Summary of challenges found with interval censoring methodsConvergenceIf failure time is a mixture of interval censored observations and exact events.How to plot the estimated survival functionIf event studied is a composite endpoint like PFSHypothesis testingTo what extend will analysis results be accepted by authoritiesPFS

SituationDate of Progression or CensoringOutcome

Progression documented between scheduled visitsDate of radiologic ass. of new lesion (if progr. is based on new lesion), orDate of last radiologic ass. (if prog. is based on increase in sum of measured lesions)Event

Death before first assessmentDate of deathEvent

Death between adequate assessments visitsDate of deathEvent

No progression (at end of trial)Date of last radiologic ass. of measured lesionsCensored

No baseline tumor assessm.Baseline time pointCensored

Treatment discontinuation for undocumented progressionDate of last radiologic ass. of measured lesionsCensored

Treatment discontinuation for toxicity or other reasonDate of last radiologic ass. of measured lesionsCensored

New anti-cancer treatment startedDate of last radiologic ass. of measured lesionsCensored

Death or progression after more than one missed visitDate of last radiologic assessment of measured lesionsCensored