The document discusses methods for analyzing recurrent event data when the number of previous episodes experienced by individuals is unknown. It compares the performance of three multivariate survival models: the Prentice, Williams, and Peterson (PWP) model; the Andersen-Gill (AG) model; and the shared frailty model (SFM). The PWP model directly handles event dependence but requires knowing the number of previous episodes, while the AG and SFM models specify a common baseline hazard. Through simulations of recurrent event data with different levels of individual heterogeneity and event dependence, the authors assess the bias, coverage, and validity of the proportional hazards assumption for parameter estimates from the three models when applied to samples from cohorts with 1, 3,

1. Use of multivariate survival models with common baseline
risk under event dependence and unknown number of
previous episodes
David Moriña, Georgina Casanovas and Albert Navarro
December 07 2014, Pisa

2. Introduction
Recurrent events
Recurrent events
• Recurrent event data refers to situations where the subject can experi-
ence repeated episodes of the same type of event
• There are many examples such as injuries, nosocomial infections, asthma
attacks . . .
• If these phenomena are studied via a cohort, the application of survival
methods would seem appropriate
• During the 1980s, at theoretical level, and during the 1990s and early
21st century in practical terms through development of software, various
methods have been proposed to tackle events of this type, both non-
parametric and parametric
2 / 28

3. Introduction
Recurrent events
Recurrent events
• Recurrent event data refers to situations where the subject can experi-
ence repeated episodes of the same type of event
• There are many examples such as injuries, nosocomial infections, asthma
attacks . . .
• If these phenomena are studied via a cohort, the application of survival
methods would seem appropriate
• During the 1980s, at theoretical level, and during the 1990s and early
21st century in practical terms through development of software, various
methods have been proposed to tackle events of this type, both non-
parametric and parametric
2 / 28

4. Introduction
Recurrent events
Recurrent events
• Recurrent event data refers to situations where the subject can experi-
ence repeated episodes of the same type of event
• There are many examples such as injuries, nosocomial infections, asthma
attacks . . .
• If these phenomena are studied via a cohort, the application of survival
methods would seem appropriate
• During the 1980s, at theoretical level, and during the 1990s and early
21st century in practical terms through development of software, various
methods have been proposed to tackle events of this type, both non-
parametric and parametric
2 / 28

5. Introduction
Recurrent events
Recurrent events
• Recurrent event data refers to situations where the subject can experi-
ence repeated episodes of the same type of event
• There are many examples such as injuries, nosocomial infections, asthma
attacks . . .
• If these phenomena are studied via a cohort, the application of survival
methods would seem appropriate
• During the 1980s, at theoretical level, and during the 1990s and early
21st century in practical terms through development of software, various
methods have been proposed to tackle events of this type, both non-
parametric and parametric
2 / 28

6. Introduction
Recurrent events
Recurrent events
• Recurrent events present two problems which cannot be handled using
the standard methods
• Individual heterogeneity (the unmeasured variability between subjects
beyond that of the measured covariates)
• Within-subject correlation, that can be specially problematic if there is also
event dependence (the risk of experiencing the event changes as a function
of the number of previous episodes presented by the individual)
• Ignoring this phenomenon and using methods not taking it into account
results in inefficient estimates
• Event dependence is tackled through the application of models employing
baseline hazards specific for the episode to which the individual is at risk
• The most widely used is that proposed by Prentice, Williams and Peterson
(PWP)
• This is an extension of the Cox proportional hazards model, which esti-
mates baseline hazards appropriate to each episode through stratification
by the number of previous episodes
3 / 28

7. Introduction
Recurrent events
Recurrent events
• Recurrent events present two problems which cannot be handled using
the standard methods
• Individual heterogeneity (the unmeasured variability between subjects
beyond that of the measured covariates)
• Within-subject correlation, that can be specially problematic if there is also
event dependence (the risk of experiencing the event changes as a function
of the number of previous episodes presented by the individual)
• Ignoring this phenomenon and using methods not taking it into account
results in inefficient estimates
• Event dependence is tackled through the application of models employing
baseline hazards specific for the episode to which the individual is at risk
• The most widely used is that proposed by Prentice, Williams and Peterson
(PWP)
• This is an extension of the Cox proportional hazards model, which esti-
mates baseline hazards appropriate to each episode through stratification
by the number of previous episodes
3 / 28

