The Analysis of Recurrent EventsA Summary of Methodology and InformativeCensoring Considerations

Professor Jennifer RogersHead of Statistical Research

29th July 2020

DeclarationThe views and opinions expressed in this presentation are my own and do notnecessarily reflect the position or views of PHASTAR

OutlineMotivation

Conventional analysesExamplesProblems

SettingRecurrent EventsScientific Questions

Existing Models for Recurrent EventsConsiderationsBut what about Informative Censoring...

CV Death as an EventGhosh and LinJoint Frailty ModelsSimulation Study

New Directions

Motivation

Composite endpoints

Standard approach in many cardiovascular trials

• Include two or more types of related clinical events• Increase event rate and avoid multiplicity• Analysis focussed on time to first event

• Examples in cardiovascular trials:− CV death, MI and stroke in hypertension trials− CV death and HF hospitalisation in heart failure trials

CHARM-PreservedYusuf et al. (2003)

• CHARM: three parallel, independent trials• Candesartan vs. placebo in 3021 patients with symptomatic heart failure• CHARM-Preserved: preserved ejection fraction ≥ 40%

• Primary outcomes− Overall programme: all-cause mortality− Component trials: composite of death from cardiovascular disease or

hospitalisation for heart failure• Analysed as time to first event using Cox proportional-hazards model

Cox P-H model

In heart failure, analysis of composite endpoints proceeds in a standard manner:

• Exploratory analysis using Kaplan-Meier− t(1) < t(2) < t(3) < . . .: ordered event times− mj: number at risk just before time t(j)− dj: number with event at time t(j)− S(t) =

∏kj=1

(mj−djmj

), t(k) ≤ t < t(k+1)

• Estimation using Cox proportional-hazards model− hi(t) = exp{βzi}h0(t)

CHARM-Preserved

Problems

What is wrong with composite endpoints?

Only first occurring endpoint is analysed

Furthermore...• HF not characterised by a single event• Chronic diseases characterised by recurrent events• Repeat, non fatal events ignored

CHARM-Preserved

Median follow-up: 37 months

HF Hospitalisations Candesartan Placebo(N=1513) (N=1508)

≥ 1 admissions 230 278≥ 2 admissions 95 114All admissions 392 547‘Unused’ admissions 162 269

Compare to time-to-event

• Time-to-event endpoints− Statistical approaches well established− Gold standard in many indications− Substantial experience in regulatory assessment− Ignores all events after the first

• Recurrent event endpoints− Statistical approaches more complex− Less regulatory experience− Good experience in some indications do exist (e.g. MS and asthma)− More efficient as information beyond the first event is used

Setting

Recurrent events

Recurrent events involve repeat occurrences of the same type of event over time

Examples include:• Heart failure hospitalisations in CV studies• Exacerbations in COPD trials• Seizures in epilepsy trails• Asthma attacks in asthma trials

Patient profiles

Focus of this seminar

• We will consider indications where recurrent events are clinically meaningful− Treatment expected to impact first event− Treatment also expected to impact subsequent events

• We shall be focussing more on analysis methods, rather than design aspects

• Events are instantaneous, i.e. they have no duration• Events do not affect trial conduct, e.g. no treatment switching after an event

Scientific questions

• Does the intervention decrease the event number over the study periodcompared to control?• How many events does the intervention prevent, on average, compared tocontrol?• What is the intervention effect on the number of higher-order events, e.g. 3rdevent, compared to control?• What is the effect of intervention on the number of subsequent events amongthose who experienced a preceding event?

Measure of effect

Need to decide which aspect of the recurrent event data process is of interest1. Cumulative number of events over a specified time period

− Number of events by end of study events2. Rate of events

− Number of events per unit time3. Time to event

− Times to successive events4. Gap times between successive events

− Times between successive events

Existing Models for Recurrent Events

Existing Methodology

• Non-parametric estimator for mean cumulative function

• Time-to-event approaches− WLW: cumulative time from randomisation to events− PWP: analyses gap times, conditional risk sets− Andersen-Gill: extension of Cox proportional-hazards model

• Methods based on event rates− Poisson: total events divided by follow-up− Negative Binomial: individual Poisson rates which vary according to Gamma

Mean Cumulative Function

• N(t): Counting process, i.e. number of events a subject has experienced bytime t• Arbitrary MCF: µ(t) = E{N(t)}

How do we estimate µ(t) = E{N(t)}?

