If another restriction on the randomization is placed ori a Latin sguare, we have a Graeco-Latin
sguare. Table 1.23b exhibits a 4 x 4 Graeco-Latin sguare.

Table 1.23b

Row

In this design the third restriction has levels a, B, y, 8. Note that a, B, y and 8 not only each
appear exactly once within each row and column, but they also appear exactly once with each level
of treatments A, B, C, D. The Graeco-Latin sguare can be constructed by superimposing an
orthogonal (same size) Latin sguare on the original Latin sguare. In other words, the third
restriction factor along with column and row is also a 4 X 4 Latin sguare. It has treatments a, B, v,
8 and is orthogonal to the original Latin sguare with treatments A, B, C, and D. Here
orthogonality means each letter in one Latin sguare appears exactly once in the same position as
each letter of the other sguare.

The analysis of variance for a Graeco-Latin sguare is very similar to that for a Latin sguare.
Let GREEK denote the third restriction factor on a 4 x 4 Graeco-Latin sguare. The MANOVA
specifications would be

MANOVA Y BY ROW(l,4), COL(l,4), GREEK(1,4), TRT(1,4)/
DESIGN<ROW, COL, GREEK, TRT/

Note that a small Graeco-Latin sguare design may not be very practical, since very few degrees of
freedom are left for the residual. PA

1.24 Factorial Designs

In a factorial design, the effects of several different factors are investigated simultaneously.
Suppose we wish to study the effects of two factors on the yield of a chemical. The first factor is
temperature at 100"F, 200"F, and 300"F. The other factor is pressure at 20 psi and 40 psi. This
experiment is a two-factor factorial design with three levels for the first factor and two levels for the
second. The treatments for this experiment are the 6 combinations of the levels of the factors. The
model for the 3 x 2 factorial design is

Yi < pota; t Bit (aB)i; t cije

where a; is the temperature effect, B; is the pressure effect, and (aB);; is the temperature-pressure
interaction.

A factorial experiment containing one observation per cell (treatment) constitutes one
replicate of the design. The design may be replicated k times in two possible ways. If each
observation has different experimental conditions for replications within cells (e.g., each replicate
is a block), the design is crossed by another factor within k levels (i.e., block effect). If the
experimental condition is the same for the replications within cells, the number of factors remains
unchanged, and the variation within cells is attributed to the error.

The following example illustrates the use of MANOVA to perform the analysis of a 4 x 4 x 3
factorial in randomized bločks (two blocks) with a covariate. The data are taken from Cochran and
Cox (1957, p. 176). |

The model contains the main effects (NTREAT, LENPER, CURRENT), all two-way
interactions (NTREAT BY LENPER,...,LENPER BY CURRENT), and the three-way interac-
tion (NTREAT BY LENPER BY CURRENT). The SPSS commands for this analysis are shown
in Figure 1.24a and the analysis of variance table in Figure 1.24b.

MANOVA 21