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D I S C U S S I O N P A P E R S E R I E S
Forschungsinstitut
zur Zukunft der Arbeit
Institute for the Study
of Labor
Inference for Shared-Frailty Survival Models
with Left-Truncated Data
IZA DP No. 6031
October 2011
Gerard J. van den Berg
Bettina Drepper
Inference for Shared-Frailty Survival
Models with Left-Truncated Data
Gerard J. van den Berg
University of Mannheim, VU University Amsterdam,
IFAU-Uppsala and IZA
Bettina Drepper
University of Mannheim
Discussion Paper No. 6031
October 2011
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IZA Discussion Paper No. 6031
October 2011
ABSTRACT
Inference for Shared-Frailty Survival Models
with Left-Truncated Data
*
Shared-frailty survival models specify that systematic unobserved determinants of duration
outcomes are identical within groups of individuals. We consider random-effects likelihood-
based statistical inference if the duration data are subject to left-truncation. Such inference
with left-truncated data can be performed in the Stata software package. We show that with
left-truncated data, the commands ignore the weeding-out process before the left-truncation
points, affecting the distribution of unobserved determinants among group members in the
data, that is, among the group members who survive until their truncation points. We critically
examine studies in the statistical literature on this issue as well as published empirical studies
that use the commands. Simulations illustrate the size of the (asymptotic) bias and its
dependence on the degree of truncation. We provide a Stata command file that maximizes
the likelihood function that properly takes account of the interplay between truncation and
dynamic selection.
JEL Classification: C41, C34
Keywords: stata, duration analysis, left-truncation, likelihood function, dynamic selection,
hazard rate, unobserved heterogeneity, twin data
Corresponding author:
Gerard J. van den Berg
Department of Economics
University of Mannheim
L7, 3-5
68131 Mannheim
Germany
E-mail:
gerard@uni-mannheim.de
*
We thank Arne Uhlendorff for helpful comments. Stata is a registered trademark of StataCorp LP.
1 Introduction
In this paper we consider inference for shared-frailty survival models. These are
Mixed Proportional Hazard (MPH) models in which systematic unobserved deter-
minants of duration outcomes are identical within units or groups of individuals.
We allow the spell durations to be subject to left-truncation, meaning that the
duration outcome is only observed if it exceeds a certain threshold value, and we
focus on random-effects likelihood-based inference. We show that the Stata soft-
ware package command to estimate shared-frailty survival models in the presence
of left-truncated duration data should not be applied, since it maximizes a like-
lihood function that does not properly take account of dynamic selection before
the truncation points.
In order to explain this and to motivate the relevance of our contribution, we
start with an introduction into the survival models with unobserved heterogeneity
(or frailty terms) that are included in Stata for statistical inference. Shared-frailty
models are an important class of such models.
Empirical survival studies or studies in duration analysis commonly adopt
some version of the Mixed Proportional Hazard (MPH) model for the hazard
rate. The MPH model stipulates that the individual hazard rate (or exit rate
out of the current state) θ depends on the elapsed duration t, on explanatory
variables x and on unobserved determinants v such that
θ(t|x, v) = λ(t)φ(x)v
at all t, x, v for some functions λ and φ (see Lancaster, 1990, and Van den Berg,
2001, for surveys). Here, φ is the function of interest although sometimes λ is also
of interest. Typically, at least some elements of the vector x are time-varying, but
for ease of exposition we ignore this in this paper. Notice that without loss of
generality v can be seen as the joint multiplicative effect of a vector of unobserved
determinants on the individual hazard rate. The term v is often called the frailty
term. It is not directly estimated from the data, as it varies across individuals.
Moreover, in contrast to linear regression analysis, ignoring unobserved hetero-
geneity leads to biased estimates of λ and φ. This is because individuals with a
high v leave the state of interest on average earlier than individuals with low v.
This phenomenon is called “weeding out” or “sorting”. It may occur at differ-
ent speeds for different x, causing the composition of survivors in terms of v to
change over time. In general, ignoring this leads to a negative bias in the estimate
of λ(t) and a bias in the estimated covariate effects (Lancaster, 1990, Van den
Berg, 2001). The most common approach for inference is to assume that v has a
2
distribution G in the population and to estimate its parameters along with (the
parameters of) λ and φ using Maximum Likeliho od Estimation, where the likeli-
hood contribution of an individual spell integrates over G. In econometrics, this is
called random-effects estimation. To ensure that identification is not fully driven
by functional form assumptions, it is assumed that x and v are independently
distributed in the population and that E(v) = 1. The population constitutes the
inflow into the state of interest (although this may be modified; see below). By
far the most common functional form for G is the gamma distribution. This can
be justified as an approximation to a wide class of frailty distributions (Abbring
and Van den Berg, 2007). The approximation improves with left-truncation of the
durations. An alternative frailty distribution is the Inverse-Gaussian distribution.
Often it is natural to assume that small subsets of different individuals or
spell durations share the same value of v. For example, different unemployment
spells of the same person may share the same unobserved determinant v. Or the
mortality rates of identical twins may assumed to depend on identical unobserved
determinants v. In general, the data may identify groups or units or strata such
that different spells within a group or unit or stratum share the same v. Data with
this feature are often called multi-spell duration data. To keep the terminology
simple, consider the case where for each unit in the sample we observe at most two
spells. The unit has a given value of v, and we assume that its spell durations
are independent drawings from the univariate duration distribution of t given
x, v, where, of course, v is unobserved, so that the durations given x are not
independent. It depends on the context whether x is also identical across spells
or individuals within a unit. For ease of exposition, we take the data to consist
of a random sample of units. We return to this below.
