Download Statistical models for estimating the effects of intermediate variables

Statistical models for estimating
the effects of intermediate variables
in the presence of
time-dependent confounders
Dissertation zur Erlangung des Doktorgrades
der
Fakultät für Mathematik und Physik
der Albert-Ludwigs-Universität
Freiburg im Breisgau
vorgelegt von
Christine Gall
geboren in Erlangen
Dezember 2011
Dekan:
Prof. Dr. Kay Königsmann
1. Referent:
Prof. Dr. Martin Schumacher
Institut für Medizinische Biometrie und Medizinische Informatik
Albert-Ludwigs-Universität Freiburg im Breisgau
Stefan-Meier-Str. 26
79104 Freiburg
2. Referent:
Prof. Dr. Odd Aalen, Oslo
Datum der Promotion:
06. Februar 2012
Summary
Estimating the effect of a time-varying exposure when time-dependent confounders are
involved is not feasible with standard statistical models. Models for causal inference
cope with time-dependent confounders, but are still controversially discussed with respect to their unverifiable assumptions and the interpretation of their effect measures. In
this thesis, two such models proposed by Robins are addressed which are defined within
the counterfactual framework. This framework facilitates the definition of the treatment effect, but requires untestable identifying assumptions. These assumptions only
implicitely pose restrictions on the observable data which means that the confounding
structure between observables cannot be illustrated straightforwardly.
Insight into the properties of these two counterfactual models is given by bringing them
together with common approaches defined within the observable framework. For illustration and to explore the applicability of the models, two data examples are considered.
The Structural Nested Failure Time Model (SNFTM) applies for survival settings. Its
modelling assumptions with respect to the observable data structure are shown by
proposing a multistate model which is conform with the SNFTM and where the causal
SNFTM parameter directly enters. Multistate models do not use counterfactual or latent
variables, but directly model the observable variables such that they arise successively
as in a prospective trial. This model is also used as simulation model for data-generation
to compare the behaviour of a typically used Cox model and the SNFTM.
Marginal Structural Models are flexible with respect to the type of outcome. To give
access to this approach to people unfamiliar with the counterfactual framework, we show
that it can be seen as an extension of a common approach developed for the handling
of missing outcomes which is based on related unverifiable assumptions. Reducing the
complexity, we regard the different components step by step before finally incorporating
the structural model.
i
Contents
Summary
i
1. Introduction
1
1.1. Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2. Data examples
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1. Nosocomial infections on intensive care units . . . . . . . . . . . .
4
1.2.2. Preoperative breast cancer therapy . . . . . . . . . . . . . . . . .
5
1.3. Time-dependent confounding . . . . . . . . . . . . . . . . . . . . . . . . .
5
2. Causal inference
7
2.1. Causal conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2. Causal approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3. Counterfactual models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3. Theoretical background
11
3.1. Theory concerning counterfactual framework . . . . . . . . . . . . . . . .
11
3.1.1. Counterfactual framework . . . . . . . . . . . . . . . . . . . . . .
11
3.1.2. Structural Nested Failure Time Model . . . . . . . . . . . . . . .
13
3.1.3. Marginal Structural Models . . . . . . . . . . . . . . . . . . . . .
18
3.1.4. Contrasting SNFTM and MSM . . . . . . . . . . . . . . . . . . .
21
3.2. Theory concerning observable framework . . . . . . . . . . . . . . . . . .
22
3.2.1. Multistate models . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.2.2. Inverse Probability of Censoring Weighting . . . . . . . . . . . . .
23
4. SNFTM: Effect of nosocomial infection on length of hospital stay
4.1. Definition and estimation of the extra stay . . . . . . . . . . . . . . . . .
25
26
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Contents
4.2. Application: SIR3 study . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. Information on the ICU data
26
. . . . . . . . . . . . . . . . . . . .
26
4.2.2. Estimation of the SNFTM parameter . . . . . . . . . . . . . . . .
28
4.2.3. Estimation of the extra stay . . . . . . . . . . . . . . . . . . . . .
29
4.2.4. Artificial ventilation as time-dependent confounder . . . . . . . .
29
5. Multistate model conform with assumptions of SNFTM
31
5.1. Idea to model the action of a time-dependent confounder . . . . . . . . .
32
5.2. Definition of the multistate model . . . . . . . . . . . . . . . . . . . . . .
32
5.2.1. Effect of INF on AV and of AV on INF and discharge . . . . . . .
33
5.2.2. Discharge hazard for infected, unventilated patients . . . . . . . .
34
5.2.3. Discharge hazard for infected, ventilated patients . . . . . . . . .
34
5.2.4. Increased risk of AV due to infection: cINFAV > 1 . . . . . . . . . .
38
5.3. Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
5.3.1. Data generation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.3.2. Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.3.3. Illustration of characteristics of infected population . . . . . . . .
42
5.3.4. Estimation of effects by Cox model . . . . . . . . . . . . . . . . .
43
5.3.5. Estimation of effects by SNFTM
. . . . . . . . . . . . . . . . . .
44
5.3.6. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
5.4. Comparison to simulation proposed by Robins . . . . . . . . . . . . . . .
46
5.5. Extension to model current AV status . . . . . . . . . . . . . . . . . . . .
46
5.5.1. The case without time-dependent confounding . . . . . . . . . . .
48
5.5.2. The case with time-dependent confounding . . . . . . . . . . . . .
49
5.5.3. Assessing the assumption of no unmeasured confounders . . . . .
49
6. Marginal Structural Models as extension of missing data approach
53
6.1. Time-dependent confounding within one treatment arm . . . . . . . . . .
54
6.2. First step: outcome after application of a fixed number of cycles . . . . .
54
6.3. Second step: dose effect by comparing two groups . . . . . . . . . . . . .
57
6.4. Third step: dose effect by comparing all groups using a Marginal Struc-
iv
tural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
6.5. Contrasting IPCW with the MSM approach . . . . . . . . . . . . . . . .
59
6.6. Therapy effect: comparison of both treatment arms . . . . . . . . . . . .
60
Contents
6.7. Application: Geparduo study . . . . . . . . . . . . . . . . . . . . . . . .
6.7.1. Information on the breast cancer data
61
. . . . . . . . . . . . . . .
61
6.7.2. Estimation of the weights . . . . . . . . . . . . . . . . . . . . . .
62
6.7.3. Dose effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.7.4. Therapy effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
7. Discussion and outlook
67
A. Appendix: Abbreviations and notation
71
A.1. Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
A.2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
B. Appendix: Multistate model meets assumption of no unmeasured confounders
75
B.1. Start in state 4 reached from 2 compared to start in state 2 . . . . . . . .
76
B.1.1. Start in state 4 reached from 2
. . . . . . . . . . . . . . . . . . .
76
B.1.2. Start in state 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
B.2. Start in state 3 compared to start in state 1 . . . . . . . . . . . . . . . .
77
B.2.1. Start in state 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
B.2.2. Start in state 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
C. Appendix: Parts of this thesis published previously or submitted for peer
review
85
v
1. Introduction
This thesis addresses the impact of a time-varying exposure on an outcome of interest. If
time-dependent covariates or even time-dependent confounders are involved, it is difficult
to specify and estimate the effect of the time-varying exposure. Whereas including timedependent confounders in standard statistical models is not feasible, statistical models
for causal inference cope with them. But they come along with untestable assumptions
and their effect measures are not straightforwardly interpretable within the observable
data structure.
The focus lies on two structural models proposed by Robins, the Structural Nested
Failure Time Model (SNFTM) and the Marginal Structural Model (MSM). They are
defined within the counterfactual framework, where, according to any possible exposure
regime, a potential outcome is defined. The outcome which belongs to the actually
observed exposure regime is set equal to the observed outcome. As the data does not
comprise any direct information on the other potential outcomes, problems of causal
inference can be viewed as a problem of missing data with respect to the counterfactual
outcomes associated with exposure regimes other than the one actually observed [1]. The
counterfactual framework facilitates the definition of the effect as it does not involve any
probability models for the occurrence of exposure [2]. However, it is controversially
discussed because it explicitely makes assumptions whose validity is untestable. For
some statisticians this is unacceptable, as e.g. for Dawid [3]. Others, like Greenland
and Morgenstern [4], appreciate that this makes aware of the limitations of empirical
research on causal effects and offers the opportunity to modify experimental design or
evaluation techniques towards plausible assumptions. The counterfactual framework
provides a formalization of the assumptions used for causal inference. However, they
only determine the confounding structure within the counterfactual model which is not
explicitely transferable to the confounding structure within the embedded observable
setting.
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1. Introduction
The aim of this thesis is to explore the characteristics and the applicability of these two
counterfactual models and to work out new aspects which give insight into the model
properties and their relevance within the observable data structure. For illustration, two
data examples are considered which are outlined below.
The SNFTM applies for time-to-event outcomes. To clarify the confounding structure
assumed by the SNFTM with respect to relations between the observable data, we
synthesize the SNFTM with multistate modelling. Multistate models apply to event
history data. Changes over time are regarded as the occurrence of events which are
described by transitions between different states. The states are defined according to the
possible types of events that occur. They are defined within the observable framework.
The observable data structure is directly modeled and all variables arise successively
as in a prospective trial. As they take into account the chronological order, causal
interpretations might be facilitated in the sense that the cause will always precede the
effect regarding the interaction between life history events [5]. We propose a simple
joint model defined by a multistate model where the causal SNFTM parameter enters
directly. The difficulty is that the SNFTM parameter refers to the total causal effect
whereas the multistate model is characterised by direct effects. We solve this problem
by defining a delayed impact of changes in covariates which are already affected by the
exposure. This illustrates that effect parameters should be compared with caution. The
multistate model is also used as simulation model for data-generation to illustrate data
characteristics and explore the behaviour of a typically used Cox model in comparison
to the SNFTM.
MSMs belong to another class of causal structural models. They are flexible with respect to the type of outcome. According to our data example, we regard a time-constant
outcome measured at the end of study. Their parameters are estimated by Inverse
Probability of Treatment Weighting. We present MSMs as an extension of the common approach of Inverse Probability of Censoring Weighting (IPCW) developed for the
handling of missing outcomes. It is based on related unverifiable assumptions about the
reasons for missingness, but, however, their necessity is rather accepted by the statistical
community. By contrasting both approaches, we give insight into the structural model
approach.
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1.1. Structure of the thesis
1.1. Structure of the thesis
In chapter 2, we shortly address the aim of causal approaches and the idea of the
counterfactual framework. Chapter 3 outlines the theoretical background used within
this thesis. Its first part refers to the counterfactual framework with focus on the SNFTM
and MSMs. The SNFTM is reviewed in detail providing all instructions for the analysis
of the ICU example (chapter 4). The MSM is introduced more shortly with focus on
the underlying ideas, as a detailed illustration is given in chapter 6 when contrasting
the MSM to IPCW. For both structural models, the description of the different steps is
similarly structured in order to prepare their comparison in section 3.1.4. The second
part of chapter 3 corresponds to the observable framework considering the non-causal
approaches used to illustrate aspects of the structural models. These are multistate
models and IPCW.
Chapters 4 and 5 address the SNFTM. The ICU data example is analysed by a SNFTM
in chapter 4 where we additionally address a quantity whose definition is facilitated by
the counterfactual framework and whose estimation is done by plugging in the SNFTM
parameter. In chapter 5, we propose a simple joint model defined by a multistate model
which is conform with the SNFTM. Its main purpose is not to facilitate data generation
using it as simulation model, but to illustrate the data structure required by the SNFTM
only with respect to observable variables.
In chapter 6, we illustrate MSMs as an extension of a missing data approach with exemplification by data on breast cancer. Thereby, we split up the deterrend complexity
of the MSM by first doing without the structural model and accessing the counterfactual framework as well as the estimating procedure by relating it to the more familiar
procedure of IPCW.
1.2. Data examples
We regard two data examples which both comprise a time-dependent confounder. One
focusses on a survival endpoint, the other on a binary outcome manifest at the end of
study. To already keep the settings in mind and get an impression of the action of a
time-dependent confounder, we shortly introduce them without quantifying any data
characteristics. The first example is used to illustrate the SNFTM which is investigated
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1. Introduction
in chapters 4 and 5. The second example is considered in chapter 6 which addresses the
MSM.
1.2.1. Nosocomial infections on intensive care units
The SIR3 (Spread of nosocomial infections and resistant pathogens) study is a cohort
study to prospectively collect data to examine the effect of hospital-acquired, i.e. nosocomial, infections. All patients admitted to certain intensive care units (ICUs) who stayed
on the unit for more than 48 hours are followed during their ICU stay. The documented
data include baseline parameters as well as time-dependent variables which are recorded
on a daily basis. The latter include clinical parameters and the use of medical devices,
e.g. artificial ventilation. Furthermore, there is information on the onset times of nosocomial infections. We focus on pneumonia which is one of the most frequent and severe
nosocomial infections. We are interested in the impact of the occurrence of a nosocomial
infection on the length of ICU stay which is given by death and discharge time, respectively. Here, the use of artificial ventilation is a candidate for being a time-dependent
confounder. Figure 1.1 shows typical courses of the ICU stay. A detailed description of
Figure 1.1.: Typical courses of ICU stay. End of ICU stay: discharge alive, death,
→ censored. . Time from nosocomial infection. – – – Use of artificial
ventilation.
the data can be found in Grundmann et al. [6] and Beyersmann et al. [7].
4
1.3. Time-dependent confounding
1.2.2. Preoperative breast cancer therapy
The Geparduo study [8] is a randomised controlled clinical trial in breast cancer to
compare two preoperatively applied chemotherapy schemes with respect to a pathologic
result on remission. The chemotherapies are given in repeated cycles where medical
values measured before each cycle may lead to stop chemotherapy prior to the last
planned cycle. Regardless of the number of given chemotherapy cycles, the study patients
are operated and parts of the breast are resected. Therefore, response assessment was
possible for every patient.
The main interest of the Geparduo study lies in the outcome after all planned cycles.
However, we use the data separately per treatment arm and analyse the outcome after
a certain number of cycles. This is possible, as due to the observed early stopping, for
some patients the outcome is known after a reduced number of cycles. As the medical
values used to assess, whether chemotherapy should be continued, are measured before
each cycle, i.e. after randomisation, we need to account for time-dependent confounding.
1.3. Time-dependent confounding
We illustrate the action of a time-dependent confounder by the ICU data example.
Here, a time-dependent confounder is characterised by not only being a risk factor for
nosocomial infection and a prognostic factor for the length of stay, but also by being
influenced by the infection after its occurrence. A potential time-dependent confounder
is the use of artificial ventilation (AV).
We denote the length of stay on ICU by T . The occurrence of infection is described by
the time-dependent status variable INF(t) which is 0 until the occurrence of infection
and then jumps to 1. Hence, INF(t) = 1 means that the patient was infected at some
time before t, but is not necessarily infected at time t. This definition encodes past
infection exposure typically investigated in hospital epidemiology [9]. The information
whether AV is switched on or off is given by the status variable AV(t).
Part of a confounding situation is shown by figure 1.2 where we assume that values can
only change on a daily basis, that the infection occurs at day k, and that the patient is
still in ICU at day k + 1. In the first period, from day 1 until day k, AV influences the
risk of infection and the length of stay. In the second period, after day k, the infection
5
1. Introduction
AV(k)
AV(k+1)
INF(k)
T
Figure 1.2.: Time-dependent confounding: situation where infection occurs at day k
and patient is still in ICU at day k + 1; AV(t) = artificial ventilation status,
INF(t) = infection status, T = length of stay
might have an effect on the need for artificial ventilation, that is on the values AV(t) for
t > k. The dashed pathway from INF(k) over AV(k + 1) towards T indicates that the
value of AV(k + 1) might partially represent an effect of the infection on the length of
stay. The contribution of the dashed part of the arrow from AV(k + 1) to T can be seen
as an indirect effect of the infection.
Note, that this illustrates why the inclusion of AV(t) into a statistical model is complicated. Conventional adjustment for AV(t) for values t > k would either diminish the
effect of the infection status or might indicate an influence of infection, even if it does
not affect the length of stay.
In general, time-dependent confounding is only possible, if confounder and exposure are
both time-varying. Only then, the relation between confounder and exposure can be
reversed.
With respect to the breast cancer example, a time-dependent confounder is a prognostic
factor which changes as a result of previous applications of treatment and in turn has an
impact on the further application of subsequent treatment. A potential time-dependent
confounder is increased leucocytes which is measured before each cycle.
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2. Causal inference
2.1. Causal conclusions
Statistical models serve to detect associations between different characteristics or exposure and outcome variables. Statisticians never affirmed that associations can be used
to identify causal relations, but often, they are utilised with the intention to achieve a
causal conclusion.
The most famous example of when a statistical model detects an astonishing relation
is probably that the more storks live in an area, the more babies are born. This result
obviously cannot be used to prove that storks deliver babies, as the information, if
the storks live on the countryside or in town is missing. It shows that statistics can
only show associations and do not allow causal conclusions without further restrictions
on the data. However, in daily life, associations are often enough. For the midwife
who considers where to open a private practice, it does not make a difference, if she
checks which places are near the countryside or where there live more storks. This
shows that association is enough for prediction in the unchanged population. Here, the
emphasis is placed on the attribute unchanged. To illustrate this further, imagine that
the government intervened such that bringing up children were much more attractive
in town than on the countryside. Then, the midwife would need a causal conclusion as
the relation between the incidence of storks and the favourable location of the private
practice would change.
In medicine, causal relations are required to optimise prevention and treatment of diseases, as the aim is e.g. to change medication or surgery techniques to maximise the
patient’s benefit. So far, only randomised experiments are overall accepted to infer on
causal relations. Within the last four decades, statisticians more and more tried to deal
with using statistics to answer causal questions.
Inference on causal relations is only possible with well defined causal criteria and cannot
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2. Causal inference
do without posing restrictions on the underlying data structure, which cannot be verified
by statistical experiments. This includes a careful investigation, if the causal question
is appropriate. For example, we cannot just ask for the effect of a reduction of the
body mass index on mobility. The answer mainly depends on the way the reduction
is achieved, e.g. either by dieting or by increasing activity. A good way to find out,
if a causal question is applicable, is to attempt to formulate a randomised trial which
answers the causal question. This also clarifies, which statement we expect from the
question.
2.2. Causal approaches
Within the last decades, the need for statistical methods to facilitate causal inference
is more and more recognized by the statistical community [10, 11]. Different formal approaches for causal inference were proposed. The most prominent are graphical models,
counterfactual models and structural-equations models. They are outlined and contrasted by e.g. Pearl [12] and Greenland and Brumback [13].
Pearl [12] summarises the requirements of a causal approach. The most important
conclusion is that for causal inference one requires properties of the data-generating
process which are not given by the data alone, even if data were collected for the whole
population. Hence, one must cope with untestable assumptions. Furthermore, this
requires new mathematical notation for expressing causal relations which is not given
by standard probability calculus.
Aalen [14] points out that causality is a dynamic concept as the cause must precede the
effect. As stated by Aalen and Frigessi [11], it appears a weakness of many models of
causality, e.g. graphical models, that time is absent. To incorporate the eminent status
of time, Fosen et al. [15, 16] proposed a special approach to graphical models which is
called dynamic path analysis.
