10

INTRODUCTION TO BASIC CONCEPTS IN EXPERIMENTAL DESIGN

where ju; — A < the minimum treatment effect an experimenter is inter-
ested in detecting, k < number of treatment levels, c, < sguare root of
population error variance, and n — size of sample. If estimates of ji; — ft
and o, can be made, the size of a sample necessary to achieve a designated
power can be determined from Table D.I4 by a process of trial-and-error.
The probability of type | and type II errors, x and B respectively, that the
experimenter is willing to accepl must also be specified.

Assume that a treatment has three levels and that the smallest
treatment effects of interest to the experimenter are —4, —1, and 45.
That is,

k
Saj — a z (-4? 4 (-1 - 6? - €.
1

Assume, also, that on the basis of previous research o, is estimated to be
six and that the investigator wishes the power of his test (1 — B) to egual
at least .80 and the probability of a type | error to egual .05. If nine subjects
are assigned to each of the three treatment levels, Table D.i4 in the appendix
can be used to determine if the specified power is achieved. For example,

(—4)? £(—-1? (S?

6/./9 6/3

with k — 1 — 2 and N — k — 24 degrees of freedom. The value of $ <
1.87 and x — .05 are entered in the table for k — 1 degrees of freedom.
The curve corresponding to N — k degrees of freedom indicates that the
power is egual to .79, which is less than that desired by the experimenter.
If the sample size is increased to 30, with ten subjects assigned to each
treatment level, the power can be estimated from

(—-4? £(-I £ (5)?

V 3

A RR A K TA
6/,/10

with k — 1 < 2 and N — k — 27 degrees of freedom. The probability of
detecting the specified treatment effects is approximately .83. Thus the
reguired sample size is found to be 30.

If reasonable estimates of the parameters can be made, the reguired
sample size should always be computed before the experiment is begun.
If these preliminary calculations indicate that the power of the experi-
mental design is inadeguate, the experimenter may choose not to conduct
the experiment or may modify it so as to increase its power. The two
most common procedures for increasing power are (1) to increase the
size of the sample and (2) to employ an experimental design that provides
a more precise estimate of treatment effects and a smaller error term. The
first procedure was illustrated in this section. The second is described in

ua

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