Conditional Association Tests Under a Fitted LMM

What Is Conditional in a Mixed-Model Association Test?

A genome-wide association analysis tests one candidate variant at a time while accounting for covariates and correlation among individuals. In a linear mixed model (LMM), that correlation is represented by a covariance structure, often informed by a genomic relationship matrix. The null-model fitting step estimates the covariate effects and covariance parameters without the candidate variant. A null-fitted conditional association test then uses those fitted values as fixed inputs when testing the variant’s effect.

“Conditional” here describes the analysis after plugging in estimates from the fitted null model. It does not mean that estimates from the phenotype have magically become known constants, nor does it mean that the test conditions on the observed phenotype in a probability-theory sense. The distinction matters: a test that freezes nuisance estimates is not identical to one that estimates them again after adding the variant.

The Model and the Hypothesis

Let \(y\) be the \(n\)-vector of quantitative phenotypes, \(X\) the \(n\)-by-\(q\) matrix of fixed covariates (including an intercept if appropriate), and \(g\) the \(n\)-vector of dosages for one variant. A dosage commonly records the number of copies of a designated allele, from 0 to 2. The LMM for this candidate variant is

Here \(\gamma\) is the vector of covariate coefficients, \(\beta\) is the per-dosage association effect, \(K\) is a specified relationship matrix, \(\tau\) is the random-effect variance component, and \(\sigma_e^2\) is the residual variance. Under these assumptions, the marginal distribution of \(y\), after accounting for the random effect, is normal with mean \(X\gamma+g\beta\) and covariance

The single-variant null hypothesis is \(H_0:\beta=0\). A two-sided alternative is \(H_a:\beta\ne 0\); a one-sided alternative is appropriate only when a directional effect was specified in advance. For a quantitative phenotype, \(\beta\) represents the change in the modeled phenotype mean per additional dosage unit, conditional on the modeled covariates and the assumed covariance structure. It is an association parameter, not by itself a causal effect.

The null model omits \(g\beta\). Suppose its fitted parameters are denoted by \(\hat{\gamma}_0\), \(\hat{\tau}_0\), and \(\hat{\sigma}_{e,0}^2\). The conditional formulation substitutes these values into the candidate-variant model:

In this formulation, \(\hat{\gamma}_0\), \(\hat{\tau}_0\), \(\hat{\sigma}_{e,0}^2\), and the specified \(K\) are fixed while the candidate effect is tested. The null sets \(\beta=0\); the alternative permits \(\beta\ne0\). This description defines what is held fixed, but does not select a particular test statistic or its calibration.

Why the Conditional Model Has This Form

The justification follows directly from the LMM assumptions. Since \(u\) and \(\varepsilon\) are independent normal vectors, their sum is normal, with mean zero and covariance equal to the sum of their covariances. Therefore \(u+\varepsilon\sim N(0,\tau K+\sigma_e^2I)\). Adding the fixed mean \(X\gamma+g\beta\) gives the stated marginal distribution for \(y\). Substituting the null-model estimates gives the plug-in conditional model.

The cross-covariance terms are zero because the random effect and residual are assumed independent. Under \(H_0\), the mean reduces to \(X\gamma\); after plugging in the null estimates, it is \(X\hat{\gamma}_0\). This argument establishes the model distribution under the stated assumptions. It does not establish that the plug-in test has exact finite-sample calibration: the fitted values were estimated from data, so their uncertainty has been set aside in this conditional calculation.

Which Parameters Stay Fixed?

It is useful to name the distinction explicitly. The candidate effect \(\beta\) is the parameter of interest; under the null it equals zero, while under the alternative it is allowed to differ from zero. The nuisance parameters describe the baseline mean and covariance rather than this variant’s effect. In the fully frozen conditional formulation, these nuisance parameters are replaced by their null-model estimates and are not updated during the variant test.

This is a definition of one conditional testing setup, not a claim that every mixed-model software implementation freezes every nuisance parameter. Some procedures keep the covariance parameters at their null estimates but re-estimate covariate coefficients when fitting the alternative. Others use a null-model calculation that does not fit the alternative mean model at all. Reports should therefore identify the actual procedure and its parameter treatment. The phrase “conditional on the fitted null model” is not enough to infer every implementation detail.

Assumptions, Diagnostics, and Failure Modes

The formulation depends on assumptions about both the mean and covariance. The covariates and genotype coding must represent the intended comparison; the LMM’s residual and random-effect distributions must be suitable for the phenotype; and \(K\) must be an appropriate representation of the modeled relatedness. The covariance matrix \(\tau K+\sigma_e^2I\) must be valid (positive semidefinite, and positive definite for an ordinary nonsingular multivariate normal density). The random effect and residual must be independent under the specified model. A test’s reference distribution also depends on its chosen statistic and calibration; merely writing the conditional model does not settle that issue.

Before interpreting a variant test, check that the null model converged, that its variance estimates are plausible, and that the relationship matrix and sample ordering match the phenotype and genotype vectors. Inspect residual patterns and the distribution of association evidence across variants for signs that the mean or covariance model is inadequate. These checks cannot prove the assumptions, but can reveal problems such as extreme outliers, an inappropriate trait scale, or residual structure left unexplained by the model.

A further limitation is uncertainty in the fitted nuisance parameters. The plug-in model treats \(\hat{\gamma}_0\) and the variance estimates as fixed for the conditional calculation, even though they were estimated. This can be a useful computational and inferential strategy, especially when the same null model serves many variant tests, but it does not automatically account for all uncertainty from null-model fitting. Whether that matters for a particular analysis depends on the sample, the model, and the test calibration.

A common mistake is to say, “The variance components were estimated under the null, so they are known and cannot affect the variant test.” The correction is that the procedure holds their fitted values fixed; it does not turn estimates into known population parameters. Another mistake is to assume that every alternative model reuses the null covariate coefficients and covariance estimates. That is true only for the explicitly frozen formulation; implementations differ, so verify what is refitted.

How This Formulation Differs from Nearby Choices

The main contrast is between a null-fitted conditional analysis and a test that refits nuisance parameters under the alternative. The conditional approach reuses null estimates and changes only the designated genotype effect in the fully frozen version. A refitted alternative allows some nuisance estimates to change when genotype is added. These approaches answer related questions, but they do not necessarily produce the same test statistic or p-value.

A second contrast is with a naive regression that treats observations as independent. The conditional LMM retains the covariance \(\hat{V}_0\), so correlated phenotypes are not represented as independent measurements. Ignoring that covariance can misstate the uncertainty of an association estimate. Conversely, specifying an LMM does not guarantee protection against every source of misspecification: an unsuitable \(K\), omitted covariates, or a poor trait model can still distort inference.

The conditional model is therefore best understood as the setup for a test, not as a complete testing recipe. The chosen statistic and its null calibration must be stated separately. In particular, a conditional formulation alone does not imply a specific standard error, reference distribution, or p-value.