### 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**