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# Dynamic Modelling and Statistical Analysis of Event Times

^{*}

## Abstract

This review article provides an overview of recent work in the modelling and analysis of recurrent events arising in engineering, reliability, public health, biomedical, and other areas. Recurrent event modelling possesses unique facets making it different and more difficult to handle than single event settings. For instance, the impact of an increasing number of event occurrences needs to be taken into account, the effects of covariates should be considered, potential association among the inter-event times within a unit cannot be ignored, and the effects of performed interventions after each event occurrence need to be factored in. A recent general class of models for recurrent events which simultaneously accommodates these aspects is described. Statistical inference methods for this class of models are presented and illustrated through applications to real data sets. Some existing open research problems are described.

**Key Words and Phrases:**Counting process, hazard function, martingale, partial likelihood, generalized Nelson-Aalen estimator, generalized product-limit estimator

## 1 Introduction

A decade ago in a *Statistical Science* article, Singpurwalla (1995) advocated the development,
adoption, and exploration of dynamic models in the theory and practice of
reliability. He also pinpointed that the use of stochastic processes in the
modelling of component and system failure times offers a rich environment to
meaningfully capture dynamic operating conditions. In this article, we review recent
research developments in dynamic failure time models, both in the context of
stochastic modelling and inference concerning model parameters. Dynamic models are
not limited in applicability and relevance to the engineering and reliability areas.
They are also relevant in other fields such as public health, biomedicine,
economics, sociology, and politics. This is because in many studies in these varied
areas, it is oftentimes of interest to monitor the occurrences of an event. Such
events could be the malfunctioning of a mechanical or electronic system,
encountering a bug in a computer software, outbreak of a disease, occurrence of
migraines, reoccurrence of a tumor, a drop of 200 points in the Dow Jones industrial
average during a trading day, commission of a criminal act in a city, serious
disagreements between a married couple, or the replacement of a cabinet
minister/secretary in an administration. These events recur and so it is of interest
to describe their recurrence behavior through a stochastic model. Supposing that
such a model has excellent predictive ability for event occurrences, then if event
occurrences lead to drastic and/or negative consequences, preventive interventions
may be attempted to minimize damages, whereas if they lead to beneficial and/or
positive outcomes, certain actions may be performed to hasten event occurrences.

It is because of performed interventions after event occurrences that dynamic models become especially pertinent, since through such interventions, or sometimes noninterventions, the stochastic structure governing future event occurrences is altered. This change in the mechanism governing the event occurrences could also be due to the induced change in the structure function in a reliability setting governing the system of interest arising from an event occurrence (cf., Hollander and Peña (1995); Kvam and Peña (2005); Peña and Slate (2005)). Furthermore, since several events may occur within a unit, it is also important to consider the association among the inter-event times which may arise because of unobserved random effects or frailties for the unit. In addition, there is a need to take into account the potential impact of environmental and other relevant covariates, possibly including the accumulated number of event occurrences, which could affect future event occurrences.

In this review article we describe a flexible and general class of dynamic stochastic models for event occurrences. This class of models was proposed by Peña and Hollander (2004). We also discuss inference methods for this class of models as developed in Peña, Strawderman, and Hollander (2001), Peña, Slate, and Gonzalez (2006), and Kvam and Peña (2005). We demonstrate its applications to real data sets and indicate some open research problems.

## 2 Background and Notation

Rapid and general progress in event time modelling and inference benefited immensely through the adoption of the framework of counting processes, martingales, and stochastic integration as introduced in Aalen (1978) pioneering work. The present review article adopts this mathematical framework. Excellent references for this framework are the monograph of Gill (1980), and the books by Fleming and Harrington (1991) and Andersen, Borgan, Gill, and Keiding (1993). We introduce in this section some notation and very minimal background in order to help the reader in the sequel, but advise the reader to consult the afore-mentioned references to gain an in-depth knowledge of this framework.

We denote by (Ω, *F*, **P**) the basic
probability space on which all random entities are defined. Since interest will be
on times between event occurrences or on the times of event occurrences,
nonnegative-valued random variables will be of concern. For a random variable
*T* with range _{+}
= [0, ∞), *F*(*t*)
= *F** _{T}*(

*t*) =

**P**{

*T*≤

*t*} and

*S*(

*t*) =

*S*

*(*

_{T}*t*) = 1 −

*F*(

*t*) =

**P**{

*T*>

*t*} will denote its distribution and survivor (also called reliability) functions, respectively. The indicator function of event

*A*will be denoted by

*I*{

*A*}. The cumulative hazard function of

*T*is defined according to

with the convention that *S*(*w*−)
= lim_{t}_{↑}_{w}*S*(*t*) =
**P**{*T* ≥
*w*} and ${\int}_{a}^{b}\cdots ={\int}_{(a,b]}\cdots $. For a nondecreasing function *G* :
_{+} →
_{+} with *G*(0) = 0
its product-integral over (0, *t*] is defined via (cf.,
Gill and Johansen (1990), Andersen et al. (1993)) ${\prod}_{w=0}^{t}[1-G(dw)]={lim}_{M\to \infty}{\prod}_{i=1}^{M}[1-(G({t}_{i})-G({t}_{i-1}))],$ where as *M* → ∞, the
partition 0 = *t*_{0} <
*t*_{1}
*<* ··· < *t** _{M}* =

*t*

_{0}satisfies max

_{1≤}

_{i}_{≤}

*|*

_{M}*t*

*−*

_{i}*t*

_{i}_{− 1}| → 0. The survivor function in terms of the cumulative hazard function becomes

For a discrete random variable *T* with jump points
{*t** _{j}*}s, the hazard λ

*at*

_{j}*t*

*is the conditional probability that*

_{j}*T*=

*t*

*, given*

_{j}*T*≥

*t*

*, so Λ(*

_{j}*t*) = ∑

_{{}

_{j: tj≤t}}λ

*. If*

_{j}*T*is an absolutely continuous random variable with density function

*f*(

*t*), its hazard rate function is λ(

*t*) =

*f*(

*t*)/

*S*(

*t*) so $\mathrm{\Lambda}(t)={\int}_{0}^{t}\lambda (w)dw=-logS(t).$ The product-integral representation of

*S*(

*t*) according to whether

*T*is purely discrete or purely continuous is

A benefit of using hazards or hazard rate functions in modelling is they provide qualitative aspects of the event occurrence process as time progresses. For instance, if the hazard rate function is increasing (decreasing) then this indicates that the event occurrences are increasing (decreasing) as time increases, and thus we have the notion of increasing (decreasing) failure rate [IFR (DFR)] distributions. For many years, it was the focus of theoretical reliability research to deal with classes of distributions such as IFR, DFR, increasing (decreasing) failure rate on average [IFRA (DFRA)], etc., specifically with regards to their closure properties under certain reliability operations (cf., Barlow and Proschan (1981)).

