  Effect modification means the magnitude or stability of a treatment effect
varies as a function of an observed covariate. Generally, larger and more
stable treatment effects are insensitive to larger biases from unmeasured
covariates, so a causal conclusion may be considerably firmer if this pattern
is noted if it occurs. We propose a new strategy, called the submax-method,
that combines exploratory and confirmatory efforts to determine whether there
is stronger evidence of causality - that is, greater insensitivity to
unmeasured confounding - in some subgroups of individuals. It uses the joint
distribution of test statistics that split the data in various ways based on
certain observed covariates. For $L$ binary covariates, the method splits the
population $L$ times into two subpopulations, perhaps first men and women,
perhaps then smokers and nonsmokers, computing a test statistic from each
subpopulation, and appends the test statistic for the whole population, making
$2L+1$ test statistics in total. Although $L$ binary covariates define $2^{L}$
interaction groups, only $2L+1$ tests are performed, and at least $L+1$ of
these tests use at least half of the data. The submax-method achieves the
highest design sensitivity and the highest Bahadur efficiency of its component
tests. Moreover, the form of the test is sufficiently tractable that its large
sample power may be studied analytically. The simulation suggests that the
submax method exhibits superior performance, in comparison with an approach
using CART, when there is effect modification of moderate size. Using data from
the NHANES I Epidemiologic Follow-Up Survey, an observational study of the
effects of physical activity on survival is used to illustrate the method. The
method is implemented in the $\texttt{R}$ package $\texttt{submax}$ which
contains the NHANES example.
