Cross factor: One factor is crossed with another when each of its levels is tested in each level of the other factor. For example, in a test of influences on crop yield, a watering regime factor (W: wet and dry) is crossed with a sowing density factor (D: high and low) when the response to the wet regime is tested at both high and low sowing density, and so is the response to the dry regime. If each of the four combinations of levels has replicate observations, then a cross-factored analysis can test for an interaction between the two treatment factors in their effect on the response. In a statistics package the fully replicated model is called by writing:
W + D + W*D
where 'W*D' refers to the interaction between W and D. Or equally:
where '|' is the command for main effects and their interaction.
An interaction between factors refers to the influence of one factor on a response depending on the level of another factor. For example, students may respond to different tutorial systems (T) according to their gender (G) and age (A), tested with model Y = T|G|A + ε. If older boys respond better to one system and younger girls to another, this may be indicated by a significant interaction effect T*G*A on the response. If one factor is a covariate, the interaction is illustrated by different regression slopes at each level of the categorical factor. Two covariates show a significant interaction in a curved plane for their combined effect on the response.
An interaction term must always be entered in the model after its constituent main effects. The significance of its effect is reported before them, however, because its impact influences interpretation of the main effects. A treatment main effect can be interpreted independently of its interactions with other factors only if the others are random factors. Thus, in the teaching example above, it would be misleading to interpret a significant main effect T as meaning that one tutorial system generally works better than another, if the student response to system depends on age and/or gender. Equally in the agricultural example, it would be unwise to report no general effect of sowing density from a non significant main effect D, if the interaction with watering regime is significant. In this case, density will have had opposite effects on yield depending on watering, with the result that it appears to have no effect on pooling the levels of watering. A fully replicated design with two cross factors can have eight alternative outcomes in terms of the significance of its main effects and 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.