In monitoring an experimental unit for the occurrence of a recurrent event,
it is convenient and advantageous to utilize a counting process
{*N*(*s*), *s*
≥ 0}, where *N*(*s*) denotes
the number of times that the event has occurred over the interval [0,
*s*]. The paths of this stochastic process are
step-functions with *N*(0) = 0 and with jumps of unity.
If we represent the calendar times of event occurrences by *S*_{1}
*< S*_{2}
*< S*_{3}
*<* … with the convention that
*S*_{0} = 0, then $N(s)={\sum}_{j=1}^{\infty}I\{{S}_{j}\le s\}$. The inter-event times are denoted by *T** _{j}* =

*S*

*−*

_{j}*S*

_{j}_{−1},

*j*= 1, 2, …. In specifying the stochastic characteristics of the event occurrence process, one either specifies all the finite-dimensional distributions of the process {

*N*(

*s*)}, or specifies the joint distributions of the

*S*

*s or the*

_{j}*T*

*s. For example, a common specification for event occurrences is the assumption of a homogeneous Poisson process (HPP) where the inter-event times*

_{j}*T*

*s are independent and identically distributed exponential random variables with scale β. This is equivalent to specifying that, for any*

_{j}*s*

_{1}<

*s*

_{2}< … <

*s*

*, the random vector (*

_{K}*N*(

*s*

_{1}),

*N*(

*s*

_{2}) −

*N*(

*s*

_{1}), … ,

*N*(

*s*

*) −*

_{K}*N*(

*s*

_{K}_{−1})) have independent components and with

*N*(

*s*

*) −*

_{j}*N*(

*s*

_{j}_{−1}) having a Poisson distribution with parameter β(

*s*

*−*

_{j}*s*

_{j}_{−1}). From this specification, the finite-dimensional distributions of (

*N*(

*s*

_{1}),

*N*(

*s*

_{2}), …,

*N*(

*s*

*)) can be obtained.*

_{K}An important concept in dynamic modelling is that of a history or a
filtration, a family of σ-fields **F** =
{*F*_{s}*: s* ≥ 0} where *F** _{s}* is a sub-α-field of

*F*with

*F*

_{s1}

*F*

_{s}_{2}for every

*s*

_{1}

*< s*

_{2}and with

*F*

*= ∩*

_{s}

_{h}_{↓0}

*F*

_{s}_{+}

*, a right-continuity property. One interprets*

_{h}*F*

*as all information that accrued over [0,*

_{s}*s*]. When augmented in (Ω,

*F*,

**P**), we obtain the filtered probability space (Ω,

*F*,

**F**,

**P**). It is with respect to such a filtered probability space that a process {

*X*(

*s*) :

*s*≥ 0} is said to be adapted [

*X*(

*s*) is measureable with respect to

*F*

*, ∀*

_{s}*s*≥ 0)]; is said to be a (sub)martingale [adapted,

*E*|

*X*(

*s*)| < ∞, and, ∀

*s*,

*t*≥ 0,

*E*{

*X*(

*s*+

*t*)|

*F*

*}(≥) =*

_{s}*X*(

*s*) almost surely]. Doob-Meyer’s decomposition guarantees that for a submartingale

**Y**= {

*Y*(

*s*) :

*s*≥ 0} there is a unique increasing predictable [measurable with respect to the σ-field generated by adapted processes with left-continuous paths] process

**A**= {

*A*(

*s*) :

*s*≥ 0}, called the compensator, with

*A*(0) = 0 such that

**M**= {

*M*(

*s*) =

*Y*(

*s*) −

*A*(

*s*) :

*s*≥ 0} is a martingale. Via Jensen’s Inequality, then for a square-integrable martingale

**X**= {

*X*(

*s*) :

*s*≥ 0}, there is a unique compensator process

**A**= {

*A*(

*s*) :

*s*≥ 0} such that

**X**

^{2}−

**A**is a martingale. This process

**A**, denoted by X = {

*X*(

*s*) :

*s*≥ 0}, is called the predictable quadratic variation (PQV) process of

**X**. A useful heuristic, though somewhat imprecise, way of presenting the main properties of martingales and PQVs are through the following differential forms. For a martingale

**M**,

*E*{

*dM*(

*s*)|

*F*

_{s}_{−}} = 0, ∀

*s*≥ 0; whereas, for the PQV

**M**,

*E*{

*dM*

^{2}(

*s*)|

*F*

_{s}_{−}} =

*Var*{

*dM*(

*s*)|

*F*

_{s}_{−}} =

*d*

*M*(

*s*), ∀

*s*≥ 0.

