Daniels M.J., Hogan J.W. Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis

Подождите немного. Документ загружается.

90 MISSING DATA MECHANISMS

and R,conditionally on covariates X,andisindexed by a ﬁnite-dimensional

parameter ω. 2

Deﬁnition 5.6. Ful l-data response model.

The full-data response model characterizes the distribution of Y conditionally

on covariates X.Itisindexed by a parameter θ = θ(ω)thatisasubset or

function of the full-data parameter ω,andisrelated to the full-data model

via

p(y | x, θ(ω)) =



r∈R

p(y, r | x, ω),

where R is the sample space of R. 2

Agivenfull-data model will give rise to only one observed-data model, but

the converse of course is not true. Our motivation for starting with a full-data

model is to force the analyst to specify, through appropriate subject-matter

motivations, a model for the data generating process and its relationship to

the missingness indicators.

Inference about θ will depend crucially on choices made in specifying the

full-data model p(y, r | x, ω); however, observed data oﬀer no information to

validate these choices as they relate to the distribution of missing responses

mis

.Assumptions made by the data analyst will generally exert consider-

able inﬂuence over ﬁnal inferences, even for very large samples of observed

data. This point can sometimes be lost when using popular models based

on assumptions like missing at random (MAR) and implemented in standard

software packages. In many cases, this approach yields valid inference about

aparticular full-data model under speciﬁc assumptions about the joint distri-

bution of Y and R and/or constraintsontheparameter ω.

As an example, many software packages now ﬁt random eﬀects models to

incomplete data using maximum likelihood, under the assumption that both

the random eﬀects and the residual error distributions are normal (e.g. PROC

MIXED in SAS or xtreg in Stata). When the data are incomplete, the analyst

is implicitly assuming that the model speciﬁed in the software routine applies

to the full-data response, and that missingness is MAR (or more speciﬁcally,

ignorable;seeDeﬁnition 5.12).

5.3.2 Missing data mechanisms

In general terms, the missing data mechanism is the stochastic mechanism

leading to missingness among elements of Y .Formally, we will use the term

missing data mechanism to refer to the conditional distribution of missing

data indicators R given the full-data response Y and covariates X,denoted

by p(r | y, x, ω). Generally speaking, speciﬁcation of p(y, r | x, ω)will imply a

missing data mechanism; in practice, the modeler often begins with a working

ASSUMPTIONS ABOUT MISSING DATA MECHANISM 91

assumption about the missing data mechanism and uses it to specify the full-

data model.

Deﬁnition 5.7. Missing data mechanism.

The missing data mechanism is the model for the joint distribution of miss-

ing data indicators R as a function of Y and X;itisindexed by a ﬁnite-

dimensional parameter ψ = ψ(ω)andwritten as p(r | y, x, ψ(ω)). 2

Any full-data model can therefore be factored as the product of a full-data

response model and the associated missing data mechanism,

p(y, r | x, ω)=p(y | x, θ(ω)) p(r | y, x, ψ(ω)). (5.1)

Being able to characterize the missing data mechanism does not depend on

whether the full-data model has been speciﬁed according to (5.1), which is

commonly referred to as a selection model factorization. The implied missing

data mechanism can, in principle, be derived from any speciﬁcation of the

full-data model, but it may not always take a closed form.

5.4 Common assumptions about the missing data mechanism

Restrictions on the missing data mechanism can be classiﬁed as missing com-

pletely at random (MCAR), missing at random (MAR), or missing not at

random (MNAR); these progressively weaker assumptions delineate the de-

pendence of the missing data indicators R on the observed and missing parts

of the full-data response vector Y (conditionally on X and possibly also on

V ). Thetaxonomy and its terminology were developed by Rubin (1976), and

generalized using the ‘coarsening’ framework by Heitjan and Rubin (1991).

Robins et al. (1995), Jacobsen and Keiding (1995), Gill and Robins (1997),

and Gill and Robins (2001) develop formalizations for stochastic process and

longitudinal data settings. Recent surveys on the characterization of missing

data mechanisms in longitudinal studies, from a variety of diﬀerent perspec-

tives, can be found in Little (1995), Little and Rubin (2002), van der Laan

and Robins (2003), Hogan et al. (2004b), Tsiatis (2006), and Molenberghs

and Kenward (2007).

To deﬁne missing data assumptions, it is useful to rewrite the missing data

mechanism as

p(r | y, x, ψ)=p(r | y

obs

, y

mis

, x, ψ)

to emphasize dependence of R on the observedandmissingcomponents of Y .

Although we write ψ,itisassumed that ψ = ψ(ω) unless stated otherwise.

