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

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260 CASE STUDIES: NONIGNORABLE MISSINGNESS

0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0 2.5

MAR

Mixture

Each expert

Figure 10.5 OASIS trial: posterior distribution of the treatment odds ratio ϕ under

priors for ∆ corresponding to MAR, expert-speciﬁc opinion, and a mixture of expert

opinion. The MAR prior places point mass at (∆

, ∆

)=(0, 0).

hand, mixture models are well suited because the parameter space is easier

to separate. The various pattern mixture models ﬁt here all have the same

observed-data likelihood.

Second, eliciting informative priors can be complex. A key component of

the elicitation is to develop tools for which expert opinion can be accurately

acquired. Here, experts were asked to condition their opinion about smoking

rate at time j among dropouts using hypothetical smoking rates for non-

dropouts, conditional on smoking status Y

at time j − 1. This helped the

experts calibrate their priors.

Finally, we saw that the (ignorable) selection model MAR analysis and

the MAR mixture model analysis were identical. This is not a general phe-

PEDIATRIC AIDS TRIAL 261

Table 10.8 OASIS trial: posterior inferences about time-speciﬁc smoking probabil-

ity, derived using informative priors elicited from experts. The MAR inferences are

shown for comparison.

Expert-Speciﬁc Posteriors

Treatment Month MAR MNAR 1 2 3 4

1.83.83.83 .83 .83 .83

ET 3 .87 .88 .87 .89 .89 .88

6.82.85.83 .87 .87 .85

12 .78 .84 .83 .87 .87 .83

1.85.85.85 .85 .85 .85

ST 3 .85 .87.86.87.87 .86

6.84.86.86 .87 .86 .86

12 .88 .90 .90 .91 .90 .90

Trt eﬀect ϕ .51 .65 .65 .62 .71 .68

(.2, 1.1) (.2, 1.4)

nomenon. The reason for their equivalence here is that with longitudinal bi-

nary data, a mixture of ﬁrst-order serial dependence models can also be repre-

sented as a single ﬁrst-order serial dependence model. Hence inferences about

the full data will be the same under MAR.

10.4 Pediatric AIDS trial: Mixture of varying coeﬃcient models

for continuous dropout

10.4.1 Overview

In this example we use a mixture of varying coeﬃcient models to analyze

data from the pediatric clinical trial described in Section 1.7. Individuals

were scheduled for follow-up every 3 months, but the actual measurement

times varied considerably from individual to individual. Dropout time was

ﬁxed at the timeofthelastobserved measurement. The observed CD4 data,

in the log scale, are shown in Figure 1.5, with several individual trajectories

highlighted. From the small random sample of individuals, we see that those

with longer follow-up tend to have higher overall CD4 counts and less steep

declines over time.

Our objective is to characterize the average CD4 count over time in both

treatment arms, and to draw inference about the diﬀerence in mean CD4 at

262 CASE STUDIES: NONIGNORABLE MISSINGNESS

the end of the study period. We assume that the average (log) CD4 is linear

in time with slope and intercept that depend on dropout time. The condi-

tional distributions (given dropout time) are averaged across the distribution

of dropout times separately by treatment, using a mixture of varying coeﬃ-

cient models (Section 8.4.4). The intercept and slope of CD4 will be given by

smooth but unknown functions β

(u)andβ

(u)ofdropouttime.

Figure 10.6 is an exploratory plot that can be used to motivate the mix-

ture model. It shows individual-speciﬁc OLS slope estimates plotted against

dropout time, and suggests a possibly nonlinear association between slope and

dropout time. However, it should be kept in mind that the slope estimates

for earlier dropoutsarebasedonfewer observations and tend to have higher

variance.

Inference from the mixture of VCM will be compared to inference under a

standard regression model that assumes MAR. Hogan et al. (2004a) analyze

these data using a similar model; the Bayesian formulation given here derives

mainly from Su and Hogan (2007), who provide a more extensive treatment,

including detailed sensitivity analysis.

050100 150 200

−15 −10 −5 0 5

050100 150 200

−15 −10 −5 0 5

Least Squares Slopes

follow-up time (weeks)follow-up time (weeks)

Figure 10.6 Pediatric AIDS study: OLS slopes vs. follow-up time for each treatment,

with unweighted lowess smoother ﬁt to each set of points.

PEDIATRIC AIDS TRIAL 263

10.4.2 Model speciﬁcation: CD4 counts

Centering and rescaling the time axis leads to a more interpretable treatment

eﬀect and more stable implementation of posterior sampling. Let t

and U

respectively, denote measurement time and dropout time in the original scale.

