BACKGROUND: Using univariate and multivariate variance components linkage analysis methods, we studied possible genotype × age interaction in cardiovascular phenotypes related to the aging process from the Framingham Heart Study. RESULTS: We found evidence for genotype × age interaction for fasting glucose and systolic blood pressure. CONCLUSIONS: There is polygenic genotype × age interaction for fasting glucose and systolic blood pressure and quantitative trait locus × age interaction for a linkage signal for systolic blood pressure phenotypes located on chromosome 17 at 67 cM

Almasy, Laura

Blangero, John

Diego, Vincent P

Dyer, Thomas D

Soler, Júlia MP

BMC Genetics

English

PubMed

Abstract\ud
            \ud
            \ud
               \ud
                  Background\ud
               \ud
               Using univariate and multivariate variance components linkage analysis methods, we studied possible genotype × age interaction in cardiovascular phenotypes related to the aging process from the Framingham Heart Study.\ud
            \ud
            \ud
               \ud
                  Results\ud
               \ud
               We found evidence for genotype × age interaction for fasting glucose and systolic blood pressure.\ud
            \ud
            \ud
               \ud
                  Conclusions\ud
               \ud
               There is polygenic genotype × age interaction for fasting glucose and systolic blood pressure and quantitative trait locus × age interaction for a linkage signal for systolic blood pressure phenotypes located on chromosome 17 at 67 cM.We thank Drs. Jean W. MacCluer and Ravindranath Duggirala for critically reviewing the manuscript. This work is supported in part by NIH R01MH59490.We thank Drs. Jean W. MacCluer and Ravindranath Duggirala for critically reviewing the manuscript. This work is supported in part by NIH R01MH59490

Diego, Vincent P

Almasy, Laura 

Dyer, Thomas D

Soler, Júlia MP

Blangero, John 

Biblioteca Digital da Produção Intelectual da Universidade de São Paulo (BDPI/USP)

  Universidade de São Paulo
 
2003-12
 
Strategy and model building in the fourth
dimension: a null model for genotype × age
interaction as a Gaussian stationary stochastic
process
 
 
 
