Modeling and Analysis of Recurrent Event Data

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1 Edsel A. Peña Department of Statistics University of South Carolina Columbia, SC New Jersey Institute of Technology Conference May 20, 2012

2 Historical Perspective: Random Censorship Model (RCM) Random Observables: T 1, T 2,...,T n IID F C 1, C 2,...,C n IID G F and G not related {T i } {C i } (Z 1, δ 1 ), (Z 2, δ 2 ),...,(Z n, δ n ) Z i = T i C i and δ i = I {T i C i } Goal: To make inference on the distribution F or the hazard Λ = df/ F, or functionals (mean, median, etc.).

3 Nonparametric Inference F F = space of continuous distributions Λ = log F; F = 1 F = exp( Λ); λ = Λ N(s) = i I {Z i s; δ i = 1} and Y (s) = i I {Z i s} NAE: ˆΛ = dn Y and PLE: ˆ F = [ 1 dn Y ] Properties of ˆΛ and ˆF well-known, e.g., biased; consistent; and when normalized, weakly convergent to Gaussian processes. Avar(ˆ F(t)) = 1 n F(t) 2 t 0 df(s) F(s) 2Ḡ(s)

4 An Informative RCM Koziol-Green (KG) Model (1976): β 0, Ḡ(t) = F(t) β Lehmann-type alternatives Proportional hazards: Z i = min(t i, C i ) F β+1 Λ G = βλ F δ i = I {T i C i } BER(1/(β + 1)) An important characterizing property: Z i δ i

5 MMC Data: First Event Times and Tau s Migratory Motor Complex (MMC) data set from Aalen and Husebye, Stat Med, MMC Data Set Unit Number Calendar Time

6 MMC Data: Hazard Estimates KG model holds if and only if Λ τ Λ T1 MMC Data Set Cumulative Hazard Estimate First Event Time and Tau

7 KG Model s Utility Chen, Hollander and Langberg (JASA, 1982): exploited independence between Z i and δ i to obtain the exact bias, variance, and MSE functions of PLE. Comparisons with asymptotic results. Cheng and Lin (1987): exploited semiparametric nature of KG model to obtain a more efficient estimator of F compared to the PLE. Hollander and Peña (1988): obtained better confidence bands for F under KG model. Csorgo & Faraway (1998): KG model not practically viable, but serves as a mathematical specimen (a lá yeast) for examining exact properties of procedures and for providing pinpoint assessment of efficiency losses/gains.

8 Recurrent Events: Some Examples admission to hospital due to chronic disease tumor re-occurrence migraine attacks alcohol or drug (eg cocaine) addiction machine failure or discovery of a bug in a software commission of a criminal act by a delinquent minor! major disagreements between a couple non-life insurance claim drop of 200 points in DJIA during trading day publication of a research paper by a professor

9 Full MMC Data: With Recurrences n = 19 subjects; event = end of migratory motor complex cycle; random length of monitoring period per subject. MMC Data Set Unit Number Calendar Time

10 Data Accrual: One Subject

11 Some Aspects in Recurrent Data random monitoring length (τ). random # of events (K) and sum-quota constraint: k K K = max k : T j τ with T j τ < j=1 j=1 Basic Observable: (K, τ, T 1, T 2,...,T K, τ S K ) always a right-censored observation. dependent and informative censoring. K+1 effects of covariates, frailties, interventions after each event, and accumulation of events. j=1 T j

12 Simplest Model: One Subject T 1, T 2,... IID F: (renewal model) perfect interventions after each event τ G F and G not related no covariates (X) no frailties (Z) F could be parametric or nonparametric Peña, Strawderman and Hollander (JASA, 01): nonparametric estimation of F.