The multi-spell MPH model was first proposed by Clayton (1978) and is nowa-
days known under the name “shared-frailty model”. Notice that it has the same
unknown functions as the single-spell MPH model, namely λ, φ and G. The em-
pirical analysis of shared-frailty models is widespread (see e.g. Hougaard, 2000,
and Van den Berg, 2001, for surveys). If the underlying modeling assumptions are
correct, multi-spell data enable identification of the MPH model under weaker as-
sumptions than single-spell data, and the estimation results are more robust with
respect to functional-form assumptions (Van den Berg, 2001). By straightforward
extension of the estimation with single-spell data, the most common estimation
methods are random-effect procedures where each unit or group provides a like-
lihood contribution that integrates over the distribution G of v across the units
and where λ, φ and G are parameterized.
1
1
If different individuals within a unit or group have different values of x then Stratified
3
The Stata software package offers a large number of pre-programmed estima-
tion routines for survival analysis. In this sense Stata is unique among the avail-
able software packages covering survival analysis, and indeed it has become pop-
ular among empirical researchers. The main survival model estimation command
streg also captures the shared-frailty model, by invoking the option shared()
to indicate which individuals share the same value of v. Gutierrez (2002) gives
an overview of parametric shared-frailty models in Stata. See Hirsch and Wienke
(2011) for an overview of software packages with estimation routines for shared-
frailty models.
Sampling schemes where durations are left-truncated are common in single-
spell as well as in multi-spell survival analysis (Guo, 1993). For example, unem-
ployment duration spells are often only recorded in register data if the duration
exceeds one month. Population register data typically follow individuals from a
given point in calendar time onwards, where the starting points of the spells that
are ongoing at the beginning of the register’s observation window are often ob-
served as well. The spells that started say t
0
time units before the beginning of the
observation window are then only observed if the duration exceeds t
0
. With the
increasing availability of such register data in socio-economic and health research,
the usage of left-truncated duration data has increased. This also applies to multi-
spell data. For example, death causes of Danish twins were only systematically
recorded as of January 1, 1943, so to study death causes among those born before
1943, it makes sense to restrict attention to both twin members being alive on
January 1, 1943.
2
If the duration from birth until death due to a specific death
cause is the relevant duration variable then this variable is left-truncated at the
age attained on January 1, 1943. Hence, the left-truncation points as measured in
the age dimension differ across twin pairs. In studies with hospital patients, only
the patients are observed who survive up to the point when the trial period at
the hospital starts. If the patient subsequently experiences remission and relapse
then subsequent illness spells may not be left-truncated.
Stata allows for left-truncation of the duration data, through the enter()
option when declaring the data as duration data by the stset command. Impor-
tantly, the value t
0
of the truncation threshold may differ across individuals (as
Partial Likelihood Estimation can be used as an alternative (fixed effects) method (Kalbfleisch
and Prentice, 1980, Chamberlain, 1985, Ridder and Tunalı, 1999). In the present paper we are
not concerned with that method.
2
After all, if a twin member is observed to have died before 1943 then it is not known
whether this was due to the cause of interest or due to another cause. In the latter case, the
moment of death due to the cause of interest is right-censored by an event with an unknown
distribution, and inference would include the estimation of this distribution.
4
well as across spells for a given unit in the case of the shared-frailty model).
Notice that left-truncation gives rise to a second selection issue, on top of
the selection generated by the dynamic weeding-out. After all, surviving up to
some threshold value is more likely if the frailty term is small. The Stata routine
for shared-frailty models
3
ignores the fact that the second selection impacts on
the first selection. Restricting the outcome to exceed a lower threshold implies
that the frailty distribution in the sample systematically differs from that in
the population upon inflow into the state of interest.
4
If the former distribution
is nevertheless assumed to equal the latter, then, as we shall see, the resulting
estimators of β and λ are inconsistent. One may redefine the population to be
the survivors at t
0
but this only makes sense if t
0
is identical across all units and
spells.
The interplay between left-truncation and dynamic selection has always been
recognized in the single-spell survival analysis literature. As we discuss b elow,
with multiple spells the role of this interplay has been obscured. However, we are
not the first to point out the importance of dealing with the above interplay and
its implication for the frailty distribution in the sample, in shared frailty models.
Jensen et al. (2004) provide a lucid account. They contrast the correct likelihood
function to the likelihood function where the interplay is ignored for the case of
gamma-distributed frailties, and they discuss the bias when using the latter. They
point out that Nielsen et al. (1992), which is a seminal paper in survival analysis,
used the latter likeliho od in the case of left-truncated data in the shared frailty
model. Elsewhere in the literature, Rondeau and Gonzalez (2005) use the correct
likelihood for their semi-parametric estimator of the shared frailty model in the
case of left-truncated data, whereas Do and Ma (2010) use the other likelihood
function for their semi-parametric estimator in the same setting.
The remainder of the paper is structured as follows. In Section 2 we discuss
left-truncation in multiple spell duration data in more detail. We show under
which conditions the likelihood function of the streg,shared() command is mis-
specified for left-truncated data, and we present the correct likelihood function.
We also discuss the analogous problem with the stcox command in Stata for the
semi-parametric estimation of the shared gamma frailty mo del. We list a number
of empirical studies that have used this stcox command to estimate shared-frailty
models with left-truncated data. In Section 3 we demonstrate in a short simu-
lation study how the magnitude of the bias resulting from the misspecification
3
This routine is available since Version 7, up to and including the current Version 12.
4
See Ridder (1984) for an account of the differences between frailty distributions in different
types of single-spell samples.
5
depends on the level of truncation and the variance of the frailty distribution. We
also examine the performance of the stcox command in this setting, and we list
published articles that use this Stata command to semi-parametrically estimate
the shared gamma frailty model with left-truncated data. Section 4 concludes. In
the Appendix we introduce a corrected Stata command called stregshared.
