Model 1: 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-1 analysis of a randomized block design with two or more treatments accommodates inconsistent variation in the response to each lower-order treatment term across blocks. To do this, each treatment term in the model must have its own error mean square, given by the mean square for the interaction of that term with the block term. In contrast, a Model-2 analysis assumes MS[S΄*A], MS[S΄*B], and MS[S΄*B*A] are estimating the same quantity, by virtue of having negligible contributions from differences between blocks in the variation due to treatment main effects.

 

The Model-1 analysis is appropriate if there are biological reasons for suspecting block-by-treatment interactions. Those interactions 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]. Note, however, that such tests often have low power. By not assuming consistent treatment effects across blocks, a Model-1 analysis achieves broad-sense inference for significant main effects, since they are found to be significant even in the event of an interaction with blocks. The associated cost, however, is reduced power to detect main effects (and lower-order interactions) compared to a Model-2 analysis with its 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/