13 Nonparametric Estimation of F Y (t) = N(t) = n K i I {T ij t} i=1 j=1 n K i I {T ij t} + I {τ i S iki t} i=1 j=1 t dn(w) GNAE : Λ(t) = 0 Y (w) t [ GPLE : F(t) = 1 dn(w) ] Y (w) 0

14 Main Asymptotic Result k kth Convolution: F (k) (t) = Pr T j t Renewal Function: ρ(t) = ν(t) = 1 Ḡ(t) σ 2 (t) = F(t) 2 t 0 t j=1 F (k) (t) k=1 ρ(w t)dg(w) df(w) F(w) 2Ḡ(w)[1 + ν(w)] Theorem (JASA, 01): n( F(t) F(t)) GP(0, σ 2 (t))

15 Extending KG Model: Recurrent Setting Wanted: a tractable model with monitoring time informative about F. Potential to refine analysis of efficiency gains/losses. Idea: Why not simply generalize the KG model for the RCM. Generalized KG Model (GKG) for Recurrent Events: β > 0, Ḡ(t) = F(t) β with β unknown, and F the common inter-event time distribution function. Remark: τ may also represent system failure/death, while the recurrent event could be shocks to the system. Remark: Association (within unit) could be modeled through a frailty.

16 Estimation Issues and Some Questions How to semiparametrically estimate β, Λ, and F? Parametric estimation in Adekpedjou, Peña, and Quiton (2010, JSPI). How much efficiency loss is incurred when the informative monitoring model structure is ignored? How much penalty is incurred with Single-event analysis relative to Recurrent-event analysis? In particular, what is the efficiency loss for estimating F when using the nonparametric estimator in PSH (2001) relative to the semiparametric estimator that exploits the informative monitoring structure?

17 Basic Processes S ij = j k=1 T ik N i (s) = I {S ij s} j=1 Y i (s) = I {τ i s} R i (s) = s S in = backward recurrence time i (s ) A i (s) = s 0 Y i (v)λ[r i (v)]dv N τ i (s) = I {τ i s} Y τ i (s) = I {τ i s}

18 Transformed Processes Z i (s, t) = I {R i (s) t} N i (s, t) = Y i (s, t) = A i (s, t) = s 0 N i (s ) j=1 s 0 N Z i (v, t)n i (s)) i (dv) = I {T ij t} j=1 I {T ij t} + I {(s τ i ) S in i (s ) t} t Z i (v, t)a i (dv) = Y i (s, w)λ(w)dw 0

19 Aggregated Processes N(s, t) = Y (s, t) = A(s, t) = n N i (s, t) i=1 n Y i (s, t) i=1 n A i (s, t) i=1 N τ (s) = Y τ (s) = n Ni τ (s) i=1 n i=1 Y τ i (s)

20 First, Assume β Known Via Method-of-Moments Approach, estimator of Λ: ˆΛ(s, t β) = t 0 { N(s, dw) + N τ } (dw) Y (s, w) + βy τ (w) Using product-integral representation of F in terms of Λ, estimator of F: ˆ F(s, t β) = t w=0 { 1 N(s, dw) + } Nτ (dw) Y (s, w) + βy τ (w)

21 Estimating β: Profile Likelihood MLE Profile Likelihood: L P (s ; β) = β Nτ (s ) {[ n s { i=1 Estimator of β: [ s v=0 { v=0 1 Y (s, v) + βy τ (v) 1 Y (s, v) + βy τ (v) ˆβ = arg max β L P(s ; β) } N τ i ( v) ] } Ni (s, v) ]}

22 Is It A Legitimate Profile Likelihood? Let C be the space of all cumulative hazard functions defined on [0, ). Theorem Let L(s ; Λ, β) be the full likelihood function given by L = { n s i=1 v=0 exp { } [ ] N Y i (v) i (v)λ[r i (v)]dv [Y τ i (v)βλ(v)dv] Nτ i (v) [ n s i=1 0 ]} s Y i (v)λ[r i (v)]dv + Yi τ (v)βλ(v)dv. 0 then L P (s ; β) = max Λ C L(s ; Λ, β).