2 Likelihood specification with left-truncated du-
ration data and shared frailties
Consider a random sample of single spells, if the MPH model applies. The ran-
dom sample consists of independent draws from the distribution of T |X for var-
ious values x of X, where T denotes the random duration variable. We consider
likelihood-based inference, and for the moment we take λ, φ and G to be para-
metric functions. The spell durations may be independently right-censored but
we are not concerned with that here. Consequently, the likelihood contribution
of a single spell is the probability density function f
u
(t|x) of T |X evaluated at
the observation (t, x), with
f
u
(t|x) = E
v
(f
c
(t|x, v)) =
Z
v
λ(t)φ(x)v exp(−Λ(t)φ(x)v)dG(v)
in which Λ(t) :=
R
t
0
λ(u)du denotes the so-called integrated baseline hazard and
f
c
is the probability density function of T |X, V .
Next, consider a random sample of units each with j = 1, 2 spells that share
their frailty term v. Throughout the paper we assume that conditional on v, the
spells are independent. The likelihood contribution of a unit with non-truncated
uncensored duration outcomes t
1
|x
1
and t
2
|x
2
then equals
R
v
f
c
(t
1
|x
1
, v)f
c
(t
2
|x
2
, v)dG(v).
Left-truncation of a single-spell duration outcome variable means that the
variable is only observed if its value exceeds a lower threshold, say t
0
. Throughout
the paper we are only concerned with deterministic t
0
. In a random sample of left-
truncated single spells, the individual likelihood contribution equals f
u
(t|x)/(1 −
F
u
(t
0
|x)) with F
u
being the distribution function associated with the density f
u
.
With multiple spells per unit (or group or stratum), left-truncation of a spell
duration outcome can be defined analogously, regardless of whether other spells
are observed for this unit where the outcome exceeds its lower threshold. However,
sometimes none of the duration outcomes of a unit is observed or used if at least
one of them is left-truncated. The study of cause-specific mortality with twin data
mentioned in Section 1 is such an example. For expositional reasons it is useful to
6
consider this case first. If the number of spells (observed or not observed) of a unit
is known then the model can be used to derive the likelihood function. Suppose
that each unit consists of two spells j = 1 , 2 and that the spells are observed
conditional on both spell durations surviving up to their truncation points t
01
and t
02
, resp ectively. This might be called “strong left-truncation”. In the simple
case of no censoring, the likelihood contribution L of the unit is now given by the
density function of t
1
, t
2
|T
1
> t
01
, T
2
> t
02
, x, which can be expressed as
L =
Z
∞
0
f
c
(t
1
|T
1
> t
01
, x
1
, v)f
c
(t
2
|T
2
> t
02
, x
2
, v) dG(v|T
1
> t
01
, T
2
> t
02
, x) (1)
with x = (x
1
, x
2
) and T
j
denoting the random duration variables. We thus av-
erage over the conditional frailty distribution G(v|T
1
> t
01
, T
2
> t
02
, x) in units
where both spells survive up to their truncation points t
0j
(and given x). This is
distribution of v in the sample of observed spells. It can be expressed in terms of
the model primitives through
dG(v|T
1
> t
01
, T
2
> t
02
, x) =
(1 − F
c
(t
01
|x
1
, v))(1 − F
c
(t
02
|x
2
, v))dG(v)
R
∞
0
(1 − F
c
(t
01
|x
1
, w))(1 − F
c
(t
02
|x
2
, w))dG(w)
where
1 − F
c
(t
0j
|x
j
, v) = exp(−Λ(t
0j
)φ(x
j
)v)
Note that even if only one of the spells j within a unit has t
0j
> 0, the distribution
G(v|T
1
> t
01
, T
2
> t
02
, x) differs from G(v).
Assuming a gamma-distributed frailty with E(v) = 1 and V ar(v) = σ
2
yields
5
L = φ(x
1
)λ(t
1
)φ(x
2
)λ(t
2
)(σ
2
+ 1)(1 + σ
2
M(t
01
, t
02
))
1/σ
2
(1 + σ
2
M(t
1
, t
2
))
−(1/σ
2
+2)
,
(2)
where M(t
1
, t
2
) = φ(x
1
)Λ(t
1
) + φ(x
2
)Λ(t
2
). Note that for ease of exposition we
omit the dependence of M on x
1
, x
2
.
Instead of the above type of left-truncation, we may consider sampling schemes
with different types of reduced observability of low spell durations in a shared-
frailty model. If only one spell per unit is not left-truncated then one may never-
theless include it in the data used for inference. However, given that the number of
spells per unit equals two, we directly infer that the other spell duration t
j
satis-
fies t
j
≤ t
0j
. In other words, t
j
is left-censored instead of left-truncated. The unit
then provides a likelihood contribution equal to
R
v
f(t
1
|x
1
, v)F (t
2
|x
2
, v)dG(v),
5
See Appendix 1 for details.
7
where we took j = 2 and where F
c
denotes the cumulative distribution function
of t
2
|x
2
, v.
Alternatively, the number of spells per unit may not be fixed and may increase
with the sample size. Jensen et al. (2004) provide a detailed formal likelihood
derivation in a rather general dynamic sampling framework where the number of
(possibly simultaneously occurring) spells per unit may increase with the time
that units are followed, and where all observed spells per unit are used for the
statistical inference. Under some assumptions, the likelihood contributions are
identical to equation (1). In particular, if two spells are observed for some unit,
then the distribution of the frailty term of this unit, conditional on the two
spell durations exceeding t
01
and t
02
, respectively, equals G(v|T
1
> t
01
, T
2
>
t
02
, x).
6
Equation (2) replicates likelihood equations in e.g. Jensen et al. (2004)
and Rondeau and Gonzalez (2005) for the shared gamma frailty model with left-
truncated data.
We now turn to the likelihood function used in Stata. The Stata Manual (e.g.
