Wald, Score, and Likelihood-Ratio Tests for Fixed Effects

Why Test Fixed Effects in a Genetic Mixed Model?

A genetic mixed model can account for covariance among individuals while estimating the effects of measured predictors. For example, a model may include age, sex, or a clinical covariate as fixed effects and represent genetic similarity through a random polygenic effect. A test of a fixed effect asks whether its coefficient, or a specified set of coefficients, is compatible with a null value after the other modeled terms are accounted for. It tests a model parameter; it does not establish that the predictor causes the outcome.

Three standard approaches are the Wald test, the score test, and the likelihood-ratio test. They use different features of the likelihood: the Wald test uses the unrestricted estimate, the score test uses the slope of the likelihood at the null fit, and the likelihood-ratio test compares the best fits under the null and alternative. Under regularity conditions they have the same large-sample reference distribution, but they need not give identical statistics in finite samples.

Model and Fixed-Effect Hypothesis

The tests below assume the mean and covariance model are correctly specified, the fixed-effect design has the required full rank, and the tested restrictions are identifiable. For a standard test of \(q\) fixed-effect restrictions, write \(H_0:C\beta=d\) against the unrestricted alternative, where \(C\) has rank \(q\). A common special case partitions \(\beta=(\beta_0,\beta_1)\) and tests \(H_0:\beta_1=0\); the remaining coefficients are nuisance fixed effects and are allowed to vary.

If \(V\) is known, generalized least squares gives \(\hat{\beta}=(X'V^{-1}X)^{-1}X'V^{-1}y\), with covariance \((X'V^{-1}X)^{-1}\). In practice, the variance components in \(V\) are usually estimated. The usual tests then plug in an estimate of \(V\); their chi-square calibration is generally an asymptotic result, not an exact finite-sample guarantee.

The Wald Test

The Wald test measures how far the unrestricted estimate is from the null restriction relative to its estimated uncertainty. Let \(\widehat{\operatorname{Var}}(\hat{\beta})\) be the estimated covariance matrix of the fixed-effect estimates. Then the Wald statistic is

Under \(H_0\), and with regularity conditions and sufficiently large information, \(W\) is approximately distributed as \(\chi^2_q\), a chi-square distribution with \(q\) degrees of freedom. For one restriction, \(W\) is the squared Wald \(z\)-statistic. The multivariate form accounts for covariance among the tested coefficient estimates.

The Score Test

The score test evaluates how strongly the likelihood would initially increase if the null restriction were relaxed. It requires fitting the model under \(H_0\), but not fitting the unrestricted model. For fixed \(V\), consider \(H_0:\beta_1=0\) in the partition \(X=(X_0,X_1)\), where \(X_0\) contains nuisance fixed effects and \(X_1\) contains the \(q\) tested columns. Define the residual precision matrix after fitting the null fixed effects as

The score for \(\beta_1\), evaluated at its null value and with the nuisance coefficients fitted, is \(U_1=X_1'P_0y\). Its information matrix is \(I_{11}=X_1'P_0X_1\). The score statistic is

When \(V\) is estimated, the null fit includes estimating the covariance parameters under the null. In a Gaussian model parameterized separately by its mean and covariance, the expected information between mean and covariance parameters is zero. This supports the usual fixed-effect score calculation using the null covariance fit; more generally, nuisance-parameter uncertainty must be handled in the score's information calculation. Under regularity conditions, \(S\) is approximately \(\chi^2_q\) under \(H_0\).

The Likelihood-Ratio Test

The likelihood-ratio test compares the maximum likelihood under the null with the maximum likelihood under the alternative. Let \(\ell_0\) and \(\ell_1\) denote the maximized log-likelihoods for the null and alternative, respectively. For nested models, the statistic is

The null model must be nested within the alternative, and both fits must use the same outcome data and compatible likelihood specifications. If covariance parameters are unknown, they are optimized separately under each model. Under regularity conditions, \(\Lambda\) has an asymptotic \(\chi^2_q\) distribution under \(H_0\), where \(q\) is the number of independent restrictions. To compare models that differ in their fixed effects, use maximum likelihood: restricted maximum likelihood likelihoods are based on error contrasts and generally are not directly comparable when the fixed-effect design changes.

Why the Three Tests Can Agree—and Differ

For known \(V\) in the Gaussian linear mixed model, the three tests of the same fixed-effect restrictions coincide. This can be seen for \(H_0:\beta_1=0\). Let \(P\) be the residual precision matrix after fitting all columns in \((X_0,X_1)\). The unrestricted estimate of the tested coefficients is \(\hat{\beta}_1=I_{11}^{-1}U_1\), and its covariance is \(I_{11}^{-1}\). Hence the Wald statistic is \(U_1'I_{11}^{-1}U_1\), exactly the score statistic.

The log-likelihood for known \(V\), after maximizing over the fixed effects in a given model, differs between the null and alternative only through the residual quadratic forms. The matrix identity

shows that the reduction in the quadratic form is \(y'(P_0-P)y=U_1'I_{11}^{-1}U_1\). Therefore \(\Lambda=2(\ell_1-\ell_0)=U_1'I_{11}^{-1}U_1\) as well. All three statistics then have an exact \(\chi^2_q\) null distribution when \(V\) is known and the stated Gaussian model holds.

When covariance parameters are estimated, the fits and plug-in covariance estimates can differ, so the three statistics need not be numerically identical. Their chi-square reference distributions are typically large-sample approximations. The Wald test is convenient after a full fit; the score test can avoid fitting the alternative; the likelihood-ratio test compares the two fitted likelihoods directly. These practical differences matter especially when information is limited or estimates are near parameter constraints.

Common Pitfall and Practical Interpretation

• Comparing restricted likelihoods across fixed-effect designs. REML likelihoods from models with different fixed-effect design matrices are not generally on a common scale for a likelihood-ratio test. Use maximum likelihood for this comparison.
• Giving the test statistic the wrong degrees of freedom. The chi-square degrees of freedom equal the number \(q\) of independent restrictions, not the number of individuals or all coefficients in the model.
• Calling a nonsignificant result proof of no effect. A large p-value means the data do not provide strong evidence against the stated null at the chosen threshold; it does not establish that the effect is exactly zero.
• Interpreting a joint test one coefficient at a time. A joint test evaluates the full restriction. Its result does not say which coefficient is responsible for a departure from the null.

In a genetic mixed model, covariance adjustment can change both coefficient precision and the evidence against a fixed-effect null. Report the tested restriction, the test and statistic, its degrees of freedom, the p-value, and—when interpreting an effect—its estimated coefficient with units and uncertainty. The test concerns the fixed-effect mean structure conditional on the specified genetic covariance model.