Interpretation of ANOVA main effect, ignoring one row

Question

I'm reading a paper which performs a 3x2 ANOVA like so:

Condition: | A    | B    | C
-------------------------------
Phase 1    | x_A1 | x_B1 | x_C1
Phase 2    | x_A2 | x_B2 | x_C2

They find a significant main effect of condition, and of phase, and no interaction between the two. I am not interested in phase 2, and only want to know if there is a difference between conditions in phase 1. Roughly, I want to know if a 3x1 ANOVA on x_A1, x_A2 and x_A3 would show an effect. Given the lack of interaction, is it possible to conclude that there was a difference between conditions in phase 1 only? Or is the result affected by the presence of the phase 2 data, even though an interaction wasn't significant?

(As I say, this is from a paper, so I don't have access to the original data. I've tried searching for previous ANOVA questions, but couldn't find this question specifically.)

Answer 1

The result is affected by the presence of phase 2 data. The idea of the ANOVA in terms of efficency is two use the additional information provided in the second phase to underpin the robustness of the discovered underlying main effects.

If you decompose your sum of squares you will see that testing for the main condition effects will involve a) the means of the single conditions over all phases ($SS_{condition}$) and b) additional information about the error term deduced from phase 2 observations ($SS_E$).

Both will effect your test results and it is not clear how they would change if you disregard the information of phase 2. In general, the additional information will improve your estimates by reducing their variance. But there are still cases conceivable where disregarding the information could change your test results. For instance if the observed condition effect in phase 2 is sufficiently stronger and the new variance of the error term is sufficiently higher, the condition effect might turn out to be insignificant.

But in doing the ANOVA you assume a model which looks like follows (disregarding the interaction effect due to insignificance):

$y_{ijk}= \mu + \gamma_i + \beta_j + \epsilon_{ijk}$

If $\gamma_i$ describes the main condition effect of condition i and you found that it is significant (i.e. at least one $\gamma_i \neq 0$ ), then there is a condition effect for every phase j (including j=1). Therefore you could conclude from the larger model that conditions are relevant in phase 1 only.