Stata, 2009, p. 383) gives a likelihood contribution for the case of two possibly
left-truncated spells and a shared gamma frailty model. This is used in the streg
command with the options frailty(gamma) and shared(). In the absence of
right-censoring, the likelihood contribution states that
7
L
Stata
= φ(x
1
)λ(t
1
)φ(x
2
)λ(t
2
)(σ
2
+1)(1+σ
2
(M(t
1
, t
2
)−M(t
01
, t
02
)))
−(1/σ
2
+2)
. (3)
which obviously differs from equation (2). In Appendix 2 it is shown that the
right hand side of equation (3) can be rewritten as
L
Stata
=
Z
∞
0
f
c
(t
1
|T
1
> t
01
, x
1
, v)f
c
(t
2
|T
2
> t
02
, x
2
, v) dG(v) (4)
where G(v) is a gamma distribution. This expression corresponds to the likelihood
contribution presented in Gutierrez (2002, p.34) for general frailty distributions.
By comparing equations (4) and (1) it is clear under which conditions equations
(3) and (2) differ, and also what is the underlying reason for them to differ. First,
6
Because of the dynamically evolving sampling scheme, where new spells per unit may start
during the observation window, they need to make an approximation to deal with changes in the
composition of the inflow during the observation window. This is an additional complication
that does not affect the issues we focus on but which does not allow us to draw on their
simulation results to assess the bias due to ignoring the interplay between left-truncation and
dynamic selection.
7
We translate the notation of the Stata Manual, as follows: S
ij
(t
ij
) = e
−φ(x
ij
)Λ(t
ij
)
and
h
ij
(t
ij
) = φ(x
ij
)λ(t
ij
), where we omit the index i.
8
they differ if and only if Var(v) > 0 and at the same time at least one of the
following inequalities applies: t
01
> 0, t
02
> 0. Secondly, they differ because the
conditional densities in equation (4) are averaged over the inflow distribution
G(v) instead of the frailty distribution G(v|T
1
> t
01
, T
2
> t
02
, x) conditional on
the spell durations being left-truncated. The critical issue is that the likelihood
in (3) treats the data as if no sorting had taken place prior to the beginning of
the observation window. So it is implicitly assumed that the inflow distribution
of frailties at t = 0 does not change until the point of truncation. But since the
subjects are at risk from t = 0 onwards, this assumption cannot hold.
The above problem carries over to the case where the frailty is assumed to
follow an Inverse-Gaussian frailty distribution in the streg command. The like-
lihood function for a shared frailty model with shared Inverse-Gaussian frailties
and left-truncated duration data is derived in Appendix 3. This may be contrasted
to the function given in the Stata Manual (Stata, 2009, p. 383).
An ad-hoc approach to deal with the discrepancy between the likelihood func-
tion and the Stata routine is to simply assume from the outset that the frailty
distribution in the sample does not depend on x and on the truncation points.
This effectively amounts to a redefinition of the population, as the inflow into
the state of interest at the moment of left-truncation, with the assumption that
in this newly defined population, v is independent of x and of the elapsed time
spent in the state of interest at the truncation point. Under this assumption,
the Stata likelihood is correct. If the truncation points are not dispersed in the
original population then such an approach may make sense. It replaces the as-
sumption that v and x are independent in the inflow into the state of interest
by the assumption that they are independent at the moment of truncation. If an
MPH model guides the exit rate between the inflow and the truncation point,
then the latter assumption in general entails that x and v are dependent in the
original population that constitutes the inflow into the state of interest.
However, if the truncation points t
0j
are dispersed then this approach does
not make much sense. For example, consider two units i, i
0
each with two spells
j. The units have identical systematic duration determinants including identical
x within and across units, but their left-truncation points differ. We take, in
obvious notation, 0 < t
0i1
= t
0i2
< t
0i
0
1
= t
0i
0
2
< ∞, so that within each unit
there is no dispersion of truncation points. The ad-hoc approach would require
the distribution of v in the first unit at t
0i1
to equal the distribution of v in the
second unit at t
0i
0
1
. But in the first unit, in between t
0i1
and t
0i
0
1
, the frailty
distribution evolves dynamically over time in accordance to the shared frailty
model, leading to a different distribution at t
0i
0
1
than at t
0i1
. By implication, the
9
distributions of v at t
0i
0
1
would differ across units, not because the units behave
differently, but because of the way in which they have been sampled.
So far, the Stata issues we discussed refer to the use of the options shared
and frailty() in the streg command, in conjunction with the use of the option
enter() in the command stset. The streg command with the options shared
and frailty() corresponds to parametric shared-frailty models. However, Stata
also offers a routine for the semi-parametric estimation of shared-frailty models,
and this routine can also be applied in the case of left-truncated data. Specifi-
cally, the stcox command with the options shared allows for the semi-parametric
estimation of a shared-frailty model where G(v) is assumed to be a gamma dis-
tribution, φ(x) = exp(x
0
β), and λ(t) is an unspecified function (Cleves, Gould
and Gutierrez, 2004). This command can be used in conjunction with the left-
truncation option enter() in the command stset.
The semi-parametric estimation method is developed by Therneau and Gramb-
sch (2000) who do not discuss left-truncation of the duration data. It maximizes
a penalized partial likelihood function, where the penalty function penalizes the
distance between the fitted gamma distribution and the estimated frailty terms.
Therneau and Grambsch (2000) show that with a particular choice of penalty
function, this estimation metho d is equivalent to maximization of a full likeli-
hood using an EM algorithm.
