1) the separate one-way ANOVAs are not the same models like a 2x2 ANOVA model.

2) in the separate ANOVAs, not the complete data is used.

3) the separate ANOVAS do not allow to account for an interaction.

As both, data and models are different, you shouldn't expect to get the same results.

A one-way ANOVA on 4 groups should be equivalent to a 2x2 ANOVA including the interaction. The one-way model fits a separate mean to each of the four groups. A 2*2 model not including an interaction fits an intercept plus two main effects, and the one combination of the two effects results from the sum of the two main effects (the interaction is the difference of this sum to the mean of the combined effects).

This is also about adjustment. You are adjusting the effect of a factor when you model a 2x2 ANOVA. On the other hand, a one-way ANOVA lacks such adjustment.

#Mehemet, would you be so kind to elaborate a bit on the differences you pointed out in your answer as I was not able to understand it.

My belief was that in a one-way ANOVA you have a model

Y=a+b*X1+error (just one model coefficient to be estimated: "b")

and in a two-way (2x2) we had

Y=a+ b1*X1 + b2*X2 + b12*X1X2 +error (three model coefficients to be estimated: "b1", "b2" and "b12" --> 2 main effect, plus one interaction).

Thanks in advance, Luis

Luis, the models you exemplify are correct. You know, we (want to) adjust for the effects of other variables (covariates, confounders etc or X2 in your example) when making inferences. I wanted to point that including a new variable (X2) into the model is just an adjustment for the effect of that variable (X2). (A two-way ANOVA is a perfect example). So you get a new and more complicated model where you have adjusted for X2 and in which the main effect of X1 could be different than those in the simple (first) model.