8. Introduction
Recurrent events
Recurrent events
• Recurrent events present two problems which cannot be handled using
the standard methods
• Individual heterogeneity (the unmeasured variability between subjects
beyond that of the measured covariates)
• Within-subject correlation, that can be specially problematic if there is also
event dependence (the risk of experiencing the event changes as a function
of the number of previous episodes presented by the individual)
• Ignoring this phenomenon and using methods not taking it into account
results in inefficient estimates
• Event dependence is tackled through the application of models employing
baseline hazards specific for the episode to which the individual is at risk
• The most widely used is that proposed by Prentice, Williams and Peterson
(PWP)
• This is an extension of the Cox proportional hazards model, which esti-
mates baseline hazards appropriate to each episode through stratification
by the number of previous episodes
3 / 28

9. Introduction
Recurrent events
Recurrent events
• Recurrent events present two problems which cannot be handled using
the standard methods
• Individual heterogeneity (the unmeasured variability between subjects
beyond that of the measured covariates)
• Within-subject correlation, that can be specially problematic if there is also
event dependence (the risk of experiencing the event changes as a function
of the number of previous episodes presented by the individual)
• Ignoring this phenomenon and using methods not taking it into account
results in inefficient estimates
• Event dependence is tackled through the application of models employing
baseline hazards specific for the episode to which the individual is at risk
• The most widely used is that proposed by Prentice, Williams and Peterson
(PWP)
• This is an extension of the Cox proportional hazards model, which esti-
mates baseline hazards appropriate to each episode through stratification
by the number of previous episodes
3 / 28

10. Introduction
Recurrent events
Recurrent events
• Recurrent events present two problems which cannot be handled using
the standard methods
• Individual heterogeneity (the unmeasured variability between subjects
beyond that of the measured covariates)
• Within-subject correlation, that can be specially problematic if there is also
event dependence (the risk of experiencing the event changes as a function
of the number of previous episodes presented by the individual)
• Ignoring this phenomenon and using methods not taking it into account
results in inefficient estimates
• Event dependence is tackled through the application of models employing
baseline hazards specific for the episode to which the individual is at risk
• The most widely used is that proposed by Prentice, Williams and Peterson
(PWP)
• This is an extension of the Cox proportional hazards model, which esti-
mates baseline hazards appropriate to each episode through stratification
by the number of previous episodes
3 / 28

11. Introduction
Recurrent events
Recurrent events
• Use of the PWP model requires knowing at every moment the number of
previous episodes suffered by each individual
• To have this detailed information would imply having the complete history
of each individual with respect to the event of interest
• With the exception of studies using specific sampling in healthy popula-
tions, or studies based on particular interventions and/or events which
significantly determine health status and are relatively infrequent (for ex-
ample cardiovascular events, cancers, etc), it is not usually possible to
have such information
• In public health contexts we are interested in estimating the marginal ef-
fect of one or several covariates (exposures) on an event, the previous
history of which is often unknown
4 / 28

12. Introduction
Recurrent events
Recurrent events
• Think of examples such as studying episodes of sickness absence in
workers of all ages (some of whom may have been working for many
years), or studying the occurrence of asthma attacks in a sample includ-
ing people who already had this problem previously
• When the number of previous episodes suffered by the individual is un-
known, we have no method to directly handle occurrence dependence,
and the usual practice in such cases is to fit models specified with a com-
mon baseline hazard, or frailty models
• The aim of the present study is to assess the performance of two models,
as possible alternatives to PWP when we want to estimate the effect of
one or several exposures on the risk of presenting a recurrent event af-
fected by event dependence, in situations where the number of previous
episodes of each individual is unknown
5 / 28

