# Longitudinal variance-components analysis of the Framingham Heart Study data

- Stuart Macgregor
^{1}Email author, - Sara A Knott
^{2}, - Ian White
^{2}and - Peter M Visscher
^{2}

**4(Suppl 1)**:S22

https://doi.org/10.1186/1471-2156-4-S1-S22

© Macgregor et al; licensee BioMed Central Ltd 2003

**Published: **31 December 2003

## Abstract

The Framingham Heart Study offspring cohort, a complex data set with irregularly spaced longitudinal phenotype data, was made available as part of Genetic Analysis Workshop 13. To allow an analysis of all of the data simultaneously, a mixed-model- based random-regression (RR) approach was used. The RR accounted for the variation in genetic effects (including marker-specific quantitative trait locus (QTL) effects) across time by fitting polynomials of age. The use of a mixed model allowed both fixed (such as sex) and random (such as familial environment) effects to be accounted for appropriately. Using this method we performed a QTL analysis of all of the available adult phenotype data (26,106 phenotypic records).

In addition to RR, conventional univariate variance component techniques were applied. The traits of interest were BMI, HDLC, total cholesterol, and height. The longitudinal method allowed the characterization of the change in QTL effects with aging. A QTL affecting BMI was shown to act mainly at early ages.

## Background

In this paper we analyze the Framingham Heart Study offspring data using univariate and multivariate variance component techniques, with particular emphasis on how inherited factors related to heart disease change over the life of an individual.

### Data available

There were 4692 individuals in the study. The data were ascertained in two cohorts. The first had up to 21 trait measures for the 40 years following 1948. The second cohort had up to 5 trait measures for the 20 years following 1971. Genotype data were available for 1702 individuals. The vast majority of individuals in the study had all their measures when they were age 20 or older; measures at younger ages were not analyzed. Phenotype data was available for 2885 individuals. In total, 26,106 phenotypic records were used in the full multivariate analysis. The traits considered were body mass index (BMI), height, fasting high density lipoprotein cholesterol (HDLC), and total cholesterol.

### Manipulation of data for analysis

Age stratified data. Age bands used for univariate analyses. The multivariate analyses use all the data simultaneously.

Age | 20–30 | 30–40 | 40–50 | 50–60 | 60–70 | 70–80 |
---|---|---|---|---|---|---|

Number of individuals | 783 | 1817 | 2263 | 1964 | 1410 | 879 |

## Methods

### Univariate analyses

For BMI and height, potential covariates were sex, cohort, cigarette consumption, and alcohol consumption. For HDLC and total cholesterol, BMI and an indicator variable for hypertension treatment were also considered.

#### Polygenic

The traits were examined for variation across time using Residual Maximum Likelihood (REML, program ASREML) [1] to calculate polygenic heritabilities in the six age bands.

#### Quantitative Trait Locus (QTL)

Standard univariate variance components (VC) analyses were done using the SOLAR program [2] and confirmed using ASREML. LODs were calculated using multipoint IBDs (identity by descent coefficients) every 1 cM.

### Longitudinal Analysis

#### Polygenic

A RR model was fitted to the full (up to 26,106 records) data set for each trait. The model allowed both the additive genetic effect and the permanent environment term to vary linearly with age. The model was therefore

y_{ij} = μ + (a_{i1} + a_{i2} × age*) + (c_{i1} + c_{i2} × age*) + f_{i} + e_{ij},

where y_{ij} is the phenotype of individual i at time point j, μ represents the fixed effects, e_{ij} is the special or temporary environmental effect, f_{i} is an effect for family or household and the terms a_{i1}, a_{i2}, c_{i1}, and c_{i2} are the coefficients of the linear polynomial linking mean corrected age (age*) to the relevant genetic and permanent environmental terms. Note that using age* instead of age means the polynomials are *orthogonal* (see [3]). The genetic and permanent environment terms were assumed to have unstructured variance-covariance matrices, denoted by matrices **G** (with entries g_{ij}) and **P** (with entries p_{ij}), respectively. These estimated (co)variances are then linked to a relevant set of n ages (in this case 20–95). For example, for the genetic effect at age x the variance contribution is

g_{11} + 2 × [x - mean(x)] × g_{12} + [x- mean(x)]^{2} × g_{22}. (1)

In matrix notation the n × n matrix, **T**, of phenotypic (co)variances is hence decomposed as

**T** = **XGX**^{T} + **XPX**^{T} + σ_{e}^{2}**I**, (2)

where **X** = (**1 age***) with **1** an n-vector of 1s and **age*** a vector of ages from age*(1) to age*(n). σ_{e} ^{2} is the e_{ij} term variance and **I** is the identity matrix. In cases where a family effect is included, an additional term, σ_{f}^{2}**11**^{T}, where σ_{f}^{2} is the variance term associated with the family effect, should be added to equation (2) (assuming no relationship between age and family effect).

