7. Continuous Trait Parameters

For a continuous trait, the phenotypes are assumed to distribute normally within genotypes. The penetrance equals the height of the normal density (including 1/√2π) as the phenotype. You can scale, standardize, or transform phenotypes by assigning transformations in the metadata document.

The effects on the genotype means of up to ten covariates may be estimated for each continuous trait when using modules qmlprmv or qmlprdd. This extension allows for genotype-specific age or measured environmental effects. Only linear effects are estimated; for cross or squared terms, include the product or square as an OV and treat it as another covariate.

The power parameter corresponds to P in the MacLean power transformation assignable in the metadata document. Set P = 1 for no transformation. Estimating P within a 1-genotype model allows you to determine the transformation to correspond to a single normal density. Estimating P within the environmental model allows you to test for a mixture of distributions. If you then assign the power transformation in the metadata document, setting its constant to the estimate of P, further analyses will consider the transformed phenotypes. Estimating P within a genetic model allows transformation of the phenotypes to obtain the best fit to each model.

Only modules qmlprdd and qmlprddp restrict the number of alleles to two and the within-genotype standard deviations to equality.

1. Means/Standard Deviations

Module qmlprmv parameterizes the model as means μ_i, standard deviations σ_i, and slope s_ji for covariate j for each genotype i. The number of loci and alleles included in the genetic model are unrestricted.

2. Dominance/Displacement

Module qmlprdd restricts the genetic model to two alleles. The parameters comprise the total mean μ_T, total standard deviation σ_T, dominance d, displacement t for the mean, mean slope s_j for covariate j, and dominance d_sj, displacement t_sj for covariate j. If p represents the frequency of allele 1, q = 1 - p, μ_i and s_ij represent the mean and slope of covariate j, respectively, for genotype i, i = 1, 2, 3, and σ represents the within-genotype standard deviation, then
μ_T = p²μ₁ + 2pqμ₂ + q²μ₃,
s_j = p²s_1j + 2pqs_2j + q²s_3j,
σ_T² = p²[μ₁² + Σs_1j²] + 2pq[μ₂² + Σs_2j²] + q²[μ₃² + Σs_3j²] + σ²,
d = (μ₂ - μ₁)/(μ₃ - μ₁),
t = (μ₃ - μ₁)/σ,
d_sj = (s_2j - s_1j)/(s_3j - s_1j),
t_sj = (s_3j - s_1j)/σ.

The reverse equations equal:
μ₁ = 2pqdt - q²t,
μ₂ = μ₁+ dt,
μ₃= μ₁+ t,
s_1j = 2pqd_sjt - q²t_sj,
s_2j = s_1j+ d_sjt_sj,
s₃= s_1j + t_sj.

Note that t differs from the definition of displacement in Morton & MacLean [1974]. For multi-locus models, d, t, d_sj, and t_sj are locus-specific and t and t_sj add across loci.

3. Means/Standard Deviations/Threshold

Module qmlprmvt parameterizes the model as means for each genotype μ_i, standard deviations for each genotype σ_i, and threshold T. Parameter T specifies the lower limit of the trait for individuals whose phenotypes are impossible to obtain because medication or the disease process has altered the level. For example, medicated hypertensives might be included in an analysis of blood pressure by specifying that their levels exceed the diagnostic threshold. Observation records would contain a missing value for the continuous trait input and "affected" status for a corresponding disease trait. The number of loci and alleles included in the genetic model are unrestricted.

4. Dominance/Displacement/Proportion

Module qmlprddp restricts the genetic model to two alleles. The parameters comprise the total mean μ_T, total standard deviation σ_T, dominance d, displacement t, and proportion P. If p represents the frequency of allele 1, q = 1 - p, μ_i represent the genotype means, i = 1, 2, 3, and σ represent the within-genotype standard deviation, then
μ_T = p²μ₁ + 2pqμ₂ + q²μ₃,
σ_T² = p²μ₁² + 2pqμ₂² + q²μ₃² + σ²,
d = (μ₂ - μ₁)/(μ₃ - μ₁), and
t = (μ₃ - μ₁)/σ.

Note that t differs from the definition of displacement in Morton & MacLean [1974]. For multi-locus models, d and t become locus-specific and t adds across loci. Parameter P specifies the proportion of individuals in the population whose phenotypes are impossible to obtain because medication or the disease process has altered the level. For example, medicated hypertensives might be included in an analysis of blood pressure by specifying that their levels exceed the 95^th percentile. Each observation record would be designated a missing value for the continuous trait and affected for a corresponding disease trait.