  This paper is about how we study statistical methods. As an example, it uses
the random regressions model, in which the intercept and slope of
cluster-specific regression lines are modeled as a bivariate random effect.
Maximizing this model's restricted likelihood often gives a boundary value for
the random effect correlation or variances. We argue that this is a problem;
that it is a problem because our discipline has little understanding of how
contemporary models and methods map data to inferential summaries; that we lack
such understanding, even for models as simple as this, because of a
near-exclusive reliance on mathematics as a means of understanding; and that
math alone is no longer sufficient. We then argue that as a discipline, we can
and should break open our black-box methods by mimicking the five steps that
molecular biologists commonly use to break open Nature's black boxes: design a
simple model system, formulate hypotheses using that system, test them in
experiments on that system, iterate as needed to reformulate and test
hypotheses, and finally test the results in an "in vivo" system. We demonstrate
this by identifying conditions under which the random-regressions restricted
likelihood is likely to be maximized at a boundary value. Resistance to this
approach seems to arise from a view that it lacks the certainty or intellectual
heft of mathematics, perhaps because simulation experiments in our literature
rarely do more than measure a new method's operating characteristics in a small
range of situations. We argue that such work can make useful contributions
including, as in molecular biology, the findings themselves and sometimes the
designs used in the five steps; that these contributions have as much practical
value as mathematical results; and that therefore they merit publication as
much as the mathematical results our discipline esteems so highly.
