  The effect of vigorous physical activity on mortality in the elderly is
difficult to estimate using conventional approaches to causal inference that
define this effect by comparing the mortality risks corresponding to
hypothetical scenarios in which all subjects in the target population engage in
a given level of vigorous physical activity. A causal effect defined on the
basis of such a static treatment intervention can only be identified from
observed data if all subjects in the target population have a positive
probability of selecting each of the candidate treatment options, an assumption
that is highly unrealistic in this case since subjects with serious health
problems will not be able to engage in higher levels of vigorous physical
activity. This problem can be addressed by focusing instead on causal effects
that are defined on the basis of realistic individualized treatment rules and
intention-to-treat rules that explicitly take into account the set of treatment
options that are available to each subject. We present a data analysis to
illustrate that estimators of static causal effects in fact tend to
overestimate the beneficial impact of high levels of vigorous physical activity
while corresponding estimators based on realistic individualized treatment
rules and intention-to-treat rules can yield unbiased estimates. We emphasize
that the problems encountered in estimating static causal effects are not
restricted to the IPTW estimator, but are also observed with the
$G$-computation estimator, the DR-IPTW estimator, and the targeted MLE. Our
analyses based on realistic individualized treatment rules and
intention-to-treat rules suggest that high levels of vigorous physical activity
may confer reductions in mortality risk on the order of 15-30%, although in
most cases the evidence for such an effect does not quite reach the 0.05 level
of significance.
