Constrained Bayes Estimates of Random Effects

when Data are Subject to a Limit of Detection

Reneé H. MooreDepartment of Biostatistics and Epidemiology

University of Pennsylvania

Robert H. Lyles, Amita K. Manatunga, Kirk A. EasleyDepartment 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

• One Indicator: high initial and/or steeply increasing HIV RNA levels over time

• Limitation: HIV RNA below a certain threshold not quantifiable

IS THIS CHILD’S HIV INFECTION AT GREATER RISK OF RAPID PROGRESSION?

• 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: 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

eZuXΨY

6

The Mixed Linear Model

Assumptions: E(u)= 0 and E(e)= 0

eZuXΨY

XΨ(Y) E

ZuXΨu)|(Y E

7

The Mixed Linear Model

V ar e( )e 2 V

V ar ( ) 'Y Z D Z Ve2

euDΔσ(u) ,2Var

eZuXΨY

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”

M S E P E ( ~ ) ( ~ )'u u A u u

E(u|Y)

)()|~( uEYuE

eZuXΨY

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

Y a b tij i i ij ij ( ) ( ) (i = 1, … , k ; j=1, …, ni)

L f f a b F d a b f a b f b da dbii

k

i

k

ij i ij

ni

Y i i i i i i ij ni

ni( ; ) ( ) [ { ( | , )}{ ( | , )} ( | ) ( ) ] .* Y Y Y

1 1 1

1

1 1

To get estimates of =

- ni1 detectable measurements: f(Yij|ai,bi)

- ni - ni1 non-detectable measurements: FY(d|ai ,bi)

T),,,,( 222

21

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

eZuXΨY

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 ui toward u

E(u|Y))()|~( uEYuE

13

Bayes with LOD

Lyles et al. (2000), using the MLEs from L( ;Y),

)(

)()|(),|()|()( *

*

i

iiiiiiiii

iiBi

Yf

ddffYfYEye

)(ii aor )(ii b

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

k

i

k

ii

Bi

Bi yEyeye

1 1

22 |)())()((

• 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 )CB B Bi iw w

12

1

2

( )( ) 1( )

H yw w yH y

1( ) ( | )1 y

kH y tr Var

1

1

kB B

iik

2

21

( ) ( )k

B B

ii

H y

(1) posterior mean matches sample mean

(2) posterior variance matches sample variance

Recall:

minimizes MSEP =

within the class of predictors of

CB

i 2

1( ) |

k

i ii

E y

(1 )CB B Bi iw w

( ) ( ,..., )T

i i ky

i

s.t.

satisfies (1) but NOT (2)B

i Bayes

Adjust

Con. Bayes

CONSTRAINED BAYES ESTIMATION

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) BayesLOD and Ghosh (1992) CB CBLOD

RANDOM INTERCEPT-SLOPE MODEL

• 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

(i = 1, … , k ; j=1, …, ni)

i ia

i ib

Y a b tij i i ij ij ( ) ( )

i

i

ij

N

~

,

3

1

2

12

12 2

2

20

00

0 0

Intercept:

Slope:

Under random intercept-slope model, Lyles et.al. (2000) get MLEs of =

• ni1 detectable measurements: f(Yij|ai,bi)

• ni - ni1 non-detectable measurements: FY(d|ai

,bi)• d = limit of detection (LOD)

T),,,,( 222

21

*

1

1

1 1 1 1

( ; ) ( )

{ ( | , )}{ ( | , )} ( | ) ( )

Y Yk

ii

n nk i i

ij i i Y i i i i i i ii j j ni

L f

f Y a b F d a b f a b f b da db

BAYES PREDICTOR FOR LOD

*

*

( )( ) ( | )

( )i i i i

B

i i i

i

f Y d de y E Y

f Y

)( ii a or )( ii b

• 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 βi toward β or αi toward α

CB PREDICTIONS OF αi AND βi

(i = 1, … , k ; j=1, …, ni)

Y a b tij i i ij ij ( ) ( )

1*

1 1 1 1

( ; ) [ { ( | , )}{ ( | , )} ( | ) ( ) ]n nk i i

ij i i Y i i i i i i ii j j ni

L Y f Y F d f f d d

)(

)()|(),|()|(~

*

*

i

iiiiiiiii

iiBi Yf

ddffYfYE

BBi

CBi ww ~)1(~~

COMPARING CONSTRAINED BAYES ESTIMATES

PARAMETER ESTIMATES BASED ON 2 METHODS:& Adjust Likelihood for LODAd Hoc Imputation

*

1

1

1 1 1 1

( ; ) ( )

{ ( | , )}{ ( | , )} ( | ) ( )

Y Yk

ii

n nk i i

ij i i Y i i i i i i ii j j ni

L f

f Y a b F d a b f a b f b da db

1( ; ) (Y ) ( | , ) ( | ) ( )

k

i ij i i i i i i ii

L f f Y a b f a b f b da db

Y

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

• Goal of Predictor is to Match

• Compare and

• Recall: Yij= (α + ai) + (β + bi)tij + εij

, ,( ) ( )i CB i CB itVar t Var Y

, , ,,

it CB i CB i CBBut Y t BECAUSE

,it BY

,it CBY

2 2 21 12 2[ ] [ ] 2it itE Y t and Var Y t t

, , ,( | )Yit i i B i B it BE Y t Y

itY

, ,( ) ( ) ( ) ( )i CB i i CB iVar Var and Var Var

PREDICTION OF Yit = αi + t βi • Our previous CB predictors set out to match

but did not enforce constraint

, ,cov( , ) cov( , )i CB i CB 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?

• calculated for all k subjects at each unique t,it B

Y2

2 , ,1

( ) ( )k

it B t Bi

H y Y Y

11

1( ) (1 ) ( | )Y

k

iti

H y k Var Y

2 2,( | ) ( | )Y Yit it i it BVar Y E Y Y

PREDICTION OF Yit = αi + t βi 2 2

,( | | )Y)= ( Yit it i it BVar Y E Y Y

2 * 2 *

2 *

*

2 *

*

( | ) ( | )

1 ( , , )( )1 ( | , ) ( | ) ( )

( )

Y Y

YY

YY

it i it it i it

it it i i it i

i

it i it i it i i it i

i

E Y y f y dy

y f y dy df

y f y f y f dy df

1 1* *

1( | , ) ( | , ) ( | , )Y

ni n ni ii it i ij it i Y it i

jwhere f y f Y y F d y

P2C2 ALL 59 PREDICTORS OF Yit AT EACH t

0.0 0.2 0.4 0.6 0.8 1.0 1.2

23

45

67

Time Reached Class A

Pre

dict

or o

f Y

it =

alp

ha +

bet

a*t

THE 59 INDIVIDUAL PREDICTORS OF Yit AT EACH CHILD’S UNIQUE t

0.0 0.2 0.4 0.6 0.8 1.0 1.2

23

45

6

Time Reached Class A

Pre

dict

or o

f Y

it =

alp

ha +

bet

a*t • Bayes

o CB

SIMULATION STUDY FOR Yit

• 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)

2 2

1 2

2

12

5, 1 , 1, 0.1,- 0.26, 0.23

-

SIMULATION STUDY FOR Yit

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

it i iY t Sample Mean

, ,,it B it CBY Y ,it CBYSample Variance

,it BY

Sample Variance

SIMULATION STUDY FOR Yit

Bayes (closed circles) and CB (open circles) estimates of 80 simulated patients. The line plotted is . .

ˆˆ ˆitY t

.

SUMMARY

• Proposed LOD-adjusted CB predictors- Intercepts and Slopes- R.V. (Yit) 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!!