  Hotelling's $T^2$-test for the mean of a multivariate normal distribution is
one of the triumphs of classical multivariate analysis. It is uniformly most
powerful among invariant tests, and admissible, proper Bayes, and locally and
asymptotically minimax among all tests. Nonetheless, investigators often prefer
non-invariant tests, especially those obtained by selecting only a small subset
of variables from which the $T^2$-statistic is to be calculated, because such
reduced statistics are more easily interpretable for their specific
application. Thus it is relevant to ask the extent to which power is lost when
variable selection is limited to very small subsets of variables, e.g. of size
one (yielding univariate Student-$t^2$ tests) or size two (yielding bivariate
$T^2$-tests). This study presents some evidence, admittedly fragmentary and
incomplete, suggesting that in some cases no power may be lost over a wide
range of alternatives.
