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Oct 2, 2013 (3 years and 10 months ago)

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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



瀠批p

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
(d|a
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
(
d|a
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!!