INTRODUCTION TO BASIC CONCEPTS IN EXPERIMENTAL DESIGN

subjects are assigned to the treatment levels so that each possible pair of
treatment levels occurs together within some block an egual number of
times. A design having these characteristics is called a balanced incomplete
block design. Partially balanced designs are those in which some pair of
treatment levels occur together within the blocks more often than do other
pairs.

FACTORIAL EXPERIMENT

A factorial experiment permits an investigator to evaluate the
combined effects of two or more treatments in a single experiment. This is
accomplished by combining building block designs so that one level from
each of two or more treatments is presented simultaneously. The most
commonly used building block designs are the completely randomized
design and the randomized block design. |

In the microwave radiation example, an investigator can, by using
a factorial experiment, evaluate the. eflects of radiation and also the
effects of a second treatment, such as room temperature. Assume that
there are two levels of ambient room temperature, 4; — 80" and a, < 65",
and three levels of radiation, b, <0,b, < 20,000, and b; < 40,000 micro-
watts. Tables 1.4-6 and 1.47 illustrate the use of two freguently used
building block designs in a factorial experiment. In Table 1.4-6 the three
subseripts designate a particular temperature level, radiation level, and
subject, in that order. The three subscripts in Table 1.4-7 designate a
particular temperature level, radiation level, and block, in that order.

In the completely randomized factorial design of Table 1.4-6 it is
assumed that 18 albino rats are randomly assigned to the six treatment
level combinations. In the randomized block factorial design example
shown in Table 1.4-7, the treatment level combinations are randomly
assigned within each block of litter mates. The models for the completely
randomized and randomized block factorial designs are, respectively,

Xijm 5 Ht tit Bj A aB;; £ Emi
Xiim zu ti; NA B; LET aB;; d- Eijm:

The effect of temperature is designated by a; radiation by B;, interaction
of temperature and radiation by 4B;; experimental error by e, and litter
by z,, Both designs permit an investigator to determine if radiation dosage
has the same effect on food consumption at 80" ambient room temperature
as it has at 65? ambient room temperature. It is conceivable that radiation
might be more detrimental at a high ambient room temperature than ata
low ambient room temperature. If such a result is found, it is called an
interaction effect.

The error effect for a completely randomized factorial design, using
the scheme described previously, can be written

Či) Z Xiim — a, — B; — ab;; —- fi.