Residual Permutation With Nuisance Covariates

Why Permute Residuals?

A genetic predictor may be associated with an outcome for reasons that are not the predictor’s conditional effect of interest. Age, ancestry components, batch, or other measured variables can be related to both the predictor and the outcome. A test that ignores these nuisance covariates can therefore answer the wrong question. The Freedman–Lane procedure constructs a permutation reference distribution for the predictor’s contribution after fitting the nuisance structure.

The method is useful when the covariates should remain fixed while the genetic predictor is tested, but the errors themselves are not directly observed and cannot simply be permuted. Its central idea is to fit the null model, permute its residuals, add them back to the null fitted values, and refit the full model. Because fitted residuals are not generally independent exchangeable observations, this procedure is usually an approximation to the desired null distribution, not an exact test.

Models and the Freedman–Lane Algorithm

Let \(y\) be an \(n\)-vector of outcomes, \(Z\) an \(n\times q\) matrix of nuisance covariates (including an intercept when appropriate), and \(g\) an \(n\)-vector of genetic predictor values. Consider the full linear model

The target is the conditional test \(H_0:\beta=0\) against a specified alternative, commonly \(H_a:\beta\ne0\). The reduced model under \(H_0\) is \(y=Z\gamma+\varepsilon\). Assume \(Z\) has full column rank and that \(g\) adds a linearly independent direction beyond the columns of \(Z\); otherwise \(\beta\) is not separately estimable.

Write \(P_Z=Z(Z'Z)^{-1}Z'\) for projection onto the columns of \(Z\), and \(M_Z=I-P_Z\) for residualization against those columns. The reduced fitted values are \(\hat y_0=P_Zy\), and the reduced residuals are \(e_0=M_Zy\). For a permutation matrix \(P_b\), the permuted response is \(y^{*(b)}=\hat y_0+P_be_0\). Fit the full model to \(y^{*(b)}\), using the same \(Z\) and \(g\), and record the chosen statistic, such as the \(t\)-statistic for \(\beta\).

For a two-sided test, a common choice is the absolute \(t\)-statistic. With \(B\) randomly sampled permutations and observed statistic \(T_{\mathrm{obs}}\), the add-one Monte Carlo p-value is

The added one counts the observed arrangement in the Monte Carlo reference set and avoids reporting a zero p-value. The p-value describes how often a statistic at least as extreme as the observed one occurs under this constructed reference distribution; it is not the probability that \(H_0\) is true.

Why the Procedure Is Approximate

If the true errors \(\varepsilon\) were known, a permutation test could be based on permuting them when their joint distribution is invariant under the allowed permutations. But under the null, the reduced residuals are \(e_0=M_Z\varepsilon\), not the errors themselves. Projection generally makes residuals dependent and gives them unequal variances, even when the errors have a common variance. Thus permuting \(e_0\) does not generally reproduce the exact distribution obtained by permuting \(\varepsilon\).

The source of the approximation can also be seen from the coefficient estimate. By residualizing the genetic predictor against \(Z\), the full-model estimate is

Under \(H_0\), the numerator is \(g'M_Z\varepsilon\). In a Freedman–Lane replicate it becomes \(g'M_ZP_be_0=g'M_ZP_bM_Z\varepsilon\). These expressions need not have the same distribution. The method can nevertheless be useful when the model is correctly specified, errors are approximately exchangeable under the permitted permutations, and no small number of observations dominates the fitted residuals. It is not a blanket guarantee of exact type I error control.

The linear-model assumptions relevant to this argument are a correctly specified nuisance mean under the null, a meaningful full-model coefficient, and errors whose distribution is sufficiently compatible with the permutation scheme. In the usual model-based justification, errors are independent with mean zero and common variance conditional on the fixed design; stronger distributional symmetry or exchangeability makes permutation reasoning more plausible. Independence alone does not make fitted residuals exchangeable. If observations have related-sample or block structure, permutation units and restrictions must reflect the design, as in the earlier material on restricted permutations.

Interpreting a Permutation Run

A practical analysis should examine whether the nuisance model captures the mean structure and whether residuals show patterns that conflict with the proposed permutation scheme. Residual-versus-fitted plots, residuals against key covariates, and leverage measures can reveal gross problems. A few high-leverage observations can make residual variances markedly unequal and allow those observations to influence many permuted statistics. A failure of these checks is a reason to reconsider the model or inferential method, not a reason to keep permuting and trust the resulting p-value. More detailed diagnosis of variance nonconstancy belongs to the subsequent material on heteroskedasticity.

Diagnostics, Alternatives, and Limitations

The procedure is most defensible when the reduced model is appropriate under the null, the outcome and design are aligned observation by observation, the genetic predictor has variation remaining after adjustment, and the error structure is compatible with the allowed permutations. If there are exchangeability blocks, permutations must respect them; a free shuffle across subjects from distinct blocks changes the reference distribution in an unjustified way. Restrictions do not remove the projection issue: permuted fitted residuals still need not behave like permuted errors.

A model-based \(t\)- or \(F\)-test uses its reference distribution under the specified linear-model assumptions instead of building a residual-permutation distribution. It may be preferable when those assumptions are credible and a direct regression test is adequate. Freedman–Lane can be attractive when a permutation calibration is desired and errors are plausibly exchangeable, but its approximation can be poor when leverage is high, the mean model is misspecified, or exchangeability restrictions are ignored. The choice is not simply “permutation is assumption-free”: every permutation procedure relies on a defensible invariance or approximation argument.

The method also depends on the statistic. A \(t\)-statistic compares the estimated coefficient with its estimated standard error; a studentized statistic can be more suitable than an unstandardized coefficient when scale varies, but it does not automatically fix an invalid permutation scheme. For a two-sided alternative, compare absolute statistics; for a one-sided alternative, define the direction in advance and compare in that direction. A large permutation count reduces Monte Carlo noise but cannot correct a systematically inappropriate reference distribution.

Common Mistakes

• Permuting the raw outcome. This destroys the fitted nuisance structure. Freedman–Lane keeps \(\hat y_0\) fixed and permutes the reduced-model residuals.
• Calling the test exact because it uses permutations. Reduced-model residuals are projected, generally dependent, and not identically distributed. The usual Freedman–Lane justification is approximate.
• Shuffling across restricted units. Related samples or blocks can invalidate unrestricted permutations. The allowed permutations must follow the design’s exchangeability structure.
• Ignoring estimability. If \(g'M_Zg=0\), no conditional genetic coefficient is identifiable; if it is nearly zero, uncertainty can be large.
• Interpreting a p-value as a probability of the null. A permutation p-value measures extremeness under the constructed null reference distribution, not the probability that \(H_0\) is true.