Estimates of the phenotypic and component variances (genetic, permanent environment, error) at any age are given by the appropriate diagonals of **T**, **XGX**^{T}, **XPX**^{T}, and σ_{e}^{2}**I**, respectively. Estimates of heritability are obtained from the relevant variances. The off-diagonals of the n × n matrices are the covariances (or correlations if standardized) between the ages. Note that although a linear polynomial is fitted, the graphs of the variances against age are quadratic, because equation (1) is quadratic in age.

#### QTL

The above model was then extended to include an additional term for an age-dependent QTL effect. The model is therefore

y_{ij} = μ + (a_{i1} + a_{i2} × age*) + (c_{i1} + c_{i2} × age*) + (l_{i1} + l_{i2} × age*) + f_{i} + e_{ij},

where the terms l_{i1} and l_{i2} are the terms of the linear polynomial linking mean corrected age (age*) to QTL effect. The QTL effect is assumed to have an unstructured variance-covariance structure, with matrix **Q** (with entries q_{ij}). The full decomposition, allowing one to calculate estimates of QTL-specific heritabilities is therefore,

**T** = **XGX**^{T} + **XPX**^{T} + **XQX**^{T} + σ_{e}^{2}**I**

## Results

### Univariate analyses

#### Polygenic

#### QTL

Univariate LOD scores

Chromosome | Position (cM) | Trait | Age band for trait | LOD |
---|---|---|---|---|

16 | 95 | BMI | 20–30 | 3.12 |

5 | 183 | Height | 60–70 | 2.61 |

10 | 23 | HDLC | 70–80 | 2.50 |

12 | 119 | HDLC | 20–30 | 2.46 |

14 | 138 | T. Chol | 50–60 | 2.57 |

19 | 101 | T. Chol | 50–60 | 3.11 |

20 | 24 | T. Chol | 40–60 | 3.03 |

### Longitudinal analysis

#### Polygenic

Phenotypic and Genotypic Correlations. Polygenic model correlations derived from the full longitudinal analyses (Equation 2).

Phenotypic correlations | Genotypic correlations | |||||
---|---|---|---|---|---|---|

Trait | Age 30–70 | Age 30–50 | Age 50–70 | Age 30–70 | Age 30–50 | Age 50–70 |

Height | 0.79 | 0.90 | 0.89 | 0.83 | 0.96 | 0.95 |

BMI | 0.42 | 0.70 | 0.84 | 0.42 | 0.75 | 0.91 |

Total Cholesterol | 0.37 | 0.57 | 0.61 | 0.60 | 0.90 | 0.88 |

HDLC | 0.41 | 0.56 | 0.64 | 0.80 | 0.94 | 0.96 |

#### QTL

We also looked at the other four QTL peaks listed in the univariate results section. However, convergence problems prevented us from obtaining reliable results. Similar problems arose when fitting higher order polynomials to the data.

## Discussion

We performed analyses that explain how the components of variance change over time. The RR model fitted is typically only used for polygenic genetic effects in animal breeding sire models. We have expanded the basic RR model to allow the analyses of both extended pedigrees and marker-specific IBD information. The agreement between the univariate and multivariate analyses performed was good and some of the larger QTL effects were more fully characterized in the longitudinal analyses.

Fitting a higher order polynomial for the relationship between age and the genetic effects may have resulted in a closer fit between the univariate and multivariate results but, in addition to the practical problems of fitting such models, the true relationship between the traits and age is unlikely to be especially complex.

As an alternative to polynomial-based RR approaches, character process models [3] may be useful for longitudinal data analyses, particularly when the correlation between trait measures at distant ages is low. However, when the correlations between trait measures over time is high (as is the case for most of the traits here) polynomial-based methods are effective [3].

The multivariate QTL analyses indicated that one of the QTL detected acted across the range of ages while the other two acted more strongly at the extremes of the age ranges. For some traits there may be correlations between trait value and survival. This may lead to biased QTL effects for QTL acting at later ages. However, maximum likelihood procedures can account for this form of "selection" under certain circumstances [5] and the selection pressure on a single QTL is likely to be small so that a bias in (co)variance estimates may be negligible.

Time constraints prevented a full longitudinal genome scan for QTL but the results shown here indicate that this may be a possibility for other large data sets. The method presented here allows all of the available data to be used in a single powerful analysis.

## Declarations

### Acknowledgments

Financial support was provided by Akzo Nobel Organon, the Biotechnology and Biological Sciences Research Council, and the Royal Society.

## Authors’ Affiliations

## References

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## Copyright

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