For the HPP N =
{*N*(*s*) : *s*
≥ 0} with rate β and with **F**
= {*F** _{s}* =
σ{

*N*(

*w*),

*w*≤

*s*} :

*s*≥ 0},

**N**is a submartingale since its paths are nondecreasing. Its compensator process is

**A**= {

*A*(

*s*) = β

*s : s*≥ 0}, so that

**M**= {

*M*(

*s*) =

*N*(

*s*) − β

*s : s*≥ 0} is a martingale. Furthermore, since

*N*(

*s*) is Poisson-distributed with rate β

*s*, so that {

*M*

^{2}(

*s*) −

*A*(

*s*) = (

*N*(

*s*) − β

*s*)

^{2}− β

*s : s*≥ 0} is a martingale, the PQV process of

**M**is also

**A**. Through the heuristic forms, we have

*E*{

*dN*(

*s*)|

*F*

_{s}_{−}} =

*dA*(

*s*),

*s*≥ 0: Since

*dN*(

*s*) {0, 1}, then we obtain the probabilistic expression

**P**{

*dN*(

*s*) = 1|

*F*

_{s}_{−}} =

*dA*(

*s*),

*s*≥ 0: Analogously,

*Var*{

*dN*(

*s*)|

*F*

_{s}_{−}} =

*E*{[

*dN*(

*s*) −

*dA*(

*s*)]

^{2}|

*F*

_{s}_{−}} =

*dA*(

*s*),

*s*≥ 0: These formulas hold for a general counting process {

*N*(

*s*) :

*s*≥ 0} with compensator process {

*A*(

*s*) :

*s*≥ 0}. In essence, conditionally on

*F*

_{s}_{−},

*dN*(

*s*) has a Bernoulli distribution with success probability

*dA*(

*s*). Over the interval [0,

*s*], following Jacod, the likelihood function could be written in product-integral form as

with the last equality holding when *A*(·) has
continuous paths.

Stochatic integrals play a crucial role in this stochastic process framework
for event time modelling. For a square-integrable martingale **X**
= {*X*(*s*) :
*s* ≥ 0} with PQV process
**X** =
{*X*(*s*) :
*s* ≥ 0}, and for a bounded predictable
process *Y* = {*Y*
(*s*) : *s* ≥ 0}, the
stochastic integral of *Y* with respect to *X*,
denoted by $\int YdX=\{{\int}_{0}^{s}Y(w)dX(w):s\ge 0\},$ is well-defined. It is also a square-integrable martingale with
PQV process $\langle \int YdX\rangle =\{{\int}_{0}^{s}{Y}^{2}(w)d\langle X\rangle (w):s\ge 0\}.$ When *X* is associated with a counting process
*N*, that is, *X* = *N*
− *A*, the paths of the stochastic integral
∫*Y dX* can be taken as pathwise Lebesgue-Stieltjes
integrals.

Martingale theory also plays a major role in obtaining asymptotic properties of estimators as first demonstrated in Aalen (1978), Gill (1980), and Andersen and Gill (1982). The main tools used in asymptotic analysis are Lenglart’s inequality (cf., Lenglart (1977); Fleming and Harrington (1991); Andersen et al. (1993)) which is used in proving consistency; Rebolledo (1980) martingale central limit theorem (MCLT) (cf., Fleming and Harrington (1991); Andersen et al. (1993)) which is used for obtaining weak convergence results. We refer the reader to Fleming and Harrington (1991) and Andersen et al. (1993) for the in-depth theory and applications of these modern tools in failure-time analysis.

## 3 Class of Dynamic Models

Let us now consider a unit being monitored over time for the occurrence of a
recurrent event. The monitoring period could be a fixed interval or it could be a
random interval, and for our purposes we denote this period by [0,
τ], where τ is assumed to have some distribution
*G*, which may be degenerate. With a slight notational change
from Section 2 we denote by
*N*^{†}(*s*) the number of events
that have occurred on or before time *s*, and by
*Y*^{†}(*s*) an indicator
process which equals 1 when the unit is still under observation at time
*s*, 0 otherwise. With *S*_{0} = 0
and *S** _{j}*,

*j*= 1, 2, … denoting the successive calendar times of event occurrences, and with

*T*

*=*

_{j}*S*

*−*

_{j}*S*

_{j}_{−1},

*j*= 1, 2, … being the inter-event or gap times, observe that

Associated with the unit is a, possibly time-dependent, 1 ×
*q* covariate vector **X** =
{**X**(*s*) : *s*
≥ 0}. In reliability engineering studies, the components of
this covariate vector could be related to environmental or operating condition
characteristics; in biomedical studies, they could be blood pressure, treatment
assigned, initial tumor size, etc.; in a sociological study of marital disharmony,
they could be length of marriage, family income level, number of children residing
with couple, ages of children, etc. Usually, after each event occurrence, some form
of intervention is applied or performed, such as replacing or repairing failed
components in a reliability system, reducing or increasing physical activity after a
heart attack in a medical setting. These interventions will typically impact the
next occurrence of the event. There is furthermore recognition that for the unit the
inter-event times are associated or correlated, possibly because of unobserved
random effects or so-called frailties. A pictorial representation of these aspects
is contained in Figure 1. Observe that because
of the finiteness of the monitoring period, which leads to a sum-quota accrual
scheme, there will always be a right-censored inter-event time. The observed number
of event occurrences over [0, τ],
*K* =
*N*^{†}(τ), is also informative about
the stochastic mechanism governing event occurrences. In fact, since $K=max\{k:{\sum}_{j=1}^{k}{T}_{j}\le \tau \}$ then, conditionally on (*K*, τ), the
vector (*T*_{1}, *T*_{2},
…, *T** _{K}*) have dependent components, even if at the outset

*T*

_{1},

*T*

_{2}, … are independent.