13. Models
Recurrent events
Models
• All the models we are considering are non-parametric and extensions of
the Cox model
• Prentice, Williams and Peterson (PWP)
• Andersen-Gill (AG)
• Shared frailty model (SFM)
6 / 28

14. Models
Recurrent events
Models
• All the models we are considering are non-parametric and extensions of
the Cox model
• Prentice, Williams and Peterson (PWP)
• Andersen-Gill (AG)
• Shared frailty model (SFM)
6 / 28

15. Models
Models
Prentice, Williams and Peterson (PWP)
• For recurrent phenomena in situations of event dependence, the survival
model of reference is PWP
• It incorporates event dependence through stratifying by the number of
previous episodes presented by each individual
• There is a specific baseline hazard for each particular episode to which
the individual is at risk
• When the i-th individual is at risk of the k-th episode, the hazard function
is defined as
hik (t) = h0k (t)eXi
ˆβ
,
where h0k (t) = e
ˆβ0k and Xi
ˆβ represent the vector of covariates and the
regression coefficients
• This model is only applicable if the episode number to which each
individual is at risk is known at all times
7 / 28

16. Models
Models
Andersen-Gill (AG)
• It’s the natural extension of the Cox model for proportional hazards
• It’s based on counting processes and assumes that the baseline risk is
common to all episodes and independent of the number of previous
episodes presented
• When the i-th individual is at risk of the k-th episode, the hazard function
is defined as
hi (t) = h0(t)eXi
ˆβ
,
where h0(t) = e
ˆβ0 and is therefore the same for all episodes
• Notice that the PWP model is a stratified AG model
8 / 28

17. Models
Models
Shared frailty model (SFM)
• May be used in contexts of recurrent events, where the different
episodes of a given individual share a frailty independent of that of other
individuals
• In addition to the observed regressors, this model also accounts for the
presence of a latent multiplicative effect on the hazard function:
hi (t) = Ui · h0(t)eXi
ˆβ
,
where the baseline hazard is specified independently of the episode k to
which the individual is exposed, h0(t) = e
ˆβ0
• Ui is an individual random effect which is not directly estimated from the
data, but instead is assumed to have unit mean and finite variance,
which is estimated
• Since Ui is a multiplicative effect, we can think of the frailty as
representing the cumulative effect of one or more omitted covariates
• Specifically, the model used in this study is the shared gamma frailty
model, with E[Ui ] = 1 and V[Ui ] = θ
9 / 28

18. Simulations
Examples
Examples
• We illustrate the application of these models reproducing two
phenomena described in the literature
• The frequency of long-term sickness absence in a cohort of Dutch workers,
with a baseline hazard of 0.0021 per worker-week
• The frequency of falls among residents of a geriatric centre, with a baseline
hazard of the first fall of 0.0361 per resident-week
10 / 28

19. Simulations
Examples
Generation of populations
• Eighteen different populations of 250,000 individuals, each with 20 years
of follow-up, were generated using the survsim package in R
• These populations are dynamic in the sense of being open on the left,
i.e. follow-up of individuals may begin before the start of the study period
• For each individual i the hazard of the next episode k has been
simulated through an exponential distribution:
hik (t) = exp (β0k + β1X1 + β2X2 + β3X3) · νi
where eβ0k is h0k (t), i.e. the baseline hazard for individuals exposed to
episode k
11 / 28

20. Simulations
Examples
Generation of populations
• The maximum number of episodes which a subject may present has not
been fixed, although the baseline hazard has been considered constant
when k ≥ 3. X1, X2 and X3 are the three covariates which represent the
exposure, with Xi ∼ Bernoulli(0.5). β1, β2 and β3 are the parameters of
the three covariates which represent the effect, and have been set,
independently of the episode k to which the subject is exposed, to:
β1 = 0.25, β2 = 0.50 and β3 = 0.75 in order to represent effects of
different magnitude
• νi is a random effect
• Event dependence has been introduced through using various values of
h0k (t) by specifying different β0k
• Individual heterogeneity was introduced through the random effect νi .
This is constant over the various episodes of a given individual but differs
between individuals
12 / 28