Causal relations are often represented by directed acyclic graphs (DAGs), see e.g. [13,17].
DAGs are a good instrument to visualise causal structures which also serves to better
communicate with clinicians. They indicate the statistical dependencies and independencies of the included variables. They demonstrate the consequences of conditioning
on certain variables and thereby support choosing the analysing strategy.
8
2.3. Counterfactual models
2.3. Counterfactual models
Counterfactual models, also called potential-outcome models, were established to infer on
the effect of an intervention. They are based on the idea of reconstructing a randomised
trial. Within the counterfactual framework, the outcome consists of a vector whose
elements are variables that are interpreted as – possibly counterfactual – outcome had
a certain exposure state been true. For example, if exposure is either no treatment or
treatment, the counterfactual outcome is given by (Y0 , Y1 ). Here Y0 refers to the outcome
after no treatment and Y1 to the outcome after treatment. Typically, only one of these
outcomes is known from the data. If, actually, treatment is given, Y1 is set equal to the
observed outcome whereas Y0 is not known and called counterfactual. This facilitates
the definition of the causal effect which is given by characteristics of the distribution of
counterfactuals, e.g. by E(Y1 − Y0 ) in case of a linear treatment effect. The definition
does not depend on the probability that a certain counterfactual outcome is actually
observed, e.g. on the probability that treatment is given. This is the major difference
between counterfactual outcomes and other frameworks for causal inference. To cope
with the unobservable outcome values, one needs identifying assumptions which describe
the confounding mechanism. They are unverifiable and facilitate the estimation of the
effect by observable variables.
If the exposure is time-independent, the identifying assumption states that given the
confounders X, the counterfactual outcome (Y0 , Y1 ) is independent of the treatment
assignment Z which is formally described by
(Y0 , Y1 )⊥⊥ Z|X
(2.1)
where the conditional independence is denoted by ⊥⊥ . It implicitely ensures that X
comprises all confounders which affect Z and the outcome. For illustration, we consider
patient’s health condition which affects the doctor’s decision to give treatment or not,
e.g. if treatment is aggressive and he supposes that it is rather suitable for patients having
a better constitution. Then, if health condition is not included in X, the assignment
Z does contain information on (Y0 , Y1 ), because Y0 as well as Y1 tend to be better,
as whatever treatment is assigned, the patient has a better prognosis due to his good
condition. Note, that Z affects Y but neither Y0 nor Y1 . However, Z indicates whether
Y0 or Y1 can be observed. Identifiability follows from (2.1), as one can rewrite E(Y1 ) in
9
2. Causal inference
terms of observable variables by
E(Y1 ) = E(E(Y1 |X)) = E(E(Y1 |X, Z = 1)) = E(E(Y |X, Z = 1))
and analogously perform with E(Y0 ).
Effect estimates can e.g. be obtained by the Propensity Score (PS) proposed by Rosenbaum and Rubin [18] which describes the conditional probability of treatment given the
confounders. There are several procedures to use the PS, among them stratification,
matching and inverse weighting by the PS.
Within this thesis, we focus on time-dependent exposures and their effect on a timeconstant or survival outcome. Here, one regards not only two but several counterfactual
outcomes according to each possible sequence of exposure. Furthermore, the identifying
assumption is sequentially defined. It consecutively considers the time points of exposure conditioning on the respective covariate and exposure history. For these settings,
Robins made proposals on two structural models, which are outlined in section 3.1, the
SNFTM and the MSM. The overall strategy is to build a structural model which relates
the outcome vector elements with respect to the exposure variable and thereby defines
the effect of interest. Possible confounders are not included in the structural model.
Their relations are considered by the estimating procedure which is deduced from the
identifying assumptions such that the parameters of the structural model are causally
interpretable.
The SNFTM applies for longitudinal data with a time-to-event endpoint. The estimating
procedure is called g-estimation where g stands for generalised treatment. The MSM can
be used for various types of outcome. Quantification of the effect of a treatment regime
is done by the estimation method Inverse Probability of Treatment Weighting. It can
also be seen as an extension of inverse weighting by the PS transferred to time-dependent
exposures.
10
3. Theoretical background
3.1. Theory concerning counterfactual framework
Within this section, we illustrate the counterfactual framework and review the structural
models SNFTM and MSM. Whereas the SNFTM only applies for survival settings, the
MSM is flexible with respect to the outcome.
We regard a time-varying exposure which only changes once. In our ICU example, this
is the occurrence of a nosocomial infection where the exposure variable changes at the
onset time of the infection. In our breast cancer example, we regard early stopping of
treatment. Here, the exposure changes when the first cycle is withheld. We choose the
notation and interpretation of variables according to the data examples we keep in mind.
3.1.1. Counterfactual framework
The counterfactual outcome not only comprises a single outcome variable, but a whole
vector of outcome variables. Each of these variables belong to a certain exposure regime
and describe the hypothetical outcome, had it been affected by this regime.
First, we regard the case of a time-independent outcome Y with observed exposure
described by ∆p = (∆1 , ∆2 , ..., ∆p ) where ∆k is either 0 or 1 and only switches once.
For example, exposure can be treatment applied in at most p cycles where ∆k equals 1,
if cycle k is given. If treatment is stopped once, it is also withheld for the rest of the
cycles. Then, the counterfactual outcome is defined as the vector of random variables
(Y0 , Y1 , Y2 , ..., Yp )
(3.1)
The component Yk is interpreted as the potential outcome, if treatment was given for
the first k cycles. Y0 refers to the outcome after no treatment.
According to this interpretation, the observable outcome Y is linked to the counterfactuals. If treatment was never given, the observed outcome Y is set equal to Y0 . Otherwise,
11
3. Theoretical background
if treatment was given for the first k cycles, Y is set equal to the outcome Yk . The
remaining outcome variables are not observed and therefore called counterfactual.
As there are many more variables defined by (3.1) than can be observed, assumptions are
needed to make the problem identifiable. These assumptions describe the confounding
mechanism in the observed data. To account for the time-dependent aspects and the
chronological order of cause and effect, the confounding mechanism is usually characterised by sequentially defined conditional independence assumptions which require for
all possible exposure values k:
Y0 , Y1 , Y2 , ..., Yp ⊥⊥ ∆k |X k , ∆k−1
(3.2)
where conditional independence is denoted by ⊥⊥ . Here, we write X k = (X1 , ..., Xk )
for the covariate history prior to cycle k. Baseline covariates are included in X1 . To
simplify notation, we define ∆0 to be 0.
The assumption in (3.2) is called the assumption of no unmeasured confounders. It can
be interpreted in two ways. Firstly, by comparison of two patients just before cycle k
who were treated so far and have the same covariate history X k . According to (3.2),
knowing that for one of them treatment is withheld from cycle k, does not enable to
better predict any of the counterfactual outcomes (Y0 , Y1 , Y2 , ..., Yp ). Thus, one can
regard the study as a virtual sequential randomised clinical trial, where at each time k
the decision whether to stop treatment is taken completely randomly conditional on the
known history X k . Secondly, (3.2) implicitely ensures that all information which affects
stopping treatment from cycle k is included in X k . For example, we regard a medical
factor Fk which influences the doctor’s decision to stop treatment and which is high,
if the patient has a bad prognosis. Then, knowing ∆k does improve the prediction on
any counterfactual outcome Yi , as ∆k provides information on Fk and therefore on the
probability that the patient has a bad prognosis.
In case of a survival setting with observed outcome T , the counterfactual outcome can
be represented by
(T ∞ , T 1 , ..., T nmax )
if we assume that the onset of exposure can only occur at discrete times, e.g. daily,
and that the onset times of exposure lie between 1 and nmax . Here, we write the index
for the hypothetical exposure as superscript in order to distinguish the counterfactual
12
3.1. Theory concerning counterfactual framework
outcome from a sequence describing a time-dependent variable. Furthermore, we write
T ∞ for the outcome, if no exposure occurred. Now, the assumption of no unmeasured
confounders includes conditioning on T ≥ k:
T ∞ , T 1 , ..., T nmax ⊥⊥ ∆k |∆k−1 , X k−1 , T ≥ k
(3.3)
3.1.2. Structural Nested Failure Time Model
The SNFTM was proposed by Robins [19, 20] for the situation of a time-varying exposure, time-dependent confounders and a time-to-event outcome. It is explained in detail
by Robins [21] and comprehensively illustrated by Keiding et al. [22]. In particular, we
regard the deterministic SNFTM where a deterministic relationship between counterfactuals and observables is assumed. Robins et al. [20] and Robins [21] describe how this
assumption can be relaxed.
The approach was initially intended to evaluate the effect of a sequentially given treatment on an event time. Then, applications arose that addressed the effect of an intermediate event, e.g. by Keiding et al. [22, 23]. The method was further modified to
fit the context of clinical trials to adjust for post-randomisation variables and compliance, e.g. by Robins [24], and illustrated by e.g. Korhonen et al. [25] and Yamaguchi and
Ohashi [26]. Hernán et al. [27] elucidated the approach introducing structural models by
distinguishing them from associational models and describing the estimating procedure
as a reconstruction of sequentially randomised groups.
Within the following sections, we explain the SNFTM in detail to illustrate its modelling
ideas and to provide all necessary steps for the analysis of the ICU data (chapter 4).
As we regard the case where the time-varying exposure only switches once, we describe
it by I, the time of its first occurrence. Keeping in mind our ICU example, where the
time-varying exposure is the onset of a nosocomial infection, we interpret I as the time
from admission to nosocomial infection. Using the notation from section 1.3, I can be
defined by I := inf{t : INF(t) = 1}. We further interpret the time-to-event outcome,
denoted by T , as the length of stay on ICU. As is usual for hospital data, our ICU
data set includes only administrative censoring due to end of documentation. Hence,
we only illustrate this aspect which is, together with the modification of the estimating
procedure to include censoring due to competing risk, clearly illustrated in Keiding et
al. [22].
13
3. Theoretical background
3.1.2.1. Model structure
The SNFTM is based on the strong version of an accelerated failure time model (as defined in e.g. Cox and Oakes [28, section 5.2.]) transferred to the context of counterfactual
variables. It relates the counterfactual event time T ∞ , had no exposure occurred, to the
observations T and I by the acceleration parameter exp(γ):

I + (T − I) exp(γ) for I < T
T∞ =
T
otherwise
(3.4)
Note, that I < T means that the exposure occurred until the terminal event happened.
The model assumes that the exposure causes a transformation of the time scale by the
factor exp(−γ) from the time of occurrence. This results in a transformation of the
time span from exposure to the counterfactual event time T ∞ . For exp(γ) lower than 1,
exposure leads to later event times as compared to the case where the exposure had not
occurred, cf. figure 3.1.
0
I
T∞
T
(T ∞ − I) · exp(−γ)
Figure 3.1.: Accelerated failure time model: situation with prolongation effect of exposure, i.e. exp(γ) < 1
In the context of our ICU example, the time from I to T ∞ can be interpreted as the
remaining time needed to recover without infection. This time is supposed to be related
to the patient’s health condition. The longer this time, the more impact the infection
has on the patient’s recovery.
Note, that the accelerated failure time model in (3.4) not only links T ∞ with T and I,
but also determines a deterministic relation between any two counterfactual outcomes
T k and T j as (3.4) can be used to calculate T k and T j from T ∞ .
3.1.2.2. Inference from the observable data
The SNFTM is a structural model which links T ∞ with T and I by the population
parameter exp(γ). Possible confounders are not included in the SNFTM because they
14
3.1. Theory concerning counterfactual framework
are a characteristic of the observed data. Their relationships are accounted for by the
assumption of no unmeasured confounders shown in (3.3). As the exposure is a status
variable which only once jumps from 0 to 1, (3.3) is rewritten for the g-estimation method
proposed by Robins [19] to derive the adequate estimating procedure for exp(γ). We
review the method by first assuming that there is no censoring.
To show the dependence of T ∞ on γ, we define

I + (T − I) exp(γ) for I < T
T ∞,γ =
T
otherwise
(3.5)
where T ∞ = T ∞,γ0 , if γ0 is the true population value.
Being the constitutive step, the assumption of no unmeasured confounders given in (3.3)
is defined in terms of the exposure hazard λI for the time until start of the exposure,
which can be interpreted as follows. λI multiplied by a very small time interval (a few
minutes, say) is the probability that the exposure starts within this small time interval
under the condition that the individual was unexposed at the beginning of this time
interval. One requires that given the confounder history just prior to time t, denoted by
Ft− , the exposure hazard λI is not affected by T ∞,γ0 :
λI (t|Ft− ) = λI (t|Ft− , T ∞,γ0 ).
(3.6)
Note, that Ft− does not include T ∞,γ0 . The covariate-information addressed in (3.6)
is restricted to the time to exposure. I.e. it only involves the period where the timedependent confounders can affect the occurrence of the exposure, but are not yet affected
by the exposure. This enables appropriate adjustment for time-dependent confounders.
In the case where the exposure might switch at several discrete times, one rewrites (3.3)
by e.g. using a pooled logistic model for the conditional probability of ∆k = 1 given the
exposure and covariate history. Here, one claims that this conditional probability should
not be affected by T ∞,γ0 . Then, appropriate adjustment for time-dependent confounders
is possible, as the exposure and covariate history is correctly taken into account.
3.1.2.3. Estimation of the SNFTM parameter
The estimator of exp(γ) can be obtained by using the step-by-step algorithm called
g-estimation which is explained in detail in Robins [21]. It is built on the fact that if
(3.6) holds, the effect of T ∞,γ should be 0. If the hazards in (3.6) follow Cox regression
15
3. Theoretical background
models, estimation can be done via standard software. Then, regarding λI (t|Ft− , T ∞,γ ),
T ∞,γ is included as a covariate. Within each step of the g-estimation algorithm, a certain
value for exp(γ) is chosen. Then, T ∞,γ is calculated from T and I for every observation
using (3.5), and the Cox regression model for the exposure hazard is fitted with all
confounders and T ∞,γ . Finally, exp(γ) is chosen so that the p-value of the score test
for T ∞,γ is largest. A 95% confidence interval is obtained by including those values of
exp(γ) where the p-value is greater or equal to 0.05.
To illustrate the g-estimation procedure, table 3.1.2.3 shows an oversimplified situation
with no time-dependent confounders where T ∞,γ0 is constantly equal to 20 and the true
exp(γ0 ) equals 0.7, i.e. exposure prolongs the time to event. For T ∞,γ0 = 20 and several
exposure times, T was calculated by (3.5). Then, for a value below and above the
true exp(γ0 ), T ∞,γ was calculated from T and I by (3.5). One sees that, if exp(γ) is
incorrectly chosen, T ∞,γ contains information about the exposure status. In detail, for
exp(γ) = 0.6, the later the exposure starts the greater is the calculated value of T ∞,γ .
Whereas T ∞,γ with exp(γ) = 0.8 decreases for later exposure times. This affects the
estimated regressor for T ∞,γ and its p-value.
counterfactual
observables
calculations for fixed γ 6= γ0
T ∞,γ0
I
T
T ∞,γ , exp(γ) = 0.6
T ∞,γ , exp(γ) = 0.8
20
5
26.4
17.8
22.1
20
8
25.1
18.3
21.7
20
10
24.3
18.6
21.4
20
15
22.1
19.3
20.7
Table 3.1.: Illustration of values regarded within g-estimation procedure for true
exp(γ0 ) = 0.7
3.1.2.4. Administrative censoring due to end of documentation
In this section, we describe how to cope with administrative censoring due to end of
documentation. Here, the potential censoring time C is already known for every patient
on admission. The idea is to replace T ∞,γ0 in (3.6) by a function of T ∞,γ0 , C, γ0 and t:
λI (t|Ft− ) = λI (t|Ft− , f (T ∞,γ0 , C, γ0 , t)).
16
(3.7)
3.1. Theory concerning counterfactual framework
As C is already known on admission, it is included in F0 . Hence, if γ = γ0 , (3.7) follows
from (3.6). The aim is to choose f (T ∞,γ0 , C, γ0 , t) such that it can be determined for
every observation in the risk set at each time t whether it is censored or not independently
of the onset of exposure I.
The focus here is not on the possibly censored outcome min(T, C), but on the possibly
censored value of T ∞,γ which belongs to a time scale transformed by exp(γ) from the
onset of exposure. Therefore, in the case of exp(γ) < 1, we consider a transformation of
C which is defined time-dependently as follows:
Cγ (s) = s + (C − s) exp(γ).
Then, as Cγ (s) increases with s and Cγ (t) < Cγ (s) for s ∈]t; C],
min(T ∞,γ , Cγ (t))
(3.8)
is our candidate for f (T ∞,γ0 , C, γ0 , t) as it can be determined for every observation
whether the outcome T is censored or not. Three other functions f (T ∞,γ0 , C, t) are
discussed in Keiding et al. [22]. We chose the one which fits most to our interpretational
line.
The definition of Cγ (t) is such that sequentially increasing censoring times are generated.
Thereby, artificial censoring which changes over time is introduced. Considering risk
sets at time t, the lower t the more artificial censoring is introduced. Note, that the
prolongation and hence the artificial censoring mechanism depends on the acceleration
factor exp(γ).
As for exp(γ) > 1, Cγ (s) decreases with s, the constant min(T ∞,γ0 , C) can be chosen for
f (T ∞,γ0 , C, γ0 , t) in this case.
3.1.2.5. Algorithm by Robins to simulate data conform with SNFTM
A data generating process, typically used for a SNFTM was proposed by Robins [21]
and further applied by Young et al. [29]. Here, individual patient data is generated
separately for each subject. A counterfactual event time is sampled first. Then, a
separate process is used to sample the interaction of exposure and covariates. The
impact of the confounders on the outcome is achieved by relating their distribution to the
initially sampled counterfactual outcome. The process is stopped at the observed event
17
3. Theoretical background
time which is obtained by subsequently transforming the counterfactual time according
to the already sampled exposure values.
For our situation, considering the time-dependent confounder CONF(t) and the start of
exposure described by the time-dependent status variable EXP(t), which is 0 until the
start of exposure and then jumps to 1, it applies as follows
Step 1: Simulate the counterfactual event time T ∞ from a failure time distribution (e.g.
Exponential or Weibull) with hazard λ0 (t).
Step 2: Simulate the time of the next change in either CONF(t) or EXP(t), denoted by
K, where the time to change is modeled by a failure time distribution with hazard
λCONF (t) and λEXP (t), respectively. Here, λCONF (t) should depend on T ∞ and the
history of EXP(t) in order that CONF(t) is a confounder and affected by the
exposure. Furthermore, λEXP (t) should depend on the history of CONF(t) such
that CONF(t) acts as confounder and must exclude T ∞ such that the assumption
of no unmeasured confounders is satisfied.
Step 3: Determine T as follows. If EXP(t) already jumped to 1, calculate T using (3.4),
i.e. by T = I + (T ∞ − I)/ exp(γ), where I = inf{t : EXP(t) = 1}. Otherwise, set
T equal to T ∞ .
If K < T repeat steps 2 and 3.
