Marginal Effects for Competing Risks Models with Piecewise Constant Hazards*
| DOI | http://doi.org/10.1111/j.1468-0084.2009.00551.x |
| Date | 01 August 2009 |
| Published date | 01 August 2009 |
| Author | Tomi Kyyrä |
539
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2009. Published by Blackwell Publishing Ltd,
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 71, 4 (2009) 0305-9049
doi: 10.1111/j.1468-0084.2009.00551.x
Marginal Effects for Competing Risks Models
with Piecewise Constant HazardsÅ
Tomi Kyyr ¨
a
Government Institute for Economic Research (VATT), Helsinki, Finland
(e-mail: tomi.kyyra@vatt.fi)
Abstract
In the competing risks context, the effect of a covariate on the hazard function for
a particular cause may be very different from its effect on the likelihood of exiting
due to that cause. The latter probability is a function of all cause-specific hazards,
and thereby potentially affected by indirect effects via hazards for competing causes.
We consider the effects of covariates on the cumulative probability of exiting from a
particular cause. These ‘marginal effects’ are decomposed into direct effects via the
hazard of interest and indirect effects via the hazards for competing causes. For the
piecewise constant hazard specification we derive simple closed-form expressions
for the marginal effects that can be computed from the standard hazard function esti-
mates. An empirical application illustrates that the marginal effects provide a useful
and coherent way of summarizing the results of competing risks analysis.
I. Introduction
Competing risks arise in the analysis of duration data on individuals subject to leave
the initial state due to one of several competing causes. For example, a person may
leave unemployment by taking a job, by withdrawing from the labour force or by
participating in a labour market programme. The conventional approach involves
the specification and estimation of hazard functions for exits caused by competing
causes. The cause-specific hazard function describes the instantaneous exit rate due
to a particular cause conditional on not having exited due to any cause previously.
*I thank Ossi Korkeam¨akifor research assistance and two anonymous referees, Pekka Ilmakunnas, Michael
Rosholm, Ilpo Suoniemi and Roope Uusitalo for helpful comments. This work has also benefited from com-
ments received at the EEA-ESEM conference in Vienna,the AEAconference in Naples, and the XXIII Summer
Meeting of Finnish Economists in Jyv¨askyl¨a.
JEL Classification numbers: C41, J64.
540 Bulletin
But it does not provide direct information about the cumulative or overall probability
of exit due to a given cause as these are functions of all cause-specific hazards. In
general, the effect of a covariate on the hazard function for a particular cause can be
very different from its effect on the likelihood of exiting due to that cause by a given
time (Gray, 1988).
For example, the goal of policy-makers may be to induce the unemployed to find
an acceptable job within a reasonable amount of time, while transitions out of the
labour force and into labour market programmes are viewed as less desired outcomes.
The effects of policy variables, such as benefit levels or the maximum length of en-
titlement periods, on the hazard rate to employment may be less interesting from
a policy perspective than their effects on the likelihood of leaving unemployment for
employment by a given time. The employment effect of a policy change that affects
the employment hazard may be reinforced or attenuated by changes in hazard rates
out of the labour force and into labour market programmes. As a consequence, a
policy change with a strong effect on the employment hazard may have a negligible
or even inverse effect on the probability that the unemployment spell will end with
employment. Alternatively, a policy change with no effect on the employment
hazard may still have a significant effect on the likelihood of exiting to employment
due to indirect effects via the competing hazards. A uniform change in the employ-
ment hazard can also decrease the employment probability of some people but in-
crease it for some others. In policy evaluations this kind of response heterogeneity
can be of considerable interest.
In general, the effects of covariates on the cause-specific hazards are non-trivially
related to their effects on the cause-specific exit probabilities, making the interpre-
tation of covariate effects in the competing risks analysis tricky. This issue seems to
be overlooked in many economic applications of competing risks data. In the con-
text of many qualitative response models, like multinominal logit and ordered probit
models, an analogous issue arises (e.g. Cameron and Trivedi, 2005, p. 646). As a
consequence, the results of such models are usually reported in terms of the mar-
ginal effects, the effects of explanatory variables on the probability of interest. This
paper argues that a similar practice is equally useful in the context of competing risks
duration models as well. More precisely, we consider the effects of covariates on the
cumulative probability of exiting due to a particular cause by a given time. These
‘marginal effects’ are decomposed into direct effects via the hazard of interest and
indirect effects via the competing hazards.1Towhat extent the effect of a covariate on
the cumulative exit probability stems from a change in a particular hazard function
can be of particular interest. For example, in assessing the effect of unemployment
benefits on the likelihood of finding a job, the indirect effect via the hazard rate into
labour market programmes may be particularly interesting from a policy perspec-
tive, as the eligibility rules for such programmes are under the direct control of the
employment authorities.
1Unlike in the multinominal logit and ordered probit models, the sum of the marginal effects of a given
covariate with respect to different exit probabilities does not need to be zero in the competing risks model.
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2009
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