MECHANISTIC: Systems analytic or mechanistic

EMPIRICAL: Empirical or often designated as statistical ’empirical’ 

When the system under study is complex and hardly anything is known about its structural connectivity and functional mechanisms, yet one has to produce hypotheses about it based on some external characteristics such as a dose-response (secondary system), one often relies on. mathematical functional forms for such a system. These mathematical functions are empirical models. They may incorporate some mechanistic assumptions so that they may look realistic. Numerically, these models are generally easier to handle as opposed to many mechanistic models. Most normal theory based statistical hypothesis testing and confidence interval procedures are based on such models.

A mechanistic (although the present author has some problem in using ‘statistical’ as a synonym for mechanistic model, as the name implies, should have as many features of the primary system built into it as observations or data will allow. Such a model should be consistent with the observed behavior of the system – retrodiction – [Rescigno and Beck, 1987]; it should further be predictive of the system’s future behavior or behavior under perturbation – prediction – [Rescigno and Beck, 1987]. One must have some knowledge of the primary system in terms of structural connectivity and functional mechanisms. Some prefer to call this type of models realistic, intrinsic, and various other names. Many great discoveries in biology, medicine, and other branches of science have been made using such models. In this context one must remember that such models do not necessarily have to have an explicit mathematical expressions; they could be just conceptualizations.

One should not get the wrong impression that mechanistic models are not useful for such statistical techniques; they may be more difficult to handle numerically from estimation standpoints. Some people would call empirical models extrinsic because they are based purely on the external behavior of the system. Some call them statistical models. As mentioned earlier, it is unfair to assume that statisticians always like to use empirical models for their purposes. The reasons why there are abundance of this type of models in literature are obvious. Our knowledge about the primary system may be inadequate-to-none to allow us the formulation of a mechanistic model or one may not be interested in understanding the inherent structure of the system. In the present author’s mind, the phrase statistical model includes both types of models. One must remember that an empirical model may be ‘retroactive’ (explaining what happened from a secondary system) and even locally ‘predictive’ (Le. interpolation may be performed within the range of observations), but it is, in general, not globally ‘predictive’ (indicating outcome of future experiments). In fact, empirical models should never be used with any authority for extrapolative purposes. According to Fisher [1925], K.F. Gauss in the early 1800’s may have been instrumental in developing empirical modeling concept with his work on maximum likelihood and least squares theories. ‘Gauss, further, perfected the systematic fitting of regression formulate, simple and multiple, by the method of least squares, which, in the cases to which it is appropriate, is a particular example of the method of maximum likelihood’ [Fisher, 1925]. A slight variation of empirical modeling idefined by Ashby [1958].