5.4.1 Missing completely at random (MCAR)

Missingness completely at random occurs when data are missing for reasons

wholly unrelated to either the observed or missing parts of Y ,conditionally

92 MISSING DATA MECHANISMS

on X.Withrepeated measurements, the data can be used to provide evidence

against the MCAR assumption through the association between R and Y

obs

;

e.g., by regressing R

on one or more observed components of (Y

,...,Y

j−1

)

(see also Chen and Little, 1999). In general, however, analyses under the

weaker MAR assumption will provide valid inference when MCAR holds. We

include a description here for the purposes of completeness and to introduce

MAR and MNAR.

Deﬁnition 5.8. Missing completely at random.

Missing responses are missing completely at random (MCAR) if, for all x

and ψ,

p(r | y, x, ψ)=p(r | x, ψ);

i.e., if p(r | y, x, ψ)isaconstantfunction of y. 2

One implication of MCAR is that missingness can be fully explained by

covariates X that are included in the full-data model. Another is that the

full-data distribution can be factored as

p(y, r | x, ω)=p(y | x, ω) p(r | x, ω),

meaning that the observed and missing response data have the same distribu-

tion, conditionally on X.Under MCAR, valid inference about the full-data

response distribution can be based solely on those with complete response

data. Even though this is a valid approach, it discards data on those without

complete response and therefore does not make optimal use of the available

data; the posterior variance of θ may be higher than necessary.

Example 5.3. Missing completely at random without covariates.

Alongitudinal cohort study of school performance will record test scores each

year for 2 years of secondary school; these are Y

and Y

.Intheﬁrst year,

1000 students are sampled and their test scores are recorded. In the second

year, budget constraints force the investigators to reduce the sample size to

800. Data on Y

are collected for a random subsample of the original 1000

students. Because the subsample is randomly drawn, missing responses on Y

are missing completely at random. 2

Example 5.4. Missing completely at random with covariates.

In the same study, suppose the investigators are interested in comparing test

scores between boys and girls, and let X denote gender. The distribution of

interest is p(y

| x, θ). Suppose further that the random subsample at time

2oversamplesgirls, so that girls are more likely to have Y

observed. Because

gender is a model covariate, the missing data on Y

are missing completely at

random. 2

The MCAR assumption usually is not realistic for longitudinal studies be-

cause unplanned missingness is so common. Consider the ﬁrst CTQ study of

ASSUMPTIONS ABOUT MISSING DATA MECHANISM 93

smoking cessation described in Section 1.4, where Y =(Y

,...,Y

)

are the

cessation indicators measured weekly over the course of the study, X is the

binary indicator of treatment, and R =(R

,...,R

)

are the binary miss-

ingness indicators. The full-data model of interest is p(y | x, θ), which charac-

terizes the joint distribution of smoking outcomes conditionally on treatment

group X.TheMCARassumption states that p(r | y

obs

, y

mis

,x,ψ)=p(r |

x, ψ); in this context, it means that probability of nonresponse can depend on

treatment group, but within treatment group, nonresponse is completely in-

dependent of cessation outcomes. Under the plausible (and testable) scenario

that within treatment group, participants observed to be heavier smokers are

more likely to drop out, the MCAR assumption would not hold.

5.4.2 Missing at random (MAR)

Amorerealistic condition for many longitudinal studies is missing at random

(MAR), which requires that missingness is independent of missing responses

mis

,conditionally on observed responses Y

obs

and model covariates X.

Deﬁnition 5.9. Missing at random.

Missing responses are missing at random (MAR) if, for all y

obs

, x and ψ,

p(r | y

obs

, y

mis

, x, ψ)=p(r | y

obs

, x, ψ);

i.e., if p(r | y

obs

, y

mis

, x, ψ)isaconstantfunction of y

mis

. 2

The MAR assumption is one component of the ignorability condition, which

allows valid inference about θ to be based on the likelihood function for Y

obs

Ignorability is discussed in detail in Section 5.7.

Example 5.5. Amissing at random mechanism.

Continuing with Examples 5.3 and 5.4, if those with lower values of the ﬁrst

test score Y

are less likely tositforthesecondtest,thenmissingness in Y

depends on Y

.Ifmissingness further depends on X, but does not depend on

any other variable including Y

,thenthe missing data mechanism is MAR. 2

Application of the MAR condition for dropout mechanisms requires some

more development and is discussed in more detail in Section 5.5.

5.4.3 Missing not at random (MNAR)

Although the MAR assumption is fairly general in that it allows the missing

data mechanism (and sometimes the missing data itself) to be explained by

observables, there are cases where MAR may fail to hold. MAR will not be

valid, for example, when the probability of missingness depends on the value of

the missing response — or on other unobservables — even after conditioning

on observed data. We refer to mechanisms of this type as missing not at

random (MNAR).

