Two quasi-factorial arrangements which are especially well adapted to the testing of differences between large numbers of varieties are described and treated in detail as to their analysis and value.

The arrangements described are balanced incomplete block and lattice square designs. Soybean variety trials are used to illustrate the analysis and the relative precision on soils of varying homogeneity.

Although the efficiency factor of these designs, because of the confounding of variety differences with block effects, is lower than that of randomized complete block designs, yet on soil of normal variability the designs permit the elimination of sufficient variability due to soil differences to more than offset this loss. The merits of the lattice square arrangement are demonstrated on extremely heterogeneous soil where use of the design results in a gain in precision of 150 percent over that of randomized complete blocks. An illustration is also presented in which the design on very uniform soil results in a loss of precision of 31.5 percent.

The use of these designs is recommended in variety trials involving large numbers of varieties when the trials are conducted on variable soil and when differences between the varieties are relatively small.