What is the right way to analyze Full factorial design (by groupe/ by factor?)

Question

I have a conceptual issue about the analysis of the variance of a full factorial experiment design. (3 Var / 2 levels each). In all papers I read, there are plenty of ways to do, some things may seem logical but conceptually causing problems.

I will illustrate with a simple example before asking the questions:

Let us suppose that we test the effect of three organic packaging to know if those that are perceived as organic are the least polluting. We have for example the variables:

Material: Aluminum Vs plastic

Color: Green Vs Red

Shape: Rounded Vs angled

We finally get 8 versions of boxes. Randomly assigned to 8 different groups of 100 individuals each (unpaired, without repetition) and then we measure perceptions ...

My questions are:

• I still do not understand the link between the analysis by group (by version which is the combination of the 3 factors) and the analysis by factor (by color, form, or material).

Some references say it is according to the needs of the study but in this case, why perform anova for each box version? Why do we not directly analyze the effect of each factor and deduce the most relevant version?

Moreover, a significant Anova test only says that there is at least one group different from the others, and does not always allow to explore results beyond this. Because pairing comparison tests (like Duncan test) uses type 2 error while Anova uses type 1.

Moreover, all the known assumptions (homogeneity of variances, choice of the covariables, homogeneity of the samples structure ... etc) are found again the doubts as soon as we pass from the plan to the factor. If we have N = 800 for the versions, we will have 8 groups of 100. But when we analyze by factor, we will get 2 groups of 400 for each factor, but in papers, nobody checks again all the assumptions cited above.

I ask you to kindly enlighten me I am lost. I especially want to know the right way to analyze a full factorial design? analysis by factor is it a deepening of the comparison of the 8 groups. How can I know which version is the most relevant? Is it through interaction effects?

Is it right to say that if we get a sig* interaction effect of 3 factors means that we are talking about the version ?

Last question, in some papers, researchers merge certain groups, can someone tell me how?

Answer 1

Starting at the most complicated end we look at the three-way interaction. If this is significant we often then find it easier to prefer to examine the effect you have called version which is a factor with eight levels. Suppose it is not signficant then we look at the two-way interactions next and interpret them. Suppose we find that one of them is significant, say material by colour. The we know that the difference between aluminium and plastic packages depends on what colour it is, it may even be reversed. If there are no two-way interactions we proceed to the main effects which directly compare aluminium with plastic irrespective of colour or shape, and so on.

So the short answer is that the analysis as a 2 by 2 by 2 factorial may lead to simpler scientific statements but in case of complicated higher order interactions we may resort to flattening out the design into an eight level factor.