94 MISSING DATA MECHANISMS

Deﬁnition 5.10. Missing not at random.

Missing responses are missing not at random (MNAR) if, for some y

mis

=



mis

p(r | y

obs

, y

mis

, x, ψ) = p(r | y

obs

, y



mis

, x, ψ);

i.e., if R depends on some part of Y

mis

,evenafter conditioning on Y

obs

and X. 2

Example 5.6. Missing not at random mechanism: dropout in a smoking ces-

sation study.

Returning to the smoking cessation trial, dropout is an inevitable complica-

tion. Consider a simpliﬁed version where the full data consist of cessation

outcomes (Y

), treatment indicator X,andmissing data indicator R.Un-

der MAR,

P (R =1| X, Y

=1)=P(R =1| X, Y

=0),

which implies that, conditional on treatment group and time-1 smoking sta-

tus Y

,the dropout probability is the same for both smokers and nonsmokers

at time 2. In smoking cessation studies,thisassumption will not hold if the

dropout probability diﬀers by Y

(or equivalently, if smoking status Y

diﬀers

between dropouts and non-dropouts having the same (X, Y

)proﬁle).Licht-

enstein and Glasgow (1992) cite empirical evidence that in smoking cessation

studies, participants followed up after dropout are more likely than not to

have experienced relapse, raising the possibility that MAR is not plausible

here. 2

5.4.4 Auxiliary variables

Recall that the full-data response model is p(y | x, θ(ω)), and primary interest

is in parameters governing this distribution. We have implicitly assumed up

to this point that (Y , R, X)constitutestheonlydatathat potentially are

available.

In many cases, however, investigators may have access to a set of auxil-

iary variables,denoted by V ,thatmaybecorrelated with one or more of

(Y , R, X). Examples of auxiliary variables include covariates that are ob-

served but not of direct interest, or other longitudinal responses that are

measured concurrently with Y .Inaclinical trial, the model covariates X

would include only treatment indicators and possibly stratiﬁcation variables;

however, other covariates might be available. In an HIV cohort study where

longitudinal CD4 counts are the response of interest, the investigator may also

have access to viral load measurements taken at the same time points.

If the primary responses are completely observed, the modeler generally

has little or nothing to gain by using these auxiliary variables to estimate

ASSUMPTIONS ABOUT MISSING DATA MECHANISM 95

covariate eﬀects or other parameters;however, when data are incomplete,

auxiliary variables can be helpful to the extent that they explain variation in

the joint distribution of Y and R.Missingness mechanisms can be deﬁned

conditionally on auxiliary variables as well.

Deﬁnition 5.11. Auxiliary variable MAR (A-MAR).

Missing responses are A-MAR if MAR holds conditionally on auxiliary vari-

ables V ; i.e., if, for all y

obs

, x, z,andψ,

p(r | y

obs

, y

mis

, x, v, ψ)=p(r | y

obs

, x, v, ψ).

Clearly, A-MAR does not imply MAR. Consequently, when primary interest

is in the full-data response model p(y | x, θ)—whichdoesnot involve v

—itisnecessary to specify a full-data model conditional on v and integrate

it out to obtain the unconditional model. Speciﬁcally, the full-data model of

(Y , R | X)is

p(y, r | x, ω)=



p(y, r, v | x, ω) dv. (5.2)

Returning to the longitudinal HIV example, suppose Y is longitudinal CD4

count and V is longitudinal viral load, and that there is appreciable miss-

ingness in CD4. The MAR condition requires the analyst to assume that

conditional on observed CD4 history, missingness is unrelated to the CD4

count that would have been measured; this may be unrealistic. Further sup-

pose that the investigator can conﬁdently specify a joint model for CD4 count

and viral load (e.g., based on knowledge of disease progression dynamics), and

that viral load is thought to explain suﬃcient variability in CD4 count that

one could use it to predict missing responses. In other words, the analyst is

willing to assume A-MAR, or MAR conditional on viral load.

This approach requires specifying the joint distribution of (Y , R, V | X)

under the integral sign in (5.2). One possibility is to write

p(y, r, v | x, ω)=p(y, v | x, ω) p(r | y, v, x, ω),

where the ﬁrst factor is the joint model for CD4 and viral load, the second

is the missing data mechanism. As we will see in Chapter 7, incorporating

auxiliary covariates can sometimes be simpler than it would appear. Under

A-MAR (and assuming ignorability, see Section 5.7), the missing data mech-

anism can be left unspeciﬁed, although the joint model for Y and V does

have to be speciﬁed. Section 6.6 describes some speciﬁc models that can be

used for this purpose; in Section 7.5, we illustrate by analyzing data from the

CTQ II trial, where the responses are smoking cessation outcomes measured

weekly for 8 weeks, and the auxiliary process is individual weight, recorded

concurrently.