Mean Cumulative Function• dNi(t): jump of Ni over a small time interval [t, t + dt)• Yi(t): indicator for subject i being at risk over [t, t + dt)• Y(t) =

∑ni=1 Yi(t): total number at risk over [t, t + dt), where n is number of

randomised subjects• dN(t) =

∑ni=1 Yi(t)dNi(t): total number of events observed over [t, t + dt)

• t(1), t(2), . . . , t(H): H distinct event times across all n patients

Nelson-Aalen estimator for the MCF is given by:

µ(t) =∑

{h|t(h)≤t}

dN(t(h))

Y(t(h))

CHARM-Preserved

CHARM-Preserved

CHARM-Preserved

WLWWei-Lin-Weissfeld Wei et al. (1989)

• Interested in first K events• Analyse each time ordered event using a Cox proportional-hazards model• Estimate test statistic or hazard ratio for each time ordered event• Combine K estimates using optimal weights or 1/variance

WLW

CHARM-PreservedHR 95% CI p-value

1st HFH 0.80 (0.68,0.96) 0.0152nd HFH 0.82 (0.62,1.07) 0.1463rd HFH 0.65 (0.43,0.97) 0.036

• 508 had at least 1 HFH• 209 had at least 2 HFH• 98 had at least 3 HFH

CHARM-PreservedCandesartan Placebo(N=1514) (N=1509)

Follow-up years 4424.62 4374.03Deaths 244 237CV deaths 170 170HF Hospitalisations:1 135 1642 56 553 23 254 9 135 4 96 1 47 2 28 0 2≥9 0 4All admissions 392 547

WLW

• Preserves randomisation• Analyses cumulative effect of treatment on hospitalisations fromrandomisation− Effect on second includes effect on first− Difficult to interpret global treatment effects

• Semi-parametric approach: no assumption on baseline hazard needed• Can’t analyse all hospitalisations due to small numbers for higher order events• Need to specify K in advance• Subjects considered to be at risk for event k, even if they haven’t experiencedevent k − 1

PWPPrentice-Williams-Peterson Prentice et al. (1981)

• Analyses gap times between different failures• Subject not at risk of second event until they’ve had a first− Conditional risk set for event k made up of all subjects who have had event k− 1

• Analyse each time ordered event using a Cox proportional-hazards model• Estimate test statistic or hazard ratio for each time ordered event• Combine K estimates using optimal weights or 1/variance

PWP

CHARM-Preserved

HR 95% CI p-value1st HFH 0.80 (0.68,0.96) 0.0152nd HFH 0.99 (0.76,1.30) 0.9593rd HFH 0.68 (0.46,1.02) 0.066

PWP

• Semi-parametric approach: no assumption on baseline hazard needed• Conditional risk sets better reflect true disease progression• Doesn’t assume common baseline hazard for each gap time• Can’t analyse all hospitalisations due to small numbers for higher order events• Need to specify K in advance• Parameters for each of the k events need to be interpreted conditionally:treatment comparisons are not protected through randomisation• Difficult to interpret global treatment effects

Andersen-GillAndersen and Gill (1982)

• Extension of Cox proportional-hazards model (proportional-intensity)− λ(t) = exp{βzi}λ0(t)− λ0(t): baseline intensity function

• Each gap time contributes to the likelihood• Gives a intensity/hazard ratio for recurrent events• Assumes that events are independent− Robust standard errors accommodate heterogeneity

CHARM-Preserved

Andersen-Gill

• Semi-parametric approach: no assumption on baseline hazard needed• Can analyse all hospitalisations for all individuals• Assumes common baseline hazard for each gap time• Proportionality assumption may be too strong in practice− Intensity/hazard ratio assumed to be constant through time and common

across recurrent events

Poisson

• Commonly used for event rates• Simple: total number of events divided by total follow-up in each group• Gives a rate ratio for recurrent events• Assumes that all events are independent• Perform a Poisson regression on the count data, adjusting for treatment andincluding an offset for time in the study

