Quality & Quantity 38: 649–651, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
An Alternative Model for the Analysis of Variance
with Random Effects
Schulstr. 54, D-69436 Schönbrunn, Germany (E-mail: firstname.lastname@example.org)
Abstract. The standard model for the analysis of variance with random effects implies, for the case
of two independent variables, that single effects must be tested not against the error, but against the
interaction mean squares. This causes, in comparison with the ﬁxed effects AV, a considerable loss
of test power, particularly for the 2 × 2 table. An alternative modelling of the interaction effect is
proposed which completely avoids the loss of power.
Key words: Analysis of variance, random effects, test power
Let us consider a relatively simple situation: two independent variables X, Y with
K = 2 classes each, and let there be m>1 individuals in each of the four cells.
The usual textbook presentation implies that the mean squares (MS) for X (or
Y ) must be tested, not against the error MS, but against the interaction MS. This
is so if the effects are considered to have come about in the following way. The
two X-levels are independent values of a normal random variable with variance
var(X) >= 0 which are added to the general mean. (For Y , things are analog-
ous, with var(Y ) >= 0.) The interaction is generated by adding four independent
normal random values with variance var(I ) >= 0 to the four cells. Finally, 4m
independent normal random variables with variance var(e) > 0 are added as an
error term to generate the individual values of the dependent variable. Then, in
fact, the expectation of the X-effect is
E(MS(X)) = var(e) + mvar(I ) + Kmvar(X)
and the expectation of the interaction effect is
E(MS(I )) = var(e) + mvar(I ),
(see, e.g., Kendall and Stuart (1976), p. 67, formula 36.33). Thus, in order to
test var(X) = 0, an F -test of MS(X)/MS(I ) is appropriate. This has grave con-
sequences, particularly for K = 2: MS(X) with df(X) = 1 is tested against MS(I )