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://southampton.likn.co/~cpd/anovas/datasets/