Negative Binomial

• Events within an individual related - naturally accommodated by negativebinomial• Each individual has their own individual Poisson hospitalisation rate• Poisson rates vary according to Gamma• Straightforward to implement• Does not require complex data files• Perform a negative binomial regression on the count data, adjusting fortreatment and including an offset for time in the study

Negative Binomial

• Simple and naturally allows for overdispersion• Correlation of events with the same individual is accounted for through theinclusion of a random effect term• Poisson process assumption for the conditional counting process may nothold• Constant baseline assumption may be too strong in practice− Could assume other parametric models for conditional counting process

• Rate ratio also assumed to be constant over time and common acrossrecurrent events

CHARM-PreservedRogers et al. (2014)

HR 95% CI p-valueAdjudicated composite 0.89 (0.77,1.03) 0.118Unadjudicated composite 0.86 (0.74,1.00) 0.050

RR 95% CI p-valuePoisson 0.71 (0.62,0.81) <0.001Negative binomial 0.68 (0.54,0.85) <0.001Andersen-Gill 0.71 (0.57,0.88) 0.002

Why?

• Treatment acts on incidence of first hospitalisations and on recurrences

• CHARM-Preserved− Poisson for firsts: 0.82 (0.69-0.97, P=0.025)− Negative binomial for repeats: 0.58 (0.39-0.87, P=0.009)

Considerations

CHARM-Preserved

HR 95% CI p-valueAdjudicated composite 0.89 (0.77,1.03) 0.118Unadjudicated composite 0.86 (0.74,1.00) 0.050WLW 1st HFH 0.80 (0.68,0.96) 0.015WLW 2nd HFH 0.82 (0.62,1.07) 0.146WLW 3rd HFH 0.65 (0.43,0.97) 0.036PWP 1st HFH 0.80 (0.68,0.96) 0.015PWP 2nd HFH 0.99 (0.76,1.30) 0.959PWP 3rd HFH 0.68 (0.46,1.02) 0.066Poisson 0.71 (0.62,0.81) <0.001Negative binomial 0.68 (0.54,0.85) <0.001Andersen-Gill 0.71 (0.57,0.88) 0.002

Scientific questions

• Does the intervention decrease the event number over the study periodcompared to control?• How many events does the intervention prevent, on average, compared tocontrol?• What is the intervention effect on the number of higher-order events, e.g. 3rdevent, compared to control?• What is the effect of intervention on the number of subsequent events amongthose who experienced a preceding event?

Statistical Considerations• Modelling framework− Fully parametric− Semi-parametric− Non-parametric

Event rate− Constant− Time-varying− Unspecified

• Overdispersion• Censoring− Informative censoring assumption

More hospitalisations→ increased risk of death− Terminal event

But what about Informative Censoring...

Incorporating CV Death

• Increase in HF hospitalisations⇒ increased risk of death• Censoring due to CV death not independent• Comparison of hospitalisation rates confounded

Informative censoring must be incorporated into analysis!

Incorporating CV Death

• Increase in HF hospitalisations⇒ increased risk of death• Censoring due to CV death not independent• Comparison of hospitalisation rates confounded

• Count death as an additional event• Marginal analysis of recurrent events accounting for death• Joint modelling strategies

WLW Alternative• Classify death as an extra (different) event

Composite endpoint

Composite of Repeat HFHs and CV Death

• CV death treated in same way as a HF hospitalisation

• Apply all standard methods

• Rate ratio for composite of HF hospitalisation and CV death

• Death that occurs during HF hospitalisation treated as single event

CHARM-Preserved

Estimate 95% CI p-valuePoisson 0.78 (0.69,0.87) <0.001Negative binomial 0.75 (0.62,0.91) 0.003Andersen-Gill 0.78 (0.65,0.93) 0.006

Note that there were 170 CV deaths in each group

Ghosh and LinGhosh and Lin (2000)• Mean frequency function: µ(t) = E{N∗(t)}

• Death is treated as a terminal event− Patient remains in risk set after death− Count remains constant

