Statistical Methods for Psychology

two-way, we took an A 3 B table of cell means, calculated , subtracted the main

effects, and were left with. In the three-way, we have several interactions, but we

will calculate them using techniques analogous to those employed earlier. Thus, to ob-

tain we will take a B 3 Ctable of cell means (averaging over A), obtain ,

subtract the main effects of Band C, and end up with. The same applies to

and. We also follow the same procedure to obtain , but here we need to be-

gin with an A 3 B 3 Ctable of cell means, obtain , and then subtract the main

effects andthe lower-order interactions to arrive at. In other words, for each

interaction we start with a different table of cell means, collapsing over the variable(s)

in which we are not at the moment interested. We then obtain an for that table

and subtract from it any main effects and lower-order interactions that involve terms

included in that interaction.

Variables Affecting Driving Performance

For an example, consider a hypothetical experiment concerning the driving ability of two

different types of drivers—inexperienced ( ) and experienced ( ). These people will

drive on one of three types of roads—first class ( ), second class ( ), or dirt ( ), under

one of two different driving conditions—day ( ) and night ( ). Thus we have a 2 33 32

factorial. The experiment will include four participants per condition (for a total of 48 par-

ticipants), and the dependent variable will be the number of steering corrections in a one-

mile section of roadway. The raw data are presented in Table 13.14a.

The lower part of Table 13.14a contains all the necessary matrices of cell means for the

subsequent calculation of the interaction sums of squares. These matrices are obtained sim-

ply by averaging across the levels of the irrelevant variable. Thus, the upper left-hand cell

of the ABsummary table contains the sum of all scores obtained under the treatment com-

bination , regardless of the level of C(i.e., ). (Note: You should be

aware that I have rounded everything to two decimals for the tables, but the computations

were based on more decimals. Beware of rounding error.^6 )

Table 13.14b shows the calculations of the sums of squares. For the main effects, the

sums of squares are obtained exactly as they would be for a one-way. For the first-order

interactions, the calculations are just as they would be for a two-way, taking two vari-

ables at a time. The only new calculation is for the second-order interaction, and the

AB 11 ABC 1111 ABC 112

C 1 C 2

B 1 B 2 B 3

A 1 A 2

SScells

SSABC

SScellsABC

SSAC SSABC

SSBC SSAB

SSBC SScellsBC

SSAB

SScells

Section 13.12 Higher-Order Factorial Designs 447

B 2

C 1

A 1

A 2

B 1 B 2

C 2

A 1

A 2

B 1

Figure 13.3 Plot of second-order interaction

(^6) The fact that substantial rounding error accumulates when you work with means is one major reason why
formulae for use with calculators worked with totals. I am using the definitional formulae in these chapters
because they are clearer, but that means that we need to put up with occasional rounding errors. Good computing
software uses very sophisticated formulae optimized to minimize rounding error.