14

INTRODUCTION TO BASIC CONCEPTS IN EXPERIMENTAL DESIGN

dd) - Xjsut Bit či

According to this model, an individual score is egual to the population
mean j, plus a treatment effect $;, plus an error effect e,;, which is unigue
for each individual subject. In a particular experiment, the parameters
u, B;, and gi; are unknown, but sample estimates of these parameters are
given by ji, B;. and £;;, respectively. It can be shown by maximum-likelihood
methods that unbiased estimates of the reguired parameters are provided
by the statistics

fi < X.. —u
B; — (X.; me X.) id B;
či — (Xi; a X.) — Eij:

The symbol > indicates that the term on the left is an estimator of the
term on the right. According to the maximum-likelihood method, the best
estimate is the one that gives the highest probability of obtaining the ob-
served data. It should be noted that a: maximum-likelihood estimator
is not necessarily unbiased, although the center of its distribution is gener-
ally close to the value of the parameter estimated. Assumptions associated
with the mathematical model for a completely randomized design are
discussed in Chapter 2 and explicitly stated in connection with the descrip-
tion of each design in subseguent chapters.

The meaning of the term error efject is somewhat elusive. An intui-
tive understanding of this term can be obtained by an examination of
Table 1.4-2 and the linear model for the design. It is obvious that the scores
for all S rats exposed to treatment level b, in this table will probably not
be identical. Variation among the five scores can be attributed to a variety
of sources —experiences of the rats prior to participation in the experiment.
unintended variation in administration of the treatment level, lack of
reliability in measuring the effect of the treatment level, ete. An error
effect is an estimate of all effects not attributable to a particular treatment
level. This can be seen from the linear model if the terms in eguation (b)
are rearranged and statistics are substituted for the parameters. The
eguation can be written

D LI

či; < X; — p;—E.
Thus the error effect is that portion of a score remaining after the treat-
ment eflect and grand mean are subtracted from it. An experimenter
attempts, by using an appropriate design and experimental controls, to
minimize the size of the error effect. Designs described in subseguent
paragraphs permit an experimenter to accomplish this by isolating addi-
tional sources of variation that affect individual scores.

RANDOMIZED BLOCK DESIGN

A randomized block design is based on the principle of assigning
subjects to blocks so that the subjects within each block are more homo-
geneous than subjects in different blocks. Assume that the 15 albino rats