Statistical Methods for Psychology

Table 13.12 contains the variance components for fixed and random variables for two-

way factorial designs, where the subscripts in the leftmost column stand for fixed (f) or

random (r) variables.^4 You simply calculate each of these terms as given, and then form the

appropriate ratio. This procedure is illustrated using the summary table from the design in

Table 13.8, where subjects were asked to identify an upper or lower case letter and the Let-

ters used were random.^5

If we let represent the fixed effect of Case and brepresent the random effect of Let-

ter, then we have (using the formulae in Table 13.9)

sN^2 b=(MSB 2 MSerror)>na

=(378.735 2 8.026)> 1035 =7.414

sN^2 a=(a 2 1)(MSA 2 MSAB)>nab

=(2 2 1)(240.25 2 47.575)>(10 323 5)=1.927

a

Section 13.9 Measures of Association and Effect Size 439

(^4) If you need such a table for higher-order designs, you can find one at http://www.uvm.edu/~dhowell/StatPages/More_
Stuff/Effect_size_components.html.
(^5) Some authors do as I do and use for effects of both random and fixed factors. Others use to refer to effects
of fixed factors and r^2 (the squared intraclass correlation coefficient) to refer to effects of random factors.
v^2 v^2
Table 13.12 Estimates of variance components in two-way
factorial designs
Model Variance Component
Af Bf
Af Br
ArBr
The summary table for Eysenck’s study is reproduced below for
convenience.
Source df SS MS F
C (Case) 1 240.25 240.250 29.94
L (Letter) 4 1514.94 378.735 47.19
CL 4 190.30 47.575 5.93
Error 90 722.30 8.026
Total 99 2667.79
p , .05
s^2 e=MSe
s^2 ab=(MSAB 2 MSe)>n
s^2 b=(MSB 2 MSAB)>na
s^2 a=(MSA 2 MSAB)>nb
s^2 e=MSe
uNab^2 =(a 2 1)(MSAB 2 MSe)>na
s^2 b=(MSB 2 MSe)>na
uN^2 a=(a 2 1)(MSA 2 MSAB)>nab
s^2 e=MSe
uNab^2 =(a 2 1)(b 2 1)(MSAB 2 MSe)>nab
uNb^2 =(b 2 1)(MSB 2 MSe)>nab
uNa^2 =(a 2 1)(MSA 2 MSe)>nab