Continuous response variable.

A, B, C

Fixed factor (e.g., Treatment A of watering regime).

A΄, B΄, C΄

Random factor (e.g., Treatment B΄ of crop genotype).

a, b, c

Number of sample levels of factor A, B, C (e.g., factor A may have a = 2 levels, corresponding to 'low' and 'high').

S΄, P΄, Q΄, R΄

Random factor representing randomly selected subjects/blocks (S΄), plots (P΄), sub-plots (Q΄), or sub-sub-plots (R΄), to which treatments are applied.

Si, Pi, Qi, Ri,

Independent and randomly chosen subject/block, plot, sub-plot or sub-sub-plot which provides a replicate observation of the response.


The size of each sample, given by the number of measures of the response in each combination of factor levels (including any repeated measures), or by the number of measures across all values of a covariate.


Total number of measures of the response across all factor levels.


Hierarchical nesting of one factor in another (here, B΄ is nested in A).


Interaction between factors in their effects on the response (here, interaction of B with A).


Cross-factoring, with analysis of interaction(s) and main effects (here, B cross factored with A, and analysis of A + B + B*A).


Residual variation left unexplained by the model, taking the form S΄(...), P΄(...), Q΄(...) or R΄(...).

Y = C|B|A + ε

Full model (here, variation in Y around the grand mean partitions amongst the three main effects A, B, C plus the three two-way interactions B*A, C*A, C*B plus the one three-way interaction C*B*A, plus the unexplained residual (error) variation ε = S΄(C*B*A) around each sample mean. This would not be a full model if only main effects were tested, or only main effects and two-way interactions).



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.