**Model 2**: ANOVA
designs without full replication, such as randomized block designs, confound the
residual unmeasured variation with the variation due to the highest order
block-by-treatment interaction. A Model-2 analysis of a randomized block
design with two or more treatments assumes negligible variation between blocks
in the response to each lower-order treatment term. All mean squares for
block-by-treatment interactions are thereby assumed to measure the same
quantity, meaning that they can be pooled into a single variance component for
the error term.

A Model-2 analysis is appropriate if there are biological
reasons for suspecting consistency across blocks in treatment main effects. The
existence of non-negligible variation between blocks in the response to
lower-order treatment term can be tested *post hoc*. For example, the randomized complete block model S΄|B|A
allows testing for the S΄*A
interaction with *F* = MS[S΄*A] / MS[S΄*B*A],
and the S΄*B
interaction with *F* = MS[S΄*B] / MS[S΄*B*A].
In the event of *P* < 0.2, it may be advisable to use a
Model-1
analysis, which
measures the effect of each term against its interaction with the block. Note,
however, that such tests often have low power. The
pooled error mean square for the Model-2 analysis of this design is given by
SS[S΄*A
+ S΄*B
+ S΄*B*A]
/ [(*s* - 1)(*a* - 1) + (*s* - 1)(*b* - 1) + (*s* - 1)(*b*
- 1)(*a* - 1)]. A Model-2 analysis will achieve narrow-sense inference for
non-significant main effects, since the negligible effect of a treatment depends
on the truth of the underlying assumption of a consistent treatment effect
across blocks. The associated benefit, however, is increased power to detect real main effects (and lower-order interactions) compared to
a
Model-1
analysis, due to
the larger error degrees of freedom for the single pooled error term.

Randomized block designs may be analysed by Model 1 or Model 2. Split plot designs are generally analysed by Model 2. Repeated measures designs are generally analysed by Model 1.

Doncaster, C. P. & Davey, A. J. H. (2007) *Analysis of Variance and Covariance: How to
Choose and Construct Models for the Life Sciences*. Cambridge: Cambridge
University Press.

http://www.southampton.ac.uk/~cpd/anovas/datasets/