96 MISSING DATA MECHANISMS

5.5 Missing at random applied to dropout processes

The deﬁnitions of MCAR, MAR, and MNAR given in the previous section

were general. In this section we describe the implications of the MAR as-

sumption for handling monotone missing data patterns caused by dropout.

Although the MAR condition is rather simple to state and interpret for cross-

sectional data, it is somewhat less transparent when applied to longitudinal

data, particularly if missingness patterns are non-monotone. When missing-

ness follows a monotone pattern, as when dropout is the sole cause of missing

data, it turns out that MAR has an intuitive representation in terms of the

hazard of dropout, with obvious analogies to the cross-sectional setting.

Our discussion assumes a discrete setofmeasurementtimes within indi-

vidual, and assumes dropout happens at one of the scheduled times, t

,...,t

Recall that the missingness indicator at time t

is R

,withR

= I(Y

observed).

Hence dropout time, denoted by U ,isU =1+



(see also Deﬁnition 5.4).

Further deﬁne, for each individual,

= {Y

,...,Y

}

to be the subset of the full-data vector Y that would be observed up to and

including measurement time j (i.e., the response history up to and includ-

ing t

Under monotone dropout, the MAR condition states that

P (U = j |

)=P (U = j | Y

j−1

or that the marginal probability of dropping out at t

cannot depend on

measurements at or beyond t

(Molenberghs et al., 1998). In practical settings,

this version of MAR can be somewhat diﬃcult to interpret because it makes

reference to the marginal probability of dropping out at t

, P (U = j), rather

than the more intuitive hazard function, P (U = j | U ≥ j). The marginal

dropout rate is the probability of dropping out at t

among all those who

beginfollow-up, while the hazard rate is the probability of dropping out at t

among those still eligible to be measured at t

At time t

and for any given response history Y

= {Y

,...,Y

},let

h(t

| Y

)=P (U = j | U ≥ j, Y

)

denote the hazard of dropout at t

conditional on Y

.Itturns out thatmono-

tone missingness caused by dropout is MAR if and only if the hazard of

dropout at t

depends only on observed response history Y

j−1

,andcondition-

ally on

j−1

is independent of present and future outcomes {Y

j+1

,...,Y

Proposition 1. Under monotone dropout, missingness is MAR if and only

if for all j,thehazard of dropout at t

is independent of current and future

responses Y

,...,Y

conditionally on pastresponses Y

,...,Y

j−1

Proof: Recall that under MAR, P (U = j |

)=P (U = j | Y

j−1

). By deﬁni-

MAR AND DROPOUT 97

tion, the hazard can be written in terms of the marginal dropout probabilities

h(t

| Y

P (U = j |

)

1 −



j−1

l=1

P (U = l | Y

)

for any (j, k)suchthat 1 ≤ j ≤ J and 1 ≤ k ≤ J.Hence, under MAR, we

have

h(t

| Y

P (U = j |

)

1 −



j−1

l=1

P (U = l | Y

)

mar

P (U = j |

j−1

)

1 −



j−1

l=1

P (U = l | Y

l−1

)

=function of Y

,...,Y

j−1

only.

Conversely, suppose the hazard of dropout at t

given Y

depends only on

j−1

;thatis,

P (R

=0| R

= ···= R

j−1

=1, Y

)

= P (R

=0| R

= ···= R

j−1

=1, Y

j−1

To show this implies MAR, we need to demonstrate that P (U = j |

)

is a constant with respect to Y

,...,Y

,orequivalently, is a function of

,...,Y

j−1

only. Writing it in terms of the hazard, we have

P (U = j |

)=P (R

= ···= R

j−1

=1,R

=0| Y

)

= P (R

=0| R

= ···= R

j−1

=1, Y

)

× P (R

j−1

=1| R

j−2

= ···= R

=1, Y

)

× ···×P (R

=1| Y

)

= h(t

| Y

)

j−1



k=1

{1 − h(t

| Y

)}. (5.3)

But if h(t

| Y

)=h(t

| Y

j−1

), then the probability of dropout at j,given

by (5.3), reduces to

P (U = j |

)=h(t

| Y

j−1

)

j−1



k=1

{1 − h(t

| Y

k−1

)}

=function of Y

,...,Y

j−1

only.

