The models you described are different:

In two-way ANOVA, you have three terms: Factor 1, Factor 2, and their interactions.

In one-way ANOVA, you include only the interaction term from the first model (Using a one-way ANOVA with a different code for each cell is equal to using a two-way ANOVA with only the interaction term included).

So, the results are different, and the df of the error term is different. The first model has three hypotheses, while the second has one. Also, in the two models, the interaction term is about two different questions. In the first model (the full factorial two-way ANOVA), the interaction is calculated after accounting for the main effect of each factor. In the second model (one-way ANOVA or two-way ANOVA with only the interaction term included) the interaction term is the only effect.

I hope this helps.

Hello Jason,

The within-cell error term for a 2-way factorial design will be exactly the same as that for a one-way design if each cell in the two-way design is treated as a separate group/cell in the one-way design (assuming both factors are fixed effects factors). Thus, you could run "post hoc" tests using one of the Tukey methods (e.g., HSD) among the a x b cells in a one-way re-expression of the 2-way design if this was important to you.

However:

1. You say there is evidence of interaction. Why not just run simple effects tests, as these would represent the comparisons of interest, would they not?

2. Among all the possible post-hoc tests, there are more powerful options than Tukey's HSD or WSD. Why opt for these?

Good luck with your work.

@David Morse I speak ANOVA almost not at all but factorial designs exist to study possible interactions so I don't understand how a factorial design can reduce completely to a complete random block design without showing that there's no interaction which it seems to me either defeats the original purpose or makes df difficult to understand. BTW why use a factorial design if you aren't interested in interaction.? I'm certainly confused would you please help me out? Thanks very much, David Booth BTW Apologies for the run on sentence.

Hello All (and David Booth),

Imagine the simplest possible two-way factorial design (levels of factor A = A1, A2; levels of factor B = B1, B2). Now, re-express that design as a one-way anova having four cells: A1B1, A1B2, A2B1, A2B2.

All of the relevant factorial effects may be replicated by the following set of contrasts:

A effect: 1, 1, -1, -1

B effect: 1, -1, 1, -1

A x B interaction: 1, -1, -1, 1

As each of these is a one df effect, they all could be generated via post hoc tests (as in the case of Scheffe tests).

Simple effects tests would also be possible via these selected pairwise comparisons (with appropriate adjustment for aggregate alpha).

A|B1: 1, 0, -1, 0

A|B2: 0, 1, 0, -1

B|A1: 1, -1, 0, 0

B|A2: 0, 0, 1, -1

That's the point I was trying to make.

Best regards to everyone.

@David Morse thanks for the explanation I understand now.i appreciate that very much. Best wishes David Booth