Wald tests for fixed effects with optional small-sample denominator df

Pseudo analysis of variance for the fixed terms of an mmes fit, based on incremental or conditional Wald statistics, with chi-square, residual, Satterthwaite or Kenward-Roger reference distributions.

Usage

wald_mmes(object, ssType = c("incremental", "conditional"),
          denDF = c("none", "residual", "satterthwaite", "kr"), terms = NULL)

Arguments

object: a model of class "mmes".
ssType: "incremental" tests each term adjusted for the terms preceding it in the fixed formula. "conditional" tests each term adjusted for every other term that does not contain it, so marginality is respected: a term is never adjusted for a term that contains it structurally (A:B contains A) or implicitly (its columns, together with the intercept, span the tested term, e.g. locations coded separately within regions).
denDF: "none" gives chi-square tests of the Wald statistic. "residual" uses F tests with n - rank(X) denominator df. "satterthwaite" and "kr" (Kenward-Roger) compute approximate denominator df; "kr" also uses the Kenward-Roger adjusted covariance of the fixed effects and scaled F statistic.
terms: optional subset of fixed terms (as named in object$Dtable).

Details

Let $\Phi$ be the covariance of the fixed effects and $M=\Phi^{-1}$. For a given order of the terms, the upper Cholesky factor $U$ of $M$ defines $z=U b$; the incremental Wald statistic of a term is the sum of the squared elements of $z$ belonging to it, i.e. a test of $L b=0$ with $L$ the corresponding rows of $U$. Conditional tests use the order [terms not containing the tested term, tested term, terms containing it].

For Satterthwaite and Kenward-Roger df the model inputs are rebuilt from the stored call and $X^\top V^{-1}X$ is re-evaluated at perturbed covariance parameters (on the reported scale of covPar); its numerical first and second derivatives give the quantities $P_i$ and the Kenward-Roger adjustment. The covariance of the covariance parameters is theta_se, the inverse average information. The second-derivative term of $V$ enters through the Hessian of $X^\top V^{-1}X$; it vanishes for variance-component models, where the method is the exact Kenward-Roger approximation. Because the average (rather than expected) information is used, df can differ slightly from software using the expected information. The re-evaluation requires the data and objects used in the original call to be available, REML estimation, and a Gaussian fit.

anova(object) with a single model returns wald_mmes(object, ...).

Value

A data frame of class "wald.mmes" with columns Df, denDF, Wald, F.value and p.value, one row per term.

References

Kenward MG and Roger JH (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53: 983-997.

Examples

data(DT_yatesoats, package = "enhancer")
m <- mmes(Y ~ V * N, random = ~ B + B:MP, rcov = ~ units,
          data = DT_yatesoats, verbose = FALSE)
wald_mmes(m)
wald_mmes(m, ssType = "conditional", denDF = "kr")