We use the shifted and rescaled time variables t =(t

− C)/L and U =

− C)/L,whereC = U

= 139 and L =max

.Thelength of the

rescaled time axis is equal to one and it is centered at the (sample) average

follow-up time.

The model used here is described more generally in Section 8.4.4; the spe-

ciﬁc formulation is similar to Example 8.8. Treatment assignment is denoted

by Z (1 = high dose AZT, 0 =lowdoseAZT).Lett denote (rescaled) time

elapsed from beginning of follow-up, let Y (t)=log{CD4(t)+1},andletU

denote (rescaled) length of follow-up (i.e., dropout time). Our objective is to

characterize the distribution of Y (t)overthetimeinterval.

Figure 1.5 shows, in the original time axis, a scatterplot of log CD4 counts

vs. time, with individual proﬁles highlighted. Even in a small sample of indi-

viduals, we see that longer follow-up time tends to be associated with greater

intercept and slope.

As with Example 8.8, weassume the process Y

(t)ispotentially observed

at a set of points T

= {t

,...,t

i,n

},giving rise to the full-data vector

=(Y

),...,Y

i,n

))

=(Y

,...,Y

i,n

)

Conditional on dropout, we assume the log CD4 trajectory is linear in time.

Let x

=(1,t

), so that X

=(x

,...,x

i,n

)

is the n

× 2designmatrix.

The full-data distribution p(y,u | z)isfactoredasp(y | u, z) p(u | z); the

ﬁrst factor follows

| U

= u, Z

= z ∼ N(µ

(u), Σ

). (10.20)

The conditional mean µ

(u)isann

× 1vector having elements

ijz

(u)=E(Y

| u, z)=β

(u)+β

(u)t

, (10.21)

where β

(u)andβ

(u)aresmooth but unspeciﬁed functions of u.When

using the rescaled time axis, the intercept term β

(u)isthefull-data mean

of log CD4 count at time t =

= 139 among those who drop out at time

u;theslopeβ

(u)isthefull-data mean change in log CD4 from baseline to

end of the study among those dropping out at u.Whenbothβ

(u)andβ

(u)

are constant in u,wehaveMAR.

The variance Σ

is parameterized using a random eﬀects structure

= Σ

(Ω,σ)=x

Ωx

+ σ

where Ω is a 2×2positive deﬁnite matrix and I

is an identity matrix having

dimension n

264 CASE STUDIES: NONIGNORABLE MISSINGNESS

The smooth functions β

(u)andβ

(u)arerepresented in terms of pe-

nalized splines. We use a low-rank thin-plate spline basis (Crainiceanu et al.,

2005). Suppressing individual-level subscripts for now, we have

β(u; α, ψ)=α

+ α

u +



l=1

|u − s

, (10.22)

where s

are ﬁxed knot points, α =(α

,α

)

,andψ =(ψ

,...,ψ

)

.In

the full-data loglikelihood, we introduce a penalty term that shrinks the co-

eﬃcients of (10.22) to induce smoothness; it takes the form λφ

Dφ,where

λ>0isasmoothingparameter,φ

=(α

, ψ

), and D is a symmetric

matrix with dimension 2 + k.Following Crainiceanu et al. (2005), we only

penalize higher-order coeﬃcients ψ,sothat

D =

2×2

2×k

k×2

k×k

where the (l, l



)entryofQ is |s

− s



.Nowwritev(u)=(1,u)andw(u)=

(|u − s

,...,|u − s

), and let



ψ = Q

ψ and



w(u)=w(u)Q

−

The penalized splines have a linear model representation as

β(u, φ)=v(u)α +



w(u)



ψ,

where the



ψ ∼ N(0,τ

)areviewed as random eﬀects with variance τ

1/(2λ). Applying this separately to each of the four functions (intercept and

slope function for each treatment group), we have, for l =0, 1andz =0, 1,

(u, φ)=β(u, φ

)=v(u)α



w(u)



with



∼ N(0,τ

The conditional distribution of CD4 counts conditional on dropout time

(10.20) can be rewritten as

| u, z) ∼ N ( X

β(u; φ

, λ

), Σ(Ω

,σ

)),

where φ

=(α

, α



)andτ

=(τ

,τ

)

.Priorsfor this part of

the model are

∼ N(0, 10



| τ

∼ N(0,τ

−2

∼ Gamma(.01, 1/.01),

Ω

−1

∼ Wishart(3, 3I),

−2

∼ Gamma(.01, 1/.01),

for l =0, 1andz =0, 1.

