OTHER EXPERIMENTAL DESIGNS

ne following sums of

,565.00

7,553.10

045,756 IN (1,970)?
10 40

5,961.50

ta for this example.
z S, |Spe, — 279, no
error term for col-
s found to be 4.04.
srees of freedom as-
vely, are 2.96 at the

column differences

a,

19.5

19.5 TWO-FACTOR EXPERIMENTS WITH REPEATED MEASUREMENTS 309

are significant at the S percent level but fall short of significance at the |
percent level.

ta

TWO-FACTOR EXPERIMENTS WITH
REPEATED MEASUREMENTS

In Sections 19.3 and 19.4 one-factor experiments with repeated measure-
ments were considered. On occasion experiments are encountered that in-
volve two factors with repeated measurements. Given R levels of one
treatment and C levels of another, each subject may be tested under each
of the RC treatments. If R — 2 and C < 2, the levels of R being R, and R,
and of C being C, and C,, there are four treatment combinations, R,C,,
R,C,, R,C,, and R,C;,. Each of N subjects might receive all the four treat-
ments, the presentations being possibly, although not necessarily, arranged
in random order for each subject.

Such data constitute an RCN block of numbers. Rows and columns
are treatments, and layers are experimental subjects. "These data are
analyzed as in the triple-classification case with one observation in each
cell. Use the computation formulas given in Section 17.8, writing n < |.
Seven sums of sguares result: rows, columns, subjects, R x C, R x S,
C XS, and RxCxS. There is, of course, no within-cells sum of
sguares.

The model here is a mixed model with z < 1. Rows and columns will
ordinarily be fixed variables. Layers, or subjects, is a random variable.
For this model the expectation of the sums of sguares and the degrees of
freedom are as follows:

o o OR nn

Mean sguare Expectation of mean sguare df

Rows, s,? o ŽA Coge t NCo,? R—I1

Columns, s, o, 4 Rovt NRo c-1i

Subjects, s? oi 4 RCo,: N—I

RXC,s,Ž o? ož, t Na, (R— IXC-— bh
RXS,s o iA CO ge (R— IMN- h
CXS, s o,? 4 Ra,,: (C—- IIN- b)
RXCXS, si,, o? ož, (R—IXIC- NN- ID

o ——— nn

Inspection of these expectations indicates that the appropriate error term
for testing row effects is the R X S mean sguare, F, — s,'/s,,. The appro-
priate error term for testing column effects is the C x S mean sguare, F, —
s.?/s«,. Vhe appropriate error term for testing R X C interaction is the
R XxC XS mean sguare, F,, — s,,2/s2,,. Unless the R XS, C X S, and
R x C XS interactions are assumed to be 0, which with most sets of data
will not be the case, no unbiased test of differences between subjects, or
R XS or C XS interactions, can be made. These are ordinarily not of
interest.