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

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220 INFORMATIVE PRIORS AND SENSITIVITY ANALYSIS

The prior for ξ

may be informed by a credible source of external infor-

mation, such as expert opinion, historical data, or some combination of the

two. The Bayesiansetupisideal for this because we can, at least in principle,

quantify uncertainty about assumptions through these priors.

Acommon practice for assessing sensitivity to missing data assumptions

is to compare inferences about a parameter of interest under several diﬀerent

full-data models that are not compatible in their ﬁt to the observed data. For

example, one might compare inferences under a selection model, a pattern

mixture model, and a shared parameter model. It is also common to com-

pare nested models that have diﬀerent assumptions about the missing data

mechanism, but also have diﬀerent observed data likelihoods (for example by

ﬁtting model (9.2) and comparing it to the nested model with ψ

=0).By

contrast, principles 1.(b) and 1.(c) emphasize that model parameterizations

should allow for exploration of sensitivity to missing data assumptions along

acontinuum of model speciﬁcations that have similar or identical ﬁts to the

observed data.

9.3 Parameterizing the full-data model

Recall that any full-data model has an associated extrapolation factorization

(9.1). The component p(y

mis

| y

obs

, r, ω

)istheextrapolation distribution,

or the distribution of the missing data given the observed data; without para-

metric assumptions, it cannot be identiﬁed by observed data. The component

p(y

obs

, r | ω

)istheobserved data distribution and is proportional in ω to

the observed data likelihood.

We seek a parameterization ξ(ω)=(ξ

, ξ

)suchthat ξ

is identiﬁed

by observed data and ξ

is a sensitivity parameter. Sensitivity analysis and

elicitation of informative priors is then be based on functions of ξ

alone

(although this may not be possible in all full-data models).

To make inferences under MNAR, we construct a prior for ξ

and draw

inference about the full-data distribution under this prior (Rubin, 1977). In

practice we will frequently introduce a redundant parameter (or vector of

parameters) ∆ that captures explicitly the departures from MAR. The prior

on ξ

can then be expressed in terms of ∆ and ξ

.Thefollowing example

provides a concrete illustration.

Example 9.1. Mixture model parameterization in terms of (ξ

, ξ

Consider a mixture model for univariate full-data response Y such that

Y | R =1 ∼ N(µ

(1)

,σ

)

Y | R =0 ∼ N(µ

(0)

,σ

)

R ∼ Ber(φ),

where, as usual, R =1indicates that Y is observed and R =0indicates

PARAMETERIZING THE FULL-DATA MODEL 221

missingness. Suppose the target of inference is θ = E(Y ); in terms of model

parameters,

θ = φµ

(1)

+(1− φ)µ

(0)

This model assumes observed and missing responses diﬀer in their mean

but share a common variance. The constraint on the variance is a modeling

assumption that cannot be veriﬁed by observed data. However, we can still

parameterize the model in terms of a sensitivity parameters ξ

that satisﬁes

Deﬁnition 8.1.

The vector of full-data parameters is

ω =(µ

(0)

,µ

(1)

,σ,φ).

Deﬁne ξ(ω)=(ξ

, ξ

)suchthat

= µ

(0)

=(µ

(1)

,σ,φ).

The sensitivity parameter ξ

does not appear in the observed data likelihood;

hence, the model has equivalent ﬁt to the observed data across all values of

.Itiseasytoshowthat MAR holds when µ

(0)

= µ

(1)

;or,interms of the

sensitivity parameter, when ξ

= µ

(1)

.To‘center’ the model at MAR, we

introduce a function h such that

= h(ξ

, ∆)

= µ

(1)

+∆. (9.3)

MAR holds when ∆ = 0. Departures from MAR are represented by varying

∆awayfrom0.Because ∆ does not appear in the observed-data likelihood,

varying the missing data mechanism by moving smoothly through values of

∆in(9.3) has no eﬀect on ﬁt to the observed data. The eﬀect of missing data

assumptions on inferences about θ = E(Y )canbeseen by expressing θ as a

function of ∆; i.e.,

θ(∆) = φµ

(1)

+(1− φ)µ

(0)

= φµ

(1)

+(1− φ)(µ

(1)

+∆)

= µ

(1)

+(1− φ)∆.

The ‘sensitivity’ to departures from MAR — or indeed from any missing

data mechanism characterized by ∆ = ∆

∗

—canbemeasured in terms of

the derivative ∂θ(∆)/∂∆evaluatednear ∆ = ∆

∗

.InExample 9.1, θ(∆) is

linear and its derivative is the proportion of missing data, 1 − φ;however, the

same principle applies in more complex models (see also Troxel et al., 2004

and Zhang and Heitjan, 2006).