**...**

Recognizing these different aspects in recurrent event settings, Peña and Hollander (2004) proposed
a general class of models that simultaneously incorporates all of these aspects. To
describe this class of models, we suppose that there is a filtration **F**
= {*F** _{s}* :

*s*≥ 0} such that for each

*s*≥ 0, σ{(

*N*

^{†}(

*v*),

*Y*

^{†}(

*v*+),

*X*(

*v*+), (

*v*+)) :

*v*≤

*s*}

*F*

*. We also assume that there exists an unobservable positive random variable*

_{s}*Z*, called a frailty, which induces the correlation among the inter-event times. The class of models of Peña and Hollander (2004) can now be described in differential form via

where λ_{0}(·) is a baseline hazard rate
function; ρ(·; ·) is a nonnegative function with
ρ(0; ·) = 1 and with α being some
parameter; ψ(·) is a nonnegative link function with
β a *q* × 1 regression parameter vector; and
*Z* is a frailty variable. The at-risk process
*Y*^{†}(*s*) indicates that
the conditional probability of an event occurring becomes zero whenever the unit is
not under observation. Possible choices of the ρ(·;
·) and ψ(·) functions are
ψ(*k*; α) = α* ^{k}* and ψ(

*w*) = exp(

*w*), respectively. For the geometric choice of ρ(·; ·), if

*α*> 1 the effect of accumulating event occurrences is to accelerate event occurrences, whereas if α < 1 the event occurrences decelerate, the latter situation appropriate in software debugging. The process (·) appearing as argument in the baseline hazard rate function, called the effective age process, is predictable, observable, nonnegative, and pathwise almost surely differentiable with derivative ′(·). This effective age process models the impact of performed interventions after each event occurrence. A pictorial depiction of this effective age process is in Figure 2. In this plot, after the first event, the performed intervention has the effect of reverting the unit to an effective age equal to the age just before the event occurrence (called a minimal repair or intervention), while after the second event, the performed intervention has the effect of reverting the effective age to that of a new unit (hence this is called a perfect intervention or repair). After the third event, the intervention was neither minimal nor perfect and it has the effect of re-starting the effective age at a value between zero and that just before the event occurred; while for the fourth event, the intervention was detrimental in that the re-starting effective age exceeded that just before the event occurred.

*s*), for a unit encountering successive occurrences of a recurrent event.

The effective age process could occur in many forms, and the idea is this should be determined in a dynamic fashion in conjunction with interventions that are performed. As such the determination of this process should preferably be through consultations with experts of the subject matter under consideration. Common forms of this effective age process are:

where in (3.8) with
*I*_{1}, *I*_{2}, …
being independent Bernoulli random variables with *I** _{k}* having success probability

*p*(

*S*

*) with*

_{k}*p*:

_{+}→ [0, 1], we define $\eta (s)={\sum}_{i=1}^{{N}^{\u2020}(s)}{I}_{i}$ and Γ

_{0}= 0, Γ

*= min{*

_{k}*j*> Γ

_{k}_{−1}:

*I*

*= 1},*

_{j}*k*= 1, 2, … Thus, in (3.8), (

*s*) represents the time measured from

*s*since the last perfect repair. This effective age is from Block, Borges, and Savits (1985), whereas when

*p*(

*s*) =

*p*for some

*p*[0, 1], we obtain the effective age process for the Brown and Proschan (1983) minimal repair model. Clearly, the effective age functions (3.6) and (3.7) are special cases of (3.8). Other effective age process that could be utilized are those associated with the general repair model of Kijima (1989), Stadje and Zuckerman (1991), Baxter, Kijima, and Tortorella (1996), Dorado, Hollander, and Sethuraman (1997) and Last and Szekli (1998). See also Lindqvist (1999) and Lindqvist, Elvebakk, and Heggland (2003) for a review of some of these models pertaining to repairable systems and Gonzalez, Peña, and Slate (2005) for an effective age process suitable for cancer studies.

A crucial property arising from the intensity specification in (3.5) is it amounts to postulating that, with

then, conditionally on *Z*, the process

is a square-integrable martingale with PQV process
*A*^{†}(·; *Z*,
λ_{0}(·), α, β). The class
of models is general and flexible and subsumes many current models for recurrent
events utilized in survival analysis and reliability. In particular, it includes
those of Jelinski and Moranda (1972), Gail, Santner, and Brown (1980), Gill (1981), Prentice, Williams, and Peterson (1981), Lawless (1987), Aalen and Husebye
(1991), Wang and Chang (1999), Peña et al. (2001), and Kvam and Peña (2005). We
demonstrate the class of models via some concrete examples.

### Example

The first example concerns a load-sharing *K*-component
parallel system with identical components. The recurrent event of interest is
component failure and failed components are not replaced. When a component
fails, a redistribution of the system load occurs among the remaining
functioning components, and to model this system in a general way, we let
α_{0} = 1 and α_{1},
…, *α*_{K}_{−1} be positive constants, referred to as
load-share parameters. One possible specification of these parameters is α* _{k}* =

*K=*(

*K*−

*k*),

*k*= 0, 1, 2, …,

*K*− 1, though they could be unknown constants, possibly ordered. The hazard rate of event occurrence at calendar time

*s*, provided that the system is still under observation, is λ(

*s*) = λ

_{0}(

*s*)[

*K*−

*N*

^{†}(

*s*−)]α

_{N†(s−)}, where λ

_{0}(·) is the common hazard rate function of each component and

*N*

^{†}(

*s*) is the number of component failures observed on or before time

*s*. This is a special case of the general class of models with (

*s*) =

*s*, ρ(

*k*; α

_{1}, …, α

_{K}_{−1}) = [

*K*−

*k*]α

*, and ψ(*

_{k}*w*) = 1. This is the equal load-sharing model in Kvam and Peña (2005). More generally, this could accommodate environmental or operating condition covariates for the system, and even an unobserved frailty component.

### Example

Assume in a software reliability model that there are α bugs at
the beginning of a debugging process and the event of interest is encountering a
bug. A possible model is these α bugs are competing to be encountered,
and if each of them has hazard rate of
λ_{0}(*s*) of being encountered at time
*s*, then the total hazard rate at time *s* of
the software failing is λ_{0}(*s*)α.
Upon encountering a bug at time *S*_{1}, this bug is
eliminated, thus decreasing the number of remaining bugs by one. The debugging
process is then re-started at time just after *S*_{1}
(assuming it takes zero time to eliminate the bug, clearly an
over-simplification). In general, suppose that just before calendar time
*s*,
*N*^{†}(*s*−) bugs
have been removed, and the last bug was encountered at calendar time
*S*_{N†(s−)}.
Then, the overall hazard of encountering a bug at calendar time
*s* with *s > S*_{N}_{†(}_{s}_{−)} is
λ_{0}(*s* − *S*_{N}_{†(}_{s}_{−)})[α −
*N*^{†}(*s*−)].
Thus, provided that the debugging process is still in progress at time
*s*, then the hazard of encountering a bug at time
*s* is λ(*s*) =
λ_{0}(*s* − *S*_{N}_{†(}_{s}_{−)}) max[0, α −
*N*^{†}(*s*−)]:
Again, this is a special case of the general class of models with
(*s*) =
*s*− *S*_{N}_{†(}_{s}_{−)}, a consequence of the re-start of debugging
process, ρ(*k*; α) =
max{0, α−*k*}, and
ψ(*w*) = 1. This software debugging
model is the model of Jelinski and Moranda
(1972) and it was also proposed by Gail et al. (1980) as a carcinogenesis model. See also Agustin and Peña (1999) for
another model in software debugging which is a special case of the general class
of models.

