Two-Way Additive Model

An experiment involving one-way layout with blocks is called randomized block design. In the development of this model agricultural experiments provide a classic example. To compare the effects of k different fertilizers (“treatments,” or “factor levels”), b plots of land (or “blocks”) are selected, and each of them is divided into k plots; this is viewed as a multi-sample generalization of paired experiment. Within each block, the assignment of fertilizers to plots is made at random, which helps to ensure against unintentional bias. Then data of size n are obtained for each combination of treatment and block, and summarized in the following table.

Treatment	Block-1	. . . . . . .	Block-b	Sample size
1	$X_{11 \cdot}$	. . . . . . .	$X_{1b \cdot}$	b
$\vdots$	$\vdots$	. . . . . . .	$\vdots$	$\vdots$
k	$X_{k1 \cdot}$	. . . . . . .	$X_{kb \cdot}$	b

Each $X_{ij \cdot}$ consists of measurements $X_{ijl}$ , $l=1,\ldots,n$ .