Estimating the bias of correlation coefficients

Abstract

One of the major problems arising in the correlation method of seasonal forecasting is the selection of factors to be used in forecast formulae. As pointed out by Savur (1935) there are two main ways in which the choice can he made—

(1) a priori selection in which the factors are chosen on theoretical or physical grounds, and (ii) reelection by correlation in which the factors are chosen because empirical relationships have been found to exist between them and the elements for which forecasts are required. It was early recognized that in testing the significance of correlations selected under •ii) sonic allowance should be made for the fact that they have in general been chosen as the highest of a large number. For this purpose Walker (1914) gave values of the " probable highest " correlation while Savur and copal Rao (1932) extended this to give values necessary for significance at the 5 per cent level. However, effects of selection should also be taken into account when esti-mating the true or long term value of a corre-lation coefficient, a matter which is of even. greater concern since the best estimate is required to determine the actual regression equation. In examining effects of selection from this point of view it is simplest to deal with the transformed correlation zs which is very nearly normally distributed about the value (similarly defined from the population correlation p) with variance.