• µ(t) =∫ t0 S(u)dR(u)

− dR(t) = E{dN∗(t)|D ≥ t}− S(t) = P(D ≥ t)

• Semi-parametric formulation Ghosh and Lin (2002):− µi(t) = exp{βzi}µ0(t)− µ0(t) unspecified continuous function

H∏h=1

(Yh − dhYh

)t(h) ≤ t < t(h+1)

∑{h|t(h)≤t}

dN(t(h))

Y(t(h))

CHARM-Preserved

Joint Frailty Model

Joint modelling strategies simultaneously analyse processes

• Each patient has their own independent frailty term ωi

• Proportionately affects heart failure hospitalisation rate and time to death

• Integrate out random effects to jointly model two event processes

• Treats death as a censoring event

Poisson Parameterisation

• Ni | ωi ∼Poisson(ϕixiωi)

• Xi | ωi ∼Exponential(γiωi)

• ϕi = exp(β1zi), γi = exp(β2zi)

• ωi ∼Gamma(1/θ, θ)

- Unconditional distribution for Ni is Negative Binomial- Unconditional distribution for Xi is Lomax- Greater θ→more heterogeneity

Likelihood

∆i: indicator for death

Li =

∫ωi

fi(ni | ωi)[fi(xi | ωi)

]∆i[Si(xi | ωi)

]1−∆ifi(ωi)dωi

Likelihood

Ni | ωi ∼Poisson(ϕixiωi)

Li =

∫ωi

fi(ni | ωi)[fi(xi | ωi)

]∆i[Si(xi | ωi)

]1−∆ifi(ωi)dωi

Likelihood

Ni | ωi ∼Poisson(ϕixiωi)

Li =

∫ωi

(ϕixiωi)ni exp(ϕixiωi)

ni!

[fi(xi | ωi)

]∆i[Si(xi | ωi)

]1−∆i

× fi(ωi)dωi

Likelihood

Xi | ωi ∼Exponential(γiωi)

Li =

∫ωi

(ϕixiωi)ni exp(ϕixiωi)

ni!

[fi(xi | ωi)

]∆i[Si(xi | ωi)

]1−∆i

× fi(ωi)dωi

Likelihood

Xi | ωi ∼Exponential(γiωi)

Li =

∫ωi

(ϕixiωi)ni exp(ϕixiωi)

ni!

[(γiωi) exp(−γixiωi)

]∆i[

exp(−γixiωi)]1−∆i

× fi(ωi)dωi

Likelihood

ωi ∼Gamma(θ, θ)

Li =

∫ωi

(ϕixiωi)ni exp(ϕixiωi)

ni!

[(γiωi) exp(−γixiωi)

]∆i[

exp(−γixiωi)]1−∆i

× fi(ωi)dωi

Likelihood

ωi ∼Gamma(θ, θ)

Li =

∫ωi

(ϕixiωi)ni exp(ϕixiωi)

ni!

[(γiωi) exp(−γixiωi)

]∆i[

exp(−γixiωi)]1−∆i

× 1/θ1/θ

Γ(1/θ)ω

1/θ−1i exp

(− ωiθ

)dωi

Likelihood

Li =

∫ωi

(ϕixiωi)ni exp(ϕixiωi)

ni!

[(γiωi) exp(−γixiωi)

]∆i[

exp(−γixiωi)]1−∆i

× (1/θ)1/θ

Γ(1/θ)ω

1/θ−1i exp

(− ωiθ

)dωi

• Integral has closed form• Maximisation is straightforward using Newton-Raphson

CHARM-Preserved

Estimate 95% CI p-valueRate ratio 0.69 (0.55,0.85) <0.001Hazard ratio 0.96 (0.73,1.26) 0.769

Marginal analysis of CV death:• 0.99 (0.80,1.22) p=0.918

Semi-Parametric JFM

What if we want no distributional assumptions on hospitalisation and death rates?

ri(t|ωi) = ωiϕir0(t) = ωiri(t)hi(t|ωi) = ωiγih0(t) = ωihi(t)