The Stata Manuals do not give likelihood expressions for the shared (gamma)
frailty model in stcox if the data are left-truncated. This means that we do
not know with certainty whether the command suffers from the same issue as
the streg command. However, we can assess the performance of the command
with simulated data. In the next section we show by way of simulations that
most likely the stcox command does suffer from the same issue as the streg
command. We should point out that, apart from the above, the stcox command
with the shared option has the disadvantage that the reported standard errors of
the estimated β coefficients are under-estimated in that they are obtained under
the assumption that the true variance of the gamma frailty distribution equals
the estimated variance (Cleves, Gould and Gutierrez, 2004).
We finish this section by revisiting the cases where the Stata likelihood func-
tion and our own likelihood function coincide. Recall that if none of the spells is
left-truncated then they coincide, and if there is no systematic unobserved het-
erogeneity (so Var(v) = 0) then they coincide as well. If a unit or group always
consists of one single spell, then the Stata likelihood and our likelihood do not
coincide, but our likelihood should then coincide to the likelihood of the MLE
estimator for a single-spell MPH setting with left-truncated data. We know that
10
the latter is correctly specified in Stata. By implication, with left-truncated data,
the Stata estimator for the shared frailty model with a single spell per unit does
not equal the Stata estimator for the corresponding MPH model with single-spell
data. This is readily verified. In the latter case the frailty distribution conditions
on survival until the truncation point whereas in the former case it does not.
According to Hirsch and Wienke (2011), none of the other software packages
with estimation routines for shared-frailty models allows for left- truncation, with
the exception of an R package called Frailtypack. This uses the semi-parametric
Rondeau and Gonzales (2005) estimator which uses a full likelihood function that
does take account of the interplay between dynamic selection and left-truncation
(their estimator penalizes non-smoothness of the baseline hazard function λ(t)).
3 Simulation results
Recall that we are not primarily interested in small-sample properties of estima-
tors but rather in the appropriate choice of likelihood function. The latter should
be visible in estimates based on a large sample. We simulate data from a shared
frailty model. The sample consists of units each comprising two spells with a
shared gamma frailty. The baseline hazard λ(t) follows either a Weibull speci-
fication (λ(t) = αt
α−1
) or a Gompertz specification (λ(t) = e
αt
). Furthermore,
φ(X) = e
Xβ
with X = (1 x) and x being a single time-constant covariate.
In a first step, the covariate x
ij
is drawn from a standard normal distribution
for each spell j of unit i, and the frailty term v
i
is drawn from a gamma dis-
tribution with E(v) = 1 and V ar(v) = σ
2
for each unit i. The unknown model
parameters are β ≡ (β
0
β
1
), α and σ
2
. These have the following possible values,
β
0
= 0, β
1
= 1, α = 1, σ
2
∈ {0.5, 1, 2}. (5)
so we run simulations for three different values of the variance σ
2
of the frailty
distribution. These values are in line with those in the simulations in Jensen et
al. (2004).
In a second step, for given covariates, frailty terms and parameter values, the
durations t
i1
and t
i2
are drawn independently from the distributions F
c
(t
j
|x
ij
v
j
),
j = 1, 2, respectively.
8
. Next, we draw the left-truncation thresholds t
0i1
and t
0i2
from a uniform distribution with range (0, b). All units with t
i1
≤ t
0i1
or t
i2
≤ t
0i2
8
We use the following transformation of the variable u drawn from a uniform distribution
U(0, 1): t
ij
= α
−1
log(1 − α log(1 − u
ij
)(e
X
ij
β
v
i
)
−1
) which is the inverse of the cumulative
distribution function F
c
(t
ij
|X
ij
, v
i
) = 1 − exp(−e
X
ij
β
α
−1
(e
αt
ij
− 1)v
i
).
11
are dropped. This way the sample only contains those units for which both spell
durations exceed their left-truncation points. The fraction c ∈ [0, 1] of data that
are dropped due to left-truncation can be fine-tuned by modifying b. Effectively,
the sample size of 50,000 units is determined by the requirement that each of the
spells of these units has a duration exceeding a left-truncation point. In fact, if
the data are sampled from the model with the Weibull specification with α = 1
and if σ
2
is large, then the estimation of the parameters β
0
, α is numerically
cumbersome.
9
This suggests that a larger sample is needed for reliable inference,
but in the light of the computational burden we opt for the alternative of assuming
that the researcher knows that β
0
= 0.
In the last step of the simulation procedure we use the stset and streg
commands to estimate a shared frailty model in Stata,
. stset duration, failure(cens==0) enter(t0)
. streg x , distribution(gompertz) frailty(gamma) shared(id) nohr
The results are summarized in Figures 1 and 2. The panels show the estimates of
the constant β
0
(in the case of the Gompertz specification), the covariate effect
β
1
, the Gompertz duration dependence parameter α, and the variance σ
2
of the
gamma frailty distribution. We performed separate simulations with 30 different
truncation rates c ∈ [0, 1), and we connect the resulting points to obtain the
displayed curves.
All estimates move away from their true value as the truncation rate c in-
creases from zero. In particular, at any positive truncation rate, the covariate
effect and the level of the hazard rate are under-estimated.
In general this is to be expected. As c increases, the simulated distributions
of t
0i1
and t
0i2
move to the right, so the difference between G(v) and G(v|T
i1
>
t
0i1
, T
i2
> t
0i2
, x) increases. Recall that E(v) = 1, whereas with truncation, units
with large v will have exited the state relatively often before having reached
the truncation point, so the mean of v|T
i1
> t
0i1
, T
i2
> t
0i2
, x decreases in t
0ij
.
The over-estimation of the mean frailty among the survivors at the truncation
points is then compensated by an under-estimation of the magnitude of the other
determinants of the level of the individual hazard rate (which by themselves have
increasing effects on the individual hazard rate).
The bias towards zero of the estimate β
1
can be explained analogously. The
true frailty distribution after truncation G(v|T
i1
> t
0i1
, T
i2
> t
0i2
, x) depends
9
More precisely, the estimation routine suffers from occasional numerical problems. This
even occurs in the absence of left-truncation (c = 0) if σ
2
≥ 4.
