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Partially Balanced Incomplete Block Designs Although we have concentrated on the balanced case, there are several other types of incomplete block designs that occasionally prove useful. BIBDs do not exist for all combinations of parameters that we might wish to employ because the constraint that λ be an integer can force the number of blocks or the block size to be excessively large. For example, if there are eight treatments and each block can accommodate three treatments, then for λ to be an integer the smallest number of replications is r = 21.

In this case the factors are temperature (A) and material type (B). Using the Design-Expert software package, we fit a model that the main effect of material type, the linear and quadratic effects of temperature, the material type by linear effect of temperature interaction, and the material type by quadratic effect of temperature interaction. Refer to Table 5-15 in the textbook. From examining this table, we observed that the quadratic effect of temperature and the material type by linear effect of temperature interaction were not significant; that is, they had fairly large P-values.

I = 1,2," , a a a i =1 i =1 anµ + n∑ τ i + anβ j + n∑ (τβ ) ij = y. j . , j = 1,2," , b RS T i = 1,2," , a nµ + nτ i + nβ j + n(τβ ) ij = yij . , j = 1,2," , b Recall that in general an estimable function must be a linear combination of the left-hand side of the normal equations. Consider a contrast comparing the effects of row treatments i and i ′ . The contrast is τ i − τ i ′ + (τβ ) i . − (τβ ) i ′. Since this is just the difference between two normal equations, it is an estimable function.

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