Otherwise, redefine CONF(K) :=CONF(K − dt) and EXP(K) :=EXP(K − dt)
such that no change of CONF(t) and EXP(t) happened at K.
3.1.3. Marginal Structural Models
Marginal Structural Models (MSMs) were proposed by Robins [30,31] to quantify the effect of a certain treatment regime. They adequately address time-dependent confounders
and apply for different types of outcome including time-to-event outcomes. In chapter
6, we address a time-constant binary outcome.
A treatment regime consists of a treatment plan which specifies how to determine the
treatment dose at the time-points where treatment is given. There are two types of
treatment regimes, static and dynamic regimes. In static treatment regimes the application doses are determined at baseline and do not change in response to medical
history of the individual patient. Dynamic treatment regimes provide rules to assess the
18
3.1. Theory concerning counterfactual framework
application dose in dependence of certain measurements of clinical variables available at
the respective time-point. The MSM applies to data where a dynamic treatment regime
was used, but infers on the causal effect of a static regime by comparing potential cases
where different static treatment regimes were used for all patients. In the meantime, this
approach was applied to certain data, e.g. by Mortimer et al. [32], Cole et al. [33] and
Tager et al. [34]. In the setting of randomised clinical trials, non-compliance can also
be seen as a dynamic strategy which allows for stopping treatment at a certain timepoint. In recent years, there have been proposals to adequately estimate the treatment
effect for different endpoints by structural models, e.g. by Robins [24], Vansteelandt and
Goetghebeur [35] and Loeys et al. [36].
3.1.3.1. Model structure
The MSM is a structural model that determines the relation of the distribution of the
counterfactual outcomes and the hypothetical exposure. The model is chosen according
to the type of outcome. For a counterfactual survival outcome T k where k describes the
onset of exposure at time k, a marginal structural Cox proportional hazards model
λT k (t) = λ0 (t) exp(β · 1{k<t} )
applies where the hazard might further depend on baseline covariates.
For a time-constant binary outcome Yk , one uses
logit P (Yk = 1) = β0 + β1 · k
(3.9)
In contrast to a standard statistical model of associational form, e.g.
logit P (Y = 1|N ) = α0 + α1 · N
(3.10)
where the outcome variable is a conditional probability and N a random variable, the
structural model addresses the counterfactual variables Yk and the dependent variable
k is not random.
3.1.3.2. Inference from the observable data
As the MSM is a structural model, it does not include confounding variables. The
confounding mechanism is defined by the assumption of no unmeasured confounders
19
3. Theoretical background
which is shown in (3.2) for a time-independent outcome. It is subsequently applied to
rewrite the marginal probability of the counterfactual outcomes in terms of probabilities
of observable variables, as shown in (6.2) in chapter 6. Then, appropriate weights,
which correspond to the dependence of confounders and the occurrence of the exposure,
are read off this term. By reweighting the observations by these weights, a pseudopopulation is created where time-dependent confounding is eliminated. This means,
within this pseudo-population, the probability to receive a certain treatment regime is
constant for all observations, i.e. treatment is unconfounded.
3.1.3.3. Estimation of the MSM parameters
Estimation of the MSM parameters is done in two steps using the method of Inverse
Probability of Treatment Weighting (IPTW). First, an adequate statistical model is
chosen and fitted for the weights. Then, one uses the standard statistical model which
corresponds to the MSM, e.g. the standard logistic model in (3.10) for the MSM in (3.9),
and estimates its parameters using the pseudo-population obtained by reweighting. As
in the pseudo-population treatment is unconfounded, they are consistent for the parameters of the MSM. Thus, the MSM parameters are determined by fitting the weighted
regression model chosen according to the considered outcome. Stabilised weights can be
used to obtain more efficient estimates.
The consistency of the IPTW estimator relies on the positivity assumption
P (∆m−1 = δ m−1 , X m = xm ) > 0 ⇒ P (∆m = δm |∆m−1 = δ m−1 , X m = xm ) > 0 (3.11)
for all possible δ m and xm , which is also called the assumption of experimental treatment
assignment [32,37]. It claims that at every level of the confounder history measured just
before cycle m, there is a positive probability of receiving the next cycle m and stopping
after cycle m − 1, respectively.
The MSM is very flexible, as both the structural model and the model for the weights
can be any statistical model. For most of the regression models, standard statistical
software packages provide a weighted fit. In case of a survival outcome, time-dependent
weights are used [38]. As most standard Cox model software programs do not allow for
subject-specific time-varying weights, Hernán et al. [38] recommends to fit a weighted
pooled logistic regression.
20
3.1. Theory concerning counterfactual framework
3.1.4. Contrasting SNFTM and MSM
The SNFTM is restricted to a time-to-event outcome and describes a deterministic
relation between the counterfactual outcomes refering to different hypothetical exposure
regimes k given by
T ∞ = k + (T k − k) exp(γ)
assuming an acceleration effect. This equation allows to calculate T ∞ from the observations T and I for exposed patients. For unexposed patients, T ∞ equals the observed
outcome T . The MSM applies to almost any type of outcome and describes the relation
of a characteristic of the marginal distribution of counterfactuals. In case of a survival
outcome, the marginal structural Cox model applies which assumes a multiplicative
effect of the exposure on the hazard of the counterfactual outcome. With respect to
the interpretation of the causal parameters, it is easier to transfer an epidemiological
hypothesis to the parameters of a SNFTM than to those of marginal structural Cox
models.
In both models, estimation from observational data is done by utilizing the assumption
of no unmeasured confounders, given by (3.3), which claims conditional independence
of counterfactual outcome and observed exposure. But, they differently transfer it to
deduce the estimation method. The SNFTM focusses on a model for the observed
exposure where the counterfactual T ∞ is included as dependent variable. To obtain the
SNFTM parameter, this model is fitted by the original data such that T ∞ contributes
no effect. The MSM uses a model for the observed exposure conditional on exposure and
covariate history without inclusion of the outcome to generate weights. These weights
are used to transform the original data into a pseudo-population by reweighting where
treatment application is unconfounded. The parameters of the marginal structural model
are then estimated by fitting the corresponding associational model using this pseudopopulation. The way of utilizing the assumption of no unmeasured confounders to
establish the estimation method induces that MSMs cannot be modified to address the
effect of a dynamic treatment regime. With regards to the SNFTM and g-estimation,
the extension is complicated but feasible by introducing interaction terms between timedependent treatment and covariates in the acceleration model.
In contrast to the MSM approach, the SNFTM is not restricted by the positivity assumption given in (3.11). A big advantage of MSMs, however, is that in contrast to the
21
3. Theoretical background
SNFTM, estimation can typically be done by standard statistical software packages.
3.2. Theory concerning observable framework
3.2.1. Multistate models
Multistate models [39] are used to describe longitudinal failure time data by a process
which at any time occupies one of a few possible states. A transition between the
states is called an event. Figure 3.2 shows a simple multistate model which describes a
patient’s stay on ICU which possibly involves the occurrence of a nosocomial infection.
At admission to ICU, the patient starts uninfected in state 0. The arrows indicate that
from state 0 he can either move to state 1 or 2, depending on whether he is infected or
discharged without being infected. State 1 is a transient state which the patient passes
and then moves on. State 2 is called an absorbing state, as the patient finally remains
in that state.
Figure 3.2.: Multistate model: description of ICU stay with possible nosocomial infection; denotation of states: 0 = uninfected, 1 = infection, 2 = discharge
Note, that changes in discrete time-dependent covariates can also be modelled as events.
Here, different states according to all possible values are defined.
Multistate models are characterised by hazard rates
Pij (t, t + ∆t)
,
∆t→0
∆t
αij (t; Ft− ) = lim
i 6= j
also called transition intensities, where
Pij (s, t) = P (state j at time t|state i at time s, Fs− )
22
(3.12)
3.2. Theory concerning observable framework
is the probability of being in state j at time t conditional on having been in state i at
time s and on the covariate history Fs− just before s.
3.2.2. Inverse Probability of Censoring Weighting
The method of Inverse Probability of Censoring Weighting (IPCW), see e.g. Robins et
al. [40, 41, 42], is used to include censored observations in the statistical analysis. The
idea is that censored observations are replaced by observations with similar covariate
history up to the censoring time. For this purpose, uncensored patients are weighted by
the probability of not being censored where the weights are deduced from assumptions
on the censoring mechanism.
A basic approach which uses IPCW is the Kaplan-Meier estimator [43] for the survival
curve. Here, one assumes that censoring is independent of any covariates and the outcome. Thus, if there are n observations at time t of which one is censored, the remaining
uncensored observations are reweighted equally by n/(n − 1) from t.
If censoring depends on covariates, the censoring mechanism is characterised by identifying assumptions which explain conditional independencies between covariates, missingness and the outcome. Then, one first sets up a model which explains the relations
between covariates and the occurrence of missing outcome values. The parameters of this
model are estimated according to the identifying assumptions. Then, one calculates the
weights by this model and uses the reweighted population for estimating the parameter
of interest. If censoring only depends on baseline covariates or the outcome of interest is
time-independent, the reweighted population reduces to the uncensored observations. If
censoring further depends on time-dependent covariates and one regards a time-to-event
outcome, the censored observations are included in the reweighted population until their
censoring time and the weights vary over time.
23
4. Applying the SNFTM to assess the
effect of a nosocomial infection on
the length of hospital stay
In this chapter, we address the effect of a nosocomial infection on the length of hospital
stay and regard our example on ICU data illustrated in section 1.2.1. Quantifying
the effect is complicated because infection status changes over time. Common, but
inadequate adhoc approaches tend to overestimation. A suitable method is based on a
time-to-event approach and accounts for patient characteristics.
We apply the SNFTM as illustrated in section 3.1.2 to quantify the effect of a nosocomial
infection on the length of hospital stay by the SNFTM parameter. This was already done
by Schulgen et al. [44,45]. There, it was part of a comparison of different approaches and
the intention was to demonstrate its applicability to real data. One innovative advantage
of Robins’ method is to include time-dependent confounders in an appropriate way. This
property, however, was not utilised by Schulgen et al. [44, 45], as no covariates were
included in the analysis. We now use this model in its full capacity and reanalyse the
data.
Furthermore, we address the effect of the nosocomial infection with respect to the change
in length of hospital stay where the extra stay is measured in days. This is considered
to be a relevant quantity in the field of infection control. It is e.g. used to increase the
efficacy of resource planning and to assess additional expenses due to the infection. We
use the definition given by Schulgen et al. [44, 45]. Here, the counterfactual framework
facilitates the definition of the effect and its interpretation. The extra stay is given
by a plug-in estimator which uses the SNFTM parameter. In accordance with the
literature [9, 46], we focus on the end of stay, i.e. equally consider death and discharge.
Our data example consists of 1656 admissions with only 10% deaths.
25
4. SNFTM: Effect of nosocomial infection on length of hospital stay
4.1. Definition and estimation of the extra stay
Estimation of the extra stay from observational data is not straightforward. An adhoc
approach to quantify the change in length of stay is to define two groups by retrospectively dividing the patients according to whether they have acquired a nosocomial
infection or not. The change in length of stay is estimated as the difference in the mean
length of stay. As patients who stay longer on ICU have a higher risk of entering the
group of infected patients, this comparison tends to overestimate the effect.
To assess the extra stay due to the nosocomial infection, we use a quantity proposed by
Schulgen et al. [44, 45]. They compare different definitions of the extra stay in different
frameworks. We focus on the term which determines the extra stay by comprising both
observable and counterfactual variables:
E(T − T ∞ |I < T )
(4.1)
The big advantage of this term using the counterfactual framework is that it clearly states
the type of comparison. It represents the medically optimal comparison to elimination
of the infection and describes the mean change in length of stay of an infected patient
from the considered population. It facilitates to deduce the overall number of extra
days, which could have been saved by complete elimination of nosocomial pneumonia,
by multiplying this quantity by the number of infected patients.
Estimation of the quantity in (4.1) is done as follows. We use the whole population
to estimate the SNFTM parameter. Then, this estimate is used to calculate T ∞ for
the infected population from T and I by (3.4). To estimate the expectation, we use
integrals of the Kaplan-Meier curve to include censored observations. A confidence
interval is obtained by drawing bootstrap samples from the whole analysis set, where
each step includes the estimation of the SNFTM parameter.
4.2. Application: SIR3 study
4.2.1. Information on the ICU data
The data of the SIR3 study were collected over a period of 18 months to examine the
effect of nosocomial infections, covering all patients admitted to five intensive care units
(see table 4.2.1 for types of ICU and number of admissions per ICU) who stayed more
26
4.2. Application: SIR3 study
than 48 hours. A detailed description of the data is given in Grundmann et al. [6] and
Beyersmann et al. [7].
Type of ICU
Frequency
%
Neurosurgical
137
8.3
Surgical
398
24.4
Interdisciplinary, Unit I
304
18.4
Interdisciplinary, Unit II
284
17.1
Medical
533
32.2
1656
100.0
All
Table 4.1.: Types of ICU centers and number
of admissions by ICU
We focus on nosocomial pneumonia which is one of the most frequent and severe nosocomial infections. It is considered to be hospital-acquired, if it occurred more than 48
hours after admission. We regard past infection exposure which means that the infection status remains 1 after infection, even if the infection was cured. This is typically
investigated in hospital epidemiology [9]. We exclude 220 patients who already had
pneumonia on admission which leaves 1656 admissions with ICU stay longer than 48
hours. Information on the occurrence of nosocomial pneumonia and the terminal event
are given in table 4.2.1.
Discharged
Died
Censored
All
118
32
7
157
No nosocomial pneumonia
1354
134
11
1499
All
1472
166
18
1656
Nosocomial pneumonia
Table 4.2.: Information on occurrence of nosocomial pneumonia and
terminal event
Possible prognostic factors for the risk of infection and the use of medical devices are
listed in table 4.2.1 which also contains information on the values of the baseline characteristics in our data. They comprise the baseline covariates age, sex and information
on patient’s health status at admission by the SAPS II Score [47], hospital stay before
admission to ICU and further admission status. Additionally, information on the use
27
4. SNFTM: Effect of nosocomial infection on length of hospital stay
of artificial ventilation, a chest drainage, a nasogastric tube and a urinary catheter,
respectively, is accounted for. For all devices there is daily information on being on
or off. They might act as time-dependent confounders with respect to the influence of
nosocomial pneumonia on the length of hospital stay.
Descriptive
Multivariate
Statistics
Analysis
Frequency
Prognostic factor
Hazard
Ratio 95% CI p-value*
absolute
%
Agea (years)
57.2
18.6
1.00
Sex (female)
685
41.4
not included
SAPS II scorea
33.6
18.2
0.99
[0.98;1.00]
0.09
Intubation on admission to ICU
714
43.1
0.96
[0.60;1.53]
0.86
1148
69.3
0.80
[0.55;1.15]
0.23
Surgical patient
412
24.9
not included
Elective surgery before ICU admission
763
46.1
1.65
[1.11;2.43]
0.01
Emergency surgery before ICU admission
428
25.9
1.43
[0.97;2.10]
0.07
Neurological underlying disease
345
20.8
not included
Metabolic or renal underlying disease
137
8.3
0.74
[0.34;1.62]
854
51.8
4.74
[2.76;8.15] ≤ 0.01
Baseline characteristic
Hospital stay before ICU admission
[0.99;1.01]
0.56
0.46
Time-dependent status variables
Use of artificial ventilationb
Use of chest
drainageb
321
19.4
1.10
[0.68;1.79]
tubeb
1063
64.2
4.24
[1.94;9.29] ≤ 0.01
Use of urinary catheterb
1400
84.5
1.25
[0.61;2.55]
Use of nasogastric
*
Wald test (two-sided)
a
Descriptive Statistics: Mean and standard deviation
b
Descriptive Statistics: With regard to use at least once
0.69
0.54
Table 4.3.: Descriptive statistics of prognostic factors and results of multivariate Cox
regression for hazard of infection
4.2.2. Estimation of the SNFTM parameter
To estimate the SNFTM parameter, we follow the description of the estimation method
given in section 3.1.2. To set up the appropriate Cox model for the infection hazard,
28
4.2. Application: SIR3 study
we first test the influence of covariates by univariate analyses. We then choose those
factors for the multivariate Cox model which showed an unadjusted p-value of p ≤ 0.157
in accordance with the Akaike criterion [48]. We further stratify this Cox model by ICU
center to account for distinct patient collectives. Parameter estimates are given in table
4.2.1. Applying g-estimation, we obtain an acceleration parameter exp(γ) of 0.763 with
95% confidence interval of [0.633; 0.877]. This indicates that infection prolongs ICU stay.
4.2.3. Estimation of the extra stay
Using the quantity given in (4.1), the estimate of the extra stay for the population addressed in the SIR3 study results in 5.71 extra days (95% CI [2.70; 8.56]). The overall
number of extra days, which could have been saved by complete elimination of nosocomial pneumonia, results in 157 · 5.71 = 896.5 days (95% CI [423.9; 1343.9]).
4.2.4. Artificial ventilation as time-dependent confounder
We now provide arguments that artificial ventilation (AV) acts as time-dependent confounder as done by Keiding et al. [22] regarding a data example on Graft versus Host
Disease. First, we assess the impact of AV on the occurrence of infection and on the
outcome. Therefore, we address the results in table 4.2.1 which show a significant effect
of AV on the infection hazard. Furthermore, a Cox model for the time until end of stay
including all covariates listed in table 4.2.1 and the status variable INF(t), defined in
section 1.3, showed that AV(t) has a prolongation effect on length of stay (hazard ratio
0.338, 95% CI [0.29; 0.39]). This indicates that AV must be treated as confounder.
Secondly, we investigate, if the need for AV is associated with prior infection, i.e. whether
AV is a confounder which is time-dependent. Therefore, we fit a univariate Cox regression model for the hazard of first use of AV stratified by ICU center where the infection
status enters as time-dependent variable. 877 patients where AV was not already used
on admission entered in this analysis and contributed 75 events. The estimate of the
hazard ratio for infection status was 10.9 (95% CI [2.46; 48.8]). This shows, that AV is
a time-dependent confounder.
29
5. Multistate model conform with
assumptions of SNFTM
In this chapter, we propose a multistate model which describes the situation with a
time-varying exposure, a time-dependent confounder and a survival outcome. It is defined such that it is conform with the assumptions of the SNFTM and the acceleration
parameter of the SNFTM enters directly. It only includes observable data which arises
in the appropriate chronological order. The survival time becomes manifest when the
patient enters the final absorbing state. The interaction of covariates and exposure
is modelled by direct effects relating the respective transition rates. The appropriate
embedding in the counterfactual framework is achieved by a partially delayed impact
of covariates on the terminal event, if they changed after exposition. To illustrate our
modelling assumptions, we characterise one transition by a mixture of transition probabilities from hidden substates. These hidden substates subclassify one of the observable
states. The probabilities of transitions between observable states do not depend on the
hidden substates. In contrast to counterfactuals, the substates are hidden but do not
refer to coexistent hypothetical variables which are coexistent in a hypothetical world
but never observable simultaneously for one individual.
For illustration, we use the ICU example, introduced in section 1.2.1, with artificial ventilation (AV) as time-dependent confounder and the infection as time-varying exposure.