21. Simulations
Examples
Generation of populations
• Individual heterogeneity was introduced through the random effect νi .
This is constant over the various episodes of a given individual but differs
between individuals
• We established three possibilities:
• Absence of any random effect
• νi ∼ Gamma with mean 1 and variance 0.1
• νi ∼ Uniform(0.5, 1.5)
13 / 28

22. Simulations
Examples
Generation of populations
• Individual heterogeneity was introduced through the random effect νi .
This is constant over the various episodes of a given individual but differs
between individuals
• We established three possibilities:
• Absence of any random effect
• νi ∼ Gamma with mean 1 and variance 0.1
• νi ∼ Uniform(0.5, 1.5)
13 / 28

23. Simulations
Examples
Generation of populations
• Individual heterogeneity was introduced through the random effect νi .
This is constant over the various episodes of a given individual but differs
between individuals
• We established three possibilities:
• Absence of any random effect
• νi ∼ Gamma with mean 1 and variance 0.1
• νi ∼ Uniform(0.5, 1.5)
13 / 28

24. Simulations
Examples
Cohort design
• In practice, follow-up is limited to 1, 3 and 5 years
• At the start of follow-up there are individuals who have been previously
exposed
• For each of the generated sub-bases, 500 random samples were drawn
with samples n1 = 500, n2 = 1000 and n3 = 3000
• For each selected individual the episodes they present within the
effective follow-up period were recorded
• Finally, the proposed models were fitted to each of these samples by
means of the coxph function in R
14 / 28

25. Results
Performance
Model assessment criteria
• Percentage bias: δ =
¯ˆβ−β
β
· 100
• Coverage: Proportion of times the 100 · (1 − α)% confidence interval
ˆβj ± z1− α
2
SE(ˆβj ) includes β, for j = 1, . . . , 500.
• Proportional hazards: Proportion of times that the assumption of
proportionality of hazards cannot be rejected, for j = 1, . . . , 500,
according to the contrast of Grambsch & Therneau (Biometrika, 1994)
15 / 28

26. Results
Results
Results
• The results appearing in this section only refer to cohorts with 5 years of
follow-up
• The results referring to 1 and 3 years of follow-up are very similar
16 / 28

27. Results
Bias
Bias
• The only differences between AG and SFM are observed in the
populations with high levels of occurrence dependence, the percentage
of bias being slightly lower for AG
• For these models the average bias is around 10-15% for populations
with lower occurrence dependence, and rises to 40-70% for those with
higher dependence
• In general there do not appear to be any changes in the effect
associated with β related to either sample size, or with whether the
population presented heterogenity or not
17 / 28

28. Results
Coverage
Coverage
• There are no differences in coverage between AG and SFM
• Both models only achieve performances close to 95% for populations
with small or moderate occurrence dependence and for β1 = 0.25
• For the other scenarios coverage falls notably, worsening with increasing
occurrence dependence, effect to estimate and sample size. For
example, when estimating β3 in the highest ocurrence dependence
cohorts, the percentage of samples where the 95%CI includes the true
value is between 0 and 7% for sample sizes of n = 1000 or n = 3000
• In populations with heterogeneity the average size of the 95%CI
increases, which often translates into a rise in level of coverage
18 / 28

29. Results
Proportional hazards
Proportional hazards
• SFM seems to present better performance in populations with low or
moderate occurrence dependence, although only slightly
• In general model performance worsens with increasing occurrence
dependence, effect to estimate and sample size, only reaching levels
near 90% for lowest occurrence dependence cohorts with n = 500 or
n = 1000
19 / 28