• r0/h0: baseline hazard functions

Semi-Parametric JFM

Similarly, what if we want to relax the assumption of common frailty?

ri(t|ωi) = ωiϕir0(t) = ωiri(t)hi(t|ωi) = ωiγih0(t) = ωihi(t)

Semi-Parametric JFMSimilarly, what if we want to relax the assumption of common frailty?

ri(t|ωi) = ωiϕir0(t) = ωiri(t)hi(t|ω) = ωαi γih0(t) = ωαi hi(t)

• α <0: frailties negatively correlated• α >0: frailties positively correlated• α =1: frailties identical• α =0: recurrent and terminal events independent

Likelihood

ti0 = 0, ti1, ti2, . . . , tini : observed recurrent event timesni: observed number of events for person i before xiδij and ∆i: indicators

Likelihood for person i:

Li =

∫ωi

ni∏j=1

[fi(tij | ωi)

]δij[Si(tij | ωi)]1−δij

×[fi(xi | ωi)

]∆i[Si(xi | ωi)

]1−∆ifi(ωi)dωi

LikelihoodLiu et al. (2004), Liu and Huang (2008)

Li =

∫ωi

ni∏j=1

[ωiri(tij)

]δij exp

{−∫ tij

0ωiri(u)du

}×[ωαi hi(xi)]∆i exp

{−∫ xi

0ωαi hi(u)du

}fi(ωi)dωi

• Estimation through Gaussian quadrature or penalised likelihood (Rondeauet al. (2020))

Simulation StudyRogers et al. (2016)• Subjects assigned to treatment group with probability 0.5• ωi ∼Gamma(1/θ, θ)• ri(t | ωi) was based on: r0(t), ϕi, ωi

• hi(t | ωi) was based on : h0(t), γi, ωi, α• Time to independent censoring, time to terminal event and recurrent eventgap times generated using an exponential distribution with rate parameters1/λC, 1/hi(t | ωi) and 1/ri(t | ωi) respectively• Specified maximum follow-up time

For each simulation run: 500 simulated datasets with 300 in each group

Model ComparisonsAnalyses carried out on the simulated datasets:• Composite time-to-first event− First HF hospitalisation or CV death− Cox P-H model− Remains ‘standard practice’

• Ghosh and Lin− Marginal model that accounts for death

• Time to CV death− Cox P-H model

• Joint frailty model− Gives distinct unconditional treatment effects

Results - θ = 1λC = 5%, h0 = 31%, ϕ = 0.67

Parameters Composite Recurrent Events Terminal Eventα r0 γ HRCOMP HRGL HRJFM1 HRCPH HRJFM23 1.2 0.67 0.79 (69) 0.72 (88) 0.67 (91) 0.84 (23) 0.66 (23)3 1.2 0.75 0.79 (64) 0.70 (91) 0.67 (91) 0.88 (14) 0.75 (14)1 1.2 0.67 0.76 (77) 0.72 (85) 0.68 (91) 0.71 (58) 0.67 (57)1 1.2 0.75 0.77 (70) 0.70 (89) 0.68 (90) 0.78 (37) 0.75 (35)0.5 1.2 0.67 0.75 (83) 0.70 (88) 0.68 (93) 0.68 (67) 0.66 (68)0.5 1.2 0.75 0.76 (77) 0.70 (88) 0.68 (92) 0.76 (42) 0.75 (42)

Results - θ = 3.5λC = 5%, h0 = 31%, ϕ = 0.67

Parameters Composite Recurrent Events Terminal Eventα r0 γ HRCOMP HRGL HRJFM1 HRCPH HRJFM23 0.9 0.67 0.87 (23) 0.76 (49) 0.68 (53) 0.91 (8) 0.68 (12)3 1.2 0.67 0.87 (26) 0.76 (53) 0.70 (48) 0.91 (8) 0.69 (10)1 0.9 0.67 0.84 (29) 0.81 (28) 0.69 (54) 0.79 (36) 0.66 (44)1 1.2 0.67 0.84 (33) 0.81 (32) 0.69 (55) 0.77 (37) 0.65 (39)0.5 0.9 0.67 0.81 (40) 0.79 (35) 0.71 (52) 0.72 (54) 0.67 (57)0.5 1.2 0.67 0.82 (40) 0.79 (37) 0.71 (53) 0.71 (55) 0.67 (58)