We assume a prolongation effect, i.e. exp(γ) < 1. To focus on the relevant modelling
aspects, we regard AV as status variable which remains one from the first use of AV. AV
can then be interpreted as an indicator for health status. Subsequently, we explain how
to modify the model to include the event ”switch off AV”. Furthermore, the concept
can easily be adopted to exp(γ) > 1 by interchanging uninfected and infected patients.
31
5. Multistate model conform with assumptions of SNFTM
5.1. Idea to model the action of a time-dependent
confounder
The time-dependent confounder AV is affected by the infection and influences the time
to discharge. Our aim is to model the impact of AV on the discharge hazard such
that the causal parameter exp(γ) of the SNFTM, which covers the direct and indirect
effects of the infection on discharge, enters directly. Therefore, we model hidden states
according to the reason of AV and distinguish between AV due to INF and AV not due
to INF. If the infection was the reason for AV, we model no additional effect of AV on
the discharge hazard.
The hidden states are a means to illustrate the modelling assumptions for the determination of the discharge probability for infected patients. It is defined as a mixture of
discharge probabilities from the hidden states. It does not depend on the pathway along
the hidden states but only on the observable information, if the patient was infected
before AV was switched on or afterwards.
5.2. Definition of the multistate model
A general definition of a multistate model is given in section 3.2.1 denoting the transition
hazards by αij (t; Ft− ) in (3.12). Our multistate model with states defined by infection
status (INF), AV and by discharge status (including death) is shown in figure 5.1. We
assume that every patient is uninfected and not ventilated on admission, i.e. starts in
state 1. Then, he moves along the transient states 1 to 4 according to the indicated
arrows until he is discharged, i.e. finally reaches the absorbing state 5.
Thus, if the patient is discharged after infection without being ventilated, he moves along
the path from 1 to 3 and then to 5. If he is ventillated after admission and subsequently
acquires the infection before discharge, he follows the path 1 → 2 → 4 → 5. Comparing
the transition hazards for switching from 2 to 4 and from 1 to 3 indicates whether AV
increases the risk of infection.
In the following sections, we characterise the impact of AV and INF by relating the respective transition hazards. Thereby, we allow arbitrary transition hazards for switching
from the uninfected and unventilated state 1, i.e. for α12 (t), α13 (t) and α15 (t). Here and
32
5.2. Definition of the multistate model
Figure 5.1.: Multistate model: description of states and possible transitions
further on, we either suppress the dependence of the transition hazards on the covariate
history Ft− in the notation or single out the essential ingredient. To model effects on
transitions to transient states, we assume proportional hazards, as typically done in multistate models. We equally proceed with the discharge hazard for uninfected patients.
However, with respect to discharge hazards for infected patients, we need to incorporate
the modelling assumptions of the SNFTM.
A formal check that our multistate model meets the assumption of no unmeasured
confounders is given in appendix B.
5.2.1. Effect of INF on AV and of AV on INF and discharge
We define the effect of INF on AV and of AV on INF and discharge as follows assuming
proportional hazards:
α34 (t) = α12 (t) · cINFAV
(5.1)
α24 (t) = α13 (t) · cAVINF
α25 (t) = α15 (t) · cAVDIS
(5.2)
33
5. Multistate model conform with assumptions of SNFTM
with constants cINFAV ≥ 1, cAVINF ≥ 1 and cAVDIS ≤ 1. Their range is chosen such that the
infection increases the need for AV and AV increases the risk of INF and decreases the
risk of discharge.
5.2.2. Discharge hazard for infected, unventilated patients
In order that the multistate model complies with the SNFTM, we model the influence of
INF on discharge for patients in state 3 by an accelerated failure time model [28, section
5.2.]:
α35 (t; k) = α15 (tγ,k ) · exp(γ)
with k the time of infection and the backtransformed time
tγ,k = k + (t − k) · exp(γ)
(5.3)
Recall that according to (3.4), T ∞ = Tγ,I .
5.2.3. Discharge hazard for infected, ventilated patients
When defining the discharge hazard for patients in state 4, we have to face two challenges.
First, we must realise that exp(γ) covers the overall impact of INF on the time to
discharge and, second, we must introduce AV as time-dependent confounder. In order
that exp(γ) represents the causal parameter of the SNFTM, the influence of AV on the
discharge hazard cannot be modelled straightforwardly for patients in state 4 which were
infected before first use of AV, i.e. came from state 3. This is due to the fact that the
effects are modelled with respect to the transition hazards which can be illustrated for
cINFAV = 1 as follows. The effect of INF on AV describes the risk ratio of AV of presently
infected and uninfected patients. It corresponds to the risk within the next infinitesimal
time span, but not to the overall effect until discharge. As, in the regarded situation,
α34 (t) = α12 (t), we say that the infection does not increase the risk of AV. This addresses
the direct effect of infection. As, for the infected patient in state 3, discharge and first
use of AV are two competing events, the infection also has an indirect effect on AV, if
it reduces the discharge hazard. Then, in comparison to a world without infection, the
patient stays longer at risk of AV which increases his risk of being ventilated during his
stay.
34
5.2. Definition of the multistate model
5.2.3.1. Modelling aspects of SNFTM and multistate model
Before explaining our modelling strategy, we first reflect the different modelling aspects
of the SNFTM and the multistate model. By the SNFTM, T and I are linked with T ∞
by (3.4) without taking account of further covariates. Within the multistate model, T
and I arise progressively depending on the covariate history. For uninfected patients,
T ∞ can be observed, as then T = T ∞ . For infected patients, T ∞ can be calculated from
T and I by (3.4) using exp(γ). We regard two different time scales: the time scale of
T ∞ and the time scale of T . For uninfected patients, they do not differ. For patients
infected from time k, they are equal until k and then linked by the transformation
tγ,k = k + (t − k) · exp(γ) as in (5.3) where t corresponds to the time scale of T and tγ,k
to the time scale of T ∞ .
Dependencies between covariates, I and T ∞ are governed by the assumption of no unmeasured confounders. It claims that for every time t, the distribution of T ∞ must be
the same for patients with the same covariate history until t− whether they get infected
or not at time t. We achieve that the multistate model complies with this assumption
by modelling the influence on the discharge hazard of changes in covariates after infection with respect to T ∞ such that on average their influence is comparable whether the
patient is infected or not.
We first address the case cINFAV = 1, where the infection has no impact on AV. Then, we
consider cINFAV > 1 and further introduce AV as time-dependent confounder following
section 5.1.
5.2.3.2. Modelling the influence of the infection on discharge for cINFAV = 1
In this section, we regard the case cINFAV = 1, i.e. the infection only influences the discharge hazards and AV does not act as time-dependent confounder. Here, the infection
does not influence the transition hazard from 3 to 4. But with respect to the time scale
of T ∞ , this happens earlier than in a world without infection. Therefore, we define a
possibly delayed influence of AV on the discharge hazard such that the effect of AV
switched on after infection does not start until a random time which depends on the
probability for switching on AV and on the acceleration factor exp(γ). The delayed
influence of AV compensates the fact that, in comparison to a world without infection,
the infected patient is on average ventilated more often and earlier with respect to T ∞ .
The idea is that, if the transition hazard from 3 to 4 were equal to the transformed
35
5. Multistate model conform with assumptions of SNFTM
transition hazard from 1 to 2, namely α12 (tγ,k ) · exp(γ), the impact of AV on T ∞ would
not differ between infected and uninfected patients. Thus, we rewrite the transition
hazard from 3 to 4 by adding zero as
α34 (t) = α12 (tγ,k ) · exp(γ) + (α12 (t) − α12 (tγ,k ) · exp(γ))
To illustrate our modelling strategy, we subclassify state 4 by the hidden substates A
and B with transition hazards
α3A (t; k) = α12 (tγ,k ) · exp(γ)
(5.4)
α3B (t; k) = α12 (t) − α12 (tγ,k ) · exp(γ)
(5.5)
where A and B can be interpreted as follows:
A: AV not due to INF, immediate impact of AV
B: AV not due to INF, no impact of AV
The corresponding discharge hazards are chosen such that the discharge hazard from A
is influenced by INF and AV whereas the discharge hazard from B is only affected by
INF:
αA5 (t; k) = α15 (tγ,k ) · exp(γ) · cAVDIS
αB5 (t; k) = α15 (tγ,k ) · exp(γ)
with cAVDIS already used in (5.2). Note, that αA5 (t; k) = α25 (tγ,k ) · exp(γ).
The patient in substate A can only switch to discharge, whereas from B, he can either
switch to A or 5, as shown in figure 5.2. Note that although AV = 1 for a patient
in B, the impact of AV on the discharge hazard does not start until he moved to A.
Thus, after transition from 3 to 4 at time s, with probability α12 (sγ,k ) · exp(γ)/α12 (s) the
impact of AV starts immediately. With probability (α12 (s)−α12 (sγ,k )·exp(γ))/α12 (s), it
only starts at a random later time. In order that in total it starts on average according
to the hazard α12 (tγ,k ) · exp(γ), we define the transition hazard from B to A by
αBA (t; k) = α12 (tγ,k ) · exp(γ)
To characterise the transition from 4 to 5, we determine the survival function as a
mixture of transition probabilities with respect to the possible pathways along the hidden
36
5.2. Definition of the multistate model
Figure 5.2.: State 4 subclassified by hidden substates and possible transitions between
them. Transition to state 5 (discharge) possible from every hidden substate.
A0 and B 0 only in case cINFAV > 1.
substates to 5. As well as hazards, the survival function can be used to define the
s
distribution of a survival time. We obtain for t ≥ s where 3 → 4 denotes the event that
the patient switched from 3 to 4 at time s.
s
S45 (t|I = k, 3 → 4) =
s
s
s
= P (3 → A|3 → 4) · P (in A until t|I = k, 3 → A) +
h
s
s
s
+P (3 → B|3 → 4) · P (in B until t|I = k, 3 → B) +
i
s
+P (B → A, in A until t|I = k, 3 → B) =
Z t
αA5 (u; k)du) +
= (α12 (sγ,k ) · exp(γ)/α12 (s)) · exp(−
s
Z t
+((α12 (s) − α12 (sγ,k ) · exp(γ))/α12 (s)) · exp(− (αBA (u; k) + αB5 (u; k))du) +
s
Z t
Z v
Z t
+
exp(−
(αBA (u; k) + αB5 (u; k))du) · αBA (v; k) · exp(−
αA5 (u; k)du)
s
s
v
If the patient switched from 2 to 4, the problem with AV having a different impact on
T and T ∞ does not arise and we define that he can only switch to A. Thus,
Z t
s
S45 (t|2 → 4) = exp(−
αA5 (u; k)du)
s
37
5. Multistate model conform with assumptions of SNFTM
We see that the transition from state 4 to 5 does not depend on the pathway along the
hidden substates but only on the observable information, if the patient switched to state
4 from state 2 or from state 3. As the transition hazards must be positive, this modelling
strategy delimits the possible transition hazards α12 (t) to those where for all t
α12 (t) > α12 (tγ,k ) · exp(γ)
This includes for example monotone increasing and constant α12 (t).
Figure 5.3.: Partially delayed impact of AV after infection shown with respect to T ∞
and T for comparison to impact of AV without infection
Figure 5.3 illustrates the impact of AV. Case 1) shows an uninfected patient which is
ventilated from time s and discharged at T ∞ . Case 2) shows the same patient, had he
been infected at time I. He was also ventilated from time s, but the influence of AV
had started immediately only with probability α12 (sγ,k ) · exp(γ)/α12 (s) and was delayed
otherwise. To illustrate the delayed influence, we show quantiles of the starting points.
Furthermore, we backtransformed the starting points to illustrate the influence of AV
with respect to T ∞ .
5.2.4. Increased risk of AV due to infection: cINFAV > 1
Now, we address the second challenge and introduce AV as time-dependent confounder,
i.e. regard cINFAV > 1, following the ideas of section 5.1. Here, the infection affects the
38
5.2. Definition of the multistate model
risk of AV, which means that there is a proportion of patients in state 4 which get
AV due to the infection. Furthermore, they remain under risk of getting AV for other
reasons. We consider this by subclassifying state 4 further into hidden substates and
obtain with the substates A and B from above:
A: AV not due to INF, immediate impact of AV
B: AV not due to INF, no impact of AV
B 0 : AV due to INF
A0 : AV due to INF and due to other reasons
The division into hidden substates enables to define different discharge hazards concerning the reason for AV. In order that the causal parameter exp(γ) represents all direct
and indirect effects of the infection, AV due to INF does not have an impact on the
discharge hazard. We denote the additional substates by A0 and B 0 to point out by
similarity that being in A or A0 means that AV has an impact on discharge, whereas
being in B or B 0 the discharge hazard is not affected by AV. We set
αB0 5 (t; k) = α15 (tγ,k ) · exp(γ)
αA0 5 (t; k) = α15 (tγ,k ) · exp(γ) · cAVDIS
The substates are reached as follows depending on if the patient switched from state 2
or 3 to 4. The patient who switches from 2 to 4 reaches state A, as AV was switched on
before infection. If the patient switches from state 3 to 4, he either reaches A, B or B 0
where we define α3A (t; k) and α3B (t; k) as in (5.4) and (5.5) and set
α3B0 (t) = α12 (t) · (cINFAV − 1)
(5.6)
as a consequence from (5.1) as α3A (t; k) + α3B (t; k) = α12 (t). State A0 cannot be reached
directly from 2 or 3, as otherwise, two events happened at the same instant of time.
Possible transitions between the hidden substates are shown in figure 5.2. The respective
transition hazards are linked such that the risk of AV due to INF and due to other reasons
remains comparable between already ventilated and unventilated patients. Together
with (5.4) and (5.6) we obtain
αB0 A0 (t; k) =
αAA0 (t)
=
α3A (t; k)
=
α12 (tγ,k ) · exp(γ)
α3B0 (t)
=
α12 (t) · (cINFAV − 1)
39
5. Multistate model conform with assumptions of SNFTM
Now, we are able to characterise the transition from state 4 to 5. Again, it does not
depend on the pathway along the hidden substates but only on the observable information, if the patient switched from state 2 to 4 or from state 3. Patients who change from
state 2 to 4 at s reach state A. They cannot pass B or B 0 and at most later switch to
A0 . As the discharge hazard from A and A0 is identical and there are no other possible
pathways, we obtain
s
Z
S45 (t|2 → 4) = exp(−
t
αA5 (u; k)du)
s
as in the case above with cINFAV = 1.
If a patient switches from 3 to 4, we obtain for t ≥ s:
s
S45 (t|I = k, 3 → 4) =
h
s
s
s
= P (3 → A|3 → 4) · P (in A until t|I = k, 3 → A) +
i
s
+P (A → A0 , in A0 until t|I = k, 3 → A) +
h
s
s
s
+P (3 → B|3 → 4) · P (in B until t|I = k, 3 → B) +
s
+P (B → A, in A until t|I = k, 3 → B) +
i
s
+P (B → A, A → A0 , in A0 until t|I = k, 3 → B) +
h
s
s
s
0
+P (3 → B |3 → 4) · P (in B 0 until t|I = k, 3 → B 0 ) +
i
s
+P (B 0 → A0 , in A0 until t|I = k, 3 → B 0 )
As the hidden substates B and B 0 only differ for interpretational but not for mathematical
aspects, this reduces to
s
S45 (t|I = k, 3 → 4) =
h
s
s
s
= P (3 → A|3 → 4) · P (in A until t|I = k, 3 → A) +
i
s
+P (A → A0 , in A0 until t|I = k, 3 → A) +
h
s
s
s
s
s
+ P (3 → B|3 → 4) + P (3 → B 0 |3 → 4) · P (in B 0 until t|I = k, 3 → B 0 ) +
i
+P (B → A , in A until t|I = k, 3 → B ) =
0
40
0
0
s
0
5.3. Simulation study
h
Z
= α12 (sγ−,k ) · exp(γ)/(α12 (s) · cINFAV ) · exp(−
t
i
αA5 du) +
s
+(α12 (s) · cINFAV − α12 (sγ−,k ) · exp(γ))/(α12 (s) · cINFAV ) ·
h
Z t
α12 (uγ−,k ) · exp(γ) + α35 (u) du +
· exp −
s
Z t
Z v
+
exp −
α12 (uγ−,k ) · exp(γ) + α35 (u) du · α12 (vγ−,k ) exp(γ) ·
s
Zs t
i
· exp(−
αA5 du)dv
v
s
We evaluate S45 (t|I = k, 3 → 4) for constant transition hazards α12 , α13 and α15 . Then,
all transition hazards between observable states are constant except the transition hazard
from 4 to 5. We obtain
s
S45 (t|I = k, 3 → 4) =
h
i
= (exp(γ)/cINFAV ) · exp(−(t − s) · αA5 ) +
h
+(cINFAV − exp(γ))/cINFAV · exp(−(t − s) · (α12 · exp(γ) + αB5 )) +
+ exp(s · (α12 · exp(γ) + αB5 )) · α12 exp(γ) · exp(−t · αA5 ) ·
·(1/(α12 · exp(γ) + αB5 − αA5 )) · exp(−s · (α12 · exp(γ) + αB5 − αA5 )) −
i
− exp(−t · (α12 · exp(γ) + αB5 − αA5 )
Note, that we might have introduced a further substate C to model AV due to INF
and due to other reasons with delayed impact of AV with possible transitions B → C,
B 0 → C and C → A0 . But as this would not affect the transition probability from state
4 to state 5 and as transition hazards between substates do not need to be linked due
to interpretational reasons, e.g. αB0 A0 (t; k) does not need to be equal to α12 (t), we did
not expand our model by a further substate.
5.3. Simulation study
Now, we use our simple joint model as simulation model using constant transition hazards α12 , α13 and α15 . With the generated data, we compare the length of stay of infected
and uninfected patients and explore the behaviour of a typically used Cox model in comparison to the SNFTM.
41
5. Multistate model conform with assumptions of SNFTM
5.3.1. Data generation
We choose a sample size of 1000 patients and perform 100 simulation runs per setting.
According to the multistate model proposed above, we simulate individual patient data
which describe the observable states and transition times until the absorbing state 5
(discharge) is reached. We assume that every patient is uninfected and not ventilated
on admission and choose state 1 as the overall initial state. Then, we subsequently
sample the next transition time. Therefore, we consider all possible transitions as competing risks and use the simulation algorithm presented by Beyersmann et al. [49]. More
precisely, we generate an exponentially random time distributed according to the sum of
the rates corresponding to all possible transitions. In case of time-dependent rates, we
use the sum over the cumulative hazards. To determine the next state, we toss a dice
where each possible state holds the probability of the transition rate at the generated
time divided by the sum of all transition rates at this time.
5.3.2. Parameter values
The chosen parameter values are motivated by our ICU data analysed in chapter 4. The
estimate for the acceleration parameter exp(γ) resulted to 0.76. The transition hazards
are estimated as constants without adjustment for further confounders which results to
α12 = 0.04, α13 = 0.008, α15 = 0.12 and cAVINF = α24 /α13 = 1.88. Furthermore, we
obtain cAVDIS = 0.34 by fitting a Cox model for discharge with AV as status variable
adjusted for further covariates and infection status. We use different values for cINFAV (1,
2.5, 5) which correspond to a different extent of time-dependent confounding.