30. Conclusions
Conclusions
• The PWP model presents much better results than the models with
common baseline risk
• The percentage of bias does not reach 10%, and is generally negative,
i.e. slightly underestimating the effect
• For populations free of heterogeneity the coverage levels are around
85-95%, but fall in populations with heterogeneity as the effect to
estimate and sample size increase
• In this model generally over 85% of the simulated samples comply with
the assumption of proportional hazards, however in certain particular
cases when β3 = 0.75 and the population is that of greatest
dependence, this percentage falls to around 70%
20 / 28

31. Conclusions
Conclusions
• The performance of the models with common baseline risk worsens as
occurrence dependence increases, producing worse coverage and
increasing overestimation of the effect
• Members of the exposed group have more events and therefore present
more recurrent episodes, and also they suffer these episodes earlier
than members of the non-exposed group
• The exposed subjects come to be at risk of a higher baseline hazard
sooner and in greater numbers
• By not using specific baseline risks, the increase in baseline hazard is
mostly attributed to the exposed group
21 / 28

32. Conclusions
Conclusions
• As the effect to be estimated increases, performance of models with
common baseline hazard worsens
• This leads to part of the effect of the baseline hazard being attributed to
exposure
• For these models, coverage is affected by sample size, worsening as
sample size increases
• Almost no differences were observed between the AG and SFM models,
not even for populations generated with heterogeneity, and regardless of
whether the SFM model specified it correctly (gamma) or not (uniform)
• SFM assumes a frailty specific to each individual which can represent a
cumulative effect of one or several unmeasured covariates
22 / 28

33. Conclusions
Conclusions
• If the interest of our analysis was not strictly the marginal estimates, but
rather we aimed to construct a prognostic model where the estimation of
individual hazard was a priority, the SFM models might perform better
than AG models
• If there was any association between the covariates of interest and the
unmeasured covariates, perhaps SFM could partly capture it and
present better performance than AG
• Regarding level of compliance with the assumption of proportionality of
hazards, this declines as occurrence dependence increases
• Although in populations with greater dependence it seems that more of
the AG models satisfy the assumptions than SFM, their performance in
this area is still not sufficient
23 / 28

34. Conclusions
Conclusions
• In situations of event dependence the performance of PWP is clearly
better than that of models with common baseline risk
• Even so, values of coverage and PH compliance do not achieve the
expected levels when event dependence is high, and the effect to be
estimated is large
• In the context of health sciences it is common for the phenomenon of
study to exhibit recurrence, and also that the risk of suffering an episode
changes depending on the number of episodes suffered previously
• Therefore, incorporating information about previous episodes into the
analysis would appear to be fundamental
• However, in certain contexts, this is not possible simply because the
number of previous episodes is unknown
24 / 28

35. Conclusions
Conclusions
• The AG and SFM models analysed in this study have achieved low, very
similar, performances, making it impossible to recommend one instead
of the other
• The only context in which it would seem reasonable to use one of them,
in situations involving occurrence dependence, would be when the level
of such dependence was low and the effect to be estimated was small
• Although this would produce a somewhat biased estimate, model
performance in terms of coverage and PH compliance might be
considered acceptable
• In other situations the use of these models is clearly inappropriate, in
general they present levels of coverage and PH compliance which are
low or extremely low, and blatantly overestimate the effect of the factor
25 / 28

36. Conclusions
Conclusions
• Currently there are no models available which allow estimating the
possible effect of occurrence dependence when the number of previous
episodes is unknown, and to incorporate this in fitting the model
• Consequently, it is important to find valid alternatives to permit tackling
analyses of this type
26 / 28

38. Centre for Research
in Environmental
Epidemiology
Parc de Recerca Biomèdica de Barcelona
Doctor Aiguader, 88
08003 Barcelona (Spain)
Tel. (+34) 93 214 70 00
Fax (+34) 93 214 73 02
info@creal.cat
www.creal.cat
Grup de Recerca d’Amèrica i Àfrica Llatines
Unitat de Bioestadística, Facultat de Medicina
Universitat Autònoma de Barcelona
www.uab.cat