In the frequentist setting, a typical sensitivity analysis (see, e.g., Scharf-

stein et al., 1999) ﬁts the full-data model at ﬁxed values over a range of ∆,

222 INFORMATIVE PRIORS AND SENSITIVITY ANALYSIS

and examines how the inference of interest varies with ∆. For example, if the

target parameter is θ,thesensitivity analysis might comprise point estimates

andconﬁdence intervals for θ(∆) over a range for ∆. Viewed in the Bayesian

context, this corresponds to summarizing conditional posteriors, where con-

ditioning is done on ﬁxed values of ∆. It can also be viewed as examining

sensitivity to the prior, where all the priors under consideration are point

masses.

More generally, one can use priors for ∆ that are centered at a speciﬁc

value and convey uncertainty about the missing data assumption by assigning

nonzero prior variance. This approach is used by Nandram and Choi (2002b)

in selection models with binary data, and by Rubin (1977) in mixture models

for univariate response. In the next two sections, we provide details on how

to specify priors in both selection and mixture models, using the principles

articulated above.

9.4 Specifying priors

Let h(ξ

, ∆)beaone-to-onefunction such that

= h(ξ

, ∆),

where ∆ captures the information about the missing data mechanism. To

anchor the model at MAR, there must exist some ∆

such that h(ξ

, ∆

)

implies MAR. Typically we can parameterize the model such that ∆

= 0;

examples can be found in Chapter 8 (Examples 8.4, 8.5, and 8.6); see also

Example 9.1.

It is frequently useful to specify the joint prior p(ξ

, ξ

)as

p(ξ

, ξ

)=p(ξ

| ξ

) p(ξ

), (9.4)

where p(ξ

| ξ

)capturesassumptions about the missing data mechanism.

In terms of the sensitivity parameter ∆,wecanre-express p(ξ

| ξ

)as

p(ξ

| ξ

)=p(ξ

| ξ

, ∆) p(∆ | ξ

), (9.5)

where

p(ξ

| ξ

, ∆)=I{ξ

= h(ξ

, ∆)}

is a point mass prior; that is, ξ

is a deterministic function of (ξ

, ∆)as

above.

The ﬁrst partoftheprior in (9.5) simply reparameterizes ξ

in terms of

departures from MAR; the second part, p(∆ | ξ

), encodes the missing data

mechanism and the uncertainty about it. A point mass prior

p(∆ | ξ

)=I{∆ = ∆

∗

}

for a ﬁxed value ∆

∗

encodes a particular missing data mechanism such as

SPECIFYING PRIORS 223

MAR with absolute certainty. For example, if ∆ = 0 implies MAR, then

speciﬁc MNAR mechanisms can be assumed by setting p(∆ | ξ

)=I{∆ =

∆

∗

} for some ∆

∗

= 0.

Alternatively, priors that convey uncertainty about the missing data mech-

anism can be used.Anon-degenerate prior such as

∆ | ξ

∼ N (d, D),

where d and D are hyperparameters giving the prior mean and variance for

∆,canbeusedtocenter the missing data mechanism at ∆ = d,withprior

uncertainty quantiﬁed through a variance matrix D.

More generally, when ∆ = ∆

implies MAR, we can use the prior mean

and variance as follows (using the case of scalar ∆):

MAR with no uncertainty E(∆ | ξ

)=∆

var(∆ | ξ

)=0

MAR with uncertainty E(∆ | ξ

)=∆

var(∆ | ξ

) > 0

MNAR with no uncertainty E(∆ | ξ

)=∆

∗

=∆

var(∆ | ξ

)=0

MNAR with uncertainty E(∆ | ξ

)=∆

∗

=∆

var(∆ | ξ

) > 0

To illustrate, we revisit Example 9.1.

Example 9.2. Prior speciﬁcations for univariate mixture of normals (con-

tinuation of Example 9.1).

Recall that the full-data model follows the mixture distribution

Y | R =1 ∼ N(µ

(1)

,σ

)

Y | R =0 ∼ N(µ

(0)

,σ

)

R ∼ Ber(φ),

and that the full-data parameters are partitioned as

=(µ

(1)

,σ,φ)

= µ

(0)

Let

= h(ξ

, ∆) = µ

(1)

+∆ (9.6)

so that ∆ = 0 coincides with MAR.

Following (9.4) and (9.5), we can factor the prior as

p(ξ

, ξ

, ∆) = p(ξ

| ξ

, ∆) p(∆ | ξ

) p(ξ

)

= p(µ

(0)

| µ

(1)

,σ,φ,∆) p(∆ | µ

(1)

,σ,φ) p(µ

(1)

,σ,φ).

