Causal inference for continuous-time processes when covariates are observed only at discrete times

Abstract

Most of the work on the structural nested model and g-estimation for causal
inference in longitudinal data assumes a discrete-time underlying data
generating process. However, in some observational studies, it is more
reasonable to assume that the data are generated from a continuous-time process
and are only observable at discrete time points. When these circumstances
arise, the sequential randomization assumption in the observed discrete-time
data, which is essential in justifying discrete-time g-estimation, may not be
reasonable. Under a deterministic model, we discuss other useful assumptions
that guarantee the consistency of discrete-time g-estimation. In more general
cases, when those assumptions are violated, we propose a controlling-the-future
method that performs at least as well as g-estimation in most scenarios and
which provides consistent estimation in some cases where g-estimation is
severely inconsistent. We apply the methods discussed in this paper to
simulated data, as well as to a data set collected following a massive flood in
Bangladesh, estimating the effect of diarrhea on children's height. Results
from different methods are compared in both simulation and the real
application.

Full text

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arXiv:1103.1472v1 [math.ST] 8 Mar 2011
The Annals of Statistics
2011, Vol. 39, No. 1, 131–173
DOI: 10.1214/10-AOS830
c ? Institute of Mathematical Statistics, 2011
CAUSAL INFERENCE FOR CONTINUOUS-TIME PROCESSES
WHEN COVARIATES ARE OBSERVED ONLY
AT DISCRETE TIMES
By Mingyuan Zhang, Marshall M. Joffe1and Dylan S. Small2
University of Pennsylvania
Most of the work on the structural nested model and g-estimation
for causal inference in longitudinal data assumes a discrete-time un-
derlying data generating process. However, in some observational
studies, it is more reasonable to assume that the data are generated
from a continuous-time process and are only observable at discrete
time points. When these circumstances arise, the sequential random-
ization assumption in the observed discrete-time data, which is essen-
tial in justifying discrete-time g-estimation, may not be reasonable.
Under a deterministic model, we discuss other useful assumptions
that guarantee the consistency of discrete-time g-estimation. In more
general cases, when those assumptions are violated, we propose a
controlling-the-future method that performs at least as well as g-
estimation in most scenarios and which provides consistent estima-
tion in some cases where g-estimation is severely inconsistent. We
apply the methods discussed in this paper to simulated data, as well
as to a data set collected following a massive flood in Bangladesh,
estimating the effect of diarrhea on children’s height. Results from
different methods are compared in both simulation and the real ap-
plication.
1. Introduction and motivation.
and methods for making causal inferences about the effect of a treatment
that varies in continuous time when its time-dependent confounders are
observed only at discrete times. Examples of settings in which this prob-
lem arises are given in Section 1.2. In such settings, standard discrete-time
methods such as g-estimation usually do not work, except when certain
In this paper, we study assumptions
Received September 2009; revised April 2010.
1Supported in part by NIH Grant CA095415.
2Supported in part by NSF Grant SES-0961971.
AMS 2000 subject classifications. Primary 62P10; secondary 62M99.
Key words and phrases. Causalinference,
model, diarrhea, g-estimation, longitudinal data, structural nested model.
continuous-timeprocess,deterministic
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Statistics,
2011, Vol. 39, No. 1, 131–173. This reprint differs from the original in pagination
and typographic detail.
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M. ZHANG, M. M. JOFFE AND D. S. SMALL
conditions are assumed for the continuous-time process. In this paper, we
formulate such conditions. When these conditions do not hold, we propose a
controlling-the-future method which can produce consistent estimates when
g-estimation is consistent and which is still consistent in some cases when
g-estimation is severely inconsistent.
First, we review the approach of James Robins and collaborators to mak-
ing causal inferences about the effect of a treatment that varies at discrete,
observed times.
1.1. Review of Robins’ causal inference approach for treatments varying
at discrete, observed times.In a cross-sectional observational study of the
effect of a treatment on an outcome, a usual assumption for making causal
inferences is that there are no unmeasured confounders, that is, that condi-
tional on the measured confounders, the data is generated as if the treatment
were assigned randomly. Under this assumption, a consistent estimate of the
average causal effect of the treatment can be obtained from a correct model
of the association between the treatment and the outcome conditional on the
measured confounders [Cochran (1965)]. In a longitudinal study, the analog
of the “no unmeasured confounders” assumption is that at the time of each
treatment assignment, there are no unmeasured confounders; this is called
the sequential randomization or sequential ignorability assumption, given as
follows.
(A1) The longitudinal data of interest are generated as if the treatment
is randomized in each period, conditional on the current values of measured
covariates and the history of the measured covariates and the treatment.
The sequential randomization assumption implies that decision on treatment
assignment is based on observable history and contemporaneous covariates,
and that people have no ability to see into the future. Robins (1986) has
shown that for a longitudinal study, unlike for a cross-sectional study, even
if the sequential randomization assumption holds, the standard method of
estimating the causal effect of the treatment by the association between
the outcome and the treatment history conditional on the confounders can
provide a biased and inconsistent estimate. This bias can occur when we are
interested in estimating the joint effects of all treatment assignments and
when the following conditions hold:
(c1) conditional on past treatment history, a time-dependent variable is
a predictor of the subsequent mean of the outcome and also a predictor of
subsequent treatment;
(c2) past treatment history is an independent predictor of the time-
dependent variable.

