The Gaussian Likelihood for Family Phenotypes

Why a Family Likelihood Is Multivariate

In a pedigree, relatives’ quantitative phenotypes are generally correlated. A likelihood that treats each relative’s phenotype as an independent observation would therefore discard part of the model: the dependence encoded by the pedigree covariance matrix. The Gaussian likelihood puts the family’s phenotype vector, its modeled mean, and its covariance together in one probability density.

This tutorial derives that density and its log form. It assumes that the covariance matrices for the pedigree have already been specified, as in the earlier material on building pedigree covariance matrices. The task here is to translate a chosen mean and covariance model into a likelihood, not to estimate its parameters or derive a particular covariance matrix from genetic data.

The mean vector specifies the expected phenotype for each person. A common fixed-effects model is \(\mu=X\beta\), where \(X\) is a known design matrix containing an intercept and any covariates, and \(\beta\) is a vector of unknown regression coefficients. A variance-components model expresses the covariance as a weighted sum of known pedigree matrices. For example,

Here \(A_k\) is a specified covariance-pattern matrix for component \(k\), \(\tau_k\) is its variance-component coefficient, \(I_n\) is the \(n\)-dimensional identity matrix, and \(\sigma_e^2\) is the residual variance. The parameter collection \(\theta\) includes the mean parameters and covariance parameters. The matrices and coefficients must yield a positive-definite \(V(\theta)\) for the ordinary nonsingular Gaussian density to apply. In particular, the residual term represents independent residual variation in this model; it does not mean the phenotypes themselves are independent.

The Multivariate Gaussian Likelihood

For an observed vector \(y\), define its residual vector relative to the modeled mean as \(r=y-\mu\). The multivariate normal density, and thus the likelihood as a function of the parameters, is

The determinant \(|V|\) describes the overall volume and scale of the modeled joint distribution. The quadratic form measures how far the observed phenotype vector lies from its mean, accounting for the covariance and hence the family relationships. It is a squared distance in covariance-scaled coordinates, not generally the sum of squared residuals divided by a common variance.

From Density to Log-Likelihood

Taking the natural logarithm of the Gaussian likelihood turns its product and power terms into sums. Because the logarithm is strictly increasing, a parameter value that gives a larger likelihood also gives a larger log-likelihood. Applying the logarithm to each factor yields

The first term depends only on the number of phenotypes. The second penalizes covariance matrices with a large determinant, while the final term penalizes a poor fit of the observed vector to the modeled mean, measured relative to \(V\). Both covariance-dependent terms are essential: retaining only the quadratic form would reward some covariance choices simply for making residuals appear small, without accounting for the distribution’s scale.

The expression follows directly from the density: \(\log(a b)=\log(a)+\log(b)\), \(\log(|V|^{-1/2})=-\tfrac12\log|V|\), and \(\log(\exp(-Q/2))=-Q/2\), where \(Q=r^\mathsf{T}V^{-1}r\). Thus the log-likelihood is not a separate model; it is an equivalent representation of the same Gaussian likelihood.

Mean Models, Pedigree Blocks, and Independent Families

Covariates enter through the mean vector, not by changing the Gaussian formula. For example, if \(x_i\) is an individual’s age, a linear mean model might specify \(\mu_i=\beta_0+\beta_1 x_i\). The likelihood then compares each observed phenotype with its modeled mean while still using the pedigree covariance to account for dependence. A covariate in the mean does not by itself remove familial correlation.

When several pedigrees are sampled independently, their phenotype vectors are independent under the model conditional on their specified covariates and parameters. The overall likelihood is the product of their pedigree-specific likelihoods, and the overall log-likelihood is the sum. Equivalently, the full covariance matrix is block diagonal, with one covariance block per pedigree. Independence is across pedigrees in this setup, not among members of a pedigree.

Assumptions and a Common Pitfall

A Gaussian pedigree likelihood is appropriate only when its assumptions describe the data and sampling setup well enough for the intended analysis. In particular:

• The phenotype vector is modeled as jointly multivariate normal with the stated mean and covariance. Marginal normality alone does not establish the required joint model.
• The covariance matrix is symmetric, finite, and positive definite for the ordinary density written here. A singular covariance requires a different treatment and cannot be inserted into the displayed inverse-and-determinant formula.
• The mean and covariance are correctly specified for the individuals and covariates included. Within-family dependence must be represented by the covariance rather than ignored.
• For a product of pedigree likelihoods, distinct pedigrees are assumed independent under the model. If observations share modeled dependence across family units, the block-diagonal factorization is not justified.

A frequent mistake is to say, “There are two relatives, so multiply two univariate normal densities.” That reasoning assumes their phenotypes are independent. With covariance \(V\) having nonzero off-diagonal entries, the correct joint density includes \(V^{-1}\) and \(|V|\); it generally cannot be factored into the product of the two marginal densities. Another mistake is to interpret a larger likelihood as the probability that a model is true. A likelihood is a function of parameter values for observed data, not a probability distribution over competing models unless an additional probability framework is specified.