12
.2 .4 .6 .8 1
0 .2 .4 .6 .8 1
Truncation rate
Var(v)=0.5 Var(v)=1
Var(v)=2
Covariate
.2 .4 .6 .8 1
0 .2 .4 .6 .8 1
Truncation rate
Var(v)=0.5 Var(v)=1
Var(v)=2
Duration dependence
0 .5 1 1.5 2 2.5
0 .2 .4 .6 .8 1
Truncation rate
Var(v)=0.5 Var(v)=1
Var(v)=2
Variance of frailty
Figure 1: Simulation results of a shared gamma frailty model with Weibull du-
ration dependence and left-truncated data using the Stata command streg with
the option shared()
on the covariates x. Spells with a large value of exp(X
ij
β as well as a large v
i
terminate on average earlier than other spells. So in the case of a positive β
1
, an
observation in the truncated sample with a large x is more likely to have a small
v
i
than observations with low x. The association between x and the observed
hazard rates right after the truncation point is therefore smaller than β
1
. If one
ignores this, by ignoring the dynamic selection before the truncation point, then
the resulting estimate of β
1
will be biased towards zero.
It may be instructive to consider some corresponding model expressions in the
case of single-spell duration data. The correct expression for the observed hazard
rate θ(t|T ≥ t
0
, x) at t ≥ t
0
equals
θ(t|T ≥ t
0
, x) =
λ(t) exp(x
0
β)
1 + σ
2
exp(x
0
β)Λ(t)
(6)
13
−2.5 −2 −1.5 −1 −.5 0
0 .2 .4 .6 .8 1
Truncation rate
Var(v)=0.5 Var(v)=1
Var(v)=2
Constant
.6 .7 .8 .9 1
0 .2 .4 .6 .8 1
Truncation rate
Var(v)=0.5 Var(v)=1
Var(v)=2
Covariate
.6 .7 .8 .9 1
0 .2 .4 .6 .8 1
Truncation rate
Var(v)=0.5 Var(v)=1
Var(v)=2
Duration dependence
.5 1 1.5 2 2.5
0 .2 .4 .6 .8 1
Truncation rate
Var(v)=0.5 Var(v)=1
Var(v)=2
Variance of frailty
Figure 2: Simulation results of a shared gamma frailty model with Gompertz
duration dependence and left-truncated data using the Stata command streg
with the option shared()
This does not dep end on t
0
because the hazard by definition conditions on T ≥ t,
which implies T ≥ t
0
. The expression for observed hazard assuming that there is
no dynamic selection before t
0
is equal to
λ(t) exp(x
0
β)
1 + σ
2
exp(x
0
β)(Λ(t) − Λ(t
0
))
(7)
The estimates that follow from the latter approach lead to an estimated observed
hazard at t = t
0
that fits the corresponding expression of (6) evaluated at the
true parameter values. Hence,
b
λ(t
0
) exp(x
0
b
β) = λ(t
0
) exp(x
0
β)(1 + σ
2
exp(x
0
β)Λ(t
0
))
−1
For t
0
> 0, σ > 0, this leads to the bias implications discussed above.
14
Figures 1 and 2 also show that the bias of the estimates depends on the
variance of the frailty distribution. As the latter increases, the estimates of the
hazard level and the covariate effect move further away from their true values.
Again, this is what would be expected. Notice that none of the biases vanishes
for the sample size n → ∞ for a given truncation rate.
It should be kept in mind that the simulation results in Figures 1 and 2 depend
on the choice of baseline hazard and on the gamma frailty distribution as well as
on the choice of the parameter values. For different models the magnitude of the
bias may differ from the presented results.
For Stata users who wish to avoid misspecification of the likelihood function
when estimating shared frailty models with left-truncated duration data, we pro-
grammed the Stata command stregshared, implementing the changes to the
likelihood discussed in Section 2. In the appendix we give a short description
of this new command. Simulations using stregshared confirm that the estima-
tor is correct and that the estimates converge to their true values as n → ∞
independent of the level of truncation.
As noted in the previous section, the Stata stcox command allows for semi-
parametric estimation of the shared gamma frailty model with left-truncated
data. We use this routine to estimate this model with the simulated data. This
does not impose the Weibull or Gompertz functional form for the duration de-
pendence λ, and hence standard errors are larger than above. However, with our
sample size, point estimates should be close to their asymptotic values. Instead,
it turns out that the estimates are similar to those obtained with the appropri-
ate streg command, for all values of c considered. This confirms our conjecture
that the stcox command in the case of the shared gamma frailty model with
left-truncated data is programmed on the basis of the L
Stata
likelihood as defined
in the previous section.
This result is of particular interest as the Stata stcox model has been fre-
quently used in the empirical literature to estimate shared gamma frailty models,
and sometimes the data are left-truncated. Gottard and Rampichini (2006) study
the effects of poverty on time to childbirth among young women in Bolivia. In
their data, individuals within a region are assumed to share their frailty term, and
individuals are only included in their sample if they have reached at least the age
of 14 at the time of the survey in 1998. Hence, left-truncation points vary across
individuals. They state that they use the stcox, shared command in their em-
pirical analysis. Studenski et al. (2011), who study the effect of gait speed on
survival among elderly individuals, provide another example. They use data from
9 different cohort studies, and in a sensitivity analysis of their main results, they
15
estimate shared gamma frailty models with Stata, where the frailty is taken to be
cohort-study-specific. The individual lifetime durations are left-truncated by the
entry age into the study. Hemmelgarn et al. (2007) study multidisciplinary care
for elderly patients with chronic kidney disease and its effect on survival. They
assume shared frailties for matched treated and untreated individuals, and they
estimate shared frailty models with Stata and/or SAS. Their data are subject to
left-truncation. Matching on age ensures that both lifetimes durations need to
exceed a left-truncation point in order for the pair to be included in the sample.