23 Estimators of Λ and F Estimator of Λ: ˆΛ(s, t) = ˆΛ(s, t ˆβ) = Estimator of F: ˆ F(s, t) = ˆ F(s, t ˆβ) = t 0 { } N(s, dw) + N τ (dw) Y (s, w) + ˆβY τ (w) { } t 1 N(s, dw) + N τ (dw) Y (s, w) + ˆβY τ (w) w=0

24 Illustrative Data (n = 30): GKG[Wei(2,.1), β =.2] Recurrent Event Data Subject Calendar Time

25 Estimates of β and F ˆβ =.2331 Estimates of SF for Illustrative Data (Blue=GKG; Red=PSH; Green=TRUE) Estimate Survivor Probability Time

26 Properties of Estimators G s (w) = G(w)I {w < s} + I {w s} E{Y 1 (s, t)} y(s, t) = F(t)Ḡ s (t) + F(t) E{Y τ 1 (t)} y τ (t) = F(t) β t ρ(w t)dg s (w) True Values = (F 0, Λ 0, β 0 ) y 0 (s, t) = y(s, t; Λ 0, β 0 ) y τ 0 (s) = y τ (s; Λ 0, β 0 )

27 Existence, Consistency, Normality Theorem There is a sequence of ˆβ that is consistent, and ˆΛ(s, ) and ˆ F(s, ) are both uniformly strongly consistent. Theorem As n, we have n(ˆβ β 0 ) N(0, [I P (s ; Λ 0, β 0 )] 1 ) with I P (s ; Λ 0, β 0 ) = 1 β 0 s 0 y τ 0 (v)y 0(s, v) y 0 (s, v) + β 0 y τ 0 (v)λ 0(v)dv.

28 Weak Convergence of ˆΛ(s, ) Theorem As n, { n[ˆλ(s, t) Λ 0 (t)] : t [0, t ]} converges weakly to a zero-mean Gaussian process with variance function σ 2ˆΛ (s, t) = [ s 0 [ t 0 t 0 Λ 0 (dv) y 0 (s, v) + β 0 y τ 0 (v)+ y 0 (s, v)y τ 0 (v) β 0 [y 0 (s, v) + β 0 y τ 0 (v)]λ 0(dv) y τ 0 (v) y 0 (s, v) + β 0 y τ 0 (v)λ 0(dv) ] 2. ] 1 Remark: The last product term is the effect of estimating β. It inflates the asymptotic variance.

29 Weak Convergence of ˆ F(s, ) and F(s, ) Corollary As n, { n[ˆ F(s, t) F 0 (t)] : t [0, t ]} converges weakly to a zero-mean Gaussian process whose variance function is σ 2ˆF (s, t) = F 0 (t) 2 σ 2ˆΛ (s, t) F 0 (t) 2 σ 2 Λ(s, t). Recall/Compare! Theorem (PSH, 2001) As n, { n[ F(s, t) F 0 (t)] : t [0, t ]} converges weakly to a zero-mean Gaussian process whose variance function is σ 2 F(s, t) = F 0 (t) 2 t 0 Λ 0 (dv) y 0 (s, v).

30 Asymptotic Relative Efficiency: β 0 Known If we know β 0 : ARE{ F(s, t) : ˆ F(s, t β 0 )} = { t } 1 Λ 0 (dw) 0 y 0 (s, w) { t } Λ 0 (dw) y 0 (s, w) + β 0 y0 τ(w) 0 Clearly, less than or equal to unity, as is to be expected.

31 Case of Exponential F: β 0 Known Theorem If F 0 (t) = exp{ θ 0 t} for t 0 and s, then Also, t 0, ARE{ F(, t) : ˆ F(, t β 0 )} = { } 1 1 du F 0 (t) (1 + β 0 )u 2+β 0 { } 1 du (1 + β 0 )u 2+β 0 + β 2 0 u 1+β. 0 F 0 (t) ARE{ F(, t) : ˆ F(, t; β 0 )} 1 + β β 0 + β0 2.

32 ARE-Plots; β 0 {.1,.3,.5,.7,.9, 1.0, 1.5} Known; F = Exponential Efficiency(PSH:GKG0) F(t)

33 Case of β 0 Unknown As to be expected, if β 0 is known, then the estimator exploiting the GKG structure is more efficient. Question: Does this dominance hold true still if β 0 is now estimated? Theorem Under the GKG model, for all ( F 0, β 0 ) with β 0 > 0, F(s, t) is asymptotically dominated by ˆ F(s, t) in the sense that ARE( F(s, t) : ˆ F(s, t)) 1. Proof. Neat application of Cauchy-Schwartz Inequality.

