Balanced Block Designs with Two Different Numbers of Replicates

Abstract

In most standard types of incomplete block designs (e.g. balanced incomplete block designs, and partially balanced designs, including lattice designs) all treatments occur the same number of times. This requirement may be inconvenlient in practice. For example, only small amounts of seed might be available for some new varieties, while seed of other varieties was abundant. To avoid such difficulties by taking only as many replicates of all treatments as the limitations on some treatments allowed would be an inefficient approach, involving a sacrifice of information on those treatments for which limitations did not exist, as well as on comparisons between the two kinds of treatment. For the last reason, also, the alternative of investigating the two groups in separate experiments is unacceptable. In situations of this sort there is an obvious practical need for incomplete block designs in which the number of replicates is not the same for all treatments. Such designs would also be of value when some treatments are of more interest than others: though interest in these other treatments might be enough to justify their inclusion in the experiment, the experimenter wou]d wish to spend less on investigating them than on the treatments of major interest. Several discussions on designs with varying numbers of replicates exist in statistical literature, but most of these are concerned with the analysis of designs that happen to have unequal replications of treatments, e.g. Anderson and Manning [1948], Corsten [1958], Federer [1961], Gomes and Guimaraes [1958], Graybill and Pruitt [1958], Justesen and Keuls [1958], Pearce [1948], Rao [1947], Youden and Connor [1953].