Cox (1972) proportional hazards
model is one of the most used models in biomedical and public health settings.
Extensions of this model have been used in recurrent event settings, and Therneau and Grambsch (2000) discusses some
of these Cox-based models such as the independent increment model of Andersen and Gill (1982), the conditional
model of Prentice et al. (1981), and the
marginal model of Wei, Lin, and Weissfeld
(1989). The independent increment model is a special case of the general
class of models obtained by taking either (*s*)
= *s* or (*s*)
= *s* − *S*_{N}_{†(}_{s}_{−)} with ρ(*k*;
α) = 1 and ψ(*w*) =
exp(*w*). The marginal model stratifies according to the
event number and assumes a Cox-type model for each of these strata, with the
*j*th inter-event time in the *i*th unit
having intensity *Y** _{ij}*(

*t*) λ

_{0}

*(*

_{j}*t*) exp{

*X*

*(*

_{i}*t*)β

*}, where*

_{j}*Y*

*(*

_{ij}*t*) equals one until the occurrence of the

*j*th event or when the unit is censored. The conditional model is similar to the marginal model except that

*Y*

*(*

_{ij}*t*) becomes one only after the (

*j*& minus; 1)th event has occurred.

## 4 Statistical Inference

The relevant parameters for the model in (3.5) are
λ_{0}(·), α, β, and the
parameter associated with the distribution of the frailty variable
*Z*. A variety of forms for this frailty distribution is possible,
but we restrict to the case where *Z* has gamma distribution with
mean one and variance 1/ξ. The parameter associated with
*G*, the distribution of τ, is usually viewed as
nuisance, though in current joint research with Akim Adekpedjou, a PhD student at
the University of South Carolina, the situation where *G* is
informative about the distributions of the inter-event times is being explored.

Knowing the values of the model parameters is important because the model
could be utilized to predict future occurrences of the event, an important issue
especially if an event occurrence is detrimental. To gain knowledge about these
parameters, a study is performed to produce sample data which is the basis of
inference about the parameters. We consider a study where *n*
independent units are observed and with the observables over (calendar) time
[0, *s**] denoted by

Observe that DATA* _{n}*(

*s**) provides information about τ

*s. More generally, we observe the filtrations {(*

_{i}*F*

*,*

_{iv}*v*≤

*s**),

*i*= 1, 2, …,

*n*} or the overall filtration ${F}_{{s}^{*}}=\{{\mathcal{F}}_{v}={\vee}_{i=1}^{n}{\mathcal{F}}_{iv,}v\le {s}^{*}\}.$ The goals of statistical inference are to obtain point or interval estimates and/or test hypotheses about model parameters, as well as predict the time of occurrence of a future event, when DATA

*(*

_{n}*s**) or

**F**

_{s}_{*}is available. We focus on the estimation problem below, though we note that the prediction problem is of great practical importance.

Conditional on **Z** = (*Z*_{1};
*Z*_{2}, …, *Z** _{n}*), from (2.3) the likelihood
process for (λ

_{0}(·); α, β) is

where ${B}_{i}(v;{\lambda}_{0}(\cdot ),\alpha ,\beta )={Y}_{i}^{\u2020}(v){\lambda}_{0}[{\epsilon}_{i}(v)]\rho [{N}_{i}^{\u2020}(v-);\alpha ]\psi [{X}_{i}(v)\beta ].$ Observe that the likelihood process when the model does not
involve any frailties is obtained from (4.12) by setting Z* _{i}* = 1;

*i*= 1, 2, …,

*n*, which is equivalent to letting ξ → ∞. The resulting no-frailty likelihood process is

This likelihood process is the basis of inference about
(λ_{0}(·), α, β) in the absence
of frailties. Going back to (4.12),
by marginalizing over **Z** under the gamma frailty assumption, the
likelihood process for (λ_{0}(·), α,
β, ξ) becomes

There are two possible specifications for the baseline hazard rate function
λ_{0}(·): parametric or nonparametric. If
parametrically specified then it is postulated to belong to some parametric family
of hazard rate functions, such as the Weibull or gamma families, that depends on an
unknown *p*×1 parameter vector θ. In this
situation, a possible estimator of (θ, α, β,
ξ), given **F*** _{s}**, is the maximum likelihood estimator $(\widehat{\theta},\widehat{\alpha},\widehat{\beta},\widehat{\xi})$, the maximizer of the mapping (θ, α,
β, ξ)

*L*(

*s**; λ

_{0}(·;θ), α, β, ξ). Stocker (2004) studied the finite-and large-sample properties, and associated computational issues, of this parametric ML estimator in his dissertation research. In particular, following the approach of Nielsen, Gill, Andersen, and Sorensen (1992) he implemented an expectation-maximization (EM) algorithm (cf., Dempster, Laird, and Rubin (1977)) for obtaining the ML estimate. In this EM implementation the frailty variates

*Z*

*s are viewed as missing and a variant of the no-frailty likelihood process in (4.13) is used for the maximization step in this algorithm. We refer to Stocker (2004) for details of this computational implementation. For large*

_{i}*n*, and under certain regularity conditions, it could also be shown that $(\widehat{\theta},\widehat{\alpha},\widehat{\beta},\widehat{\xi})$ is approximately multivariate normally distributed with mean (θ, α, β, ξ) and covariance matrix ${\scriptstyle \frac{1}{n}}{I}^{-1}(\widehat{\theta},\widehat{\alpha},\widehat{\beta},\widehat{\xi})$, where

