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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
OUTLINE
•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
P2C2 HIV INFECTION STUDY:
IS THIS CHILD’S HIV INFECTION AT GREATER RISK OF
RAPID PROGRESSION?
•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 3-6mos 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
IS THIS CHILD’S HIV INFECTION AT
GREATER RISK OF RAPID PROGRESSION?
• One Indicator: high initial and/or steeply
increasing HIV RNA levels over time
• Limitation: HIV RNA below a certain threshold
not quantifiable
• 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?
THE MIXED LINEAR MODEL
Y  XΨ  Zu  e
Y: N by 1 outcome variable
X: known N by p fixed effects design matrix
: p by 1 vector of fixed effects
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
The Mixed Linear Model
Y  XΨ  Zu  e
Assumptions: E(u)= 0 and E(e)= 0

E(Y)  XΨ
E(Y | u)  XΨ Zu
6
The Mixed Linear Model
Y  XΨ  Zu  e
Var(u)  σ  Δ  D,
2
u e
Var(e )  2e V
   Var( Y)  ZDZ '  e2 V
7
Y  XΨ  Zu  e
BP (best predictor, Searle et.al. 1992):


'
~
- minimizes MSEP  E (u  u ) A(u~  u )
- invariant to the choice of A, any pos. symmetric matrix
- holds regardless of joint distribution of (u, Y)
- unbiased, i.e. E (u~ | Y )  E (u )
- linear in Y
E(u|Y)
“Bayes Predictor”
8
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

Left-censoring under the mixed model
Lyles et.al. (2000) work under framework of
Yij  (  ai )  (  bi )tij  ij
(i = 1, … , k ; j=1, …, ni)
To get estimates of  = ( ,  ,12 , 22 , 2 )T
k
k
i 1
i 1
L(; Y )   f * ( Yi )   [ 




ni1
ni
j 1
j  ni11
 { f ( Yij | ai , bi )}{  FY (d | ai , bi )} f (ai |bi ) f (bi )dai dbi ].
- ni1 detectable measurements: f(Yij|ai,bi)
- ni - ni1 non-detectable measurements: FY(d|ai ,bi)
10
Y  XΨ  Zu  e
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
11
Bayes Predictor (posterior mean)
E(u|Y)
~ | Y )  E (u )
• minimizes MSEP s.t. E (u
 tends to overshrink individual ui toward u
- 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
Bayes with LOD
Lyles et al. (2000), using the MLEs from L( ;Y),
 
eiB ( y )  E (i | Yi ) 
*

f
  i (Yi | i , i ) f (i | i ) f ( i )di di
 
f * (Yi )
i  (  ai ) or i  (  bi )
13
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!
14
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
k
k


B
B
2
2
 (ei ( y)  ei ( y))  E   (i   ) | y
i 1
i 1

 Shrinkage of the Bayes estimate
• Reduces shrinkage by matching first two moments of estimates
with corresp. moments from posterior histogram of k normal means
15
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
16
Constrained Bayes Estimation
Ghosh (1992)

CB
i
 wi  (1  w)
B
  k 
where
1
B
k
i 1
B
i
 H1 ( y ) 
w  w( y )  1 

H
(
y
)
2


1
2
H ( y )  tr Var (   1 | y ) 
1
k
H ( y )   (   )
k
2
i 1
B
i
B
2
B
CONSTRAINED BAYES ESTIMATION
Ghosh (1992)

CB
i

CB
i
 wi  (1  w)
B
B
minimizes MSEP = E   (   ) | y 


k
2
i 1
i
i
within the class of predictors  ( y )    ( ,...,  )
i
i
k
s.t.
T
of

i
(1) posterior mean matches sample mean
(2) posterior variance matches sample variance
Recall:

B
i
satisfies (1) but NOT (2)
Bayes
Con.
Bayes
Constrained Bayes (CB) Estimation
CB Predictors have been shown to
reduce the shrinkage of the Bayes estimate in an appealing way
We Do!
Moore, Lyles, Manatunga (2010). Empirical constrained
Bayes predictors accounting for non-detects among repeated
Measures. Statistics in Medicine.
19
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) BayesLOD and
Ghosh (1992) CB  CBLOD
RANDOM INTERCEPT-SLOPE MODEL
Yij  (  ai )  (  bi )tij  ij
Intercept:
   a
Slope:
  b
i
i
i
i
(i = 1, … , k ; j=1, …, ni)
     2 
 i 

0
1
12
  
 

2
 i  ~ N 3      12  2 0  
  
 
2 
 
0
0
0

 ij 
 ,
•Yij : Observed HIV RNA measurement at jth time point (tij) for ith child
• ai : ith child’s random intercept deviation
• bi : ith child’s random slope deviation
Under random intercept-slope model,
2 2 2 T
Lyles et.al. (2000) get MLEs of  = ( ,  ,1 , 2 , )
k
L( ; Y)   f (Y )
*
i
i 1
     { f (Y | a , b )}{  F (d | a , b )} f (a | b ) f (b )da db 


k
i 1




ni 1
j 1
ni
ij
i
i
j ni 11
Y
i
i
i
i
i
i
• ni1 detectable measurements: f(Yij|ai,bi)
• ni - ni1 non-detectable measurements: FY(d|ai ,bi)
• d = limit of detection (LOD)
i
BAYES PREDICTOR FOR LOD
 