This result is useful for making the MAR assumption more transparent;

dropout is really an event process, anditismoreintuitive to think about

imposing conditions on hazard of dropout rather than on its marginal prob-

ability. It also is helpful in writing down the observed data likelihood under

MAR, which we discuss in Section 5.7

98 MISSING DATA MECHANISMS

In the foregoing, we have not discussed the role of covariates, and in par-

ticular time-varying covariates. Theseresultswill be expected to hold when

exogenous covariates are included in the full-data model; i.e., covariates meet-

ing the following conditions (see also Section 2.7):

1. Time-invariant covariates observed at baseline;

2. Time-varying covariates that are a deterministic function of baseline

value (e.g., age);

3. Stochastic time-varying covariates that are external to the response pro-

cess, and can be expected to be observed throughout the measurement

period (e.g., when air pollution level measured from a monitoring station

is a covariate, and health status of an individual is the response process;

the individual may drop out of the study, but air pollution will continue

to be measured.).

5.6 Observed data posterior of full-data parameters

Thus far, we have described processes that cause data to be missing, distin-

guished full from observed data, described the Rubin taxonomy for missing

data mechanisms, and showed how it applies to monotone missingness in-

duced by dropout. In particular we showed that in monotone patterns, the

MAR condition is equivalent to assuming the hazard of dropout at some time

depends only on observed response data up to and including time t

j−1

We are now ready to describe formal methods fordrawingposterior inference

about full-data parameters from observed (incomplete) data.

When using incomplete data to draw posterior inference about aspects

of the full data, constraints or model speciﬁcations must be imposed on a

full-data model and on priors for parameters indexing that model. The full-

data model p(y, r | ω), including its assumptions and constraints, will encode

both the full-data response model p(y | θ(ω)) and the missing data mecha-

nism p(r | y, ψ(ω)). Because all components of ω may not be identiﬁed, the

prior distribution p(ω)will directly inform certain aspects of the posterior

distribution p(ω | y

obs

, r). We do not conﬁne ourselves in this discussion to

aparticularfactorization or speciﬁcation ofthefull-data model; the setup

described here applies to any full-data model. Posterior inference under a va-

riety of full-data model speciﬁcations is described and discussed throughout

the remainder of this chapter.

By deﬁnition, the posterior distribution of the full-data parameter ω,given

observed data (y

obs

, r), is

p(ω | y

obs

, r)=

p(y

obs

, r | ω) p(ω)



p(y

obs

, r | ω) p(ω)dω

p(y

obs

, r | ω) p(ω)

p(y

obs

, r)

. (5.4)

IGNORABILITY ASSUMPTION 99

(For clarity we will drop covariate dependence, but it is implied throughout.)

The observed-data likelihood,proportional in ω to p(y

obs

, r | ω), is obtained

by averaging the full-data model over all possible realizations of the missing

data,

L(ω | y

obs

, r) ∝ p(y

obs

, r | ω)



p(y

obs

, y

mis

, r | ω) dy

mis

Because the marginal distribution p(y

obs

, r)inthedenominator of (5.4) is a

constant with respect to ω,theobserved-data posterior is proportional in ω

to the product of the full data prior p(ω)andtheobserved-data likelihood,

p(ω | y

obs

, r) ∝ p(ω) L(ω | y

obs

, r).

In the following sections we describe speciﬁc strategies for observed-data pos-

terior inference about full-data parameters. In general the missing data distri-

bution is not identiﬁable, and assumptions are needed for the full-data model

and informative priors need to be speciﬁed. When missingness is MAR, the

ignorability assumption provides considerable simpliﬁcation of the observed-

data posterior. We deﬁne this assumption in the next section.

5.7 The ignorability assumption

5.7.1 Likelihood and posterior under ignorability

The ignorability condition, ﬁrst described by Rubin (1976), can be used to

facilitate posterior inference without having to specify the missing data mech-

anism. An ignorable missing data mechanism meets three conditions, given

here in its deﬁnition.

Deﬁnition 5.12. Ignorable missing data mechanism (Little and Rubin, 2002).

Amissing data mechanism is said to be ignorable for the purposes of posterior

inference if

1. The missing data mechanism is MAR.

2. The full data parameter ω can be decomposed as ω =(θ, ψ), where θ

indexes the full-data response model p(y | θ)andψ indexes the missing

data mechanism p(r | y, ψ).

3. The parameters θ and ψ are aprioriindependent; i.e.,

p(θ, ψ)=p(θ)p(ψ).

An important consequence of ignorability is that inference about θ —typ-

ically of direct interest — can be based on a likelihood function that is pro-

portional in θ to the observed-data response model,

p(y

obs

| θ)=



p(y

obs

, y

mis

| θ) dy

mis