The estimation of parameters by least squares from unbalanced experiments

Summary

Data from experimental and observational studies, in agriculture as in many other contexts, are often unbalanced in respect of important classifications. Treatments may be unequally replicated, some combinations of factors may be omitted, animals may die for reasons unconnected with an experiment. Unless means are adjusted in some manner that eliminates disturbance from unequal representation of different categories, comparisons between treatments may be thoroughly misleading. Optimal procedures for the simpler situations have been familiar to statisticians for a long time. They have been little used by other scientists analysing their own data, in part because of the computational labour and in part because their nature has not been properly understood.

Modern computing power removes all excuse for the retention of methods that may be actively misleading because they bias summaries of data, or that are at best inefficient in their failure to estimate comparisons as precisely as is possible. The only remaining barrier is the mistaken belief that good methods are either so complicated that they can be comprehended only by professional statisticians or so devious that truth is destroyed rather than exhibited. This paper attempts to show good methods as inherently rational, to explain their main properties, and to illustrate them on examples small enough for the processes to be clear. The paper neither expounds statistical theory nor instructs in computing practice.