To center the model at MAR with uncertainty about MAR governed by σ,

we can use the priors

(0)

| µ

(1)

,σ,φ,∆ ∼ I{µ

(0)

= µ

(1)

+∆}

∆ | µ

(1)

,σ,φ ∼ N (0,σ

)(9.7)

224 INFORMATIVE PRIORS AND SENSITIVITY ANALYSIS

and let p(µ

(1)

,σ,φ)=p(µ

(1)

)p(σ)p(φ), with vague priors for each.

To assume MAR with absolute certainty, we replace the N(0,σ

)priorin

(9.7) with the point mass I{∆=0}.Weseefrom(9.6)thatdepartures from

MAR are measured in terms of µ

(1)

− µ

(0)

; i.e., diﬀerences in mean response

between observed and missing Y .MNAR mechanisms can be represented in

(9.7) by using a distribution or point mass centered away from zero. For

example, setting

∆ | µ

(1)

,σ,φ ∼ I{∆=σ}

states that µ

(0)

diﬀers from µ

(1)

by one standard deviation. Fixed constants

also can be used, as can prior distributions with nonzero variance. 2

In the next two sections, we discuss how to apply this framework in mixture

models and in selection models.

9.5 Pattern mixture models

9.5.1 General parameterization

The factorization that deﬁnes pattern mixture models makes them a natural

class of models for partitioning ω into its identiﬁed and nonidentiﬁed compo-

nents. To see this, recall that the PMM factorization is

p(y, r | ω)=p(y | r, α(ω)) p(r | φ(ω)). (9.8)

From here,weassume ω =(α, φ)anddrop dependence of α and φ on ω.

Then,

p(y, r | α, φ)=p(y | r, α) p(r | φ)

= p(y

mis

| y

obs

, r, α) p(y

obs

| r, α) p(r | φ). (9.9)

Notice that the extrapolation model depends only on parameters α indexing

the mixture components p(y | r, α). This property makes the reparameteri-

zation of ω =(α, φ)asξ(ω)=(ξ

, ξ

)relatively straightforward in many

practical settings. To illustrate in a simple case, we use the mixture of bivariate

normals from Examples 5.9 and 5.11.

Example 9.3. Sensitivity parameterizations for mixture of bivariate normals.

From Example 5.11, recall that the full data are Y =(Y

)

and R =

I{Y

observed}.Assume

Y | R = r ∼ N (µ

(r)

, Σ

(r)

R ∼ Ber(φ).

Within pattern r,wecanreparameterize the distribution of Y as

| Y

,R= r ∼ N(β

(r)

+ β

(r)

,σ

(r)

2|1

| R = r ∼ N(µ

(r)

,σ

(r)

PATTERN MIXTURE MODELS 225

Hence, in termsof(9.9),

α =



(r)

,β

(r)

,σ

(r)

2|1

,µ

(r)

,σ

(r)

: r =0, 1



Now deﬁne ξ(ω)=ξ(α,φ)=(ξ

, ξ

)suchthat

=(β

(0)

,β

(0)

,σ

(0)

2|1

=(β

(1)

,β

(1)

,σ

(1)

2|1

,µ

(0)

,µ

(1)

,σ

(1)

,σ

(0)

,φ).

Notice that ξ

is exclusively a function of α.

The model can be centered at MAR as follows. Let ∆ =(∆

, ∆

)

with ∆

> 0, and deﬁne the function h(ξ

, ∆)as

(ξ

, ∆)=β

(1)

+∆

(ξ

, ∆)=β

(1)

+∆

(ξ

, ∆)=∆

(1)

2|1

By setting ξ

= h(ξ

, ∆), we have

(0)

= β

(1)

+∆

(0)

= β

(1)

+∆

(0)

2|1

=∆

(1)

2|1

Setting (∆

, ∆

)=(0, 0, 1) yields MAR. Priors on ∆ can be used to

encode alternate missing data assumptions. 2

9.5.2 Using model constraints to reduce dimensionality of sensitivity

parameters

In the previous example, a relatively simple model yields 3 sensitivity param-

eters. As the dimension increases, the number of sensitivity parameters will

grow quickly. In the next example, weshowhowtouse a simple modeling

constraint to reduce the dimension of ∆ to 1. Our analysis of the Growth

Hormone data in Chapter 10 takes a similar approach for a trivariate normal

distribution.

Example 9.4. Parameterization and prior speciﬁcations for mixture of bi-

variate normals having common variance structure.