5.3.3. Illustration of characteristics of infected population
Infected patients tend to stay longer on ICU, no matter if the infection has an impact
on the time to discharge. This comes from the fact that a patient who stays longer, is
longer at risk of infection and thus, the probability that he is infected during his overall
stay is higher. This can be seen as infection induces selection of patients with longer
stay. We illustrate this by comparing the distribution of T ∞ of the infected population
and the uninfected population. Recall, that T ∞ represents the length of stay, had the
patient not been infected, i.e. a quantity which does not vary according to the influence
of infection. Figure 5.4 shows the respective Kaplan-Meier curves for a sample of 2000
42
5.3. Simulation study
0.0
0.2
0.4
^
S(t)
0.6
0.8
1.0
patients with the parameters given in 5.3.2 and cINFAV = 2.5.
0
10
20
30
40
50
time t
Figure 5.4.: Simulation results: Kaplan-Meier curves for T ∞ of subpopulation of uninfected and infected patients, dashed line for infected patients
5.3.4. Estimation of effects by Cox model
We use the simulated data to explore the behaviour of the Cox model for the discharge
hazard without
λDIS (t|INF(t), AV(t)) = λ0 (t) · exp(β1 · INF(t) + β2 · AV(t))
and with interaction term:
λDIS (t|INF(t), AV(t)) = λ0 (t) · exp(β1 · INF(t) + β2 · AV(t) + β3 · INF(t) × AV(t))
43
5. Multistate model conform with assumptions of SNFTM
where
λDIS (t|Ft− )dt = P (T ∈ [t; t + dt)|T ≥ t, Ft− )
Here, exp(β1 ) is taken as estimate for exp(γ).
As the transition rates were chosen to be constant, the acceleration of the time to
discharge due to the infection reduces to a multiplication of the discharge hazard by
exp(γ). Thus, the proportional hazards assumption of the Cox model concerning the
impact of the infection is satisfied for unventilated patients. However, the Cox model is
not appropriate to infer on the causal parameter exp(γ) for two reasons. First, it models
the direct effects of infection and AV on discharge and second, as conventional statistical
models, the Cox model cannot cope with time-dependent confounders.
In terms of our multistate model, this means the following. For uninfected patients,
the Cox model correctly models the influence of AV and we would put β2 = cAVDIS .
Also, for ventilated patients which were infected only afterwards, the influence of AV
is undelayed and correctly modelled. Then, β2 should equal cAVDIS and, moreover, β1
should coincide with exp(γ). But, if AV is first used after infection, the Cox model does
not account for the delayed influence of AV. In addition, if cINFAV > 1 and thus, AV
acts as time-dependent confounder, it does not model the impact of AV as a mixture of
effects depending on whether AV is due to INF or not. Consequently, the Cox model
is misspecified for infected patients where AV first started after infection. This leads to
estimates of β1 and β2 which deviate from the causal parameter exp(γ) and cAVDIS , the
impact of AV, respectively.
5.3.5. Estimation of effects by SNFTM
We further address the SNFTM estimates for exp(γ) which are consistent and γ̂ is
asymptotically normal as theoretically known [21]. The SNFTM copes with the timedependent confounder AV, which can be explained as follows in terms of our multistate
model. The estimating procedure uses T ∞ and the covariate history until the time to
infection. Thus, it does not include data after the first transmission to either state 3 or
4. Therefore, the results are not affected by the modelling of the discharge probability
from state 4 as long as it is such that the assumption of no unmeasured confounders is
satisfied.
44
5.3. Simulation study
5.3.6. Results
The results of our simulation study are summarised in table 5.1. The Cox model without
interaction term only performs well, if cINFAV = 1, i.e. if AV is never due to INF. Then,
the estimates for exp(γ) are only slightly influenced by the delayed impact of AV. If
cINFAV > 1, the discharge hazard for patients in state 4, where AV is due to INF, is in truth
only affected by exp(γ), but the Cox model ascribes the effect exp(β1 + β2 ). As cAVDIS
is smaller than 1, the Cox model on average estimates an effect exp(β1 ) which is closer
to 1 than the true exp(γ). The Cox model with interaction term, however, compensates
the falsely ascribed impact of AV by the additional parameter exp(β3 ) which is hence on
average estimated larger than 1. In our simulation study, the mean of the estimates of
exp(β3 ) is 1.086 ([0.686;1.579]), 1.065 ([0.659;1.916]) and 1.241 ([0.623;2.073]) for cINFAV
equal to 1, 2.5 and 5, respectively. Including the interaction term is helpful, as the
simple joint model and the Cox model have the same structure for a sufficiently long
period. But, the false modelling assumptions lead to a wide confidence interval.
Model
cINFAV = 1
cINFAV = 2.5
cINFAV = 5
without interaction
0.754, [0.609;0.988]
0.795, [0.595;0.975]
0.884, [0.719;1.099]
with interaction
0.733, [0.531;1.056]
0.757, [0.499;1.096]
0.728, [0.521;1.228]
0.753, [0.564;1.105]
0.740, [0.587;0.966]
0.745, [0.602;0.995]
Cox model
SNFTM
Table 5.1.: Simulation study with exp(γ) = 0.76 for different cINFAV : mean of estimates
for exp(γ) and 95% confidence interval
The main problem with using a standard model, which only associatively adjusts for
time-dependent confounders, is, that it shows an effect, even if the infection has no
impact on the length of hospital stay, i.e. if exp(γ) = 1. Therefore, we additionally
performed a simulation study with the above setting, cINFAV = 5, but exp(γ) = 1 and
400 runs. In the Cox analysis without interaction term, 18.25 % of the estimates for
exp(β1 ) differed significantly (p < 0.05) from 1, whereas the SNFTM only led to 6.5%
significant results.
45
5. Multistate model conform with assumptions of SNFTM
5.4. Comparison to simulation proposed by Robins
We now compare the simulation algorithm given by our multistate model to the one
proposed by Robins which is outlined in section 3.1.2.5. Here, the counterfactual time
T ∞ is modelled first. T ∞ is then influenced by INF and transformed to T , the survival
time. Already at the time the infection occurs, the survival time becomes manifest. The
interaction between AV and INF is modelled separately by a time-dependent process
which is stopped according to T . The impact of AV on T is achieved indirectly by
sampling AV dependent on T ∞ .
In contrast, by our multistate model, the changes in covariates and exposure and the
survival process are modelled simultaneously. The data arises as in a prospective trial,
which means the covariate history develops progressively and the survival time only
becomes manifest when the endpoint is reached.
5.5. Extension to model current AV status
We illustrate a possible extension of the multistate model to include AV as current
ventilation status such that the SNFTM still applies. Now, AV is 0 for unventilated
patients, AV jumps to 1 at the time of switching on AV and returns to 0, if AV is
switched off again. This means, the patient can return to the unventilated states from
2 to 1 and from 4 to 3, respectively. If the patient switches from 4 to 3, the change
in covariates happens after infection. We proceed analogously to the definition of the
transition from 3 to 4 and model a partially delayed inuence of switching off AV such that
on average its inuence is comparable whether the patient is infected or not. Therefore,
we subclassify state 3 into two hidden substates C and D which can be interpreted as
follows:
C: INF, no impact of AV
D: INF, still impact of AV
This means that both discharge hazards are influenced by INF and the discharge hazard
from D is still affected by AV:
αC5 (t; k) = α15 (tγ,k ) · exp(γ)
αD5 (t; k) = α15 (tγ,k ) · exp(γ) · cAVDIS
46
5.5. Extension to model current AV status
Moving from 1 to 3, the patient can only reach state C.
Figure 5.5.: Possible transitions between hidden substates subclassifying state 3 and
state 4
The possible transitions between the hidden substates are shown in figure 5.5. When
interpreting the transitions, we must distinguish between switching off AV due to INF
and switching off AV due to other reasons. Moving from A0 to A and from B 0 to C
corresponds to switching off AV due to INF. We first concentrate on the case without
time-dependent confounding where AV is never due to INF. Secondly, we expand the
model to the hidden substates A0 and B 0 .
In the following, we only characterise the multistate model by transition hazards from
both, observable and hidden substates. By determining the survival function analogously
47
5. Multistate model conform with assumptions of SNFTM
to section 5.2.3, it is possible to derive the transition probability from 3 to 4 and 4 to
3 as well as the discharge probability after infection such that they only depend on
pathways along observable states. But in general, this does not lead to a closed-form
expression. Then, the transition probabilities can only be approximatively evaluated
by the respective term which only addresses a finite number of transitions. This term
converges to the product integral.
5.5.1. The case without time-dependent confounding
We first address the case without time-dependent confounding, i.e. cINFAV = 1, where the
infection has no impact on the risk of AV. Thus, we neither assume an impact of the
infection on switching off AV and define
α43 (t) = α21 (t)
With respect to the transition from A to 3, we define analogously to the considerations
in section 5.2.3.2
αAC (t; k) = α21 (tγ,k ) · exp(γ)
αAD (t; k) = α21 (t) − α21 (tγ,k ) · exp(γ)
and
αDC (t; k) = α21 (tγ,k ) · exp(γ)
Note, that as the transition hazards must be positive, this modelling strategy delimits
the possible transition hazards α21 (t) to those where for all t
α21 (t) > α21 (tγ,k ) · exp(γ)
In order that the model does not get more complicated than necessary, we define
αBC (t) = α21 (t)
and disallow a transition from B to D.
Now, we regard the transition from C and D to 4. The transitions from C are identified
by the transitions from 3 considered in section 5.2.3.2 which means
αCA (t; k) = α12 (tγ,k ) · exp(γ)
αCB (t; k) = α12 (t) − α12 (tγ,k ) · exp(γ)
48
5.5. Extension to model current AV status
where
αBA (t; k) = α12 (tγ,k ) · exp(γ)
still holds. Furthermore, to achieve that α34 (t) = α12 (t), we define
αDA (t) = α12 (t)
5.5.2. The case with time-dependent confounding
If cINFAV > 1, switching off AV due to INF is additionally possible which is represented
by moving from A0 to A and from B 0 to C. We set
αA0 A (t) = αB0 C (t)
Switching off AV due to INF obviously does not happen for uninfected patients. Thus,
we do not need to relate αA0 A (t) and αB0 C (t), respectively, to another transition hazard
for interpretational reasons with respect to a world without infection.
The transition from A0 to B 0 means switching off AV necessary for other reasons. Thus,
we set
αA0 B0 (t; k) = αAC (t; k) = α21 (tγ,k ) · exp(γ)
If there are neither restrictions on the relation between the overall transition from 4 to 3
and the transition from 2 to 1, i.e. between α43 (t) and α21 (t), we can choose any function
for αB0 C (t). Claiming proportional hazards such that α43 (t)/α21 (t) is constant does not
seem to be appropriate. But, it might be appropriate to set αB0 C (t) = α21 (t) for medical
aspects.
5.5.3. Assessing the assumption of no unmeasured confounders
For the case where AV was considered as status variable which only indicates time from
first use of AV, we formally proved that the assumption of no unmeasured confounders
is fulfilled within our multistate model by comparing conditional distribution functions,
cf. appendix B. This gets very complicated in the case which addresses the current
status of AV, as many transitions are possible before reaching the absorbing state discharge. Therefore, to support that the assumption of no unmeasured confounders is
49
5. Multistate model conform with assumptions of SNFTM
also fulfilled in this case, we compare the conditional distribution functions by a simulak
tion study. Particularly, we compare the Kaplan-Meier estimates of P (T ∞ > x|2 → 4)
k
with that of P (T ∞ > x|in 2 at k) and proceed likewise for P (T ∞ > x|1 → 3) and
P (T ∞ > x|in 1 at k). We restrict the model to time-constant transition hazards α12 ,
α13 and α15 . Then, the model is homogen markov for the time until infection and we
can restrict the comparison to k = 0. According to each conditional distribution function, we sample 5000 patients which start in the respective state at time 0. We use
a more extreme acceleration parameter than in section 5.3 of exp(γ) = 0.4. Further
parameter values are chosen such that more patients get ventilated and infected. We set
α12 = 0.07, α13 = 0.09, α15 = 0.12, α21 = 0.04, αA0 A = 0.04, cAVINF = 1.88, cAVDIS = 0.34
0.0
0.2
0.4
^
S(t)
0.6
0.8
1.0
and cINFAV = 2.5. The Kaplan-Meier estimates are shown in figures 5.6 and 5.7.
0
10
20
30
40
50
time t
Figure 5.6.: Simulation results: Kaplan-Meier curves for comparison of start in 1 at time
0 with start in 3 at 0, dashed line for start in 3
50
0.0
0.2
0.4
^
S(t)
0.6
0.8
1.0
5.5. Extension to model current AV status
0
10
20
30
40
50
time t
Figure 5.7.: Simulation results: Kaplan-Meier curves for P (T ∞ > x|in 2 at 0) and
P (T ∞ > x|in A at 0), dashed line for infected patients
51
6. Illustrating Marginal Structural
Models as extension of a missing
data approach
In this chapter, we provide insight into MSMs by illustrating them as an extension of
the missing data approach called Inverse Probability of Censoring Weighting (IPCW).
In particular, we demonstrate the MSM procedure as concurrent use of IPCW per counterfactual outcome where the relation between the counterfactual outcomes is defined
by parametric assumptions through a structural model.
For illustration, we consider the breast cancer example, given in section 1.2.2. Here,
two preoperatively applied chemotherapy schemes which are given in repeated cycles
are compared within a randomised trial where early stopping might occur due to actual
measurements taken before each cycle.
Primarily, we consider only one treatment arm as if it would have been a prospective
observational study where the number of given cycles develops according to the patient’s
condition and the doctor’s decision. First, we focus on the outcome after the application
of a certain fixed number of cycles and apply the method of IPCW. Secondly, we compare two treatment regimes comprising a different fixed number of application cycles.
Thereby, we can infer on the dose effect. Thirdly, we use the MSM to evaluate these
differences simultaneously for all possible numbers of application cycles.
In addition, we reanalyse the therapy effect adjusting for time-dependent confounders
by using the first step with respect to the number of all planned cycles separately for
both treatment arms. We define the therapy effect by a statistical model of associational
form to point out the differences to the structural model.
53
6. Marginal Structural Models as extension of missing data approach
6.1. Time-dependent confounding within one
treatment arm
The chemotherapy scheme consists of the application of a certain planned number of
cycles. Clinical variables measured before each cycle might lead to stop chemotherapy
early. As early stopping does not happen by chance, the number of given cycles is
affected by baseline and time-dependent confounders.
In our setting, a time-dependent confounder is a prognostic factor for tumor status which
is characterised as a possible reason to stop treatment and furthermore as being affected
by previously given cycles. One of the possible time-dependent confounders in our data
set is increased leucocyte counts measured in WHO toxicity grades 1 to 4. Part of the
confounding situation is shown in figure 6.1. Application of treatment cycle m is given
by the status variable ∆m . If cycle m is given, ∆m equals 1, otherwise it is set to 0.
The covariates measured just before cycle m are denoted by Xm . Baseline covariates are
included in X1 . The outcome is determined by Y . For given m, the variables Xm are
known just before ∆m . We write (∆m ) and (Xm ) to address treatment application and
covariates in general without respect to a particular time point. To simplify the graph,
the figure only shows two cycles and does not contain the impact of baseline covariates.
We see, that the number of leucocytes influences the doctor’s decision to continue or
stop treatment, i.e. Xm−1 → ∆m−1 and Xm → ∆m . But also, by ∆m−1 → Xm , there
is a relation the other way round, as depending on whether cycle m − 1 is given, the
number of leucocytes is affected. As this pathway carries on to Y , the value of Xm
partially resembles a treatment effect. Therefore, usual adjustment for time-dependent
confounders is unsuitable.
6.2. First step: outcome after application of a fixed
number of cycles
In this section, we focus on the outcome after the application of a fixed number n of
cycles. To indicate this, we define a new outcome variable Yn which is equal to Y , if n
cycles are given and missing otherwise. This means, if the outcome is measured after
less or more than n cycles, it is considered as censored. Censoring is induced by the
54
6.2. First step: outcome after application of a fixed number of cycles
Xm−1
Xm XX
XXX
XXX
Q
3
Q
XXX
Q
Q
j
s
Q
XXX
s
Q
X
z
- ∆
- Y
∆m−1
m
-
*
Figure 6.1.: Time-dependent confounding within one treatment arm: relations between
time-dependent covariates (Xm ), application of treatment (∆m ) and outcome Y
relations between (Xm ) and (∆m ) shown in figure 6.1. We can interpret (∆m ) as the
censoring indicator which manifests in time. To account for the time-dependent aspects
and the chronological order of cause and effect, the censoring mechanism is usually characterised by sequentially defined conditional independence assumptions which require for
all possible numbers of cycles m:
Yn ⊥⊥ ∆m |X m , ∆m−1
(6.1)
where conditional independence is denoted by ⊥⊥ . Here, we write ∆m for (∆1 , ..., ∆m )
where ∆0 is defined to be 0 to simplify notation. X m = (X1 , ..., Xm ) is the covariate
history prior to cycle m. Baseline covariates are included in X1 .
The assumptions in (6.1) can be interpreted in two ways. Firstly, by comparison of
two patients just before cycle m < n who received chemotherapy so far and have the
same covariate history X m . According to (6.1), knowing that for one of them cycle m is
withheld, does not imply that complete remission, had n cycles been given, is more or less
likely than for the other patient where cycle m is given. This means, the fact that cycle
m is given or not does not improve the prediction on the outcome after n cycles based on
the covariate history X m . Secondly, (6.1) implicitely ensures that all information on the
disease status which influences the doctor’s decision to stop chemotherapy is included in
X m . For example, if the number of leucocytes, which has a prognostic effect on response,
is not included in X m , a patient who discontinues chemotherapy after cycle m − 1 due
to a high leucocyte count is more likely to have complete remission after n cycles than a
patient with a low leucocyte count. Then, knowing ∆m = 0 does improve the prediction
55
6. Marginal Structural Models as extension of missing data approach
on the outcome.
Now, we use (6.1) to transform the parameter of interest, E(Yn ) = P (Yn = 1), into a
term that only includes probabilities of uncensored outcomes. These probabilities can
then be estimated by weighting the observed data. The weights are read off from the
resulting term. The transformation is done by iterative multiplication by a factor which
is equal to 1. For the first cycle, (6.1) implies
P (Yn = 1|X1 = x1 ) · P (∆1 = 1|X1 = x1 ) = P (Yn = 1, ∆1 = 1|X1 = x1 )
which leads to the following:
P (Yn = 1) =
X
P (X1 = x1 ) · P (Yn = 1|X1 = x1 ) =
x1
=1
=
X
=
X
}|
{(6.1)
z
P (X1 = x1 ) · P (Yn = 1|X1 = x1 )· P (∆1 = 1|X1 = x1 )/P (∆1 = 1|X1 = x1 ) =
x1
P (X1 = x1 ) · P (Yn = 1, ∆1 = 1|X1 = x1 )/P (∆1 = 1|X1 = x1 ) =
x1
=
X
P (Yn = 1, ∆1 = 1, X1 = x1 )/P (∆1 = 1|X1 = x1 )
x1
Applying this procedure iteratively, we obtain, with 1n the vector of length n with all
elements equal to 1,
P (Yn = 1) =
X
P (Yn = 1, ∆n = 1n , ∆n+1 = 0, X n+1 = xn+1 )/P (n,xn+1 )
(6.2)
xn+1
where we sum over all possible covariate vectors xn+1 = (x1 , ..., xn+1 ) with
P (n,xn+1 ) =
n+1
Y
P (∆i = 1{i≤n} |∆i−1 = 1i−1 , X i = xi )
i=1
If n equals the number of all planned cycles, Xn+1 is defined to be 0 to simplify notation.
