Symmetry and Scale Dependence in Functional Relationship Regression

Abstract

?Two complaints against the linear functional regression model have been that the estimated regression lines are not symmetrical in x and y and that they depend on the scale of the observed data in unpredictable ways. If one takes into account the ratio of measurement X, then the estimates from linear functional regression are symmetrical (i.e., the regression of y on x and x on y are the same line). Using nonlinear least squares estimation, functional regression is easily applied to fit both nonlinear and linear models. Again taking into account X, linear and nonlinear models can be naturally rescaled (as with ordinary regression estimates) according to the units of the observed variables. The log-transformed functional regression model and the relationships between principal methods and the linear functional regression model are considered. Controversies surrounding the major axis are discussed; sim? ulation results are presented, and the validity of the method is questioned. Treatment of the linear, allometry, polynomial, and von Bertalanffy models is discussed, and a numerical example is presented. [Functional regression; in variables; major axis; reduced major axis; standard major axis; nonlinear regression; allometry.] The critical work of systematic biologists often deals with measuring ontogenetic changes and differentiating entities using regression. These problems have been made more interesting and difficult due to the realization that has grown since the days of Kermack and Haldane (1950) that ordinary least squares (OLS) is often not a suitable regression method; in biological contexts the estimated relationship be? tween any two can be biased by the existence of substantial measurement error and/or biological variation (Ricker, 1973). Alternatives to OLS usually fall under the heading of functional or structural re? gression. Kendall and Stuart (1973) pro? vided a clear introduction to these some? times confusing methods. In this paper I deal with only the func? tional relationship regression model be? cause it seems the simplest and most gen? erally applicable of the errors in variables models. However, there is a clear distinc? tion between the functional relationships posited between the mathematical vari? ables (e.g., linear, quadratic, allometric, etc.) and the functional regression method, which estimates parameters assuming that mathematical x and y are both observed with error. Least Squares Estimates Let two mathematical {xiry^,i = 1, . . ., n, be related through some func? tional relationship yi = i(xif fi), where xi and y{ are scalar but fi may be a vector of p variables. We are not able to observe xi and l/i directly but instead observe the ran? dom Xi = xi + 8{ and Yi = f(x{, fi) + ?i, where 5{ and ei are independent, nor? mal random with mean zero and variances a25 and a\, respectively. The mi? nus log likelihood for estimating the n + p unknowns (xlf..., x?, fiu ..., fip) is then (Fuller, 1987; Seber and Wild, 1989) -L = S {[Vi ffe ??/(2a2f) + (X, x{)2/(2a28)}