*I*(θ, α, β, ξ) is the observed Fisher information associated with the likelihood function

*L*(

*s**; λ

_{0}(·;θ), α, β, ξ). That is, with Θ = (θ, α, β, ξ)

^{t},

*I*(θ, α, β, ξ) = −{

^{2}/ΘΘ

^{t}}

*l*(

*s**; λ

_{0}(·;θ), α, β, ξ), where

*l*(

*s*; λ

_{0}(·;θ), α, β, ξ) is the log-likelihood process given by

Tests of hypotheses and construction of confidence intervals about model
parameters could be developed using the asymptotic properties of the ML estimators.
For small samples, they could be based on their approximate sampling distributions
obtained through computer-intensive methods such as bootstrapping. It is usually the
case that a parametric specification of λ_{0}(·) is
more suitable in the reliability and engineering situations.

In biomedical and public health settings, it is typical to specify
λ_{0}(·) nonparametrically, that is, to simply
assume that λ_{0}(·) belongs to the class of hazard
rate functions with support [0, ∞). This leads to a
semiparametric model, with infinite-dimensional parameter
λ_{0}(·) and finite-dimensional parameters
(α, β, ξ). Inference for this semiparametric model
was considered in Peña et al.
(2006). In this setting, interest is on the finite-dimensional parameters
(α, β, ξ) and the infinite-dimensional parameters ${\mathrm{\Lambda}}_{0}(\cdot )={\int}_{0}^{\cdot}{\lambda}_{0}(w)dw$ and its survivor function ${S}_{0}(\cdot )={\mathrm{\Pi}}_{w=0}^{\cdot}\hspace{0.17em}[1-{\mathrm{\Lambda}}_{0}(dw)].$ A difficulty encountered in estimating
Λ_{0}(·) is that in the intensity specification in
(3.5), the argument of
λ_{0}(·) is the effective age
(*s*), not *s*, and our interest is
on Λ_{0}(*t*), *t* ≥
0. This poses difficulties, especially in establishing asymptotic properties,
because the usual martingale approach as pioneered by Aalen (1978), Gill (1980), and
Andersen and Gill (1982) (see also Andersen et al. (1993) and Fleming and Harrington (1991)) does not directly carry
through. In the simple IID renewal setting where (*s*)
= *s* − *S*_{N}_{†(s–)},
ρ(*k*; α) = 1, and
ψ(*w*) = 1, Peña, Strawderman, and Hollander (2000) and
Peña et al. (2001), following
ideas of Gill (1981) and Sellke (1988), implemented an approach using
time-transformations to obtain estimators of
Λ_{0}(·) and
*S*_{0}(·). In an indirect way, with partial use
of Lenglart's inequality and Rebolledo’s MCLT, they obtained
asymptotic properties of these estimators, such as their consistency and their weak
convergence to Gaussian processes. This approach in Peña et al. (2001) was also utilized in Peña et al. (2006) to obtain the estimators
of the model parameters in the more general model.

The idea behind this approach is to define the predictable (with respect to
*s* for fixed *t*) doubly-indexed process
*C** _{i}*(

*s; t*) =

*I*{

*(*

_{i}*s*) ≤

*t*},

*i*= 1, 2; …,

*n*, which indicates whether at calendar time

*s*the unit's effective age is at most

*t*. We then define the processes ${N}_{i}(s,t)={\int}_{0}^{s}{C}_{i}(v,t){N}_{i}^{\u2020}(dv);\hspace{0.17em}{A}_{i}(s,t)={\int}_{0}^{s}{C}_{i}(v,t){A}_{i}^{\u2020}(dv);$ and ${M}_{i}(s,t)={N}_{i}(s,t)-{A}_{i}(s,t)={\int}_{0}^{s}{C}_{i}(v,t){M}_{i}^{\u2020}(dv)$. Because for each

*t*≥ 0,

*C*

*(·;*

_{i}*t*) is a predictable and a {0, 1}-valued process, then

*M*

*(·,*

_{i}*t*) is a square-integrable martingale with PQV

*A*

*(·,*

_{i}*t*). However, observe that for fixed

*s*,

*M*

*(*

_{i}*s*, ·) is not a martingale though it still satisfies

*E*{

*M*

*(*

_{i}*s*,

*t*)} = 0 for every

*t*. The next step is to have an alternative expression for

*A*

*(*

_{i}*s*,

*t*) such that Λ(·) appears with an argument of

*t*instead of

*(*

_{i}*v*). With

_{ij}_{−1}(

*v*) =

*(*

_{i}*v*)

*I*{

*S*

_{ij}_{−1}<

*v*≤

*S*

*} on $I\{{Y}_{i}^{\u2020}(v)=0\}$, this is achieved as follows:*

_{ij}Letting

and defining the new ‘at-risk’ process

then, after a variable transformation *w* = _{ij}_{−1}(*v*) for each summand in (4.16), we obtain an alternative form
of *A** _{i}*(

*s*,

*t*) given by ${A}_{i}(s,t;{\mathrm{\Lambda}}_{0}(\cdot ),\alpha ,\beta )={\int}_{0}^{t}{Y}_{i}(s,w;\alpha ,\beta ){\mathrm{\Lambda}}_{0}(dw).$ The utility of this alternative form is that Λ

_{0}(·) appears with the correct argument for estimating it. If, for the moment, we assume that we know α and β, by virtue of the fact that

*M*

*(*

_{i}*s; t*; α, β) has zero mean, then using the idea of Aalen (1978), a method-of-moments ‘estimator; of Λ

_{0}(

*t*) is

where ${S}_{0}(s,t)={\sum}_{i=1}^{n}{Y}_{i}(s,t;\alpha ,\beta )$. This ‘estimator’ of
Λ_{0}(·) could be plugged into the likelihood
function over [0,
*s*^{*}] to obtain the profile
likelihood of (α, β), given by

