Constrained Bayes Estimates
of Random Effects
when Data are Subject
to a Limit of Detection
Reneé
H.
Moore
Department of Biostatistics and Epidemiology
University
of Pennsylvania
Robert H. Lyles,
Amita
K.
Manatunga
,
Kirk A. Easley
Department of Biostatistics and Bioinformatics
Emory University
O
UTLINE
•
Motivating Example
•
Background
•
Review the Mixed Linear Model
•
Bayes predictor
•
Censoring under the Mixed Model
•
CB Predictors
•
Application of Methodology for CB adjusted for
LOD
•
Motivating Example
•
Simulation Studies
P
2
C
2
HIV I
NFECTION
S
TUDY
:
I
S
THIS
C
HILD
’
S
HIV I
NFECTION
AT
G
REATER
R
ISK
OF
R
APID
P
ROGRESSION
?
•
1990

1993
HIV transmitted from mother to child
in utero
•
Children in this dataset enrolled at birth or by
28
days of life
•
HIV RNA Data at
3

6
mos through
5
years of age
•
Rapid Progression is defined as the occurrence of
AIDS (Class C) or death before
18
months of age
•
One goal of the study was to identify children
with RP of disease because they may benefit from
early and intense antiretroviral therapy
•
One Indicator: high initial and/or steeply
increasing HIV RNA levels over time
•
Limitation: HIV RNA below a certain threshold
not quantifiable
I
S
THIS
C
HILD
’
S
HIV I
NFECTION
AT
G
REATER
R
ISK
OF
R
APID
P
ROGRESSION
?
•
Given non

detects, how do we predict each child’s
HIV RNA intercept and slope?
•
Given non

detects, how do we predict each
child’s HIV RNA level at a meaningful time point
associated with RP?
T
HE
M
IXED
L
INEAR
M
ODEL
Y
: N by
1
outcome variable
X
: known N by p fixed effects design matrix
㨠
瀠批p
ㄠ
癥捴潲o潦楸敤晥捴
Z
: known N by q random effects design matrix
u
: q by
1
vector of random effects
e
: N by
1
vector of random error terms
6
The Mixed Linear Model
Assumptions: E(
u
)=
0
and E(
e
)=
0
7
The Mixed Linear Model
8
BP
(best predictor, Searle et.al.
1992
)
:

minimizes

invariant to the choice of
A
, any pos. symmetric matrix

holds regardless of joint distribution of (
u
, Y
)

unbiased, i.e.

linear in Y
“Bayes Predictor”
E(
u
Y
)
Censoring under the mixed model
*
common feature of
HIV data is that some values fall below a
LOD
Ad hoc approach: substitute the LOD or a fraction of it for all
values below the limit (
Hornung
and Reed,
1990
)
Other Approaches:

Likelihood using the EM algorithm (Pettitt
1986
, Hughes
1991
)

Bayesian Methods (Carriquiry
1987
)

Likelihood based approach using algortihms (Jacgmin

Gadda
et.al.
2000
)
Lyles et. al. (
2000
) maximize an integrated joint log

likelihood
directly to handle informative drop

out and left censoring
10
Left

censoring under the mixed model
Lyles et.al. (
2000
) work under framework of
(i =
1
, … , k ; j=
1
, …, n
i
)
To get estimates of
=

n
i
1
detectable measurements:
f(Y
ij
a
i
,b
i
)

n
i

n
i
1
non

detectable measurements:
F
Y
(da
i
,b
i
)
11
E(
u
Y
)
can’t be calculated in practice!
Why?

knowledge of all parameters in the joint distribution of (
u,Y
)
What do we do?

develop predictors based on their theoretical properties for known
parameters

evaluate effect of estimating unknown parameters via simulation
studies
Bayes Predictor (posterior mean)
•
minimizes MSEP s.t.

Prediction Properties (bias, MSEP) deteriorate for individuals
whose random effects put them in tails of distribution
•
Motivated research for alternatives to Bayes

Limited translation rules (Efron and Morris,
1971
)

Constrained Bayes
tends to overshrink individual
u
i
toward
u
E(
u
Y
)
13
Bayes with LOD
Lyles et al. (
2000
), using
the MLEs from
L(
;
Y
),
14
Censoring under the mixed model
None
of the references cited for
dealing with left

censored longitudinal or repeated measures data
considered
alternatives to the Bayes predictors for random effects
We Do!
15
Constrained Bayes Estimation
Louis (
1984
)
•
Expectation of sample variance of Bayes estimates is only a
fraction of expected variance of unobserved parameters derived
from the prior
Shrinkage of the Bayes estimate
•
Reduces shrinkage by matching first two moments of estimates
with corresp. moments from posterior histogram of k normal means
16
Constrained Bayes Estimation
•
Ghosh
(
1992
): “recipe” to generalize Louis’ modified
Bayes predictor for use with any distribution
•
Lyles and
Xu
(
1999
): match predictor’s mean and
variance with prior mean and variance of random effect
Ghosh (
1992
)
where
Constrained
Bayes
Estimation
(
1
) posterior mean matches sample mean
(
2
) posterior variance matches sample variance
Recall:
minimizes MSEP =
within the class of predictors
of
s.t.
satisfies (
1
) but NOT (
2
)
Bayes
Adjust
Con.
Bayes
C
ONSTRAINED
B
AYES
E
STIMATION
Ghosh (
1992
)
19
Constrained Bayes (CB) Estimation
We Do
!
Moore, Lyles,
Manatunga
(
2010
). Empirical constrained
Bayes predictors accounting for non

detects among repeated
Measures.
Statistics in Medicine.
CB Predictors have been shown to
reduce the shrinkage of the Bayes estimate in an appealing way
BUT none
had
been adapted to account for censored data
CB Predictors with LOD
•
Lyles
(
2000
): adjusted Bayes estimate to accommodate
data subject to a LOD but did not consider CB
•
Moore (
2010
): combine Lyles (
2000
)
Bayes
LOD
and
Ghosh
(
1992
) CB
CB
LOD
R
ANDOM
I
NTERCEPT

