Preliminary study
At the first stage of model fitting, we adjusted SBP by known effective nongenetic factors of gender, age, total cholesterol, smoking, fasting glucose, hypertension treatment, and weight, and high blood pressure from Cohort 1 and 2. We regressed SBP on all these covariates mentioned above and obtained the residual of SBP referred to as SBP*. Our adjustment was initially done on each of all 100 replicates, respectively, consisting of around n = 99,300 observations from about n = 2747 sib pairs in each sample. Additionally, we adjusted on a larger sample by pooling two replicates randomly selected (replicate 43 and 47) that included the 199,536 observations from n = 5512 sib pairs.
Sib pair linkage analysis
In linkage analysis, we investigated the revised Haseman and Elston linkage statistic [2]. For the second stage of model, the mean-corrected cross-product of SBP* was used as a dependent variable, defined by
C(SBP
j
*) = (SBPj 1* - m) (SBPj 2* - m), (1)
where SBPj 1* and SBPj 2* are the residual of the observed SBP s for the first and second sibs, respectively, in the jth pair, and m is the mean of SBP
ji
* for all i and j. We considered as independent variables the number of alleles IBD at the locus in the sib pair. As similarly described in Suh et al. [7], we denote I
k
for k = 1, 2, ..., 6 as the number of alleles IBD at six markers closest to b34, b35, b36, s10, s11, and s12, which determine SBP. We also denote U
l
for l = 1, 2, ..., 5 as the number of alleles IBD at five genes closest to b5, b14, b16, b18, and b21, which are unrelated to any of these loci.
The mixed model
We considered three different models to analyze longitudinal data. First, we fitted an independence model (Model 1) which is defined as
C(SBP
j
*) = α + Σβ
k
I
jk
+ Σγ
l
U
jl
+ ε
j
,
where β
k
for k = 1, 2, ..., 6 and γ
l
for l = 1, 2, ..., 5 are parameters to be estimated.
Our second approach of the mixed model was a random effects model (Model 2). We considered the correlation between sib pairs in the model, assuming random effects to account for correlation between two sib pairs that share a common sibling.
C(SBP
j
*) = α + Σβ
k
I
jk
+ Σγ
l
U
jl
+ Σδ
m
R
jm
+ ε
j
, (2)
where E(δ
m
) = 0 and Var(δ
m
) = σ2δmfor which the mth (m = 1, 2) sibling is in common. If the mth sibling is in common, then R
jm
= 1, otherwise R
jm
= 0 for each of m = 1, 2.
Third, we considered one more random effect when different sib pairs are obtained from the same parents (Model 3). We added to the model equation (2) m = 0 when sib pairs have the same parents.