One of the most popular designs in experimental work is the randomized block. These designs can be put into three broad classes viz. complete block design, balanced incomplete block design, and the partially balanced incomplete block design. These designs are all special cases of the general two way classification with unequal numbers in the subclasses, but since the analysis of this general classification is quite complex, these special cases have evolved which are adequate to fit most needs and the analysis of these special designs is relatively easy. , , , . However, most of the block designs considered to date have one feature in common--they require each block to contain an equal number of experimental units. The exceptions are given in , , where designs are considered in which the number of experimental units in blocks differ by one. The purpose of this paper is to extend the randomized block design to include the case where all blocks do not contain the same number of experimental units. We have called this the staircase design. Suppose an experimenter, wishing to run an experiment using $N$ treatments, decides to use a randomized block design, but after arranging his material into homogeneous groups he finds that he has blocks available which have varying number of experimental units. The experimenter has various courses open to him: (1) If enough blocks are available with $N$ or more experimental units he can discard the extra units in these blocks, discard all the blocks which have less than $N$ units, and use a randomized complete block design; (2) He can discard units in the blocks until he has enough units and blocks for a balanced incomplete block or a partially balanced incomplete block design; (3) He can use all the experimental units and use the staircase design proposed in this paper. For example, if an experimenter has $N$ treatments with which he wishes to experiment using a randomized block design, and if he has blocks of unequal size, then he must rank his $N$ treatments in the order of their importance, i.e., $T_1, T_2, \cdots, T_N,$ where he considers $T_1$ the most important and $T_N$ the least important. Now suppose he has at his disposal $b_1$ blocks which each contain $N$ experimental units. Then all $N$ treatments are randomized in each of the $b_1$ blocks. Suppose further that he has $b_2$ blocks which each contain $N_1$ experimental units $(N_1 < N).$ Then the first $N_1$ treatments are arranged at random in each of the $b_2$ blocks. This process is continued until all the blocks are used. A particular example where this would be useful is an experiment involving animals as experimental units where a block consists of litter mates. Let us suppose that we have two litters of size seven, three of size five, and one of size four. Using the staircase design we can include seven treatments and still have the four we are most interested in replicated six times.
"The Staircase Design: Theory." Ann. Math. Statist. 29 (2) 523 - 533, June, 1958. https://doi.org/10.1214/aoms/1177706627