Thus, the expected outcome per treatment arm, E(Yn ), can be estimated by the mean
of Yn in the weighted uncensored subset. The weights can be deduced from (6.2) as
w = 1/P (n,xn+1 )
Assumption (6.1) makes sure that this estimate is unbiased.
56
6.3. Second step: dose effect by comparing two groups
Stabilised weights are used to obtain more efficient estimates [31]:
sw =
n+1
Y
,
P (∆i = 1{i≤n} |∆i−1 = 1i−1 )
P (n,xn+1 ) .
(6.3)
i=1
To estimate the stabilised weights, an appropriate statistical model is chosen for the
denominator of (6.3) according to the mode of treatment application. In our case,
skipping one chemotherapy cycle is not allowed. Thus, we regard the failure time variable
min{m : ∆m = 0}
and fit a discrete proportional hazards model [50] which we adjust for the time-dependent
and baseline covariates (Xm ). The estimate of the nominator is just the relative frequency of cycles applied.
6.3. Second step: dose effect by comparing two
groups
In the previous section, we estimated the mean outcome after a fixed number n of application cycles. Now, to decide whether the outcome was affected by the chemotherapy or
not, we compare two groups where n and n − 1 cycles were applied, respectively. Therefore, we apply the procedure repeatedly for different n. Then, we determine the dose
effect quantified by the odds ratio of the weighted mean outcomes. This comparison is
facilitated by the counterfactual framework due to its convenient definition of outcome
variables.
To distinguish the outcome after a differential number of cycles by notation, we again
regard the variable Yn which represents the outcome after the application of n cycles.
But now, we do not focus on a certain n, but equally consider all possible values n and
define a whole vector (Y1 , Y2 , ..., Yp ) of outcome variables. As one patient can only be
treated according to one chemotherapy scheme, there are more outcome variables than
can be observed. They are linked to the observed outcome Y as follows. If actually
n cycles were given, the observed outcome Y is equal to Yn . In other respects, we do
not know the outcome after n cycles and call Yn counterfactual. Analogously to (6.1),
we claim the following sequentially defined conditional independence assumptions for all
57
6. Marginal Structural Models as extension of missing data approach
m ≤ p:
(Y1 , Y2 , ..., Yp )⊥⊥ ∆m |X m , ∆m−1
(6.4)
It is called the assumption of no unmeasured confounders and used as identifying assumption to make estimation feasible. The consistency of the estimating procedure relies
on the positivity assumption given in (3.11).
The dose effect is quantified by the odds ratio of the outcome after application of n
versus n − 1 cycles:
Dose Effect =
P (Yn = 1)/(1 − P (Yn = 1))
P (Yn−1 = 1)/(1 − P (Yn−1 = 1))
(6.5)
To estimate the dose effect, we consecutively estimate P (Yn = 1) = E(Yn ) and P (Yn−1 =
1) = E(Yn−1 ) as done in section 6.2 by the mean of Yn and Yn−1 in the reweighted
subset of patients who received n and n − 1 cycles, respectively. This reweighting of
counterfactual variables is called Inverse Probability of Treatment Weighting (IPTW).
6.4. Third step: dose effect by comparing all groups
using a Marginal Structural Model
In order to assess the impact of the number of given cycles on the outcome simultaneously for all n, we now use a Marginal Structural Model which requires a parametric
assumption on the differences in outcome.
For our data situation with binary outcome, a simple model which links the marginal
distributions of the counterfactual variables Yn is given by the logistic model
logit P (Yn = 1) = β0 + β1 · n
(6.6)
It assumes that the odds ratios in (6.5) for different n are constant and equal to β1 .
This is a Marginal Structural Model as proposed by Robins [30, 31]. The parameter n is
not a random variable but indicates the counterfactual outcome. The estimation of the
parameters β0 and β1 is done by IPTW within two steps. First, stabilised weights as in
(6.3) are assigned to every observation according to the observed number of cycles n and
the observed covariate history X n+1 . Then, the model is fitted by weighted regression
using all observations. This typically can be done by standard statistical software.
58
6.5. Contrasting IPCW with the MSM approach
6.5. Contrasting IPCW with the MSM approach
In the previous sections, we demonstrated that the MSM procedure can be seen as
using IPCW repeatedly for all possible numbers of cycles and additionally establishing
a structural model which defines parametric assumptions on the relation between the
outcomes after a different number of cycles.
Both methods address a population parameter. This means, they estimate a marginal
parameter which depends on the characteristics of the overall population. E.g. for n
equal to the number of all planned cycles, the missing data approach allows inference
on an optimal study conduct where everybody received all cycles and no early stopping,
i.e. censoring, occurs. While the missing data approach focusses on a fixed number of
cycles, the MSM approach simultaneously predicts the marginal effect of all potential
outcomes which are coexistent and of equal interest. Here, we not only regard Yn as the
outcome variable of interest, but a whole vector (Y1 , Y2 , ..., Yp ) of outcome variables.
Within both approaches, the outcome variables allow the definition of the effects without
relation to the way on how patients are actually assigned to treatment regimes. The
assignment mechanism that controls, whether Yn is censored and which outcome variables are counterfactual, respectively, belongs to the data structure and is considered by
the assumptions (6.1) and (6.4), where (6.4) contains (6.1). Both assumptions are not
verifiable by the data but can only be made plausible by sensitivity analyses [51]. The
weights, deduced from these assumptions, are derived in the same way and their estimates use the information on covariates (Xm ) and on (∆m ) of all observations in both
analyses. However, with IPCW, the information on the outcome is only used for observations where the regarded number of cycles is given, whereas with respect to the dose
effect, all outcomes are considered and weighted differently according to the observed
number of given cycles.
We face the mentioned points in table 6.1
59
6. Marginal Structural Models as extension of missing data approach
IPCW
MSM
Measure
Population parameter
Population effect
Estimate
Outcome after n cycles:
Dose effect:
Focus on one therapy scheme
Comparison of outcomes after
with n cycles, no comparison
different number of cycles
Outcome
Yn
(Y1 , Y2 , ..., Yp )
Assumptions
Yn ⊥⊥ ∆m |X m , ∆m−1
(Y1 , Y2 , ..., Yp )⊥⊥ ∆m |Xm , ∆m−1
Structural model
Weights
Estimated by discrete proportional hazards model
using all observations, no information on outcome included
Information on outcome
Used for subpopulation
Used for whole population
who actually received n cycles
Table 6.1.: Contrasting IPCW with MSM
6.6. Therapy effect: comparison of both treatment
arms
This section aims at contrasting the structural model used by the MSM procedure with
a standard statistical model of associational form. Therefore, we not only regard one
treatment arm as in the previous sections, but address both treatment arms to assess the
therapy effect by comparing the outcome after the fully applied chemotherapy schemes.
We utilise that the treatment arms consist of randomised subgroups of the population
sample which are expected to be similar with respect to baseline characteristics and
apply the IPCW procedure outlined in section 6.2 separately for both treatment arms
choosing n equal to the number of all planned cycles p. To indicate the treatment arm,
we denote the expected outcomes by E(Yp |Z) = P (Yp = 1|Z) with the random variable
Z which equals 0 or 1. As the outcome is binary, the therapy effect is quantified by the
odds ratio:
Therapy Effect =
60
P (Yp = 1|Z = 1)/(1 − P (Yp = 1|Z = 1))
P (Yp = 1|Z = 0)/(1 − P (Yp = 1|Z = 0))
(6.7)
6.7. Application: Geparduo study
For comparison with the MSM, we also write down a statistical model for it:
logitP (Yp = 1|Z) = α0 + α1 · Z
(6.8)
The parameters α0 and α1 , where α1 equals the odds ratio, can be estimated by weighted
logistic regression fitted by the subset of observations who received all planned cycles
using the weights in (6.3).
By (6.8), the therapy effect is described by a conventional associational model where Z
is a random variable and the outcome variable is a conditional probability. In contrast,
the MSM given in (6.6) is a structural model which defines the relation between the
marginal probabilities of the counterfactual variables Yn . Here, the dependent variable
n is not random. Note, that in both cases, time-dependent confounding is accounted for
by reweighting, such that none of the models contains any covariates for adjustment.
6.7. Application: Geparduo study
6.7.1. Information on the breast cancer data
The Geparduo study [8] is a randomised controlled clinical trial run by the German
Breast Group to compare two chemotherapy schemes which are applied preoperatively
in breast cancer patients. There is a short and a long treatment arm which involve the
application of a chemotherapy scheme of four and eight cycles, respectively.
The primary endpoint is pathological complete remission (pCR) in the breast and axillary nodes. It was measured by the resected breast specimen and axillary lymph nodes
at subsequent surgery. Reasons for early stopping were partly foreseen in the study
design and consisted amongst others of toxicity and progress of disease diagnosed by
palpation carried out before each cycle. If chemotherapy was discontinued, immediate
surgery was performed. As we focus on the binary outcome complete remission, we do
not consider, that after early stopping, the endpoint was observed prior to the end of
study.
We investigate the full-efficacy population of the Geparduo study chosen by von Minckwitz et al. [8] for the primary endpoint analysis. This data set involves 855 patients of
which 441 were randomised to the long treatment arm consisting of eight cycles. We
61
6. Marginal Structural Models as extension of missing data approach
number of cycles received
1
2
Short arm (319 Obs)
2
7
Long arm (311 Obs)
2
3
3
4
5
6
7
8
14 296
-
-
-
-
23
11
12
245
2
13
Table 6.2.: Analysis set per treatment arm: number of patients by number of cycles
received
use a subset of 630 patients shown in table 6.2 where the outcome was documented and
covariate information used to model the weights is not missing.
As possible time-independent confounders between early stopping and the primary endpoint pCR at surgery, we consider the predictors of pCR identified as significant from
multivariate logistic regression analysis in [8]: tumor grade (1 and 2 vs 3) and hormone
receptor status (HR+ vs HR-). As possible time-dependent confounders, we regard palpation result and WHO toxicities where at least 5% of the patients showed grade 3 or 4
and less than 30% of the patients have missing values. These are alopecia, fatigue and
increased leucocytes.
The original analysis in [8] yields a pCR rate of 14.3% and 7.0% for the long and the
short arm, respectively. Within the multivariate analysis, the odds ratio of the long
versus the short arm results to 2.42. These results are based on an Intention-to-treat
analysis where no adjustment for early stopping is performed.
6.7.2. Estimation of the weights
The stabilised weights in (6.3) are estimated in two steps separately for each treatment
arm. To obtain the nominator, we calculate the relative frequencies of cycles applied.
For the denominator, we fit a discrete proportional hazards model for the time until the
first cycle which is not given. Patients where all planned cycles were given are censored
after the last cycle. With respect to the time-dependent variables, we first perform model
selection by univariate analyses. According to the Akaike criterion [48], we include only
those time-dependent covariates where the unadjusted p-value of the two-sided Wald
test is p ≤ 0.157. Patients with missing values are excluded in those analyses where the
missing covariate is considered. Estimates per treatment arm are given in table 6.7.2.
62
Short treatment arm
Long treatment arm
Hazard
Characteristics
Hazard
ratio
95% CI
p-value
ratio
95% CI
p-value
Tumor grade (1 and 2 vs 3)
0.29
[0.09;0.89]
0.03
0.71 [0.41;1.26]
0.24
Hormone receptor (HR+ vs HR-)
1.28 [0.41;3.96]
0.67
0.80 [0.44;1.45]
0.47
Baseline characteristics
Time-dependent characteristics
Palpation result
not included
not included
WHO toxicities
Alopecia
not included
0.81 [0.50;1.32]
Fatigue
1.64
[1.05;2.54]
0.03
Increased leucocytes
1.78 [1.28;2.47]
≤ 0.001
0.40
1.71 [1.31;2.24] ≤ 0.001
1.18 [0.96;1.44]
0.12
CI: confidence interval,
Table 6.3.: Results of multivariate discrete proportional hazards models per treatment arm for denominator of the
stabilised weights
63
6.7. Application: Geparduo study
p-value from two-sided Wald test
6. Marginal Structural Models as extension of missing data approach
As recommended by Cole and Hernán [37], we check that the stabilised weights have a
mean of one (table 6.4), which is a necessary condition for correct model specification.
stabilised weights
Mean
Max
Min
Std Dev
Short arm (4 cycles, 319 Obs)
1.017 5.680 0.093
0.476
Long arm (8 cycles, 311 Obs)
1.008 4.179 0.268
0.355
Table 6.4.: Summary of stabilised weights per treatment arm
6.7.3. Dose effect
The trial was not designed to assess the dose effect and most of the patients received
all planned cycles. Thus, its analysis can only be used for illustrating the method. We
restrict the data to the long treatment arm with eight cycles, as there is more variation
with respect to early stopping. For estimating the parameters of the MSM by the
weighted logistic model shown in (6.6), we use the SAS program Proc Genmod. To
obtain robust variances, we follow Robins et al. [31] and choose the option ”repeated“
and specify an independence working correlation matrix.
Figure 6.2 shows the probability of pCR with respect to the number of cycles applied.
The dots show the values of P (Yn = 1) = E(Yn ) estimated by the weighted means in
the subset of patients who received n cycles. The line indicates the estimate of the
MSM which assumes that the logistic model in (6.6) holds. Nobs indicates the number
of patients who actually received n cycles. We see, that the MSM mainly adjusts for the
value at n = 8 where the data gives most of the information.
Recall, that the results for the dose effect can only be used for illustration, as there
are not enough observations with early stopping to obtain reliable estimates. As this
analysis is not meant to gain knowledge about breast cancer and its rehabilitation, the
choice of prognostic factors and time-dependent confounders, as well as the assumptions
made by the model for the weights, was not agreed with medical experts. Furthermore,
we did not respect the change in chemotherapeutic decomposition and used a simple
structural model. If one is in doubt whether the model for the treatment assignment
64
6.7. Application: Geparduo study
mechanism or the model for the counterfactual data might be misspecified, one might
0.35
turn to doubly robust estimators [52].
0.30
●
0.20
0.15
●
●
0.10
P(Yn = 1)
0.25
●
0.00
0.05
●
●
1
●
2
3
●
4
5
6
7
8
12
245
Number of given cycles n
Nobs
2
3
2
13
23
11
Figure 6.2.: Probability of pCR by number of given cycles n estimated by MSM (line)
and by weighted means (dots) with Nobs number of patients who received
n cycles
6.7.4. Therapy effect
The therapy effect after the application of all planned cycles quantified by (6.7), with
Z = 1 indicating the long arm, results to an odds ratio of 2.62 with confidence interval
[1.48; 4.64]. It is calculated from the estimates of E(Yp |Z) of the long and the short
treatment arm which equal the pCR rates. They result to 16.0% (Std Error 0.0235) and
6.8% (Std Error 0.0146), respectively. In comparison to the original analysis, the pCR
rate of the long arm is higher whereas it is almost the same for the short arm, resulting
in a higher odds ratio. This confirms our expectations as in the long arm 21% of the
patients discontinue therapy which diminishes the therapy effect. In the short arm, only
7% do not receive all planned cycles which hardly affects the estimates.
65
7. Discussion and outlook
Within this thesis, we regarded models for the analysis of observational data from complex longitudinal settings. Although the necessity of randomised trials to finally deduce
causal conclusions is without controversy, the development of such models is very important as there will always be situations where costs and ethical concerns rule out
the conduct of randomised trials. Especially, if the interest lies in time-varying exposures, sequentially randomised trials might not be affordable or impracticable due to a
large sample size. Furthermore, if observational data with considerable information on
confounders and extensive knowledge on confounding structures is available, all the information contained on interesting effects should be extracted as adequately as possible.
A major problem is the correct inclusion of time-dependent confounders which are typically included in such data and which cannot be appropriately addressed by standard
statistical models.
We focussed on the structural models SNFTM and MSM which cope with time-dependent
confounders. They are defined within the counterfactual framework which is why people
unfamiliar with this setting are often sceptical about them. We delivered insight into
these models by contrasting them to common approaches defined within the observable
framework and by presenting new aspects with respect to the impact of the model properties on the observable data structure. Furthermore, we applied these counterfactual
models to real data concerning the impact of nosocomial infections on the length of
ICU stay and preoperative chemotherapies applied in case of breast cancer. In the first
example, the time-varying exposure describes the onset of infection, the second example concerns a time-varying treatment regime for which the counterfactual models were
originally proposed. In the case where a time-varying exposure is considered which is
not a treatment regime, this must be respected when interpreting the effects defined by
counterfactuals. Here, the onset of exposure is considered as being driven by an external
force which means that the counterfactual outcome belongs to a predetermined exposure
67
7. Discussion and outlook
regime where the onset of exposure is fixed at study start.
The effect of an intermediate event on the timing of a terminal event was examined with
respect to the ICU setting. This question also arises in other disciplines. For example,
in economics, the concern is about the usefulness of active labour market programs
offered to unemployees to shorten the time until reemployment. Or, in social science,
one deals with the question whether having a child is a reason for unwedded couples to
advance marriage. In clinical epidemiology, the number of extra days due to a nosocomial
infection is considered to be a relevant quantity in order to increase the efficacy of
resource planning and to assess additional expenses due to the infection. We determined
this quantity as proposed by Schulgen et al. [44, 45]. The counterfactual framework
facilitates its definition and allows a clear interpretation. The estimate is achieved
by plugging in the SNFTM parameter. Being a standard issue in clinical epidemiology
[9,46], we have regarded the combined endpoint end of stay which includes discharge and
death. By the SNFTM, the influence of the infection on the length of stay is modeled as a
rescaling of the time span from infection to the counterfactual end of stay, if no infection
occurred. The rescaling parameter does not differ according to the endpoints discharge
and death. This represents a limitation of the model. However, distinguishing those
competing events is not easy. One idea is to enter a second acceleration parameter in
the SNFTM to allow for a different behavior for the two endpoints, i.e. for the calculation
of T ∞ from T and I according to (3.4) we use exp(γ) for discharge and replace it by
exp(δ) for death. The information about the observed endpoint then enters directly in
the computation. But, it is hard to define, if the counterfactual stay T ∞ was ended by
discharge or death. We would need data that indicates whether the patient died from
the infection or whether he would have died regardless of whether the infection had
happened or not. Such a statement is difficult for medical reasons and corresponding
data will be difficult to obtain, at least from routine data bases. Furthermore, applying
g-estimation to obtain two parameters is not straightforward. Another idea which still
needs further examination is to use principal stratification [53].
