Incomplete Pedigree Phenotypes in Variance-Component Analysis

Why Incomplete Pedigrees Need an Observed-Data Likelihood

A pedigree may contain relatives whose phenotypes were not measured, could not be recovered, or were unavailable for analysis. The relevant likelihood is not the full-data likelihood with invented values inserted for those relatives. It is the probability density of the phenotypes that were observed, together with any information about how the observed subset came to be observed when that process matters.

In a variance-component analysis, relatives are linked through their covariance. A missing phenotype removes that relative’s direct phenotype contribution, but it does not erase the pedigree relationships among the remaining relatives. The observed members’ covariance must still reflect their positions in the pedigree and the model’s variance components.

The Gaussian Likelihood for the Observed Subset

Suppose the full pedigree phenotype vector follows the Gaussian variance-component model \(Y\sim N(X\beta,V(\theta))\). Here \(X\) is the fixed-effects design matrix, \(\beta\) is its coefficient vector, and \(V(\theta)\) is the covariance matrix determined by variance-component parameters \(\theta\). Let \(S_O\) select the rows for observed relatives. Then \(Y_O=S_OY\), \(X_O=S_OX\), and the observed covariance is \(V_{OO}(\theta)=S_OV(\theta)S_O^\mathsf{T}\). Thus \(Y_O\sim N(X_O\beta,V_{OO}(\theta))\).

For \(n_O\) observed phenotypes, the contribution to the likelihood is

The rows and columns selected for \(V_{OO}\) must correspond to the same observed relatives and in the same order as the entries of \(y_O\). In general, this is not equivalent to fitting the model as if the observed relatives formed a new, unrelated sample. Their covariances remain part of \(V_{OO}\).

Why Marginalizing Gives the Correct Likelihood

Partition the full phenotype vector and its mean and covariance into observed and missing components: \(Y=(Y_O^\mathsf{T},Y_M^\mathsf{T})^\mathsf{T}\), with mean \((\mu_O^\mathsf{T},\mu_M^\mathsf{T})^\mathsf{T}\) and covariance blocks \(V_{OO},V_{OM},V_{MO},V_{MM}\). The density of the observed data follows by integrating out the missing vector:

A multivariate normal distribution has normal marginal distributions. In particular, the integral above is the density of \(N(\mu_O,V_{OO})\). This establishes why the observed likelihood uses the observed covariance block, rather than a conditional distribution of observed phenotypes given the missing ones. The missing values are not conditioned on because they are unknown.

The result requires the stated full-data Gaussian model and a valid covariance matrix. Any covariance parameters that affect entries of \(V_{OO}\) can be informed by the observed data. Parameters that affect only relationships with missing phenotypes may be weakly informed or not identifiable from that pedigree’s observed phenotypes alone.

When the Observation Process Is Ignorable

Let \(R\) record which phenotypes are observed. Missingness is called missing completely at random if, conditional on the design, the distribution of \(R\) does not depend on any phenotypes. It is called missing at random if that distribution can depend on observed phenotypes and fully observed covariates, but not on the unobserved phenotypes after conditioning on those quantities.

For likelihood inference about \(\theta\), the observation process is ignorable under the missing-at-random condition when its parameters are distinct from the phenotype-model parameters. In that case, the joint likelihood factors as \(f_\theta(y_O)\,g_\psi(r\mid y_O,\text{covariates})\), where \(\psi\) denotes observation-process parameters. The second factor does not involve \(\theta\), so maximizing the observed phenotype likelihood \(f_\theta(y_O)\) gives the same likelihood-based inference for \(\theta\) as maximizing the joint likelihood. This is a condition for omitting the missingness model from inference; it is not a claim that the observation pattern is irrelevant in every sense.

When Phenotype-Dependent Observation Must Be Modeled

The key distinction is whether observation depends on a phenotype that remains unobserved. If it does, missingness is not missing at random under the stated conditioning set. The observed phenotype likelihood alone generally no longer describes the data-generating process. The likelihood must include the probability of the observed pattern, integrated over the missing phenotypes:

The observation probability inside the integral depends on \(y_M\), so it cannot generally be pulled outside the integral or discarded as a factor that is constant in \(\theta\). A complete analysis must specify the phenotype model and the observation model, including any unknown parameters. In practice, conclusions can depend on assumptions about this observation mechanism; data on a validation subset or other external information can help assess it.

Practical Checks and Common Pitfalls

For an incomplete pedigree, first identify the observed set and construct the matching covariance submatrix. Then determine whether the missingness mechanism depends on unobserved phenotypes. These are separate questions: correct covariance subsetting does not by itself make an informative observation process ignorable.

• Wrong: “A missing phenotype can be replaced by zero because the model mean is zero.” Correction: A missing value is integrated out in the likelihood; zero is a phenotype value, not a missing-data rule.
• Wrong: “A missing relative can be removed from the pedigree covariance calculation.” Correction: The relative’s phenotype is absent, but the covariance among observed relatives must still reflect their pedigree relationships.
• Wrong: “Any phenotype-related observation makes missingness nonignorable.” Correction: Dependence on observed phenotypes can satisfy missing at random. Dependence on unobserved phenotypes is the key obstacle to omitting the mechanism.
• Wrong: “Using the observed covariance block automatically handles every missingness process.” Correction: It gives the correct marginal phenotype likelihood, but phenotype-dependent observation may require the additional integrated likelihood contribution.

Finally, missing phenotype handling is distinct from how families entered a study. If recruitment or family selection conditions inclusion on phenotypes, that selection may also affect the likelihood. The within-pedigree observation model described here does not replace an appropriate treatment of study ascertainment.