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CAUSAL INFERENCE FOR CONTINUOUS-TIME PROCESSES
3
Here, “independent predictor” means that prior treatment predicts current
levels of the covariate, even after conditioning on other covariates. An exam-
ple in which the standard methods are biased is the estimation of the causal
effect of the drug AZT (zidovudine) on CD4 counts in AIDS patients. Past
CD4 count is a time-dependent confounder for the effect of AZT on future
CD4 count since it not only predicts future CD4 count, but also subsequent
initiation of AZT therapy. Also, AZT history is an independent predictor of
subsequent CD4 count [e.g., Hern´ an, Brumback and Robins (2002)].
To eliminate the bias of standard methods for estimating the causal effect
of treatment in longitudinal studies where sequential randomization holds
but there are time-dependent confounders satisfying conditions (c1) and (c2)
(e.g., past CD4 counts), Robins (1986, 1992, 1994, 1998, 2000) developed a
number of innovative methods. We focus here on structural nested models
(SNMs) and their associated methods of g-testing and g-estimation. The ba-
sic idea of the g-test is the following. Given a hypothesized treatment effect
and a deterministic model of the treatment effect, we can calculate the po-
tential outcome that a subject would have had if she never received the treat-
ment. Such an outcome is also known as a counterfactual outcome, which is
the outcome under a treatment history that might be contrary to the real-
ized treatment history. If the hypothesized treatment effect is the true treat-
ment effect, then this potential outcome will be independent of the actual
treatment the subject received conditional on the confounder and treatment
history, under the sequential randomization assumption (A1). g-estimation
involves finding the treatment effect that makes the g-test statistic have its
expected null value. For simplicity, our exposition focuses on determinis-
tic rank-preserving structural nested distribution models; g-estimation also
works for nondeterministic structural nested distribution models.
The SNM and g-estimation were developed for settings in which treatment
decisions are being made at discrete times at which all the confounders are
observed. In some settings, the treatment is varying in continuous time, but
confounders are only observed at discrete times.
1.2. Examples of treatments varying in continuous time where covariates
are observed only at discrete times.
Example 1 (The effect of diarrhea on children’s height).
ease is one of the leading causes of childhood illness in developing regions
[Kosek, Bern and Guerrant (2003)]. Consequently, there is considerable con-
cern about the effects of diarrhea on a child’s physical and cognitive de-
velopment [Moore et al. (2001), Guerrant et al. (2002)]. A data set which
provides the opportunity to study the impact of diarrhea on a child’s height
is a longitudinal household survey conducted in Bangladesh in 1998–1999
Diarrheal dis-

