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SECTION 1.4 OVERVIEW OF TYPES OF EXPERIMENTAL DESIGNS 15
> OI — NI IOOOOII o oo ——

in the previous example were taken from five different litters. Rats from the
same litter can be expected to be more homogeneous with respect to genetic
characteristics than rats from different litters. In Table 1.4-3 the 3 rats in

lation

igue h : z
 OLETS each row"that comprise a block are from the same litter. Differences
rs are among the litters can be regarded as a nuisance variable that is experi-
>. mentally isolated through the use of a randomized block design. The
wvided subscripts of X,; designate a particular litter and treatment level, in

that order. Diferences among the column means reflect treatment
effects, whereas differences among the row means reflect litter effects.
TABLE 1.4-3 Randomized Biock Design
ae a ra A II ŽE rž an nn rr nn
Treatment Levels
of the Block means
t
he bes Block (litter) p, ;
he ob- Block (litter) p X..
imator Block liner P3 o
- ock (litter) p, s:
Ena Block (liter) ps X,.
arrip- Treatment means < X., X., X., Grand mean < X..
n intui- Assignment of the three treatment levels to the rats is randomized indepen-
tion of dently for each row. The linear model for this design is
e SCOTES Xj<utB; tn; t čij.
bly not | NH

variety Unbiased estimates of the parameters are given by the statistics
eriment, a< X za
lack of P a di
n error B; <(X.;— X..) o B;
eatment fi — (X,. —- X..) o A;
ation (1) MA MA IH

E, se (X;; — X.; — A; 4 X..) — €ij:
The term z, represents an effect attributable to the ith block of 3 rats.

It can be shown, by regrouping terms in the linear model and substituting
statistics for parameters, that the error effect in a randomized block design
is egual to

čij < Xij- Ba -ta-aA.

U
The error effect for a completely randomized design was given earlier as
či; — Xi; — B, — A.
Thus the error effect č;; for a randomized block design is egual to the
completely randomized design error effect minus a block eflect fi. It is
apparent from this that the error effect for a randomized block design

will be smaller than the error effect for a completely randomized design if
the block effect rt, is appreciably greater than zero.