Two-Way Between-Subjects Design - Analysis of Variance Designs - page 181

Ta b l e 8 . 2 . Notation for a 3 × 2factorialdesign

Design Factor A

a 1 a 2 a 3

Factor B b 1

b 2

Data Set ( Y ij k )

a 1 b 1 a 2 b 1 a 3 b 1 a 1 b 2 a 2 b 2 a 3 b 2

Y 111 Y 121 Y 131 Y 112 Y 122 Y 132

Y 211 Y 221 Y 231 Y 212 Y 222 Y 232

Y 311 Y 321 Y 331 Y 312 Y 322 Y 332

Y 411 Y 421 Y 431 Y 412 Y 422 Y 432

Y 511 Y 521 Y 531 Y 512 Y 522 Y 532

AB 11 AB 21 AB 31 AB 12 AB 22 AB 32

AB Matrix of Sums Factor A

a 1 a 2 a 3 Sum

Factor Bb 1 AB 11 AB 21 AB 31 B 1

b 2 AB 12 AB 22 AB 32 B 2

Sum A 1 A 2 A 3 T

in the second matrix of Table 8.2, labeled Data Set ( Y ijk ) .Werefertothe
series or observations in this matrix collectively as Y ijk to denote three
pieces of information. The subscript i designates the ordinal position of
each participant or case within a treatment combination. The subscript
j denotes the level of the A treatment and the subscript k represents the
level of the B treatment. This notational device reminds us of the exact
treatment combination each participant receives.
The six columns of raw data in the Y ijk matrix can each be summed
to produce treatment combination totals referred to collectively as AB jk
(again, the j subscript refers to the level of the A treatments and the k
subscript signifies the level of the B treatments). These six AB jk treat-
ment combination sums can be configured into their own matrix, which
we depict in Table 8.2 as the AB Matrix of Sums .These AB treatment
combination sums can be summed vertically (to obtain the A treatment
sums) or horizontally (to obtain the B treatment sums) and will pro-
vide the ingredients for our future main effects analyses. The AB matrix
of sums can be converted into an AB matrix of means by dividing each
AB jk treatment combination sum by n .Wehavedonesoatthebottomof
Ta b l e 8 . 3 .
Before proceeding with the computation of sums of squares, degrees
of freedom, mean squares, and F ratios,weneedtocomputesomepre-
liminary calculations (e.g., treatment combination sums, squared sums
of the treatment combinations, and treatment combination means, and
standard deviations), which are depicted at the bottom of the Y ijk matrix