On the Significance of Slopes and Other Parameters Estimated by Least Squares

On the assumption that the errors of observation in both $x$ and $y$ coordinates are normally distributed at each point, and that the errors ${\epsilon }_a$, ${\epsilon }_b$, ${\epsilon }_c$ in the estimated parameters $a$, $b$, $c$ of the empirical formula $F(x,y;a,b,c)=0\mathrm{}$ are also normal, it is found that the ratio $\frac{{\epsilon }_{a}w_a^{}{}_{}{}^{\frac{1}{2}}}{\phi }$ is distributed on a Student (Pearson type VII) curve. $W_a$ is the weight of $a$, and ${\phi }^2$ is the weighted sum of the squares of the residuals. The general case is found to be the same problem as the simple one discussed by Deming and Birge and others. When $\sigma$ is known, the test of significance is direct with the normal probability integral. When $\sigma$ is not known, the classical procedure is to use an estimate derived from the fit of the curve, and, as Student and Fisher and others have insisted in other work, it is better to use the Student integral (the $z$ test) rather than the normal integral (the $u$ test). However, as is now pointed out, the classical procedure involves the assumption that the fit of the curve is an average fit, on which basis the numerical difference between the two integrals is not apt to be serious. It is important to remember nevertheless that the test thus made is the $z$ test and makes use of the diverging $z$ contours on the $\epsilon$, $s$ plane. Some distinctions between the $u$ and $z$ tests are drawn and discussed.