Consider a four-factor factorial experiment where factor A is at a levels, factor B is at b levels, factor C is at c levels, factor D is at d levels, and there are n replicates. Write down the sums of squares, the degrees of freedom, and the expected mean squares for the following cases. Assume the restricted model for all mixed models. You may use a computer package such as Minitab.

(a) A, B, C, and D are fixed factors.

(b) A, B, C, and D are random factors.

(c) A is fixed and B, C, and D are random.

(d) A and B are fixed and C and D are random.

(e) A, B, and C are fixed and D is random. Do exact tests exist for all effects? If not, propose test statistics for those effects that cannot be directly tested.