S
LOPE
M
ODEL
•
Y
ij
: Observed HIV RNA measurement at
j
th
time point (
t
ij
)
for
i
th
child
•
a
i
:
i
th
child’s r
andom intercept deviation
•
b
i
:
i
th
child’s random slope deviation
(i =
1
, … , k ; j=
1
, …, n
i
)
Intercept:
Slope:
Under random intercept

slope model,
Lyles et.al. (
2000
) get MLEs of
=
•
n
i
1
detectable measurements:
f(
Y
ij
a
i
,b
i
)
•
n
i

n
i
1
non

detectable measurements:
F
Y
(
da
i
,b
i
)
•
d
= limit of detection (LOD)
B
AYES
P
REDICTOR
FOR
LOD
•
minimizes MSEP s.t. posterior mean matches sample mean
Prediction properties (bias, MSEP) deteriorate for individuals
whose random effects put them in the tail of the distribution
strongly shrinks predicted
β
i
toward
β
or
α
i
toward
α
CB P
REDICTIONS
OF
α
i
AND
β
i
(i =
1
, … , k ; j=
1
, …, n
i
)
C
OMPARING
C
ONSTRAINED
B
AYES
E
STIMATES
P
ARAMETER
E
STIMATES
B
ASED
ON
2
M
ETHODS
:
& Adjust Likelihood for LOD
Ad Hoc Imputation
E
XAMPLE
S
IMULATION
S
TUDY
Table IV. (Moore et al.
Statistics in Medicine
,
2010
)
E
XAMPLE
S
IMULATION
S
TUDY
I
S
THIS
I
NFANT
’
S
HIV I
NFECTION
AT
G
REATER
R
ISK
OF
R
APID
P
ROGRESSION
?
•
Given non

detects, how do we predict each
patient’s HIV RNA intercept and slope?
Viable option now available
•
Given non

detects, how do we predict each
patient’s HIV RNA level at a meaningful time
point?
Extending our
Stat in Med
2010
work
I
S
THIS
C
HILD
’
S
HIV I
NFECTION
AT
G
REATER
R
ISK
OF
R
APID
P
ROGRESSION
?
P
2
C
2
HIV Data
(Chinen, J., Easley, K. et.al.,
J. Allergy Clin. Immunol.
2001
)
•
343
HIV RNA measurements from
59
kids (range:
2

11
,
median=
6
)
•
detection limit=
2.6
=log(
400
copies/mL)
•
6
% (
21
/
343
) of measurements < LOD
•
19
% (
11
/
59
) kids have at least one meas. < LOD
•
59
unique times (
t
) reached Class A HIV
*
Goal: Predict
Y
it
: HIV RNA level at time reached Class A
P
REDICTION
OF
Y
it
=
α
i
+ t
β
i
•
Goal of Predictor is to Match
•
Compare and
•
Recall:
Y
ij
= (
α
+ a
i
) + (
β
+ b
i
)t
ij
+
ε
ij
P
REDICTION
OF
Y
it
=
α
i
+ t
β
i
•
Our previous CB predictors set out to match
but did not enforce constraint
•
We develop a CB predictor for the scalar R.V.
Y
it
O
BJECTIVE
1
:
P
REDICTION
OF
Y
it
=
α
i
+ t
β
i
What is new in adapting this extension of Ghosh’s CB?
•
calculated for all k subjects at each unique
t
P
REDICTION
OF
Y
it
=
α
i
+ t
β
i
P
2
C
2
A
LL
59
P
REDICTORS
OF
Y
it
AT
EACH
t
T
HE
59
I
NDIVIDUAL
P
REDICTORS
OF
Y
it
AT
EACH
C
HILD
’
S
U
NIQUE
t
•
Bayes
o
CB
S
IMULATION
S
TUDY
FOR
Y
it
•
Parameter Assumptions:
•
1500
subjects, each with five HIV RNA values
taken every six months for
2
years
•
15
% (
1
,
089
/
7
,
500
) values < LOD =
2.8
•
8
times
(t)
of interest = (
0.03
,
0.16
,
0.36
,
0.66
,
0.85
,
1.17
,
1.32
,
1.60
)
S
IMULATION
S
TUDY
FOR
Y
it
Time Reached
Class A HIV
Status (years)
Mean
Variance
0.03
4.98
0.98
4.99
0.86
0.95
0.16
4.85
0.92
4.86
0.80
0.89
0.36
4.65
0.82
4.66
0.73
0.79
0.66
4.35
0.69
4.36
0.62
0.68
0.85
4.16
0.62
4.17
0.56
0.61
1.17
3.84
0.52
3.85
0.47
0.51
1.32
3.69
0.47
3.70
0.43
0.47
1.60
3.41
0.41
3.41
0.36
0.41
Sample Mean
Sample
Variance
Sample
Variance
S
IMULATION
S
TUDY
FOR
Y
it
Bayes (closed circles) and CB (open circles) estimates of
80
simulated patients. The line plotted is .
.
.
S
UMMARY
•
Proposed LOD

adjusted CB predictors

Intercepts and Slopes

R.V. (
Y
it
) at a meaningful time point
Relative to ad hoc and Bayes predictors:
“CBs Attenuate the Shrinkage”
Better Match True Distribution of Random Effects
Thank You!!
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