Determination of the straight line of best fit to observational data of two related variates when both sets of values are subject to error
✍ Scribed by W. A. Morgan
- Publisher
- John Wiley and Sons
- Year
- 1960
- Tongue
- English
- Weight
- 316 KB
- Volume
- 86
- Category
- Article
- ISSN
- 0035-9009
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✦ Synopsis
Abstract
The general equation to a straight line of best fit to observational data of two related variates x and y is obtained by minimizing the general expression \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 1}^n {(y_i - ax_i - b)^2 } $\end{document} ϕ(a) involving the departures of both variates from the time. It is shown that the only expression for ϕ (a) which produces consistent results with change of unit is ϕ (a) = constant. a^k^, where k depends on the relative errors (e~x~, e~y~) in measurements of the two variates.
The line of best fit is shown to be
where c is obtained from the equation (k + 2) c^2^ – 2 (k + 1) rc + k = 0, in which r is the coefficient of correlation between x and y, and k is a given function of e~x~ and e~y~. The usual regression lines and line with a slope equal to the geometric mean of the slopes of the two regression lines, arise as special cases of the general equation. Examples are given to show general application.