PEDIATRIC AIDS TRIAL 265

10.4.3 Model speciﬁcation: dropout times

For treatment groups z =0, 1, the distributions p(u | z)areleftessentially

unspeciﬁed, and we use Bayesian nonparametric approaches to characterize

them. In particular, we use a Dirichlet mixture of normal distributions (Sec-

tion 3.6), where

| z ∼ N (α

,ν

∼ G

∼ DP(F

,a),

with F

anormaldistribution. Technical details on sampling can be found in

MacEachern and M¨uller(1998).

10.4.4 Summary of analyses under MAR and MNAR

The primary objective is to draw inference about the diﬀerence in mean change

in log CD4 cell count at the end of follow-up; denote this by θ.Inthemixture

model, this corresponds to the diﬀerence in integrated slope functions β

(u)

between treatment groups,

θ =



(u) p(u | z =1)du −



(u) p(u | z =0)du.

Treatment eﬀect is summarized using the posterior mean and standard devi-

ation for θ andthe posterior probability that θ>0.

In addition to the mixture model, we ﬁt a standard random eﬀects model

that assumes MAR; the latter corresponds to a special case of the mixture

model where the β

(u)areconstantfunctions of u.

Figure 10.7 shows posterior means for the smooth functions β

(u)(for

l =0, 1; z =0, 1), together with posterior estimates of individual-speciﬁc in-

tercepts and slopes. The plots of intercept vs. dropout time indicate that mean

CD4 is higher among those with longer follow-up, with a more pronounced

eﬀect in the low dose arm; likewise, CD4slopesarehigheramong those with

longer follow-up times.

Inferences about treatment-speciﬁc intercepts and slopes, and treatment

eﬀect, are summarized in Table 10.9. In general, inferences under the mix-

ture model indicate that overall mean (intercept term) and slope over time

arelower under the MNAR assumption. The eﬀect of the MNAR assump-

tion (relative to MAR) ismostpronounced in the low dose arm, where the

association between CD4 count and dropout time is stronger (Figure 10.7).

In fact, the shift in intercepts and slopes from MAR to MNAR exceeds two

posterior standard deviations (PSD) for all parameters except the intercept

on high dose. The PSD is generally greater for the mixture model.

The MNAR assumption also inﬂuences the treatment eﬀect; under MAR,

the posterior mean of θ is −.37 (PSD = .21) with P (θ>0) = .96, suggesting

266 CASE STUDIES: NONIGNORABLE MISSINGNESS

superiority of high dose; under MNAR we have posterior mean .01 (PSD =

.25) and P(θ>0) = .49, which is far more equivocal.

10.4.5 Summary

Studies like the Pediatric AIDS trial require methods that can handle con-

tinuous time dropout. The mixture model given here induces an MNAR

dropout process by assuming the full-data response distribution, conditional

on dropout time u,hasmean that is linear in time; the intercept and slope

parameters are smooth but unspeciﬁed functions of u.

−0.6 −0.4 −0.2 0.0 0.2

−4 −2 0 2 4 6 8

−0.6 −0.4 −0.2 0.0 0.2

−15 −10 −5 0 5

−0.6 −0.4 −0.2 0.0 0.2

−4 −2 0 2 4 6 8

−0.6 −0.4 −0.2 0.0 0.2

−15 −10 −5 0 5

intercept (low dose)intercept (high dose)

slope (lowdose)slope (high dose)

follow-up time (standardized)follow-up time (standardized)

Figure 10.7 Pediatric AIDS study: posterior (pointwise) mean of intercept function

(u) and slope function β

(u),separately by treatment (z =0, 1). Points are

individual-speciﬁc posterior mean intercepts and slopes. Tick marks on top edge of

each plot are deciles of the dropout distribution.

PEDIATRIC AIDS TRIAL 267

Table 10.9 Pediatric AIDS trial: posterior means and standard deviations for key pa-

rameters from random eﬀects model (REM) ﬁt under MAR, and mixture of varying

coeﬃcient model (VCM), which assumes MNAR.

Model

Dose Parameter REM (MAR) VCM (MNAR)

Low Intercept 5.6 (.13) 5.2 (.11)

Slope –1.6 (.14) –2.1 (.13)

High Intercept 5.4 (.14) 5.3 (.14)

Slope –1.9 (.16) –2.1 (.19)

Diﬀerence (θ)–.37(.21) .01 (.25)

P (θ>0) .96 .49

An advantage of the model is that the functional form of the missingness

mechanism does not have to be known; it is characterized by functions that can

be left unspeciﬁed. However, it is necessary to specify the form of E{Y (t) | u}

as a function of t;hereweassume it is linear with intercept β

(u)andintercept

(u). A key consequence is that conditionally on U = u,themean of Y (t)

has the same functional form before and after dropout; for the linear case, the

implication is that the slope is the same prior to and after u.

