On the Application of Probability Theory to Agricultural Experiments. Essay on Principles. Section 9

In the portion of the paper translated here, Neyman introduces a model for the analysis of field experiments conducted for the purpose of comparing a number of crop varieties, which makes use of a double-indexed array of unknown potential yields, one index corresponding to varieties and the other to plots. The yield corresponding to only one variety will be observed on any given plot, but through an urn model embodying sampling without replacement from this doubly indexed array, Neyman obtains a formula for the variance of the difference between the averages of the observed yields of two varieties. This variance involves the variance over all plots of the potential yields and the correlation coefficient r between the potential yields of the two varieties on the same plot. Since it is impossible to estimate r directly, Neyman advises taking r = 1, observing that in practice this may lead to using too large an estimated standard deviation, when comparing two variety means.