Tests of significance of the differences between regression coefficients derived from two sets of correlated variates

The calculation of the partial regression of different, but correlated, dependent variates on the same set of independent variates is of common occurrence. Frequently, also, the differences between the resultant regression coefficients are of interest, and tests of significance for these differences are consequently required. In the classical wheat experiments on Broadbalk field at Rothamsted, for example, the influence of rainfall on yield has been investigated for the various plots, which receive different manurial treatments (Fisher, 1924). An extension of the method of partial regressions was used, which analytically amounted to the evaluation, for each plot separately, of the regression of yield on quantities representing the amount and distribution of rain in each year, these quantities being the same for all plots, since the plots were situated on the same field. One of the most interesting aspects of these results was the differences which were revealed between the effects of rainfall on the differently manured plots. (See also Cochran, 1935.)