CHARM-Preserved

Estimate 95% CI p-valueRate ratio 0.67 (0.54,0.85) <0.001Hazard ratio 0.98 (0.79,1.22) 0.873α 0.60 se: 0.06 <0.001

Frailties are negatively correlated

Model Comparisons

Rate ratio Hazard ratioMarginal models 0.68 (0.54,0.85) p< 0.001 0.99 (0.80,1.22) p=0.918Composite 0.75 (0.62,0.91) p= 0.003Parametric JFM 0.69 (0.55,0.85) p< 0.001 0.96 (0.73,1.26) p=0.769Semi-parametric JFM 0.67 (0.54,0.85) p< 0.001 0.98 (0.79,1.22) p=0.873

New Directions

Future Work• But what if we want to explicitly model a non constant rate?− Hospitalisations may cluster− Rate may increase just prior to death

• Deviations from proportional hazards− Flexible parametric survival models− Restricted mean survival time

• Incorporating length of stay− Multi-state models− Simple model, 3 states: in or out of hospital and death

• Numbers needed to treat considerations Rogers et al. (2015)

References IAndersen, P. K. and R. D. Gill (1982, 12).Cox’s regression model for counting processes: A large sample study.Ann. Statist. 10(4), 1100–1120.

Ghosh, D. and D. Y. Lin (2000).Nonparametric analysis of recurrent events and death.Biometrics 56(2), 554–562.

Ghosh, D. and D. Y. Lin (2002).Marginal regression models for recurrent and terminal events.Statistica Sinica 12, 663–688.

Liu, L. and X. Huang (2008).The use of gaussian quadrature for estimation in frailty proportional hazards

models.Statistics in Medicine 27(14), 2665–2683.

References IILiu, L., R. A. Wolfe, and X. Huang (2004).Shared frailty models for recurrent events and a terminal event.Biometrics 60(3), 747–756.

Prentice, R. L., B. J. Williams, and A. V. Peterson (1981, 08).On the regression analysis of multivariate failure time data.Biometrika 68(2), 373–379.

Rogers, J. K., A. Kielhorn, J. S. Borer, I. Ford, and S. J. Pocock (2015).Effect of ivabradine on numbers needed to treat for the prevention of recurrent

hospitalizations in heart failure patients.Current Medical Research and Opinion 31(10), 1903–1909.

References IIIRogers, J. K., S. J. Pocock, J. J. McMurray, C. B. Granger, E. L. Michelson,

J. Östergren, M. A. Pfeffer, S. D. Solomon, K. Swedberg, and S. Yusuf (2014).Analysing recurrent hospitalizations in heart failure: a review of statistical

methodology, with application to charm-preserved.European Journal of Heart Failure 16(1), 33–40.

Rogers, J. K., A. Yaroshinsky, S. J. Pocock, D. Stokar, and J. Pogoda (2016).Analysis of recurrent events with an associated informative dropout time:

Application of the joint frailty model.Statistics in Medicine 35(13), 2195–2205.

Rondeau, V., J. R. Gonzalez, Y. Mazroui, A. Mauguen, A. Diakite, A. Laurent,M. Lopez, A. Krol, C. L. Sofeu, J. Dumerc, and D. Rustand (2020).

frailtypackGeneral Frailty Models: Shared, Joint and Nested Frailty Models withPrediction; Evaluation of Failure-Time Surrogate Endpoints.

R package version 3.3.0.

References IV

Wei, L. J., D. Y. Lin, and L. Weissfeld (1989).Regression analysis of multivariate incomplete failure time data by modeling

marginal distributions.Journal of the American Statistical Association 84(408), 1065–1073.

Yusuf, S., M. A. Pfeffer, K. Swedberg, C. B. Granger, P. Held, J. J. McMurray, E. L.Michelson, B. Olofsson, and J. Östergren (2003).

Effects of candesartan in patients with chronic heart failure and preservedleft-ventricular ejection fraction: the charm-preserved trial.

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