A Statistical Method for AdjustingCovariates in Linkage Analysis

With Sib Pairs

Colin O. Wu, Gang Zheng, JingPing Lin, Eric Leifer and Dean Follmann

Office of Biostatistics Research, DECA, NHLBI

A Statistical Method for AdjustingCovariates in Linkage Analysis

With Sib Pairs

Colin O. Wu, Gang Zheng, JingPing Lin, Eric Leifer and Dean Follmann

Office of Biostatistics Research, DECA, NHLBI

Presenter’s Name: Colin O. WuNational Heart, Lung, and Blood InstituteNational Institutes of HealthDepartment of Health and Human Services

Presenter’s Name: Colin O. WuNational Heart, Lung, and Blood InstituteNational Institutes of HealthDepartment of Health and Human Services

1. Framingham Heart Study (GAW13)

1.1 First Generation: 5209 subjects (2336 men & 2873 women); 29 to 62 years old when recruited; 1644 spouse pairs; Continuously examined every 2 years since

1948 – medical history – physical exams – laboratory tests.

1.2 Second Generation (full dataset): 5124 of the original participants’ adult

children & spouses of these adult children; 2616 subjects are offspring of original spouse

pairs; 34 are stepchildren; 898 offspring are children with only one

parent in the study; 1576 are spouses of the offspring; Offspring cohort followed every 4 years; Interval between Exams 1&2 is 8 years.

1.3 Second Generation (sib-pair subset) 482 multi-sib families

– from 330 pedigrees; Observed trait:

systolic blood pressure; Covariates:

1. age (in years), 2. gender (0=female, 1=male), 3. drinking (average daily alcohol consumption in ml).

Genotype data: 398 random markers with an average of 10cM apart.

2. Methods for Linkage Analysis

2.1 Methods based on identity by descent (IBD) Association in pedigrees between phenotype

and IBD sharing at loci linked to trait loci; Linkage for qualitative traits

– IBD sharing conditional on phenotypes: e.g. affected sib-pair methods (Hauser & Boehnke, 1998).

Linkage for quantitative trait loci (QTL) – phenotypes conditional on IBD sharing, e.g. Haseman & Elston (1972), Amos (1994); – extremely discordant sib-pairs, e.g. Risch & Zhang (1995, 1996).

2.2 The Haseman-Elston method

1 2

2

1 2

, : observed traits for 1st and 2nd sibs;

squared trait difference in jth pair;

,

: the overall mean trait value,

: the genetic effect on the , -th sib,

: the en

j j

j j j

ij ij ij

ij

ij

X X

Y X X

X g e

g i j

e

vironmental effect on the , -th sib.i j

HE Model without Covariate Adjustment:Assumptions: (i) one locus determines , (ii) two

alleles, and , (iii) gene frequencies and .

Genotypic values:

for a individual;

for a individual;

for a i

ij

ij

g

B b p q

a BB

g d Bb

a bb

ndividual.

| ,

proportion of genes IBD for the -th pair,

, : unknown parameters.

j j j

j

E Y

j

Linkage:

Limitations: Covariate effects are not included. Genetic and environmental effects are additive. Method may not have sufficient power.

0

1

Negative potential linkage between

QTL & marker locus.

Hypothesis Testing Problem:

: 0 (no linkage)

: 0 (linkage)

H

H

2.3 HE Method with Linear Covariate Adjustment (SAGE SIBPAL)

Involve families with more than 2 sibs. Can use other measures of trait difference

– e.g. the mean-corrected cross-product. Include covariate effects in linear regression

– e.g. Elston, Buxbaum, Jacobs and Olson (2000) “Haseman and Elston Revisted”.

Linear generalization:

1

1

1 2 1 2

0 0

1

,..., : covariates for , -th sib,

, ,..., : covariate vector,

: covariate for the -th sib pair,

e.g. or ,

,..., , .

| , ,...,

pij ij

Tp

ij j ij ij

lj

l l l l lj j j j j

Tp

j j j j j

pj j j j

Z Z i j

Z Z Z

Z j

Z Z Z Z Z

Z Z Z Z

E Y Z Z

0

.p

ll j

l

Z

Linkage:

Covariate effects:

Limitation:

0 0 linka e. g

0, 1,..., effect of the -th covariate. l l p l

1 2

2

1 2

Only the information in is used

, is reduced to .

For example, use age age .

j

l l lj j j

j j j

Z

Z Z Z

Z

3. The Proposed Method

3.1. Modeling the covariates

1 2

Goal: To generalize the HE regression model

that includes the covariates , .

Assumptions:

(i) The covariates are not affected by the gene

and the environment.

(ii) The effects of gene

l lj jZ Z

and environment are

additive.

Cross-sectional data:

1

1 1

1*

,..., ,

,..., mean of given ,..., .

Equivalent form:

,...,

covariate adjusted trait

.

pij ij ij ij ij

p pij ij ij ij ij

pij ij ij ij

ij ij

X Z Z g e

Z Z X Z Z

X X Z Z

g e

Regression models for covariates:

10

1

*0

1

0

Linear model:

,..., .