4 Conclusion
This paper analyzes the implications of ignoring the effect of left-truncation of
duration data on the distribution of unit-specific unobserved determinants in the
sample, if multiple durations are observed per unit. In the presence of unobserved
heterogeneity, it is vital to correctly account for the truncation that influences
the composition of survivors in the sample, especially if the truncation thresholds
vary across units.
Stata users estimating shared frailty models with the streg or stcox com-
mand need to be aware that with left-truncated data, the estimators of the covari-
ate effects, the duration dependence and the variance of the frailty distribution
may be inconsistent. The magnitude of the bias depends on the level of trunca-
tion and also on the variance of the frailty distribution of the data generating
process. The good news is the fact that the parameter estimates for the covariate
effects are typically biased towards zero. So in the worst case, effects have been
underestimated by Stata.
16
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18
Appendix
First, note that the gamma and Inverse-Gaussian distributions are both special
cases of the non-negative exponential family with density
f(v) = v
δ
e
−λv
m(v)φ(δ, λ)
−1
. (8)
A shared frailty model with a frailty distribution of this family has the following
survival function (see Hougaard, 2000):
S(t
1
, t
2
|x) =
Z
∞
0
v
δ
e
−(λ+M(t
1
,t
2
))v
m(v) dv
1
φ(δ, λ)
=
φ(δ, λ + M(t
1
, t
2
))
φ(δ, λ)
, (9)
with M(t
1
, t
2
) = φ(x
1
)Λ(t
1
) + φ(x
2
)Λ(t
2
). The second equality follows from the
fact that (8) is equivalent to φ(δ, λ ) =
R
∞
0
v
δ
e
−λv
m(v) dv and therefore φ(δ, λ +
M(t
1
, t
2
)) =
R
∞
0
v
δ
e
−(λ+M(t
1
,t
2
))v
m(v) dv.
A Gamma frailty
Let us assume a gamma distributed frailty with E(v) = 1 and V ar(v) = σ
2
. This
implies the following restrictions on the density function in (8)
δ = 1/σ
2
− 1, λ = 1/σ
2
, m(v) = 1, φ(δ, λ) = λ
−(δ+1)
Γ(δ + 1), (10)
where Γ(σ
2
) is the gamma function. Substituting the expression for φ(δ, λ) into
the right hand side of equation (9) leads to
S(t
1
, t
2
|x) =
(1/σ
2
+ M(t
1
, t
2
))
−1/σ
2
Γ(1/σ
2
)
1/σ
2
−1/σ
2
Γ(1/σ
2
)
= (1 + σ
2
M(t
1
, t
2
))
−1/σ
2
. (11)
Since f(t
1
, t
2
|x) =
∂
2
(1−S(t
1
,t
2
|x))
∂t
1
∂t
2
it follows
f(t
1
, t
2
|x) =
∂M(t
1
, t
2
)
∂t
1
∂M(t
1
, t
2
)
∂t
2
(σ
2
+ 1)(1 + σ
2
M(t
1
, t
2
))
−(1/σ
2
+2)
. (12)
Finally, let us consider the likelihood contribution of a group i comprising two
subjects with truncation points t
01
and t
02
and no censoring. Combining the
results from equation (11) and (12) leads to
f(t
1
, t
2
|T
1
> t
01
, T
2
> t
02
, x) =
f(t
1
, t
2
|x)
S(t
01
, t
02
|x)
= φ(x
1
)λ(t
1
)φ(x
2
)λ(t
2
)(σ
2
+ 1)(1 + σ
2
M(t
01
, t
02
))
1/σ
2
(1 + σ
2
M(t
1
, t
2
))
−(1/σ
2
+2)
which is equation (2) from section 2.
19
B Likelihood in the Stata Manual
The Stata Manual (Stata, 2009, p. 383) presents the following likelihood contri-
bution for a group i of a shared frailty model with a gamma frailty in the case of
no censoring
L = φ(x
1
)λ(t
1
)φ(x
2
)λ(t
2
)(σ
2
+ 1)(1 + σ
2
(M(t
1
, t
2
) − M(t
01
, t
02
)))
−(1/σ
2
+2)
.
Rearranging and choosing δ = 1/σ
2
− 1 and λ = 1/σ
2
according to (10) yields
L = φ(x
1
)λ(t
1
)φ(x
2
)λ(t
2
)
(λ + M(t
1
, t
2
) − M(t
01
, t
02
))
−(δ+3)
Γ(δ + 3)
(λ)
−(δ+1)
Γ(δ + 1)
.
Since we know that φ(δ + 2, λ + x) = (λ + x)
−(δ+3)
Γ(δ + 3) from (10) and that
φ(δ + 2, λ + x) =
R
∞
0
v
δ+2
e
−(λ+x)v
m(v) dv from equation (9) it follows
L = φ(x
1
)λ(t
1
)φ(x
2
)λ(t
2
)
Z
∞
0
v
2
e
−(M(t
1
,t
2
)−M(t
01
,t
02
))v
v
δ
e
−λv
m(v)
λ
−(δ+1)
Γ(δ + 1)
dv
and once the restrictions (10) for the gamma distribution are imposed again
L =
Z
∞
0
f(t
1
, t
2
|T
1
> t
01
, T
2
> t
02
, x, v) dG(v) .
C Inverse-Gaussian frailty
Let us assume Inverse-Gaussian distributed frailty terms. Like with the gamma
frailty, this imposes restrictions on the density in (8)
δ = −1/2, m(v) = ψ
1/2
π
−1/2
e
−
ψ
v
v
−1
, φ(−1/2, λ) = e
−(4ψλ)
1/2
.
