The Barrier to Human Understanding of MARS

Abstract

The barrier of the title is that the terms a MARS program prints do not read as rates. A supplement to the four statistical entries of this issue's Foundations department. The four statistical entries (Conditional Expectation, Omitted Variable Bias, Endogeneity and Parameter Identification) reason about the rate a regression assigns to a feature; the Journal's own modeling uses multivariate adaptive regression splines (MARS), which report one rate per range rather than one per feature. This supplement shows, on one worked model of living area in a coastal market, how the terms a MARS program prints become a rate table, the value contribution of a quantity of the feature, and the adjustment from a comparable to a subject, defined as the amount added to the comparable's contribution to make it equal to the subject's. It states the one rule hand-built tables get wrong: a coefficient of -600 on the reflected hinge h(1200 - gla) is a rate of +600 a square foot below 1,200 square feet, not -600, because read from left to right that hinge decreases, so the term increases. It shows that the single rate a straight line would need for a given comparable and subject, the pair's secant rate, is an average of the model's rates that changes with the pair, which is why an ordinary regression with one rate per feature cannot reproduce them. A Prolog program, printed in full and published with its 69 tests, reads the coefficient table of the R package earth, prints the rate table over the observed data, the fitted values at the knots, the adjustment range by range and the pair's secant rate, and checks that the ranges sum to the adjustment; an R rendering reads an earth object directly. The tests pass under Trealla, SWI-Prolog and Scryer Prolog.