To provide insights into the observable data structure assumed by the SNFTM, we
proposed a multistate model which is conform with the SNFTM. Thereby, we established
a model where no counterfactual or latent variables are involved and where the data
arise successively as in a prospective trial. To our knowledge, models conform with
the SNFTM proposed so far only aim at generating data for simulation studies. They
68
do not get by without modelling a counterfactual outcome and furthermore model the
interaction of the time-varying exposure and time-dependent confounders separately
from the impact on the outcome.
A characteristic of a time-dependent confounder is that, in probability, it is affected
by the exposure. In terms of our ICU example, this means that the need for artificial
ventilation is increased after the nosocomial infection occurred. We model the action of a
time-dependent confounder by implicitely distinguishing, whether the exposure was the
reason for the change in the confounder variable, i.e. if AV was due to INF or not. This
additional information can be seen as an unobserved underlying process which might
alternatively be handled by a Hidden Markov Model [54] which is subject to future
investigations.
Whereas the SNFTM focusses on the total causal effect, multistate models consider
direct effects which are modelled by relating the respective transition rates. To introduce
the causal parameter of the SNFTM, we defined a delayed impact of changes in covariates
on the survival time, if the survival time is already affected by the exposure. Thereby, the
impact of covariates and exposure does not always correspond to proportional hazards.
Further research is needed to find out, if proportional hazards, possibly with adequate
interaction term, can be achieved, e.g. by other ideas on how to generate the average
start of the impact of the time-dependent confounder after exposure.
Our simulation results comparing the behaviour of a Cox model and the SNFTM might
give the impression that the SNFTM is only slightly advantageous. But, we need to
recognise that we only considered a rather simple situation of time-dependent confounding. The impression changes already in the situation with an unmeasured baseline
variable which affects the time-dependent confounder and the survival time, but has no
impact on the exposure. It can be addressed by our multistate model by defining the
respective transition rates conditional on this baseline variable. Then, the simulation
still gives unbiased estimates for the SNFTM. However, the Cox model which adjusts
for artificial ventilation, shows an association between infection and time to discharge,
even if neither the infection nor artificial ventilation affect the outcome.
In order to deliver insight into MSMs to people unfamiliar with the counterfactual framework, we demonstrated that the MSM procedure can be interpreted as a repeated use
of IPCW supplemented by a structural model which makes parametric assumptions on
how treatment affects the quantity of interest. MSMs make the assumption of exper-
69
7. Discussion and outlook
imental treatment assignment shown in (3.11) which claims that at every time-point
every patient has a chance to get further treatment. This is not absolutely satisfied in
our data example, as sometimes treatment must be stopped due to severe adverse events
such as toxicity of high grade. Van der Laan and Petersen [55] describe how to relax
this assumption and propose causal effect models which only consider possible treatment
options. To apply such models to our data and explore their properties remains subject
to further considerations.
The effect quantified by the MSM applies to the effect of a static treatment regime
where the application doses are determined at baseline and do not change in response
to past history of the individual patient. In contrast to SNFTMs, due to the concept of
MSMs, it is not feasible to extend them to quantify the effect of a dynamic treatment
regime. The sequential Cox approach proposed by Gran et al. [56] as an alternative
to MSMs addressing a survival outcome is also based on the idea to mimic randomized
controlled trials. In contrast to the MSM, when comparing individuals starting treatment
at a certain time with those who do not, it conditions on the covariates at that time.
This fact might facilitate the modification of the model to study the effect of dynamic
treatment regimes. In general, effects of dynamic treatment regimes are challenging even
with respect to their definition and interpretation. They are increasingly addressed in
the literature, e.g. by Robins and Hernán [57], Cain et al. [58] and Orellana et al. [59].
70
A. Appendix: Abbreviations and
notation
A.1. Abbreviations
AV
artificial ventilation
CI
confidence interval
DAG
directed acyclic graph
HR+, HR-
hormone receptor status
ICU
intensive care unit
IPCW
Inverse Probability of Censoring Weighting
IPTW
Inverse Probability of Treatment Weighting
MSM
Marginal Structural Model
pCR
pathological complete remission
PS
propensity score
SNFTM
Structural Nested Failure Time Model
A.2. Notation
This list only represents the relevant and general notation and may not be complete.
The given pages or equation numbers indicate first appearance. Some notation used
only once and defined there may not be found here.
71
A. Appendix: Abbreviations and notation
α0 , α1
parameters in associational model (eq. 6.8)
αij (t; Ft− )
hazard rate (p.22)
AV(t)
time-dependent status variable
describing use of artificial ventilation (p.5)
β0 , β1
parameters in MSM (eq. 6.6)
cAVDIS
constant ≤ 1 to determine effect of AV on discharge (p.33)
cAVINF
constant ≥ 1 to determine effect of AV on INF (p.33)
cINFAV
constant ≥ 1 to determine effect of INF on AV (p.33)
∆k
indicator, if cycle k is given (p.11)
∆n = (∆1 , ∆2 , ..., ∆n )
exposure history up to cycle n (p.11)
exp(γ)
acceleration parameter (p.14)
Ft−
confounder history just prior to time t (p.15)
I
time from admission to nosocomial infection,
I := inf{t : INF(t) = 1} (p.13)
INF(t)
time-dependent status variable
describing exposure status, e.g. infection status (p.5)
λDIS (t|INF(t), AV(t))
discharge hazard (p.43)
λI (t|Ft− )
exposure hazard (p.15)
n
fixed number of cycles (p.54)
P (n,xn+1 )
product of conditional probabilities (p.56)
Pij (s, t)
probability of being in state j at time t
conditional on having been in state i at time s
and on the covariate history Fs− just before s (p.22)
sγ + ,k = k + (s − k)/ exp(γ)
retransformed time (p.73)
sw
stabilised weight (p.57)
T
time to terminal event (observable outcome),
e.g. length of stay on ICU (p.5)
tγ,k = k + (t − k) · exp(γ)
T
∞
backtransformed time (eq. 5.3)
counterfactual survival outcome,
if no exposure occurred (p.12)
(T ∞ , T 1 , ..., T nmax )
counterfactual survival outcome for time-dependent
treatment for onset of exposure at discrete times and
onset times between 1 and nmax
72
A.2. Notation
T ∞,γ
counterfactual survival outcome, if no exposure
occurred calculated from eq. 3.5 for arbitrary γ
where T ∞ = T ∞,γ0 with γ0 the true population
value (p.15)
w
weight (p.56)
X
confounder (p.9)
Xk
time-dependent covariate / confounder (p.12)
X k = (X1 , ..., Xk )
covariate history prior to cycle k (p.12)
Y
observable outcome (p.11)
Y0
outcome after no treatment (p.11)
(Y0 , Y1 )
counterfactual outcome
for time-independent treatment (p.9)
(Y0 , Y1 , Y2 , ..., Yp )
counterfactual time-constant outcome
for time-dependent treatment,
if at most p cycles are applied (p.11)
Yk
potential outcome,
if treatment was given for the first k cycles (p.11)
Z
treatment assignment (p.9)
73
B. Appendix: Multistate model meets
assumption of no unmeasured
confounders
We now present a formal check that the multistate model defined in section 5.2 for the
case where AV was considered as status variable which only indicates time from first
use of AV meets the assumption of no unmeasured confounders. Therefore, we use the
discrete form of the assumption of no unmeasured confounders given in (3.3) which is
characterised by sequentially defined conditional independence assumptions. Using our
notation, it claims that for all times k:
T ∞ ⊥⊥ INF(k) | INF(k − 1), AV(k − 1), T ≥ k
where ⊥⊥ denotes conditional independence. We write X(k − 1) for (X(1), ..., X(k − 1))
where X(0) is defined to be 0 to simplify notation.
We consider time-dependent transition hazards, but mainly suppress this in the notation
for lack of space. Again, we use the backtransformed time tγ,k = k + (t − k) · exp(γ) and
further denote the retransformed time by sγ + ,k = k + (s − k)/ exp(γ).
We show that the distribution of T ∞ for patients with the same covariate history until
k− is equal whether they are infected or uninfected at time k. Note, that the uninfected
patient might be infected at later times. Recall that for uninfected patients T ∞ = T
and for infected patients T ∞ = I + (T − I) · exp(γ). Furthermore, T ∞ < T as we regard
the case exp(γ) < 1. We regard the probability of T ∞ > x in different settings. If no
infection occurred until x, we can equally consider T > x. If the infection time is u,
T ∞ > x is equivalent to T > xγ + ,u .
We use the description of our model including hidden substates. This does not delimit
our check due to the following arguments. We compare the distribution of T ∞ for
75
B. Appendix: Multistate model meets assumption of no unmeasured confounders
patients infected at k and uninfected at k which have the same covariate history until
k−, in particular INF(k − 1) = 0. Thus, if we condition on patients being in state 4 at
k, we know that they just switched to state 4 and therefore still remain in the hidden
substate that they reached. As they were not infected before k, we know that they came
from state 2 to 4 and not from state 3. Therefore, we know by the observable data that
they are in substate A at k.
Transformation of the terms is mainly done by substitution with w = tγ,k , by interchanging integrals and by using the following equality
Z x
Z u
Z
exp(−
α(t)dt) · α(u)du = (1 − exp(−
k
k
x
α(t)dt))
k
B.1. Start in state 4 reached from 2 compared to start
in state 2
First we compare patients ventilated from time s which are infected from time k > s or
uninfected at k. As k > s, the infected patient switched from state 2 to 4 at k, which we
k
denote by 2 → 4. As he switched from 2, we know that he reached substate A and that
his discharge hazard equals αA5 (t; k) = exp(γ) · α15 (tγ,k ) · cAVDIS . We can disregard, if he
switches to state A0 before being discharged as there is no other transient state than A0
which can be reached from A and as αA5 (t; k) = αA0 5 (t; k). The uninfected patient is in
state 2 at k.
B.1.1. Start in state 4 reached from 2
k
k
P (T ∞ > x|2 → 4) = P (T > xγ + ,k |2 → A) =
Z xγ + ,k
= exp(−
αA5 (t; k)dt) =
k
Z xγ + ,k
= exp(−
exp(γ) · α25 ((t − k) exp(γ) + k)dt) =
k
Z x
= exp(−
α25 (w)dw)
k
with substitution w = tγ−,k .
76
B.2. Start in state 3 compared to start in state 1
B.1.2. Start in state 2
P (T ∞ > x|in 2 at k) =
Z
x
u
P (2 → 4, in 4 until xγ + ,u |in 2 at k)du =
P (in 2 until x|in 2 at k) +
k
Z x
= exp(−
(α25 (t) + α24 (t))dt) +
Z xγ + ,u
Z u
Z x k
αA5 (t; u)dt)du =
(α25 (t) + α24 (t))dt) · α24 (u) · exp(−
exp(−
+
u
k
k
Z x
α25 (w)dw)
= exp(−
k
as
Z
x
Z
u
Z
x
exp(−
(α25 (t) + α24 (t))dt) · α24 (u) · exp(−
α25 (w)dw)du =
k
k
u
Z x
Z x
Z u
= exp(−
α25 (t)dt) ·
exp(−
α24 (t)dt) · α24 (u)du =
k
k
k
Z x
Z x
α24 (t)dt))
α25 (t)dt) · (1 − exp(−
= exp(−
k
k
k
Thus, P (T ∞ > x|in 2 at k) = P (T ∞ > x|2 → 4).
B.2. Start in state 3 compared to start in state 1
Now, we compare patients unventilated until time k− which are infected from time k or
uninfected at k. I.e. the infected patient switched to state 3 at k, whereas the uninfected
patient is in state 1 at k.
77
k
P (T ∞ > x|1 → 3) =
k
= P (T > xγ + ,k |1 → 3) =
k
= P (in 3 until xγ + ,k |1 → 3) +
u
k
+P (3 → A, u ∈ [k; xγ + ,k ], in A until xγ + ,k |1 → 3) +
u
w
k
+P (3 → A, u ∈ [k; xγ + ,k ], A → A0 , w ∈ [u; xγ + ,k ], in A0 until xγ + ,k |1 → 3) +
u
k
+P (3 → B, u ∈ [k; xγ + ,k ], in B until xγ + ,k |1 → 3) +
u
v
u
v
k
+P (3 → B, u ∈ [k; xγ + ,k ], B → A, v ∈ [u; xγ + ,k ], in A until xγ + ,k |1 → 3) +
w
k
+P (3 → B, u ∈ [k; xγ + ,k ], B → A, v ∈ [u; xγ + ,k ], A → A0 , w ∈ [v; xγ + ,k ], in A0 until xγ + ,k |1 → 3) +
u
k
+P (3 → B 0 , u ∈ [k; xγ + ,k ], in B 0 until xγ + ,k |1 → 3) +
u
w
k
+P (3 → B 0 , u ∈ [k; xγ + ,k ], B 0 → A0 , w ∈ [u; xγ + ,k ], in A0 until xγ + ,k |1 → 3)
We now rewrite parts of the sum above and then put them together again.
u
k
P (3 → A, u ∈ [k; xγ + ,k ], in A until xγ + ,k |1 → 3) +
u
v
k
+P (3 → A, u ∈ [k; xγ + ,k ], A → A0 , v ∈ [u; xγ + ,k ], in A0 until xγ + ,k |1 → 3) =
Z xγ + ,k
Z u
Z xγ + ,k
=
exp(−
(α3A + α3B + α3B0 + α35 )dt) · α3A (u) · exp(−
(αAA0 + αA5 )dt)du +
k
k
u
Z xγ + ,k Z xγ + ,k
Z u
Z v
+
exp(−
(α3A + α3B + α3B0 + α35 )dt) · α3A (u) · exp(−
(αAA0 + αA5 )dt) · αAA0 (v) ·
k
k
u
Zu xγ + ,k
· exp(−
αA0 5 dt)dvdu =
v
Z xγ + ,k
Z u
Z xγ + ,k
=
exp(−
(α3A + α3B + α3B0 + α35 )dt) · α3A (u) · exp(−
αA5 dt)du
k
k
u
B. Appendix: Multistate model meets assumption of no unmeasured confounders
78
B.2.1. Start in state 3
with, as αA5 (t; k) = αA0 5 (t; k),
Z xγ + ,k
Z
v
Z
xγ + ,k
αA0 5 dt)dv =
(αAA0 + αA5 )dt) · αAA0 (v) · exp(−
exp(−
v
u
u
Z xγ + ,k
Z xγ + ,k
Z v
αA5 dt)
exp(−
αAA0 dt) · αAA0 (v) · dv =
= exp(−
u
Z xγu+ ,k
Zu xγ + ,k
αAA0 dt))
αA5 dt)(1 − exp(−
= exp(−
u
u
The same works for
u
v
k
P (3 → B, u ∈ [k; xγ + ,k ], B → A, v ∈ [u; xγ + ,k ], in A until xγ + ,k |1 → 3) +
u
v
w
k
v
Thus, as αBA (t; k) = αB0 A0 (t; k) = α3A (t; k) and αB5 (t; k) = αB0 5 (t; k) = α35 (t; k), and by interchanging the integrals
u
v
k
P (3 → B, u ∈ [k; xγ + ,k ], B → A, v ∈ [u; xγ + ,k ], in A until xγ + ,k |1 → 3) +
u
v
w
k
+P (3 → B, u ∈ [k; xγ + ,k ], B → A, v ∈ [u; xγ + ,k ], A → A0 , w ∈ [v; xγ + ,k ], in A0 until xγ + ,k |1 → 3) +
u
v
k
79
+P (3 → B 0 , u ∈ [k; xγ + ,k ], B 0 → A0 , v ∈ [u; xγ + ,k ], in A0 until xγ + ,k |1 → 3) =
Z xγ + ,k Z xγ + ,k
Z u
Z v
=
exp(−
(α3A + α3B + α3B0 + α35 )dt) · α3B (u) · exp(−
(αBA + αB5 )dt) · αBA (v) ·
k
u
k
u
Z xγ + ,k
· exp(−
αA5 dt)dvdu +
v
Z xγ + ,k Z xγ + ,k
Z u
Z v
+
exp(−
(α3A + α3B + α3B0 + α35 )dt) · α3B0 (u) · exp(−
(αB0 A0 + αB0 5 )dt) · αB0 A0 (v) ·
k
k
u
Zu xγ + ,k
· exp(−
αA0 5 dt)dvdu =
v
B.2. Start in state 3 compared to start in state 1
+P (3 → B, u ∈ [k; xγ + ,k ], B → A, v ∈ [u; xγ + ,k ], A → A0 , w ∈ [v; xγ + ,k ], in A0 until xγ + ,k |1 → 3) =
Z u
Z v
Z xγ + ,k Z xγ + ,k
0
exp(−
(α3A + α3B + α3B + α35 )dt) · αBA (v) · α3B (u) · exp(−
(αBA + αB5 )dt) ·
=
u
k
u
k
Z xγ + ,k
αA5 dt)dvdu
· exp(−
Z
xγ + ,k
Z
k
u
k
Z
k
Z
Z
αA5 dt)dvdu =
Z u
Z v
exp(−
(α3B + α3B0 )dt) · (α3B (u) + α3B0 (u)) · exp(−
(α3A + α35 )dt) · α3A (v) ·
u
k
· exp(−
xγ + ,k Z
v
v
=
k
k
Z
(αBA + αB5 )dt) · αBA (v) ·
u
xγ + ,k
· exp(−
Z xγ + ,k Z xγv+ ,k
=
v
Z
(α3A + α3B + α3B0 + α35 )dt) · (α3B (u) + α3B0 (u)) · exp(−
exp(−
=
u
k
xγ + ,k
αA5 dt)dvdu =
Z v
Z u
(α3A + α35 )dt) · α3A (v) ·
(α3B + α3B0 )dt) · (α3B (u) + α3B0 (u))du · exp(−
exp(−
k
k
xγ + ,k
· exp(−
αA5 dt)dv =
Z v
Z v
Z
1 − exp(−
(α3B + α3B0 )dt) · exp(−
(α3A + α35 )dt) · α3A (v) · exp(−
v
Z
=
xγ + ,k
k
k
xγ + ,k
αA5 dt)dv
v
k
Furthermore, as αB0 A0 (t; k) = α3A (t; k) and αB0 5 (t; k) = α35 (t; k)
u
k
P (3 → B 0 , u ∈ [k; xγ + ,k ], in B 0 until xγ + ,k |1 → 3) =
Z xγ + ,k
Z u
Z xγ + ,k
=
exp(−
(α3A + α3B + α3B0 + α35 )dt) · α3B0 (u) · exp(−
(αB0 A0 + αB0 5 )dt)du =
k
k
u
Z xγ + ,k
Z xγ + ,k
Z u
= exp(−
(α3A + α35 )dt)
exp(−
(α3B + α3B0 )dt) · α3B0 (u)du
k
k
k
B. Appendix: Multistate model meets assumption of no unmeasured confounders
80
xγ + ,k
Z
We obtain
k
k
k
k
k
v
B.2. Start in state 3 compared to start in state 1
81
P (T ∞ > x|1 → 3) = P (T > xγ + ,k |1 → 3) =
Z xγ + ,k
(α3A + α3B + α3B0 + α35 )dt) +
= exp(−
k
Z xγ + ,k
Z u
Z xγ + ,k
αA5 dt)du +
(α3A + α3B + α3B0 + α35 )dt) · α3A (u) · exp(−
exp(−
+
u
k
k
Z xγ + ,k
Z u
Z xγ + ,k
+
exp(−
(α3A + α3B + α3B0 + α35 )dt) · α3B (u) · exp(−
(αBA + αB5 )dt)du +
k
k
u
Z xγ + ,k
Z v
Z v
Z xγ + ,k
+
1 − exp(−
(α3B + α3B0 )dt) · exp(−
(α3A + α35 )dt) · α3A (v) · exp(−
αA5 dt)dv +
k
k
k
v
Z u
Z xγ + ,k
Z xγ + ,k
exp(−
(α3B + α3B0 )dt) · α3B0 (u)du =
(α3A + α35 )dt)
+ exp(−
k
k
k
Z xγ + ,k
(α3A + α3B + α3B0 + α35 )dt) +
= exp(−
k
Z u
Z xγ + ,k
Z xγ + ,k
(α3B + α3B0 )dt) · (α3B (u) + α3B0 (u))du +
exp(−
(α3A + α35 )dt)
+ exp(−
k
k
k
Z xγ + ,k
Z v
Z xγ + ,k
+
exp(−
(α3A + α35 )dt) · α3A (v) · exp(−
αA5 dt)dv =
k
k
v
Z xγ + ,k
= exp(−
(α3A + α3B + α3B0 + α35 )dt) +
k
Z xγ + ,k
Z xγ + ,k
+ exp(−
(α3A + α35 )dt)(1 − exp(−
(α3B + α3B0 )dt)) +
k
k
Z xγ + ,k
Z v
Z xγ + ,k
exp(−
(α3A + α35 )dt) · α3A (v) · exp(−
αA5 dt)dv =
+
k
k
v
Z xγ + ,k
Z xγ + ,k
Z v
Z xγ + ,k
= exp(−
(α3A + α35 )dt) +
exp(−
(α3A + α35 )dt) · α3A (v) · exp(−
αA5 dt)dv =
k
v
Zk x
Z x k
Z v
Z x
= exp(−
(α12 + α15 )dt) +
exp(−
(α12 + α15 )dt) · α12 (v) · exp(−
α25 dt)dv
α25 (tγ−,k ) · exp(γ).
