D OTHER EXPERIMENTAL DESIGNS

, row, column, and in-
degree of freedom as-
edom associated with
peated measurements
lumber of degrees of
ts, because repeated
'hus the row effect is
) associated with the

ratio. Adjustments
used in the F ratios
g the Box procedure,
grees of freedom as-
linator of the F ratio.
| R(n — |) degrees of
ature of these tests, if
1 adjusted degrees of
f the homogeneity of

le type described in
dom to A treatment
th zn subjects in each
sums of sguares are
1, respectively. The
netimes substantially,
ing a variable that is
For example, assume
nguage are under in-
al one-way classifica-
to the four methods

Measures of scho-
this variable may be
»jects in learning the
groups, or blocks, of
her a low, scholastic
nay then be assigned
Jups of five subjects.
riment. The essence
hat subjects can be
on variable, which is
thin blocks are then
s of data for such an
bove the analysis of

NEON OKO "JEVA

PVO MNA REPE ROA di

i IA

19.14 RANDOMIZED BLOCK DESIGNS 323

data is the same as for a double-classification factorial experiment as
described in Chapter 16. Four sums of sguares are obtained. Given k
treatments and 8 blocks the degrees of freedom associated with treat-
ments, blocks, interaction, and within-cells sums of sguares are A— |,
B—1,(k—1I)(B—1I), and N — BA, respectively.

What is the purpose of a randomized block experiment? The primary
purpose is to reduce the size of the error term used in the denominator of
the F ratio, which for the fixed model is the within-cells mean sguare. The
relative efficiency of the experiment is thereby increased in relation to the
one-way classification experiment. If the blocking variable has a substan-
tial correlation with the dependent variable, the sums of sguares associated
with blocks may prove to be of some appreciable size; also an interaction
term of some magnitude may be found. The effect of this will be to reduce
the size of the within-group sum of sguares and the within-group mean
sguare and, thereby, increase the likelihood of obtaining a significant dif-
ference for the main effect.

The reader should note that in the one-way classification experiment
the number of degrees of freedom associated with the error term, the
within-groups mean sguare, is N — A, whereas in the randomized block
experiment the number of degrees of freedom associated with the error
term is N — Bk. Thus in the randomized block experiment a loss in
degrees of freedom associated with the error term occurs, which must be
compensated for by the sum of sguares associated with blocks and interac-
tion. An informative discussion of this point will be found in Myers
(1972). Myers' treatment of the subject shows that the relative efficiency
of the randomized block experiment in relation to the usual one-way clas-
sification experiment will be greater than 1 whenever the F test of the com-
bined block and interaction effects exceeds |. The relative efficiency will
increase as the sum of sguares associated with blocks and interaction ef-
fects increases.

Because the degrees of freedom associated with the error term in a ran-
domized block design are N — BA, the power of the F test will decrease as
the number of blocks increases. Also, as the number of blocks increase
the within-cells sum of sguares decreases. These are opposing effects
which suggest that in a randomized block experiment some optimum
number of blocks exists. This topic has been investigated by Feldt and
Mahmoud (1958). The reader will find Myers' (1972) discussion of this
topic helpful. The gist of the matter is that the optimum number of blocks
is related to the correlation between the blocking variable and the depen-
dent variable, sample size N, and the number of treatment levels k. The
optimum number of blocks increases with increase in the correlation and
sample size N and decreases with increase in the number of treatment
levels. In the design of a randomized block experiment investigators
should inform themselves of these matters and keep them in mind.

The blocking variable is usually a classification variable which is char-
acteristic of the subjects and is in no way under the control of the inves-

(nani nRmonorn mi ini ei Ni