   f (Y )d d 
*
e ( y )  E ( | Y ) 
B
i
i
i
 
i
i
i
i
*
f (Y )
i
i  (  ai ) or i  (   bi )
• minimizes MSEP s.t. posterior mean matches sample mean
strongly shrinks predicted βi toward β or αi toward α
Prediction properties (bias, MSEP) deteriorate for individuals
whose random effects put them in the tail of the distribution
CB PREDICTIONS OF αi AND βi
Yij  (  ai )  (  bi )tij  ij
(i = 1, … , k ; j=1, …, ni)
ni 1
ni
L( ; Y )   [  { f (Y |  ,  )}{  F (d |  ,  )} f ( |  ) f ( )d d  ]
k
*
i 1


 
j 1
ij
i
i
j ni 11
Y
i
i
i
 
i
i
i
i
*

f
  i (Yi | i , i ) f (i | i ) f (i )di di
~iB  E ( i | Yi )  
f * (Yi )
CB
B
B
~
~
~
i  wi  (1  w)
COMPARING CONSTRAINED BAYES ESTIMATES
PARAMETER ESTIMATES BASED ON 2 METHODS:
k
L( ; Y)   f (Y )   f (Y | a , b ) f (a | b ) f (b )da db 

k

i
i 1
ij
i
i
i
i
i
i
i
L( ; Y)   f (Y )
*
i
i 1
     { f (Y | a , b )}{  F (d | a , b )} f (a | b ) f (b )da db 


k
i 1




ni 1
j 1
ni
ij
i
i
j ni 11
Y
i
i
i
i
i
i
i
EXAMPLE SIMULATION STUDY
Table IV. (Moore et al. Statistics in Medicine, 2010)
EXAMPLE SIMULATION STUDY
IS THIS INFANT’S HIV INFECTION AT
GREATER RISK OF RAPID PROGRESSION?
• 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
IS THIS CHILD’S HIV INFECTION AT
GREATER RISK OF RAPID PROGRESSION?
P2C2 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 Yit: HIV RNA level at time reached Class A
PREDICTION OF Yit = αi + t βi
•
Recall: Yij= (α + ai) + (β + bi)tij + εij
•Goal of Predictor
Y is to Match
it
E[Yit ]    t  and Var[Yit ]    2t12  t 
2
1
•
Y
Compare
it , B
and
Y
it ,CB
E(Yit | Yi )  i ,B  t i , B  Yit ,B
But , Y
it ,CB
 
i ,CB
 t
i ,CB
BECAUSE
Var (i ,CB  t i ,CB )  Var (Yit )
2
2
2
PREDICTION OF Yit = αi + t βi
• Our previous CB predictors set out to match
Var (i ,CB )  Var (i ) and Var ( i ,CB )  Var ( i )
but did not enforce constraint
cov(i ,CB , i ,CB )  cov(i , i )
•We develop a CB predictor for the scalar R.V. Yit
OBJECTIVE 1:
PREDICTION OF Yit = αi + t βi
What is new in adapting this extension of Ghosh’s CB?
•
Y
it , B
calculated for all k subjects at each unique t
k
H 2 ( y )   (Yit , B  Yt , B )
2
i 1
k
H 1 ( y )  (1  k )  Var (Yit | Y)
1
i 1
Var (Yit | Y)  E (Yit | Yi )  Yit , B
2
2
PREDICTION OF Yit = αi + t βi
Var (Yit | Y)= E(Yit2 | Yi )  Yit2,B

E (Yit | Yi )   yit2 f ( yit |Yi* )dyit
2
*

1   2
*

y
f
(
y
,

,
Y
)dyit d  i
  it
it
i
i
*
f (Yi )  
1   2
*

y
f
(
Y
| yit ,  i ) f ( yit |  i ) f (  i ) dyit d  i
  it
i
*
f (Yi )  
n n
 ni1

where f (Y | yit , i )   f (Yij* | yit , i )   FY (d | yit , i ) i i1
 j 1

*
i
AT EACH
t
6
5
4
3
2
Predictor of Yit = alpha + beta*t
7
P2C2 ALL 59 PREDICTORS OF Yit
0.0
0.2
0.4
0.6
0.8
Time Reached Class A
1.0
1.2
THE 59 INDIVIDUAL PREDICTORS OF
CHILD’S UNIQUE t
• Bayes
6
AT EACH
3
4
5
o CB
2
Predictor of Yit = alpha + beta*t
Yit
0.0
0.2
0.4
0.6
0.8
Time Reached Class A
1.0
1.2
SIMULATION STUDY FOR Yit
• Parameter Assumptions:
  5,   -1 ,   1,   0.1,
2
2
1

12
 - 0.26, 
2
2
 0.23
•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)
SIMULATION STUDY FOR Yit
Time Reached
Class A HIV
Status (years)
Yit  i  t i
Sample Mean
Sample
Variance
Yit , B , Yit ,CB
Yit , B
Sample
Variance
Yit ,CB
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
SIMULATION STUDY FOR Yit
.
Bayes (closed circles) and CB (open circles) estimates of 80 simulated patients. The line plotted is
.
Yˆit  ˆ  t ˆ
.
SUMMARY