B.2.2. Start in state 1
P (T ∞ > x|in 1 at k) =
= P (in 1 until x|in 1 at k) +
u
+P (1 → 2, u ∈ [k; x], in 2 until x|in 1 at k) +
u
w
+P (1 → 2, u ∈ [k; x], 2 → 4, w ∈ [u; x], in 4 until xγ + ,w |in 1 at k) +
u
u
+P (1 → 3, u ∈ [k; x]|in 1 at k) · P (T > xγ + ,u |1 → 3) =
Z x
(α12 + α13 + α15 )dt) +
= exp(−
k
Z x
Z u
Z x
+
exp(−
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
(α24 + α25 )dt)du +
k
k
u
Z xZ x
Z u
Z w
Z xγ + ,w
+
exp(−
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
(α24 + α25 )dt) · α24 (w) · exp(−
αA5 dt)dwdu +
u
w
Zk x u
Z u k
+
exp(−
(α12 + α13 + α15 )dt) · α13 (u) ·
k
k
Z x
Z x
Z v
Z x
· exp(−
(α12 + α15 )dt) +
exp(−
(α12 + α15 )dt) · α12 (v) · exp(−
α25 dt)dv du =
u
u
u
v
B. Appendix: Multistate model meets assumption of no unmeasured confounders
82
The last transformation is valid as α3A (t; k) = α12 (tγ−,k ) · exp(γ), α35 (t; k) = α15 (tγ−,k ) · exp(γ) and αA5 (t; k) =
Z
x
k
k
v
83
B.2. Start in state 3 compared to start in state 1
(α12 + α13 + α15 )dt) +
= exp(−
k
Z x
Z u
Z x
α25 dt)du +
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
exp(−
+
u
k
k
Z u
Z x
Z x
α13 dt)α13 (u)du +
exp(−
(α12 + α15 )dt) ·
+ exp(−
k
k
k
Z x
Z u
Z x
Z v
Z x
+
exp(−
(α12 + α13 + α15 )dt) · α13 (u) ·
exp(−
(α12 + α15 )dt) · α12 (v) · exp(−
α25 dt)dvdu =
k
k
u
u
v
Z x
= exp(−
(α12 + α13 + α15 )dt) +
Z x k
Z u
Z x
+
exp(−
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
α25 dt)du +
k
k
u
Z x
Z x
α13 dt) +
(α12 + α15 )dt) · 1 − exp(−
+ exp(−
k
Z x Z vk
Z u
Z v
Z x
+
exp(−
α13 dt)α13 (u)du · exp(−
(α12 + α15 )dt) · α12 (v) · exp(−
α25 dt)dv =
k
k
k
k
v
Z x
Z u
Z x
=
exp(−
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
α25 dt)du +
k
k
u
Z x
+ exp(−
(α12 + α15 )dt) +
k
Z x
Z v
Z v
Z x
α25 dt)dv =
(α12 + α15 )dt) · α12 (v) · exp(−
α13 dt) · exp(−
1 − exp(−
+
v
k
k
k
Z x
= exp(−
(α12 + α15 )dt) +
k
Z x
Z v
Z x
+
exp(−
(α12 + α15 )dt) · α12 (v) · exp(−
α25 dt)dv
x
Z
x
Z
u
Z
w
Z
xγ + ,w
αA5 dt)dwdu =
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
(α24 + α25 )dt) · α24 (w) · exp(−
w
k
u
k
u
Z xZ x
Z u
Z w
Z x
exp(−
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
(α24 + α25 )dt) · α24 (w) · exp(−
α25 dt)dwdu =
=
k
u
k
u
w
Z x
Z w
Z x
Z u
Z x
exp(−
α24 dt)α24 (w)dw du =
=
exp(−
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
α25 dt) ·
u
Zk x
Zk u
Zu x
Z ux
=
exp(−
(α12 + α13 + α15 )dt) · α12 (u) · exp(−
α25 dt) · 1 − exp(−
α24 dt) du
exp(−
k
k
u
k
Thus, P (T ∞ > x|in 1 at k) = P (T ∞ > x|1 → 3).
u
B. Appendix: Multistate model meets assumption of no unmeasured confounders
84
with
Z
C. Appendix: Parts of this thesis
published previously or submitted
for peer review
A manuscript based on chapter 6 was published previously as the article of Gall C,
Caputo A and Schumacher M. Comparison of Marginal Structural Models to a missing data approach illustrated by data on breast cancer chemotherapies, FDM-Preprint
No. 103, University of Freiburg. A revised version was submitted to the peer-reviewed
journal Statistics in Medicine and is still under review. It is entitled Marginal Structural
Models illustrated as extension of a missing data approach with application on breast
cancer data. The co-authors are Angelika Caputo and Martin Schumacher. I am the
first author which means that I made the main contributions to conception and design,
wrote all drafts and the final version of the manuscript and carried out all analyses.
Some aspects of this thesis can also be found in Schmoor C, Gall C, Stampf S and Graf
E. Correction for confounding bias in non-randomized studies by appropriate weighting,
Biometrical Journal, 53:1–19, 2011.
85
Bibliography
[1] Rubin DB. Direct and indirect causal effects via potential outcomes. Scandinavian
Journal of Statistics, 31:161–170, 2004
[2] Rubin DB. Bayesian inference for causal effects: The role of randomization. The
Annals of Statistics, 6:34–58, 1978
[3] Dawid A. Causal inference without counterfactuals. Journal of the American Statistical Association, 95(450):407–424, 2000
[4] Greenland S and Morgenstern H. Confounding in health research. Annual Review
of Public Health, 22:189–212, 2001
[5] Aalen OO, Borgan Ø, Keiding N and Thorman J. Interaction between life history
events. nonparametric analysis for prospective and retrospective data in the presence
of censoring. Scandinavian Journal of Statistics, 7:161–171, 1980
[6] Grundmann H, Bärwolff S, Schwab F, Tami A, Behnke M, Geffers C, Halle E, Göbel
U, Schiller R, Jonas D, Klare I, Weist K, Witte W, Beck-Beilecke K, Schumacher
M, Rüden H and Gastmeier P. How many infections are caused by patient-to-patient
transmission in intensive care units? Critical Care Medicine, 33:946–951, 2005
[7] Beyersmann J, Gastmeier P, Grundmann H, Bärwolff S, Geffers C, Behnke M,
Rüden H and Schumacher M. Use of multistate models to assess prolongation of
intensive care unit stay due to nosocomial infection. Infection Control and Hospital
Epidemiology, 27(5):493–499, 2006
[8] von Minckwitz G, Raab G, Caputo A, Schütte M, Hilfrich J, Blohmer J, Gerber B,
Costa S, Merkle E, Eidtmann H, Lampe D, Jackisch C, du Bois A and Kaufmann
M. Doxorubicin with cyclophosphamide followed by docetaxel every 21 days compared with doxorubicin and docetaxel 14 days as preoperative treatment in operable
87
Bibliography
breast cancer: The geparduo study of the German Breast Group. Journal of Clinical
Oncology, 23(12):2676–2685, 2005
[9] Samore M and Harbarth S. A methodologically focused review of the literature in
hospital epidemiology and infection control, chapter 93, pages 1645–1656. Lippincott
Williams & Wilkins, Philadelphia, third edition, 2004
[10] Pearl J. Causality: models, reasoning, and inference. Cambridge University Press,
2000
[11] Aalen OO and Frigessi A. What can statistics contribute to a causal understanding?
Scandinavian Journal of Statistics, 34(1):155–168, 2007
[12] Pearl J. Causal inference in statistics: An overview. Statistics Surveys, 3:96–146,
2009
[13] Greenland S and Brumback B. An overview of relations among causal modelling
methods. International Journal of Epidemiology, 31:1030–1037, 2002
[14] Aalen OO. Dynamic modelling and causality. Scandinavian Actuarial Journal,
pages 177–190, 1987
[15] Fosen J, Ferkingstad E, Borgan O and Aalen OO. Dynamic path analysis – a new
approach to analyzing time-dependent covariates. Lifetime Data Analysis, 12:143–
167, 2006
[16] Fosen J, Borgan Ø, Weedon-Fekjaer H and Aalen OO. Dynamic analysis of recurrent
event data using the additive hazard model. Biometrical Journal, 48:381–398, 2006
[17] Greenland S, Pearl J and Robins JM. Causal diagrams for epidemiologic research.
Epidemiology, 10(1):37–48, 1999
[18] Rosenbaum PR and Rubin DB. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):41–55, 1983
[19] Robins JM. Estimation of the time-dependent accelerated failure time model in the
presence of confounding factors. Biometrika, 79:321–334, 1992
88
Bibliography
[20] Robins JM, Blevins D, Ritter G and Wulfsohn M. G-estimation of the effect of
prophylaxis therapy for pneumocystis carinii pneumonia on the survival of AIDS
patients. Epidemiology, 3:319–336, 1992
[21] Robins JM. Structural nested failure time models. In P Armitage and T Colton
(editors), Survival Analysis, P.K. Andersen and N. Keiding, The Encyclopedia of
Biostatistics, pages 4372–4389. John Wiley & Sons, Chichester, UK, second edition,
1997
[22] Keiding N, Filiberti M, Esbjerg S, Robins JM and Jacobsen N. The graft versus
leukemia effect after bone marrow transplantation: A case study using structural
nested failure time models. Biometrics, 55(1):23–28, 1999
[23] Keiding N. Event history analysis and inference from observational epidemiology.
Statistics in Medicine, 18:2353–2363, 1999
[24] Robins JM. Correcting for non-compliance in randomized trials using structural
nested mean models. Communications in Statistics: Theory and Methods, 23:2379–
2412, 1994
[25] Korhonen PA, Laird NM and Palmgren J. Correcting for non-compliance in randomized trials: an application to the ATBC study. Statistics in Medicine, 18:2879–
2897, 1999
[26] Yamaguchi T and Ohashi Y. Adjusting for differential proportions of second-line
treatment in cancer clinical trials. Part i: Structural nested models and marginal
structural models to test and estimate treatment arm effects. Statistics in Medicine,
23:1991–2003, 2004
[27] Hernán MA, Cole SR, Margolick J, Cohen M and Robins JM. Structural accelerated
failure time models for survival analysis in studies with time-varying treatments.
Pharmacoepidemiology and drug safety, 14(7):477–491, 2005
[28] Cox D and Oakes D. Analysis of Survival Data. Chapman & Hall, London, New
York, 1984
89
Bibliography
[29] Young JG, Hernán MA, Picciotto S and Robins JM. Relation between three classes
of structural models for the effect of a time-varying exposure on survival. Lifetime
Data Analysis, 16(1):71–84, 2010
[30] Robins JM. Marginal structural models. In 1997 Proceedings of the Section on
Bayesian Statistical Science, pages 1–10. American Statistical Association, 1998
[31] Robins JM, Hernán MA and Brumback BA. Marginal structural models and causal
inference in epidemiology. Epidemiology, 11(5):550–560, 2000
[32] Mortimer KM, Neugebauer RS, van der Laan MJ and Tager IB. An application
of model-fitting procedures for marginal structural models. American Journal of
Epidemiology, 162(4):382–388, 2005
[33] Cole SR, Hernán MA, Margolick JB, Cohen MH and Robins JM. Marginal structural
models for estimating the effect of highly active antiretroviral therapy initiation on
CD4 cell count. American Journal of Epidemiology, 162(5):471–478, 2005
[34] Tager IB, Haight T, Sternfeld B, Yu Z and van der Laan MJ. Effects of physical
activity and body composition on functional limitation in the elderly: Application
of the marginal structural model. Epidemiology, 15(4):479–493, 2004
[35] Vansteelandt S and Goetghebeur E. Causal inference with generalized structural
mean models. Journal of the Royal Statistical Society, Series B: Statistical Methodology, 65(4):817–835, 2003
[36] Loeys T, Goetghebeur E and Vandebosch A. Causal proportional hazards models
and time-constant exposure in randomized clinical trials. Lifetime data analysis,
11(4):435–449, 2005
[37] Cole SR and Hernán MA. Constructing inverse probability weights for marginal
structural models. American Journal of Epidemiology, 168(6):656–664, 2008
[38] Hernán MA, Brumback BA and Robins JM. Marginal structural models to estimate
the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology,
11(5):561–570, 2000
[39] Andersen P and Keiding N. Multi-state models for event history analysis. Statistical
Methods in Medical Research, 11(2):91–115, 2002
90
Bibliography
[40] Robins JM, Rotnitzky A and Zhao LP. Estimation of regression coefficients when
some regressors are not always observed. Journal of the American Statistical Association, 89(427):846–866, 1994
[41] Robins JM and Rotnitzky A. Recovery of information and adjustment for dependent
censoring using surrogate markers. In N Jewell, K Dietz and V Farewell (editors),
AIDS Epidemiology — methodological issues, pages 24–33. Birkhauser, Boston, 1992
[42] Robins JM and Rotnitzky A. Inverse probability weighted estimation in survival
analysis. In P Armitage and T Colton (editors), Encyclopedia of Biostatistics,
pages 2619–2625. John Wiley & Sons, New York, second edition, 2005
[43] Kaplan E and Meier P. Nonparametric estimation from incomplete observations.
Journal of the American Statistical Association, 53:457–481, 1958
[44] Schulgen G and Schumacher M. Estimation of prolongation of hospital stay attributable to nosocomial infections: New approaches based on multistate models.
Lifetime Data Analysis, 2:219–240, 1996
[45] Schulgen G, Kropec A, Kappstein I, Daschner F and Schumacher M. Estimation of
extra hospital stay attributable to nosocomial infections: Heterogeneity and timing
of events. Journal of Clinical Epidemiology, 53:409–417, 2000
[46] Irala-Estévez J, Martı́nez-Concha D, Dı́az-Molina C, Masa-Calles J, del Castillo AS
and Navajas RFC. Comparison of different methodological approaches to identify
risk factors of nosocomial infection in intensive care units. Intensive Care Medicine,
27(8):1254–1262, 2001
[47] Le Gall JR, Lemeshow S and Saulnier F. A new simplified acute physiology score
(SAPS II) based on a European/North American multicenter study. Journal of the
American Medical Association, 270(24):2957–2963, 1993
[48] Teräsvirta T and Mellin I. Model selection criteria and model selection tests in
regression models. Scandinavian Journal of Statistics, 13:159–171, 1986
[49] Beyersmann J, Latouche A, Buchholz A and Schumacher M. Simulating competing
risks data in survival analysis. Statistics in Medicine, 28(6):956–971, 2009
91
Bibliography
[50] Kalbfleisch JD and Prentice RL. The statistical analysis of failure time data. Wiley
Series in Probability and Mathematical Statistics. John Wiley and Sons, second
edition edition, 2002
[51] Brumback BA, Hernán MA, Haneuse SJ and Robins JM. Sensitivity analyses for
unmeasured confounding assuming a marginal structural model for repeated measures. Statistics in medicine, 23:749–767, 2004
[52] Robins JM and Bang H. Doubly robust estimation in missing data and causal
inference models. Biometrics, 61:962–972, 2005
[53] Frangakis CE and Rubin DB. Principal stratification in causal inference. Biometrics, 58:21–29, 2002
[54] MacDonald IL and Zucchini W. Hidden Markov and other models for discrete-valued
time series. Chapman and Hall, London, 1997
[55] van der Laan MJ and Petersen ML. Causal effect models for realistic individualized
treatment and intention to treat rules. The International Journal of Biostatistics,
3(1, Article 3), 2007. Available at: http://www.bepress.com/ijb/vol3/iss1/3
[56] Gran JM, Røysland K, Wolbers M, Didelez V, Sterne JAC, Ledergerber B, Furrer
H, von Wylf V and Aalen OO. A sequential cox approach for estimating the causal
effect of treatment in the presence of time-dependent confounding applied to data
from the swiss hiv cohort study. Statistics in Medicine, 29:2757–2768, 2010
[57] Robins JM and Hernán MA. Estimation of the causal effects of time-varying exposures. In G Fitzmaurice, M Davidian, G Verbeke and G Molenberghs (editors),
Longitudinal Data Analysis, chapter 23, pages 553–599. Chapman and Hall/CRC
Press, New York, 2009
[58] Cain LE, Robins JM, Lanoy E, Logan R, Costagliola D and Hernán MA. When
to start treatment? A systematic approach to the comparison of dynamic regimes
using observational data. The International Journal of Biostatistics, 6(2, Article
18), 2010. Available at: http://www.bepress.com/ijb/vol6/iss2/18
[59] Orellana L, Rotnitzky A and Robins JM. Dynamic regime marginal structural
mean models for estimation of optimal dynamic treatment regimes, Part i: Main
92
Bibliography
content. The International Journal of Biostatistics, 6(2, Article 8), 2010. Available
at: http://www.bepress.com/ijb/vol6/iss2/8
93