Equivalently,

;

,..., : linear coefficients.

pp l

ij ij l ijl

pl

ij ij l ijl

ij ij

T

p

Z Z Z

X X Z

g e

1 1

1

General parametric models

e.g. nonlinear models :

,..., ,..., ; .

Nonparametric models Hardle, 1991 :

,..., smooth function of .

Semiparametric models (Bickel et

p pij ij ij ij

p lij ij ij

Z Z Z Z

Z Z Z

al., 2000).

Longitudinal data:

(Repeated measurements over time)

1 2

1*

For -th sib pair:

number of repeated measurements,

time of the -th measurement,

1,..., .

, ,...,

= .

j j j

ijk

ij

pijk ijk ijk ijk ijk

ij ij

j

n n n

T k

k n

X X T Z Z

g e

Notation:

Linear model (Verbeke & Molenberghs, 2000):

1

1

*

observed trait,

,..., covariates at time ,

, ,..., conditional mean of ,

covariate ad

justed trait.

ijk

pijk ijk ijk

pijk ijk ijk ijk

ijk

X

Z Z T

T Z Z X

X

*0 00

1

0 00 1

,

, , ,..., : linear coefficients.

pl

ijk ijk ijk l ijkl

p

X X T Z

3.2 Covariate adjusted linkage detection General Procedure:Select a regression model for the covariates.Estimate the covariate adjusted trait based on

the above regression model.Apply the linkage procedures, such as the HE

model or the variance-components model, using the estimated adjusted trait values and genotypic values.

3.3 Cross-sectional data

Covariate adjusted HE model:

2* * *1 2

*

0

: adjusted squared trait difference.

Same derivation in HE (1972)

| .

, : unknown parameters.

Testing problem:

: 0 (no linkage

j j j

j j j

Y X X

E Y

H

1

),

: 0 (linkage). H

Estimation of adjusted trait values:

Data from sib pairs are correlated Existing estimation methods for independent

data can not be directly applied.

Two approaches:1) Use methods for correlated data, such as

GEE – treat each family as a subject, each member as a single observation.

2) Resample independent observations:

i. Randomly sample one member from each family.

ii. Estimate the parameters and adjusted trait values using the re-sampled data and procedures for independent data, such as LSE, MLE, etc.

iii. Repeat the previous steps many times and compute the estimates using the average of the estimates from the re-sampled data.

This leads to consistent estimates when the sample size (number of families) is large (Hoffman et al., 2001).

Procedure for linear adjustment model:

* *

1*

2* * *

1 2

*

ˆStep 1: Estimate by , a consistent estimator.

Step 2: Estimate and by

ˆˆ ,..., ; ,

ˆ ˆ ˆ .

ˆStep 3: Fit the HE model using and test

=0 vs. <0.

ij j

pij ij ij ij

j j j

j

X Y

X X Z Z

Y X X

Y

3.5 Longitudinal data

2* * *1 2

* *

1

*

adjusted squared difference in -th pair

at -th measurement;

/ : mean adjusted difference;

| .

Linkage detection:

0 (no linkage); 0 (lin

j

jk jk jk

n

j jk jk

j j j

Y X X

j

k

Y Y n

E Y

kage).

Two sources of potential correlations in the estimation of adjusted trait values:

i. Correlation within a sib intra-subject correlation.

ii. Correlation between sibs within a family intra-family correlation.

Nested longitudinal data.

Methods for longitudinal estimation can not be directly applied (Morris, Vannucci, Brown and Carroll, 2003, JASA).

Resampling approach:i. Randomly select one sib from each family

Resampled data contain repeated measurements of independent sibs.

ii. Estimate the covariate adjusted trait values from the above resampled data based on longitudinal estimation methods (GEE, MLE, REMLE, etc.).

iii. Repeat the above steps many times and estimate the parameters using the averages of the estimates from the resampled data.

iv. Fit the HE model using existing procedure.

4. Framingham Heart Study

Features of the data: Clustered data from families; Repeated measurements; Multi-sib families; Continuous and categorical covariates.

Variables: Quantitative trait: SBP; Covariates: age, gender (0=female, 1=male),

drinking (average daily consumption).

2

1 21

211 2

21 2

231 2

SAGE HE Model:

Use / in place of ;

age age : age difference between sibs;

gender gender : 0 if same sex, 1 if different sex;

drinking drinking ;

Fit regressi

jn

j jk jk j jk

j j j

j j j

j j j

Y X X n Y

Z

Z

Z

1 2 31 2 3

on:

| , .

No linkage: 0; Linkage: 0.

j j j j j j jE Y Z Z Z Z

0 1

2 3

*

2

* * *1 2

1

New HE Model:

Use linear model with 1000 resampling replications;

age, gender, drinking ; age

gender + drinking;

ˆage, gender, drinking ; ;

/ : averj

ijk ijk

n

j jk jk jk

X X

Y X X n

*

age squared trait difference.

Fit regression:

| , .

No linkage: =0; Linkage: <0.

j j j jE Y Z

5. Discussion Advantages for covariate adjustment: small variation for the estimates; better interpretation for the model. Directions of further research:Non-additive models, e.g. covariate-gene and

covariate-environment interactions;Covariate adjustment with other measures of

the trait difference;Methods of model selection;Models with general pedigrees.