  Testing for the significance of a subset of regression coefficients in a
linear model, a staple of statistical analysis, goes back at least to the work
of Fisher who introduced the analysis of variance (ANOVA). We study this
problem under the assumption that the coefficient vector is sparse, a common
situation in modern high-dimensional settings. Suppose we have $p$ covariates
and that under the alternative, the response only depends upon the order of
$p^{1-\alpha}$ of those, $0\le\alpha\le1$. Under moderate sparsity levels, that
is, $0\le\alpha\le1/2$, we show that ANOVA is essentially optimal under some
conditions on the design. This is no longer the case under strong sparsity
constraints, that is, $\alpha>1/2$. In such settings, a multiple comparison
procedure is often preferred and we establish its optimality when
$\alpha\geq3/4$. However, these two very popular methods are suboptimal, and
sometimes powerless, under moderately strong sparsity where $1/2<\alpha<3/4$.
We suggest a method based on the higher criticism that is powerful in the whole
range $\alpha>1/2$. This optimality property is true for a variety of designs,
including the classical (balanced) multi-way designs and more modern "$p>n$"
designs arising in genetics and signal processing. In addition to the standard
fixed effects model, we establish similar results for a random effects model
where the nonzero coefficients of the regression vector are normally
distributed.
