Health Standards and Quality Bureau
Health Care Financing Administration
At the symposium, Andrew Kirsch noted that looking at the physical context of some statistical problems sometimes exposes the fact that certain additive functional forms can be contrary to known physical relationships. In his example, a product was instead the natural construct. Wilson (1952) discusses some examples such as symmetry (§11.5, pp. 308–312), limiting cases (p. 308), and dimensional analysis (§11.12, pp. 322–328). I would note that such considerations quickly lead one to consider functional relationships that are not linear in the parameters. The supporting statistical software has been of limited utility in dealing with these functional relationships. The picture becomes more complex when one introduces stochastic components to the models and complex sample designs for collecting data.
Nonlinear regression deals nicely with the problems when the stochastic element is limited to a measurement error. However, many formulations do not readily permit the use of implicitly defined functions. However, for those of us who must perform operational analyses without the luxury of developing special tools to deal with diverse problems, adequate tools have not been easily accessible.
Expert systems are designed to prevent the incorrect application of a procedure. In addition to asking whether we have correctly used a procedure according to accepted standards, we must also ask whether the methods are useful for our solving problems. One of the very strengths of statistical methods derived for designed experiments is the very weakness of these methods. The strength is to have methods that work without regard to the underlying functional relationships. We really can design methods that work in this way by controlling the design. These methods answer the preliminary questions such as whether there is a difference or not. Any subject-matter specialist quickly wants to go beyond these questions to understand the structure of a problem. Furthermore, there are many important data sources other than designed studies or experiments. Graphical techniques are so appealing partly because our analytical tools to reveal the more complex structures are so limited and underdeveloped. I hope that graphical techniques will drive analysts to consider more complex structural relationships in the analytical context.
As I said following Paul Velleman's talk, it would be nice if statisticians would think more globally. For example, in using the results of a survival analysis package for a specific analysis, it would be useful to know how the likelihood was formulated. Better yet, some agreement on the formulation would permit comparison across various models and packages. For example, a log-likelihood for a Weibel, constant hazard, or other model form could be compared if there were some agreement in how these are formulated. Some packages do not provide this comparability, even within a given procedure. Part of the reason is that some drop constants that are not relevant for the