Fitting the Rectangular Hyperbola

Abstract

When a dependent response (y[image]) is related to an independent or controlled variable (x[image]) by the rectangular hyperbola, (x[image] - xo[image]) (y[image] - yo[image]) + c, with asymptotes x [image] and y [image] and constant c, the predicted response Y may be estimated f[image]om the [image]traight line, Y = a[image] + b {l/(x[image] +d)} . If either the customary response (y[image]) or its reciprocal (l/y[image]) varies uniformly about this line, each observation can be weighted equally in computing the 3 parameters of the hyperbola by maximum likelihood solutions of a[image], b and d (= -xo[image] or c/yo[image]), as illustrated by data on pulmonary ventilation and on the complement, antiserum reaction. When c, xo[image] or yo[image] is zero or is known a priori, the calculation depends upon which parameters require estimation, leading to a simple ML [maximum likelihood] solution for d with x[image] transformed to x + 1/(x[image] + d) or to x = x[image]/(x[image] + d), as illustrated by numerical examples of an enzyme reaction and of a DDT test on house flies. Confidence intervals are given for the regression coefficients, making allowance for limited curvature of the solution locus and any curvature of coordinates within this locus. The theoretical background of the method completes the paper.