## ABSTRACT

Consider an experiment involving two fixed-eﬀects factors; call these factors and . Suppose factor has levels and has levels. Let represent the th randomly observed response for the th level of factor and th level of , or the th observation in the th cell. Let represent the number of observations within the th cell and =P=1P=1 the total number of observed responses. Denote each cell mean by the unknown parameter ; then, for =

1 2 = 1 2 and = 1 2 each observed response may be expressed as the sum of the corresponding cell mean and a random error term , giving the cell means model,

= + Alternatively, let denote the overall mean response; then decompose the cell means into the sum of main eﬀects parameters, and , and interaction eﬀects parameters, ,

eﬀects model,

= + + + + If the interaction eﬀects are absent, the appropriate model is the additive eﬀects model

= + + + All three models mentioned above have a matrix representation,

y = Xβ + ε with terms being as previously defined. Only the cell means model has a design matrix with full column rank. Assumptions for the models discussed in this chapter are as for earlier

models. The error terms are independent, and are identically and normally distributed with ∼ N(0 2). Data for designs involving two fixed-eﬀects factors are coded in the same

manner as for block designs discussed in the previous chapter. This chapter addresses only complete designs, those for which every cell contains at least one observed response. In practice, the preference would be complete designs in which at least one cell has more than one observation. Two illustrations are given: One for which the data are appropriate for an

additive model (Table 14.1) and the other for which the data are appropriate for a nonadditive model (Table 14.2). Both datasets were generated using code of the form given in Section 13.7.2.