This profile likelihood is reminiscent of the partial likelihood of Cox (1972) and Andersen and Gill (1982) for making inference about the
finite-dimensional parameters in the Cox proportional hazards model and the
multiplicative intensity model. The estimator of (α, β),
denoted by $(\widehat{\alpha},\widehat{\beta})$ is the maximizer of the mapping (α, β)
*L** _{P}*(

*s**; α, β). Algorithms and software for computing the estimate $(\widehat{\alpha},\widehat{\beta})$ were developed in Peña et al. (2006). The estimator of Λ

_{0}(

*t*) is obtained by substituting $(\widehat{\alpha},\widehat{\beta})$ for (α, β) in ${\widehat{\mathrm{\Lambda}}}_{0}(s*,t;\alpha ,\beta )$ to yield the generalized Aalen-Breslow-Nelson estimator

By invoking the product-integral representation of a survivor function, a
generalized product-limit estimator of the baseline survivor function
*S*_{0}(*t*) is ${\widehat{S}}_{0}(s*,t)={\prod}_{w=0}^{t}[1-{\widehat{\mathrm{\Lambda}}}_{0}(s*,dw)].$

Peña et al. (2006) also
discussed the estimation of Λ_{0}(·) and the
finite-dimensional parameters (α, β, ξ) in the
presence of gamma-distributed frailties. The ML estimators of these parameters
maximizes the likelihood *L*(*s**;
Λ_{0}(·), α β,
ξ) in (4.14), with the
proviso that the maximizing Λ_{0}(·) jumps only at
observed values of * _{i}*(

*S*

*)s. An EM algorithm finds these maximizers and its implementation is described in detail in Peña et al. (2006). We briefly describe the basic ingredients of this algorithm here.*

_{ij}With θ = (Λ_{0}(·),
α, β ξ), in the expectation step of the algorithm,
expressions for *E*{*Z** _{i}*|θ,

**F**

_{s}_{*}} and

*E*{log

*Z*

*|θ,*

_{i}**F**

_{s}_{*}}, which are easy to obtain under gamma frailties, are needed. For the maximization step, with

*E*

_{Z}_{|θ(0)}denoting expectation with respect to

**Z**when the parameter vector θ equals ${\theta}^{(0)}=({\mathrm{\Lambda}}_{0}^{(0)}(\cdot ),{\alpha}^{(0)},{\beta}^{(0)},{\xi}^{(0)}),$ we require

*Q*(θ θ

^{(0)},

**F**

_{s}_{*}) =

*E*

_{Z}_{|θ(0)}{log

*L*

*(*

_{C}*s**;

**Z**, θ

^{(0)}}, where LC (

*s*;

**Z**, θ) is in (4.12). This Q(θ; θ

^{(0)},

**F**

_{s}_{*}) is maximized with respect to θ = (Λ

_{0}(·), α, β, ξ). This maximization could be performed in two steps: first, maximize with respect to (Λ

_{0}(·), α, β) using the procedure in the case without frailties except that

*S*

_{0}(

*s*,

*t*; α, β) is replaced by ${S}_{0}(s,t;Z,\alpha ,\beta )={\sum}_{i=1}^{n}{Z}_{i}{Y}_{i}(s,t;\alpha ,\beta );$ and second, maximize with respect to ξ a gamma log-likelihood with estimated log

*Z*

*and*

_{i}*Z*

*. To start the iteration process, a seed value for Λ*

_{i}_{0}(·) is needed, which could be the estimate of Λ

_{0}(·) with no frailties. Seed values for (α, β, ξ) are also required. Through this EM implementation, estimates of (Λ

_{0}(·), α, β, ξ) are obtained and, through the product-integral representation, an estimate of the baseline survivor function

*S*

_{0}(·).

## 5 Illustrative Examples

The applicability of these dynamic models still needs further and deeper investigations. We provide in this section illustrative examples to demonstrate their potential applicability.

### Example

The first example deals with a data set in Kumar and Klefsjo (1992) which consists of failure
times for hydraulic systems of load-haul-dump (LHD) machines used in mining. The
data set has six machines with two machines each classified into old, medium,
and new. For each machine the successive failure times were observed and the
resulting data is depicted in Figure 3.
Using an effective age process (*s*) =
*s* − S_{N}_{†(}_{s}_{−)} this was analyzed in Stocker (2004) (see also Stocker and Peña (2006)) using the
general class of models when the baseline hazard function is postulated to be a
two-parameter Weibull hazard function
Λ_{0}(*t*; θ_{1},
θ_{2}) =
(*t*/θ_{2})^{θ1},
and in Peña et al. (2006)
when the baseline hazard function is nonparametrically specified. The age
covariate was coded according to the dummy variables
(*X*_{1}, *X*_{2}) taking values
(0, 0) for the oldest machines, (1, 0) for the medium-age machines, and (0, 1)
for the newest machines. The parameter estimates obtained for a nonparametric
and a parametric baseline hazard function specifications are contained in Table 1 where the estimates for the
parametric specification were from Stocker
(2004). The estimates of the baseline survivor function ${\dot{S}}_{0}(\cdot )$ under the nonparametric and parametric Weibull specifications
are overlaid in Figure 4. From this table
of estimates, observe that a frailty component is not needed for both
nonparametric and parametric specifications since the estimates of the frailty
parameter ξ are very large in both cases. Both estimates of the
β_{1} and β_{2} coefficients are
negative, indicating a potential improvement in the lengths of the working
period of the machines when they are of medium age or newer, though an
examination of the standard errors reveals that we could not conclude that the
β-coefficients are significantly different from zeros. On the other
hand, the estimate of α for both specifications are significantly
greater than one, indicating the potential weakening effects of accumulating
event occurrences. From Figure 4 we also
observe that the two-parameter Weibull appears to be a good parametric model for
the baseline hazard function as the nonparametric and parametric baseline hazard
function estimates are quite close to each other. However, a formal procedure
for validating this claim still needs to be developed. This is a problem in
goodness-of-fit which is currently being pursued.