http://www.producao.usp.br/handle/BDPI/32765
 
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BMC Genetics
Open AccessProceedings
Strategy and model building in the fourth dimension: a null model 
for genotype × age interaction as a Gaussian stationary stochastic 
process
Vincent P Diego*1,2, Laura Almasy1, Thomas D Dyer1, Júlia MP Soler3 and 
John Blangero1
Address: 1Department of Genetics, Southwest Foundation for Biomedical Research, San Antonio, Texas, USA, 2Department of Anthropology, 
Binghamton University, State University of New York, Binghamton, New York, USA and 3Department of Statistics, University of Sao Paulo, Sao 
Paulo SP, Brazil
Email: Vincent P Diego* - vdiego@darwin.sfbr.org; Laura Almasy - almasy@darwin.sfbr.org; Thomas D Dyer - tdyer@darwin.sfbr.org; 
Júlia MP Soler - pavan@ime.usp.br; John Blangero - john@darwin.sfbr.org
* Corresponding author    
Abstract
Background: Using univariate and multivariate variance components linkage analysis methods, we
studied possible genotype × age interaction in cardiovascular phenotypes related to the aging
process from the Framingham Heart Study.
Results: We found evidence for genotype × age interaction for fasting glucose and systolic blood
pressure.
Conclusions: There is polygenic genotype × age interaction for fasting glucose and systolic blood
pressure and quantitative trait locus × age interaction for a linkage signal for systolic blood pressure
phenotypes located on chromosome 17 at 67 cM.
Background
The Framingham Heart Study (FHS) [1] offers a unique
opportunity to assess possible genotype × age (G × age)
interaction in the genetic architecture of complex traits. To
study G × age interaction in the FHS, we built on the var-
iance components model [2], which, assuming negligible
dominance, may be given as:
where Σ is the variance-covariance matrix of a pedigree,
 is a matrix of estimated elements, , giving the pro-
portion of alleles identical by descent (IBD) in individuals
i and j at a quantitative trait locus (QTL), Φ is the kinship
matrix of the pedigree, I is the identity matrix, and ,
, and  are variances of the additive QTL, polygenic
and environmental factors, respectively.
Our study was carried out in two phases. In Phase 1, we
made inferences on polygenic G × age interaction using
what we call the G × age model [3]. Also, we experiment
with correlation functions in the G × age model. In Phase
2, we implement the QTL version of this model. Multivar-
iate variance components linkage analysis [4] can also be
used to detect G × age interaction. We show how the bivar-
iate variance components model can be used to detect
QTL × age interaction. Given that the G × age and bivariate
from Genetic Analysis Workshop 13: Analysis of Longitudinal Family Data for Complex Diseases and Related Risk Factors
New Orleans Marriott Hotel, New Orleans, LA, USA, November 11–14, 2002
Published: 31 December 2003
BMC Genetics 2003, 4(Suppl 1):S34
<supplement> <title> <p>Genetic Analysis Workshop 13: Analysis of Longitudinal Family Data for Complex Diseases and Related Risk Factors</p> </title> <editor>Laura Almasy, Christopher I Amos, Joan E Bailey-Wilson, Rita M Cantor, Cashell E Jaquish, Maria Martinez, Rosalind J Neuman, Jane M Olson, Lyle J Palmer, Stephen S Rich, M Anne Spence, Jean W MacCluer</editor> </supplement>
This article is available from: http://www.biomedcentral.com/1471-2156/4/s1/S34
= + + ( )∑ ˆ ,Π Φ Ισ σ σq g e2 2 22 1
Πˆ πˆij
σq
2
σg
2 σe
2
BMC Genetics 2003, 4 http://www.biomedcentral.com/1471-2156/4/s1/S34
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models apply to cross-sectional and longitudinal data,
respectively, we compare their relative abilities to detect G
× age interaction.
Models and Methods
Using the bivariate mixed model [5], Blangero [6] derived
the following expression for the genetic variance in phe-
notype response to two different environments:
where  and  are the additive genetic variances of
the trait in Environments 1 and 2, in this case two ages,
 is the additive genetic variance of the phenotypic dif-
ference between environments (i.e., trait response), and
ρG is the genetic correlation between environments, in this
case ages. The term environment, denoted by E, is used in
accordance with standard statistical genetic theory [6].
There is no G × E interaction (i.e.,  = 0) if  = 
=  and ρG = 1 [6]. To model  = 0,  and ρG are
parameterized as functions of age, which is the environ-
ment of interest in our analyses:
where  is parameterized as an exponential function [7]
and ρG as exponential decay [8], and α, β, and λ are
parameters to be estimated, and m and n are two ages.
Thus, the null hypotheses are β = 0 and λ = 0, which reflect
a stationary covariance function.
In our analyses, we used the expected covariance matrix
for a given pedigree [3], which has elements giving the
covariance in trait value y for any two relatives:
where x and z index individuals at ages m and n, respec-
tively, Φxz gives their kinship status, π is the probability
that a random allele is IBD at the QTL, δxz is 1 for x = z and
0 otherwise, subscripts g, q, and e denote the polygenic,
QTL, and environmental components, respectively, and
average age is taken over the sample population, defined
as all individuals measured for the trait of interest. Some
points of clarification are needed. This is a cross-sectional
model that applies generally to three types of pair-wise
comparisons of individuals. In one type, let x = z such that