Assuming ψ = λ gives a mean frailty of 1 and choosing σ
2
= 1/(2λ) yields
V ar(v) = σ
2
. Substituting the expression for φ(δ, λ) into the right hand side of
equation (9) leads to
S(t
1
, t
2
|x) =
exp(−(4(
1
2σ
2
)(
1
2σ
2
+ M(t
1
, t
2
)))
1/2
)
exp(−(4(
1
2σ
2
)
2
)
1
2
)
= exp(1/σ
2
− 1/σ
2
(1 + 2σ
2
M(t
1
, t
2
))
1/2
). (13)
Since f(t
1
, t
2
|x) =
∂
2
(1−S(t
1
,t
2
|x))
∂t
1
∂t
2
it follows
f(t
1
, t
2
|x) =
∂M(t
1
, t
2
)
∂t
1
∂M(t
1
, t
2
)
∂t
2
(1 + σ
2
(1 + 2σ
2
M(t
1
, t
2
))
−
1
2
)S(t
1
, t
2
|x)
1 + 2σ
2
M(t
1
, t
2
)
. (14)
20
Finally, let us consider the likelihood contribution of a group i comprising two
subjects with truncation points t
01
and t
02
and no censoring. Combining the
results from equation (13) and (14) leads to
f(t
1
, t
2
|T
1
> t
01
, T
2
> t
02
, x) =
f(t
1
, t
2
|x)
S(t
01
, t
02
|x)
= φ(x
1
)λ(t
1
)φ(x
2
)λ(t
2
)
(1 + σ
2
(1 + 2σ
2
M(t
1
, t
2
))
−
1
2
) exp(1/σ
2
− 1/σ
2
(1 + 2σ
2
M(t
1
, t
2
))
1/2
)
(1 + 2σ
2
M(t
1
, t
2
)) exp(1/σ
2
− 1/σ
2
(1 + 2σ
2
M(t
01
, t
02
))
1/2
)
D The command stregshared
D.1 Syntax
The command stregshared (see http://www.ceee-mannheim.de) is designed as
an alternative to streg when fitting a shared gamma frailty model to left-
truncated duration data. The size of the units over which the frailties are shared
should not exceed two when using stregshared. The functional form of the
baseline hazard can be specified as piecewise constant, Weibull, exponential or
Gompertz. The command has a similar syntax to streg:
stregshared varlist [if] [in], shared(varname) [ noconstant distribution(baseline)
cuts(numlist) ]
D.2 Description
stregshared is implemented as a v0 evaluator and uses Stata’s modified Newton-
Raphson maximization algorithm. The command fits the same shared frailty
model as the streg command with the shared() option. The only difference
is the adjusted likelihood function described in Section 2. Like streg it requires
the data to be defined as duration data by stset and it uses the same variables
in the same format as input arguments as streg.
D.3 Options
noconstant suppresses the constant term. The default is to include a constant in
the model. Note that varlist should not include a constant term, when the option
noconstant is not used.
distribution(baseline) sets the baseline hazard function to be of the type
baseline, where baseline can be specified as weibull, exponential or gompertz.
21
If this option is not used, a Weibull model is estimated. Note that the piecewise
constant model requires this option to be specified as d(exponential).
cuts(numlist) specifies the cutoff points of a piecewise constant baseline haz-
ard. When the options noconstant and d(exponential) are used, the option
cuts(numlist ) allows to estimate a piecewise constant model. Here, numlist
holds the list of cutoff points, where the numbers have to be in strictly ascending
order. For example, if the baseline function should be piecewise constant on the
intervals [0, 5.5), [5.5, 10) and [10, ∞] use: nocon d(exponential) cuts(5.5,10).
The option cuts() can not be used with d(weibull) or d(gompertz).
shared(varname) specifies a variable defining the units within which the
frailty is shared. The variable in varname is the same variable used in the option
shared of streg. Recall that stregshared can only deal with a unit size of one
or two spells. It is not a problem for the command if some (but not all) of the
units have only one spell and others have two. But it cannot deal with units
holding more than two spells. The shared() option has to be specified.
D.4 Comparison to streg
Since the stregshared command was designed as an alternative to streg, it is
intended to work in a very similar way. So if one uses the original streg Stata
command after stset to estimate a shared gamma-frailty model with a Weibull
distribution
. stset duration, failure(fail == 1) enter(truncation)
. streg x1 x2 x3, shared(id) d(weibull) frailty(gamma) nohr
the same arguments can be used with the stregshared command in order to
estimate the same model with the adjustment in the likelihood function from
Section 2:
. stset duration, failure(fail == 1) enter(truncation)
. stregshared x1 x2 x3, shared(id) d(weibull )
Here, id is the variable that identifies the unit. The same variable is used in the
option shared() in streg. Note that the option nohr which causes streg to
display the estimated parameter values instead of the hazard ratios is not used
in our command. stregshared will display the parameter values as well as the
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hazard ratios in the estimation results.
In this example the data are left-truncated and therefore the enter(truncation)
option in stset is used, where truncation is the variable that holds the left-
truncation points for each spell. If the enter() option is not used in stset,
stregshared and streg will yield the same estimation results.
D.5 Saved results
When an estimation is run with stregshared, the command shows the choice of
baseline function, the starting values, the number of units and total observations
used in the estimation and finally the estimation results. These results include
the parameter estimates, standard deviations, values of the test statistics and the
hazard ratios.
stregshared saves the following in e():
Scalars :
est base ancillary parameter (for Weibull or Gompertz function)
est theta
frailty parameter
Matrices :
est b coefficient vector
est matrix complete matrix of estimation results
(estimates, std. err. and test statistics)
To display the matrix of estimation results after running stregshared, type:
matrix list e(est matrix)
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