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M. ZHANG, M. M. JOFFE AND D. S. SMALL
after Bangladesh was struck by its worst flood in over a century in the sum-
mer of 1998 [del Ninno et al. (2001), del Ninno and Lundberg (2005)]. The
survey was fielded in three waves from a sample of 757 households: round
1 in November, 1998; round 2 in March–April, 1999; round 3 in Novem-
ber, 1999. The survey recorded all episodes of diarrhea for each child in the
household in the past six months or since the last interview by asking the
families at the time of each interview. In addition, the survey recorded at
each of the three interview times several important time-dependent covari-
ates for the effect of diarrhea on a child’s future height: the child’s current
height and weight; the amount of flooding in the child’s home and village;
the household’s economic and sanitation status. In particular, the child’s
current height and weight are time-dependent confounders that satisfy con-
ditions (c1) and (c2), making standard longitudinal data analysis methods
biased [see Martorell and Ho (1984) and Moore et al. (2001) for discussion
of evidence for and reasons why current height and weight satisfy conditions
(c1) and (c2)]. The time-dependent confounders of current height and weight
are available only at the time of the interview, and changes in their value
that might affect the exposure of the child to the “treatment” of diarrhea,
which varies in continuous time, are not recorded in continuous time.
Example 2 [The effect of AZT (Zidovudine) on CD4 counts].
ticenter AIDS Cohort Study [MACS, Kaslow et al. (1987)] has been used
to study the effect of AZT on CD4 counts [Hern´ an, Brumback and Robins
(2002), Brumback et al. (2004)]. Participants in the study are asked to come
semi-annually for visits at which they are asked to complete a detailed in-
terview, including a complete history of their AZT use, as well as to take a
physical examination. Decisions on AZT use are made by subjects and their
physicians, and switches of treatment might happen at any time between
two visits. These decisions are based on the values of diagnostic variables,
possibly including CD4 and CD8 counts, and the presence of certain symp-
toms. However, these covariates are only measured by MACS at the time
of visits; the values of these covariates at the exact times that treatment
decisions are made between visits are not available.
The Mul-
1.3. A model data generating process.
diarrhea, the exposure or treatment process happens continuously in time
and a complete record of the process is available, but the time-dependent
confounders are only observed at discrete times. There could be various
interpretations of the relationship between the data at the treatment decision
level and the data at the observational time level. To clarify the problem
of interest in this paper, we consider a model data generating process that
satisfies all of the following assumptions:
In both the examples of AZT and

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CAUSAL INFERENCE FOR CONTINUOUS-TIME PROCESSES
5
(a1) a patient takes a certain medicine under the advice of a doctor;
(a2) a doctor continuously monitors and records a list of health indicators
of her patient and decides the initiation and cessation of the medicine solely
based on current and historical records of these conditions, the historical
use of the medicine and possibly random factors unrelated to the patient’s
health;
(a3) a third party organization asks a collection of patients from vari-
ous doctors to visit the organization’s office semi-annually; the organization
measures the same list of health indicators for the patients during their vis-
its and asks the patients to report the detailed history of the use of the
medicine between two visits;
(a4) we are only provided with the third party’s data.
Note that in (a2), we assume the sequential randomization assumption (A1)
at the treatment decision level.
The AZT example can be approximated by the above data generating
process. In the AZT example, (a1) and (a2) approximately describe the joint
decision-making process by the patient and the doctor in the real world.
(a3) can be justified by reasonably assuming that the staff at the MACS
receive similar medical training and use similar medical equipment as the
patients’ doctors. In the diarrhea example, the patient’s body, rather than
a doctor, determines whether the patient gets diarrhea. Assumption (a3),
then, is saying that the third party organization (the survey organization)
collects enough health data and that if all the histories of such health data
are available, the occurrence of diarrhea is conditionally independent of the
potential height.
1.4. Difficulties posed by treatments varying in continuous time when co-
variates are observed only at discrete times.
ated as in the previous section and we apply discrete-time g-estimation at the
discrete times at which the time-dependent covariates are observed; we will
denote these observation times by 0,...,K. In discrete-time g-estimation,
we are testing whether the observed treatment at time t (t = 0,...,K) is,
conditional on the observed treatments at times 0,1,...,t−1 and observed
covariates at times 0,...,t, independent of the putative potential outcomes
at times t + 1,...,K, calculated under the hypothesized treatment effect,
where the putative potential outcomes considered are what the subject’s
outcome would be at times t + 1,...,K if the subject never received treat-
ment at any time point. The difficulty with this procedure is that even if
sequential randomization holds when the measured confounders are mea-
sured in continuous time [as is assumed in (a2)], it may not hold when the
measured confounders are measured only at discrete times. For the discrete-
time data, there can be unmeasured confounders. In the MACS example,
Suppose our data are gener-