To illustrate this approach I will discuss a design that requires the
treatment of subjects as random effects -
both at a statistical and a
conceptual level: A single factor repeated measurements design with two
treatment levels. It is assumed, that sequence of the factor levels and
subjects are randomly combined. The respective randomization test informs
about the probability of empirical mean differences, given the 
that
these differences are exclusively the result of the random combination
of the sequence of the factor levels and the subjects. That is, the
probability that the effect is the result of chance fluctuations in the
reactions of the subjects across the two levels. The notion of "chance
fluctuation" subsumes the variation of the influences of all PCVs at the
two points of measurement. Consequently, the distribution of the test
statistic is generated by the permutation of the combinations of reactions
and treatment levels within subjects (e.g., Edgington, 1995).
In the ANOVA evaluation of this design, subjects are treated as random
effects in a mixed-model approach. This is reflected by the fact that the
F-ratio for the expected mean squares of treatment variability is compared
to the expected mean squares of the interaction of treatment by subjects,
consisting of the variance components 
and
(see Table 1). These
variance components (which are confounded in case of only one observation
per stimulus-subject combination) can be considered to be estimations of
the influence of pure chance fluctuations across the points of measurement,
since these fluctuations will cause differential treatment effects. Note
however, that the variance component
entails also "true" or manifest
differential effectiveness of the treatment. As a consequence, the expected
mean square will overestimate the amount of pure chance
fluctuations across
the points of measurement. Moreover, the hypothesis tested usually does not
predict that the treatment has numerically the same
effect on all subjects
and therefore allows for ordinal
treatment-by-subject interactions. In the
present design, however, chance variation and ordinal treatment-by-subject
interactions are not separable.
| Source | df | E(MS) |
| T | p-1 |  |
| Su | n-1 |  |
 | (n-1) (p-1) | 
(Residual) |
As a result, the mixed-model ANOVA resembles the randomization test account
of chance fluctuations by comparing treatment variance to an upper bound
estimation of pure chance fluctuations across the points of measurement
(including error variance ) in the critical F-value.
Based on these arguments, the revised question now is: Under which circumstances does the treatment of stimuli as random effects serve the same purpose as the treatment of subjects in the present design does? Three different design are distinguished.