m = n. Equation (5) gives the variances in this situation, in
accord with the standard variance components model. In
a second type, it may be such that x ≠ z while m ≠ n, and,
in a third type, it may be such that x ≠ z while m ≠ n. Note
that none of these types are longitudinal comparisons,
which would be the case where x = z while m ≠ n (i.e., the
same individual is measured at different ages). The goal of
this approach lies in estimation of the change parameters,
namely β and λ, so that we can test the null hypotheses
stated above.
For the bivariate model, a trait measured at two different
time points is treated as a bivariate phenotype. The pedi-
gree covariance matrix model for a bivariate phenotype K,
with constituent phenotypes I and J, may be written as [4]:
ΣK = 2Φ G + Q + I E,  (6)
where the new matrices G, Q, and E convey the polygenic,
QTL, and environmental variance components, respec-
tively, and  is the Kronecker-product operator. The var-
iance components for this model are those for the
univariate model for traits I and J, given by ΣI and ΣJ (cf.
equation (1)), and the trait cross-covariances, which may
be parameterized as ρgijσgiσgj, ρqijσqiσqj, and ρeijσeiσej for
the polygenic, QTL, and environmental components,
respectively [4]. The null hypotheses of no polygenic gen-
otype or QTL × age interaction are expressed as ρgij = 1 and
ρqij = 1, respectively.
Scatter plots were examined for traits showing increasing
or decreasing variance in trait values with age, suggesting
potential for G × age interaction. Traits meeting these cri-
teria, namely systolic blood pressure (SBP) and fasting
glucose (GLUC), were analyzed using SOLAR [2], with
age, sex, hypertension medication, and body mass index
(BMI) as covariates. Phase 1 analyses were conducted on
an augmented data set. Exams 12 (1970), 16 (1978), 18
(1982), and 20 (1986) in Cohort 1 were combined with
Exams 1 (1971), 2 (1979), 3 (1983), and 4 (1987) in
Cohort 2, respectively. To reduce kurtosis after combin-
ing, a few outliers were removed for SBP (1–6 per exam,
all with values > the mean) but many for GLUC (51–101
per exam, all but one > the mean). No data transforma-
tion was made. Since diabetic status was not available, we
could not control for it. This gave combined measurement
periods 1–4.
Based on genome scans that we conducted for both SBP
and GLUC, we decided to focus on SBP. For Phase 2, data
σ σ σ ρ σ σ σ σ
σ σ ρ σ
g g g G g g g g
g g G g
a∆
∆
= + − ≠ ( )
= −( )
2
1
2
2
2
1 2 1
2
2
2
2 2
1
2 2
2 1
; ,
; 2 2
2 2 2= = ( )σ σg g b,
σg1
2 σg2
2
σg∆
2
σg∆
2 σg1
2 σg2
2
σg
2 σg∆
2 σg
2
σ α β
ρ λ
g age
G m n
2 3
4
= +( ) ( )
= − −( ) ( )
exp ;
exp ,
σg
2
Cov y y m age n agex z xz G g g g g, exp exp( ) = + −( ) { } + −( ) 2
1
2Φ ρ α β α β { }
+ + −( ) { } + −( ) { } ( )
1
2
1
2
1
2 5π α β α βexp exp ,q q q qm age n age
+ + −( ) { } + −( ) { }δ α β α βxz e e e em age n ageexp exp
1
2
1
2
⊗ Πˆ ⊗ ⊗
⊗
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for SBP values corrected for hypertension treatment for
Cohorts 1 and 2, imputed following Levy et al. [9] (see
Soler and Blangero [10]), were analyzed. Bivariate analy-
ses were performed on residuals from multiple regressions
with corrected SBP values as the dependent variable and
age, sex and BMI as independent variables. For the bivari-
ate analyses, analysis of residuals was necessary due to the
unbalanced and incomplete structure of the FHS longitu-
dinal data, which made covariate specification unrepre-
sentative across measurements. For this SBP data set
(corrected values and residuals), we experimented with
combined exams from Cohorts 1 and 2 and with exams
from Cohort 2 taken separately.
Results
Phase 1
We implemented equation (5) using three different corre-
lation functions, specifically the standard normal,
Cauchy, and exponential. These gave consistent results
(not shown). Only selected results for the exponential
function given by equation (4) are reported. For SBP, only
combined measurement period 3 gave strong evidence for
polygenic G × age interaction (Fig. 1). For GLUC, com-
bined measurement periods 2–4 all gave strong evidence
for polygenic G × age interaction and we believe com-
bined measurement period 3 to be representative (Fig. 1).
The p-values reported in Figure 1 and Tables 1,2 are for the
likelihood ratio statistic: Λ = -2 [ln L(θN)- ln L(θA)], where
the null, HN (parameter constrained to 0 or 1 as appropri-
ate), is compared to the alternative, HA (parameter esti-
mated). In general, Λ is distributed as a χ2 random
variable with degrees of freedom (d.f.) equal to the differ-
ence in the number of parameters under the null and
alternative hypotheses [11]. If parameter values are con-
strained to a boundary under the null hypothesis, the
asymptotic distribution of Λ is given by a mixture of 
random variables, where n denotes the d.f. and where the
mixture may include n = 0 (point mass at 0) [11]. How-
ever, the traditional criterion (i.e., difference in parame-
ters) is conservative [3].
Genetic parameters, combined measurement period 3Figure 1
Genetic parameters, combined measurement period 3. A, Variance functions for SBP (α = 4.412 ± 0.213, p < 0.001; β 
= 0.024 ± 0.009, p = 0.019) and GLUC (α = 3.937 ± 0.202, p < 0.001; β = 0.048 ± 0.010, p = 0.003). B, Correlationfunctions for 
SBP (λ = 0.030 ± 0.016, p = 0.016) and GLUC (λ = 0.067 ± 0.024, p < 0.001).
Linkage on chromosome 17 (x-axis in cM). Left panel, Univariate analyses for corrected SBP values for combined Cohorts 1 and 2, Exams 20 and 4,