Continuing with Example 9.3, if we constrain Σ

(r)

= Σ,partofthefull-data

is identiﬁed by parameter constraints. It is easy to show that

(1)

= β

(0)

= β

(1)

2|1

= σ

(0)

2|1

= σ

2|1

(1)

= σ

(0)

= σ

226 INFORMATIVE PRIORS AND SENSITIVITY ANALYSIS

Hence,

= β

(0)

=(β

(1)

,β

,σ

2|1

,σ

,φ).

To center the model at MAR, let ξ

= β

(0)

= h(ξ

, ∆), where

h(ξ

, ∆) = β

(1)

+∆.

The prior speciﬁcation can follow (9.4) and (9.5); i.e.,

p(ξ

, ξ

, ∆) = p(ξ

| ξ

, ∆) p(∆ | ξ

) p(ξ

For the ﬁrst factor, we have

(0)

| ∆, ξ

∼ I{β

(0)

= β

(1)

+∆}.

It may be useful (but not necessary) tocalibrate the prior for ∆ given ξ

using identiﬁed model parameters; for example,

∆ | ξ

∼ N(d, σ

2|1

If d =0,theprior is centered at MAR. The conditional variance σ

2|1

var(Y

| Y

)ischosenbecause the sensitivity parameter ξ

= β

(0)

characterizes

the conditional mean E(Y

| Y

,R=0).Finally, priorsforcomponents of ξ

can be speciﬁed in the usual ways (e.g., by assuming independence between

components and assigning vague or ﬂat priors). 2

The general strategy illustrated by Example 9.4 — to specify a constrained

full-data model in conjunction with a parameterization that admits sensitivity

parameters — is used for analysis of the Growth Hormone study and the

OASIS study in Chapter 10.

9.6 Selection models

The extrapolation factorization (9.1) is not generally easy to derive in closed

form for parametric selection models. In most parametric selection models

(Section 8.3), all full-data parameters are identiﬁed, including the parameters

of the extrapolation distribution. The parameters ω

are complex func-

tions of the missing data mechanism parameters ψ and the full-data response

model parameters θ,anditis not generally possibletoﬁnd a reparameteri-

zation ξ(ω)=ξ(θ, ψ)=(ξ

, ξ

)thatsatisﬁesthe criteria of Deﬁnition 8.1.

An alternative is to use semiparametric selection models. By semiparamet-

ric we mean thatthefull-data distribution p(y | θ)isleftunspeciﬁed, but

the missing data mechanism p(r | y, ψ)mayhaveaparametric component.

If the distribution p(y | θ)hasdiscrete support(aswhenY is categorical),

then semiparametric models will usually admit sensitivity parameters. When

SELECTION MODELS 227

Y is continuous, the situation is somewhat more complicated, but candidate

sensitivity parameters can often still be found.

In either case, model parameterization has in important bearing on prior

speciﬁcation and posterior inference. We use three examples to illustrate. The

ﬁrst two illustrate the importance of model parameterization with binary re-

sponse. The third shows how to parameterize a selection model for a contin-

uous univariate response.

Example 9.5. Prior speciﬁcation in nonparametric and semiparametric se-

lection models for longitudinal binary data.

In this example we revisit the nonparametric selection model for bivariate

binary data with missingness in the second observation; this model was intro-

duced in Section 8.3.7. Let Y =(Y

)

denote the full-data response, with

R =1ifY

is observed and R =0ifmissing. The entire full-data distribution

p(y,r | ω)canbeenumerated as in Table 9.1. The parameters ω

(1)

,...,ω

(1)

corresponding to R =1,allareidentiﬁable from the observeddata. The sums

(0)

= ω

(0)

+ ω

(0)

= ω

(0)

+ ω

(0)

corresponding to R =0,alsoareidentiﬁable. Let

=(ω

(1)

,ω

(1)

,ω

(1)

,ω

(1)

,ω

(0)

,ω

(0)

denote the set of identiﬁed parameters in this multinomial distribution. Before

moving to the discussion of the selectionmodel,notethatthis model admits

asetofsensitivity parameters. Referring to Deﬁnition 8.1, if we set ξ

= ω

then either ξ

=(ω

(0)

,ω

(0)

)orξ

=(ω

(0)

,ω

(0)

)meets the criteria for a

Table 9.1 Multinomial parameterization of full-data distribution for bivariate binary

data with possibly missing Y

p(y

,r | ω)

00 0 ω

(0)

00 1 ω

(0)

01 0 ω

(0)

01 1 ω

(0)

10 0 ω

(1)

10 1 ω

(1)

11 0 ω

(1)

11 1 ω

(1)

228 INFORMATIVE PRIORS AND SENSITIVITY ANALYSIS

sensitivity parameter. Essentially, the full-data distribution has two degrees

of freedom unaccounted for by observed data.