This property can be relaxed by expanding the model to accommodate

sensitivity parameters that allow (for example) a diﬀerent slope following

dropout. For example, we can expand (10.21) by introducing a sensitivity

parameter ξ =(ξ

,ξ

), so that

E(Y

| u, z)=β

(u)+β

(u)t

+ ξ

− u)

where a

= aI{a>0} is the positive part of a.Inthisformulation, β

(u)

is the slope prior to dropout time u and β

(u)+ξ

is the slope after u.

More generally, ξ

can depend on u.Anothergeneralization of the model

allows the variance-covariance structure to vary smoothly with u;thiscan

be accomplished using an appropriate parameterization of Σ.Details can be

found in Su and Hogan (2007).

Distributions

Discrete Distributions

Binomial Y | n, θ ∼ Bin(n, θ)

The binomial model characterizes the number of successes in n independent

trials, where success probability on each trial is θ.Ithasmassfunction

p(y | θ, n)=



(1 − θ)

n−y

where y ∈{0, 1,...,n} and 0 <θ<1.

Multinomial with k categories Y | n, θ ∼ Mult(n, θ)

The multinomial model characterizes the distribution of n events across k

mutually exclusive categories labeled j =1,...,k,whereY

is the count and

is the probability for category j.Themultinomial randomvariable has

mass function

p(y | n, θ)=



···y

···θ

where y =(y

,...,y

)

, y

≥ 0,



j=1

= n,0<θ

< 1, and



j=1

=1.

Continuous distributions

Dirichlet with k categories θ ∼ Dir(β)

The Dirichlet distribution characterizes the joint distribution of a vector of

random variables θ =(θ

,...,θ

)havingθ

> 0and



=1.Itisfre-

quently used as a prior for multinomial probabilities. It density function is

p(θ | β)=

Γ(β

+ ···+ β

)

j=1

Γ(β

)

−1

···θ

−1

where β

> 0andΓ(·)isthegamma function.

268

DISTRIBUTIONS 269

Gamma Y ∼ Gamma(α, β)

Arandom variable Y having the gamma distribution has density function

p(y | α, β)=

Γ(α)β

exp(−y/β)y

α−1

with y>0, α>0, and β>0. Its mean is E(Y )=αβ.

Normal Y ∼ N(µ, σ

)

Arandom variable Y having the normal (Gaussian) distribution with mean

µ and variance σ

has density function

p(y | µ, σ)=(2π)

−1/2

−1

exp



−(x − µ)

/2σ



with −∞ <y<∞, −∞ <µ<∞,andσ>0.

Multivariate normal (q-dimensional) Y ∼ N(µ, Σ)

A q-dimensional normal random variable Y =(Y

,...,Y

)

with mean µ =

(µ

,...,µ

)

and q × q variance matrix Σ has density function

p(y | µ, Σ)=(2π)

−q/2

|Σ|

−1/2

exp



−

(y − µ)

−1

(y − µ)



where −∞ <y

< ∞, −∞ <µ

< ∞,andΣ is positive deﬁnite.

t distribution Y ∼ T

(µ, σ

)

Arandom variable Y having the t distribution with mean µ,scaleσ

and ν

degrees of freedom has density

p(y | µ, σ

,ν)=

Γ((ν +1)/2)

Γ(ν/2)(νσπ)

1/2



1+(y − µ)

/νσ



−(ν+1)/2

where −∞ <y<∞, −∞ <µ<∞, σ

> 0, and ν>0.

Multivariate t (q-dimensional) Y ∼ T

(µ, Σ)

A q dimensional vector Y having the multivariate t distribution has density

function

p(y | µ, Σ,ν)=

Γ((ν + p)/2)

Γ(ν/2)(νπ)

p/2

|Σ|

−1/2



1+(y − µ)

−1

(y − µ)/ν



−(ν+p)/2

with −∞ <Y

< ∞, −∞ <µ

< ∞, ν>0, and Σ positive deﬁnite.

Wishart (q-dimensional covariance matrix) S ∼ Wishart(ν, A)

A q dimensional covariance matrix from the Wishart distribution with scale

matrix A and degrees of freedom ν has density function

p(S | A,ν)=c(ν, q) |A|

−ν/2

|S|

(ν−q−1)/2

exp



−tr(A

−1