respectively (solid line), and for Cohort 2, Exam 4 alone (dot-dash).  Right panel, Univariate analysis for corrected SBP residuals for Cohort 2, Exam 4 (dot-dash)
and bivariate analyses for correct SBP residuals for Exams 3 and 4 (long dash) and 4 and 5 (solid line). 
χn2
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Phase 2
We recovered the linkage signal for SBP on chromosome
17 at 67 cM reported by the FHS group [9] (Fig. 2; all max-
imum LOD scores > 3). We then applied the G × age and
bivariate models in search of QTL × age interaction. Table
1 contains model selection criteria [12] and selected like-
lihood ratio tests for univariate analyses of corrected SBP
values using the G × age model for combined Cohorts 1
and 2, Exams 20 and 4, respectively. Table 2 contains sim-
ilar information for bivariate analyses of corrected SBP
residuals using the bivariate model for Cohort 2, Exams 4
and 5.
Conclusion
Our results suggest that there is polygenic G × age interac-
tion for both GLUC and SBP (Fig. 1; Tables 1,2) and QTL
× age interaction for the putative QTL for SBP phenotypes
on chromosome 17 at 67 cM (linkage: Fig. 2; interaction:
Table 1). To our knowledge, this is the first demonstration
of QTL × age interaction in humans using linkage analysis
methods.
The cross-sectional and longitudinal analyses do not give
the same results. The difference may mean that cross-sec-
tional data are better than longitudinal data at capturing
G × age interaction. However, based on the received wis-
dom regarding the relative utility of cross-sectional and
longitudinal analyses [13], this explanation does not
seem tenable. Alternatively, that the bivariate model did
not detect QTL × age interaction (Table 2) may simply be
due to the loss in power with an overly parameterized
model [12]. Another explanation is that the bivariate
model is really operating on time rather than on age and
so perhaps the brief time span between Cohort 2, Exams
4 and 5, was too short to capture an interaction effect. Yet
another explanation of the difference between the cross-
sectional and longitudinal analyses is that the latter was
carried out on Cohort 2 only, which, taken by itself,
yielded a smaller sample size. Lastly, there is the possibil-
ity of some mixture of the last three problems.
Given that equations (5) and (6) derive from the same
underlying modeling framework – namely variance
components – they can be conceptualized as a strategy for
testing the null hypothesis that a given phenotype "trans-
lated along the time axis" is a Gaussian covariance station-
ary stochastic process [14]. This is now an established
approach in statistical genetics [14], and is readily
Table 1: Univariate Analyses for Combined Cohort 1, Exam 20 and Cohort 2, Exam 4
ModelsA Ln Likelihood AICB Evidence RatioC p-Value Ratio TestD
1. Polygenic (2) -6714.557 13433.11 5.7 × 1077 1.35 × 10-5 1 vs. 6
2. P × age (5) -6537.371 13084.74 128.8433 1.69 × 10-76 1 vs. 2
3. Conα G (4) -6583.193 13174.39 3.77 × 1021 1.04 × 10-21 3 vs. 2
4. Conβ G (4) -6542.711 13093.42 9888.592 0.00108 4 vs. 2
5. Conλ (4) -6538.801 13085.6 198.1198 0.09078 5 vs. 2
6. QTL (3) -6705.086 13416.17 1.2 × 1074 2.68 × 10-74 6 vs. 7
7. QTL × age (7) -6530.512 13075.02 1 0.00105 2 vs. 7
8. Conα QTL (6) -6534.916 13081.83 30.09494 0.00300 8 vs. 7
9. Conβ QTL (6) -6533.796 13079.59 9.819076 0.01038 9 vs. 7
AModels: QTL: model for polygenic + QTL component. P × age model: polygenic G × age. Con: Constrained parameter for P × age, Q × age, or 
bivariate model, where the parameters may be α, β, λ, or ρ and where the suffix G indicates the polygenic component. The numbers of parameters 
per model are in parentheses following each one. BAIC: Akaike's Information Criterion = -2 Ln L (θ| data) + 2K, where θ is a parameter vector and 
K is the number of parameters. CEvidence Ratio: wmin/wi = exp(∆i/2), where wmin is set to 1, wi = [exp(-∆i /2)]/Σr = 1exp(-∆r/2) and ∆i = AICi - AICmin. 
DModels compared.
Table 2: Bivariate Analyses – Cohort 2, Exams 4 and 5 ResidualsA
Models Ln Likelihood AIC Evidence Ratio p-Value; Ratio Test Ratio Test
1. Biv-polyg (6) -8940.4 17888.81 15248.48 0.00024 1 vs. 2
2. Biv-QTL (9) -8930.77 17869.54 1 ----- -----
3. Conρ G (8) -8940.4 17888.81 15248.48 1.14 × 10-5 3 vs. 2
4. Conρ Q (8) -8931.25 17870.49 1.608439 0.32815 4 vs. 2
ASee footnotes to Table 1.
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extended to the genetics of aging. It is our hope that the
analyses herein contribute to a better understanding of
the genetic architecture of the complex traits associated
with the aging process and associated complex diseases
(e.g., cardiovascular disease). We conclude that the G ×
age and bivariate models offer a feasible system for model
building in the fourth dimension.
Acknowledgments
We thank Drs. Jean W. MacCluer and Ravindranath Duggirala for critically 
reviewing the manuscript. This work is supported in part by NIH 
R01MH59490.
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Linkage on chromosome 17 (x-axis in cM)Figur 2
Linkage on chromosome 17 (x-axis in cM). Left panel, Univariate analyses for corrected SBP values for combined 
Cohorts 1 and 2, Exams 20 and 4, respectively (solid line), and for Cohort 2, Exam 4 alone (dot-dash). Right panel, Univariate 
analysis for corrected SBP residuals for Cohort 2, Exam 4 (dot-dash) and bivariate analyses for corrected SBP residuals for 
Exams 3 and 4 (long dash) and 4 and 5 (solid line).


