A Significance Test for Exponential Regression

Abstract

A general method of testing the significance of nonlinear regression, suggested by Hotelling, is adapted to the regression equations $Y = be^{px}$ and $Y = a + be^{px}$. The values of $x$ are taken to be in arithmetic progression, and the standard deviation of the observed $y$ is supposed constant for all $x$. This is in contrast to the assumption, implicit in the usual procedure of fitting a straight line to $\log y$, that the standard deviation of $\log y$ is constant. It will be observed that the distribution of $y_1, y_2, \cdots, y_n$ must be such that the joint probability density for $y_1, y_2, \cdots, y_n$ is a function of $x^2_1 + x^2_2 + \cdots + x^2_n$, and this condition implies the assumption of normality. The null hypothesis is that $be^{px} = 0$ for all $x$, while the alternative hypotheses are specified by $b \neq 0, p \neq - \infty$. The method involves the calculation of the volume of a "tube" on a hypersphere in $n$-dimensional space. An asymptotic expression for the length of the tube is developed, and it is shown that the curvature of the axis is everywhere finite. From this expression, for values of the correlation coefficient $R$ between observed and fitted values of $y$ at least as great as 0.894, a function of $R$ is obtained giving the probability that a random sample would yield at least as great a value of $R$. A short table giving $R$ for various significance levels and various sizes of sample is calculated for each of the equations mentioned, and the application to certain experimental data is discussed.