*X*

_{1}

*; X*

_{2})

**...**

*S*

_{0}(

*t*) under a nonparametric and a parametric (Weibull) specification for the LHD hydraulic data of Kumar and Klefsjo (1992).

### Example

The second example is provided by fitting the general class of models to
the bladder cancer data in Wei et al.
(1989). A pictorial depiction of this data set can be found in Peña et al. (2006), where it
was analyzed using a nonparametric specification of the baseline hazard
function. This data set consists of 85 subjects and provide times to recurrence
of bladder cancer. The covariates included in the analysis are
*X*_{1}, the treatment indicator (1 =
placebo; 2 = thiotepa); *X*_{2} , the size
(in cm) of the largest initial tumor; and *X*_{3}, the
number of initial tumors. In fitting the general model in (3.5) we used
ρ(*k*; α) = α* ^{k}*. Furthermore, since the data set does not contain information about
the effective age, we considered two situations for demonstration purposes: (i)
perfect intervention is always performed $({\epsilon}_{i}(s)=s-{S}_{i{N}_{i}^{\u2020}(s-)});$ and (ii) minimal intervention is always performed (

*(*

_{i}*s*) =

*s*). The parameter estimates obtained by fitting the model with frailties, and other estimates using procedures discussed in the literature are presented in Table 2. The estimates of their standard errors (s.e.) are in parentheses, with the s.e.s under minimal repair intervention model obtained through a jacknife procedure. The other parameter estimates in the table are those from the marginal method of Wei et al. (1989), the conditional method of Prentice et al. (1981), and the Andersen and Gill (1982) method, which were mentioned earlier (cf., Therneau and Grambsch (2000)). Estimates of the survivor functions at the mean covariate values are in Figure 5.

**...**

**...**

Observe from this figure that the thiotepa group possesses higher survivor probability estimates compared to the placebo group in either specifications of the effective age process. Examining Table 2, note the important role the effective age process provides in reconciling differences among the estimates from the other methods. When the effective age process corresponds to perfect intervention, the resulting estimates from the general model are quite close to those obtained from the Prentice et al. (1981) conditional method; whereas when a minimal intervention effective age is assumed, then the general model estimates are close to those from the marginal method of Wei et al. (1989). Thus, the differences among these existing methods could perhaps be attributable to the type of effective age process used. This indicates the importance of the effective age in modelling recurrent event data. If possible, it therefore behooves to monitor and assess this effective process in real applications.

## 6 Open Problems and Concluding Remarks

There are several open research issues pertaining to this general model for
recurrent events. First is to ascertain asymptotic properties of the estimators of
model parameters under the frailty model when the baseline hazard rate function
Λ_{0}(·) is nonparametrically specified. A
second problem, which arise after fitting this general class of models, is to
validate its appropriateness and to determine the presence of outlying and/or
influential observations. This is currently being performed jointly with Jonathan
Quiton, a PhD student at the University of South Carolina. Of particular issue is
the impact of the sum-quota accrual scheme, leading to the issue of determining the
proper sampling distribution for assessing values of test statistics. This
validation issue also leads to goodness-of-fit problems. It might for instance be of
interest to test the hypothesis that the unknown baseline hazard function
Λ_{0}(·) belongs to the Weibull class of hazard
functions. In current research we are exploring smooth goodness-of-fit tests
paralleling those in Peña
(1998b,a) and Agustin and Peña (2005) which build on work
by Neyman (1937). This will lead to notions
of generalized residuals from this general class of models. Another problem is a
nonparametric Bayesian approach to failure time modelling. Not much has been done
for this approach in this area, though the comprehensive paper of Hjort (1990) provides a solid contribution for the
multiplicative intensity model. It is certainly of interest to implement this
Bayesian paradigm for the general class of models for recurrent events.

To conclude, this article provides a selective review of recent research developments in the modelling and analysis of recurrent events. A general class of models accounting for important facets in recurrent event modelling was described. Methods of inference for this class of models were also described, and illustrative examples were presented. Some open research problems were also mentioned.

## Acknowledgments

The author acknowledges the research support provided by NIH Grant 2 R01 GM56182 and
NIH COBRE Grant RR17698. He is very grateful to Dr. Sallie Keller-McNulty, Dr.
Alyson Wilson, and Dr. Christine Anderson-Cook for inviting him to contribute an
article to this special issue of *Statistical Science*. He
acknowledges his research collaborators, Mr. J. Gonzalez, Dr. M. Hollander, Dr. P.
Kvam, Dr. E. Slate, Dr. R. Stocker, and Dr. R. Strawderman, for their contributions
in joint research papers from which some portions of this review article were based.
He thanks Dr. M. Peña for the improved artwork and Mr. Akim Adekpedjou
and Mr. J. Quiton for their careful reading of the manuscript.

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- Semiparametric Inference for a General Class of Models for Recurrent Events.[J Stat Plan Inference. 2007]
*Peña EA, Slate EH, González JR.**J Stat Plan Inference. 2007 Jun 1; 137(6):1727-1747.* - [Volume and health outcomes: an overview of systematic reviews].[Epidemiol Prev. 2005]
*Davoli M, Amato L, Minozzi S, Bargagli AM, Vecchi S, Perucci CA.**Epidemiol Prev. 2005 May-Aug; 29(3-4 Suppl):3-63.* - Nonparametric Methods in Reliability.[Stat Sci. 2004]
*Hollander M, Peña EA.**Stat Sci. 2004 Nov; 19(4):644-651.* - The measurement and monitoring of surgical adverse events.[Health Technol Assess. 2001]
*Bruce J, Russell EM, Mollison J, Krukowski ZH.**Health Technol Assess. 2001; 5(22):1-194.* - Bayesian ranking of sites for engineering safety improvements: decision parameter, treatability concept, statistical criterion, and spatial dependence.[Accid Anal Prev. 2005]
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- Dynamic Modelling and Statistical Analysis of Event
TimesDynamic Modelling and Statistical Analysis of Event TimesNIHPA Author Manuscripts. Nov 2006; 21(4)1PMC

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