Strategy and model building in the fourth dimension: a null model for genotype × age interaction as a Gaussian stationary stochastic process

Vincent P Diego

Laura Almasy

Thomas D Dyer

Júlia MP Soler

John Blangero

Springer - Publisher Connector

Universidade de São Paulo

Crossref

Bivariate quantitative trait linkage analysis: pleiotropy versus co-incident linkages. Genet Epidemiol

Cohabitation, convergence, and environmental covariances.

Detecting genotype × age interaction. Genet Epidemiol

DR: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach.

for the Framingham Heart Study Investigators: Description of the Framingham Heart Study Data for Genetics Analysis Workshop

Konigsberg LW: Multivariate segregation analysis using the mixed model. Genet Epidemiol

Liang K-Y: Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions.

Linear models for the analysis of longitudinal studies. Am Statist

Longitudinal familial analysis of blood pressure involving parametric (co)variance functions.

Multipoint quantitative-trait linkage analysisin general pedigrees.

RH: Evidence for a gene influencing blood pressure on chromosome 17: genome scan linkage results for longitudinal blood pressure phenotypes in subjects from the Framingham Heart Study. Hypertension

RJ: Variance function estimation.

Statistical genetic approaches to human adaptability. Hum Biol

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Strategy and model building in the fourth dimension: a null model for genotype × age interaction as a Gaussian stationary stochastic process

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