p-value of a likelihood-ratio statistic LRT = 2 * (ll_full - ll_reduced)
for the null hypothesis that q variance-type parameters are
simultaneously at the boundary of their parameter space (typically 0),
when those q components are asymptotically INDEPENDENT of each other and
of the remaining (regular, interior) parameters. This is the Self & Liang
(1987) / Stram & Lee (1994) chi-bar-square mixture
$$p = \sum_{j=1}^{q} \binom{q}{j} 2^{-q} \, P(\chi^2_j \ge \mathrm{LRT})$$
(the j = 0 atom, a point mass at LRT = 0, contributes nothing once
LRT > 0). For q = 1 this is the familiar
0.5 * P(chi^2_1 >= LRT) – half the naive chi-square_1 p-value, because
half of the mixture's probability mass sits at the boundary atom.
This formula assumes the q boundary components are mutually
independent (their asymptotic score covariance is diagonal) and that the
rest of the model's Fisher information is regular (non-singular) at the
null. Neither assumption is checked by this function – it computes the
formula exactly as specified. The caller (in gllvmTMB,
anova.gllvmTMB_multi()) is responsible for deciding whether those
assumptions are defensible for the comparison at hand, and for refusing to
call this function (rather than calling it and reporting a wrong p-value)
when they are not. See anova.gllvmTMB_multi()'s documentation for the
specific case gllvmTMB refuses: a latent-rank ("number of factors") test
spanning more than one new loading column, where the added parameters are
not simple independent scalar variances and the correct reference
distribution is not known in closed form.
References
Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82(398), 605-610.
Stram, D. O. and Lee, J. W. (1994). Variance components testing in the longitudinal mixed effects model. Biometrics, 50(4), 1171-1177.