The corresponding selection model can be written as

∼ Ber(θ

)

| Y

∼ Ber(θ

2|1

)

R | Y

∼ Ber(π),

with

logit(θ

)=α

logit(θ

2|1

)=β

+ β

logit(π)=ψ

+ ψ

Priors are usually speciﬁed as

p(ψ, θ)=p(ψ)p(θ), (9.10)

where here, θ =(α, β)(Scharfstein et al., 2003). It is often further assumed

that

p(ψ)=p(ψ

,ψ

)p(ψ

,ψ

), (9.11)

i.e., the sensitivity parameters in the MDM are aprioriindependent of the

other MDM parameters. These independent priors do not (in general) im-

ply that p(ψ

,ψ

, ω

)—thecorresponding joint prior on the sensitivity pa-

rameters (ψ

,ψ

)andtheidentiﬁed parameters ω

—canbefactored as

p(ψ

,ψ

)p(ω

). The prior dependence between (ψ

,ψ

)andω

has implica-

tions for posterior inference. In particular, the prior and posterior for the

sensitivity parameters, (ψ

,ψ

)arenot(ingeneral) equivalent unlike in mix-

ture models. Before exploring the aprioridependence and the implications for

posterior inference, we note that this aprioridependence is desirable. Scharf-

stein et al. (2003) (p. 501) argue that it makes little sense to construct (or

elicit) independent priors on ω

and (ψ

,ψ

). But, by specifying independent

priors on θ and ψ,thedependence between ω

and (ψ

,ψ

)isinduced.

To see how ω

and (ψ

,ψ

)areaprioridependent, we explore the reparam-

eterization of the model from (ω

,ψ

)to(θ,ψ

,ψ

). The function

to move between these parameterizations is indexed by (ψ

,ψ

= h

,ψ

(θ,ψ

,ψ

We derive this function for the simple cross-sectional setting of a binary re-

sponse. Consider the selection model

Y ∼ Ber(θ)

logit{P (R =1| y)} = ψ

+ ψ

where R =1meansY is observed. In this model, ψ

is the sensitivity param-

SELECTION MODELS 229

eter. We denote the identiﬁed parameters as ω

=(θ

,α), where

= P (Y =1| R =1)

α = P (R =1).

With a little algebra, it can be shown that



= h

(θ

,α)=





αθ

1 − α

1+θ

exp(−ψ

)/(1 − θ

)

logit(α)+log{(1 − θ

)+exp(−ψ

)θ

}





The aprioridependence can be seen explicitly by deriving the Jacobian using

these results. For a related discussion, see Gustafson (2006).

As a result, aprioriindependence between the full data response model

parameters and the missing data mechanism parameters given in (9.10) and

between the sensitivity parameters and the other parameters in the missing

data mechanism given in (9.11) does not imply aprioriindependence between

and (ψ

,ψ

) (the sensitivity parameters). This lack of aprioriindepen-

dence between ω

and (ψ

,ψ

)implies that the posterior for the sensitivity

parameters is not equal to the prior; i.e.,

p(ψ

,ψ

| y

obs

, r) = p(ψ

,ψ

However, it is often still the case that

p(ψ

,ψ

| y, r) ≈ p(ψ

,ψ

). (9.12)

To see why the equality between prior and posterior does not hold here, we

write

p(ψ

,ψ

| y

obs

, r) ∝



p(y

obs

, r | ψ

,ψ

, ω

)p(ω

| ψ

,ψ

)p(ψ

,ψ

)dω



p(y

obs

, r | ω

)p(ω

| ψ

,ψ

)p(ψ

,ψ

)dω

= p(y

obs

, r | ψ

,ψ

)p(ψ

,ψ

The ﬁrst equality holds given that ω

parameterizes the observed data distri-

bution. When we integrate out ω

,weareleftwith an observed data distribu-

tion that depends on (ψ

,ψ

). However, in a nonparametric selection model,

it will typically be the case that

p(y

obs

, r | ψ

,ψ

)

p(y

obs

, r)

≈ 1.

That is, p(y

obs

, r | ψ

,ψ

)isalmostconstant with respect to (ψ

,ψ

). How

well (9.12) holds for diﬀerent semiparametric speciﬁcations is an area of on-

going work. 2

We illustrate the approximate equality given in (9.12) using the OASIS

data (described in Section1.6)inthenext example.