34 ARE-Plots; β 0 {.1,.3,.5,.7,.9, 1.0, 1.5} Unknown; F = Exponential Efficiency(PSH:GKG) F(t)

35 Assessing Asymptotic Approximations under Exponential n = 50 n = 100 β 0 Mean SE ASE Mean SE ASE Mean: mean of the simulated values of ˆβ. SE: standard error of the simulated values of ˆβ. ASE: asymptotic standard error obtained from earlier formula replications were performed.

36 Simulations under Weibull F: ˆβ Results n = 30,α = 2 n = 50,α = 2 n = 30,α =.9 n = 50,α =.9 β 0 Mean SE Mean SE Mean SE Mean SE

37 Simulated Rel Eff of F : ˆ F under a Weibull F with α = 2 Simulated Efficiencies of PLEPSH relative to PLEGKG Effi(PSH:GKG) F(t)

38 Simulated Rel Eff of F : ˆ F under a Weibull F with α = 0.9 Simulated Efficiencies of PLEPSH relative to PLEGKG Effi(PSH:GKG) F(t)

39 Mis-Specified Model: F 0 is Exp(1) and G 0 is Uniform[0, θ] Simulated Efficiencies of PLEPSH relative to PLEGKG Effi(PSH:GKG) Theta = 1 Theta = 2 Theta = 4 Theta = F(t)

40 Mis-Specified Model: F 0 is Weibull(2, 1) and G 0 is Exp(θ) Simulated Efficiencies of PLEPSH relative to PLEGKG Effi(PSH:GKG) Theta = 1.2 Theta = 1 Theta = 0.5 Theta = F(t)

41 Briefly: On More General Recurrent Event Models Must be able to model the following: comparing treatment groups. effect of covariates or regressor variables. effect of interventions after each event. effect of accumulating event occurrences. modeling association of inter-event times within a unit or subject.

42 Recall: Data Accrual for One Subject

43 Bladder Cancer Data Set: Groups [control (red), thiotepa (blue)]; Covariates; in WLW (89) Subject Calendar Time

44 A General Class of Dynamic Recurrent Event Models P{dN i (s) = 1 F s, Z i } = Z i λ 0 [E i (s)]ρ[(s, N i (s ));α]ψ[x i(s)β]ds λ 0 ( ): baseline hazard rate function; nonparametric. E i ( ): predictable effective (real) age process; effect of interventions. Ei (s) = s: minimal intervention. Ei (s) = s S in i (s ): perfect intervention. ρ(, ; α): known function; effect of event occurrences. ψ( ): known link function; effect of covariate X i. Z i : unobserved frailty variable; distribution H( ; η). Model Parameters: (λ 0 ( ), α, β, η).

45 Some Inferential Aspects for General Model Peña and Hollander (2004): proposed and discussed model, especially its generality. Peña, Slate and Gonzalez (2007): semiparametric estimation of model parameters. Stocker and Peña (2007): parametric estimation of model parameters. Peña (2012?): asymptotic properties of the estimators of semiparametric estimators of model parameters. Other inference methods: goodness-of-fit; group comparisons; model validation; estimating parameters in effective age process; Bayesian Approach. Challenge: need to gather effective age or real age of subjects/units/components in real studies.

46 Some Concluding Thoughts Revisited RCM with one event and Koziol-Green (KG) model. Revisited nonparametric estimation of inter-event distribution with recurrent event data. Extended KG model to recurrent event settings. Semiparametric estimation under this GKG recurrent event model. Examined properties of estimators under GKG structure. Efficiency losses of fully nonparametric estimator relative to GKG-based estimator. How about a Bayesian Approach to Inference? Touched on general dynamic models for recurrent event data.

47 Acknowledgements Work on GKG model joint with Akim Adekpedjou (Missouri University of Science and Technology). Research support from the National Science Foundation and National Institutes of Health. Thanks to Professor Sundar Subramanian and the organizers for inviting me to this conference.

Semiparametric Estimation for a Generalized KG Model with R. Model with Recurrent Event Data

Semiparametric Estimation for a Generalized KG Model with Recurrent Event Data Edsel A. Peña (pena@stat.sc.edu) Department of Statistics University of South Carolina Columbia, SC 29208 MMR Conference Beijing,