Date: Fri, 11 Feb 2005 08:02:03 -0500 Yiftach Gordoni "SAS(r) Discussion" Yiftach Gordoni marginal means' comparison and main effects' test in anova

In the SAS onlinedoc on chapter 12, pp. 168 it is written that: "for models involving interactions..., in the absence of a priori parametric restrictions, it is not possible to obtain a test of a hypothesis for the main effect free of parameters of higher-level effects with which the main effect is involved". But, as I understand it, in a 2x2 design, for example, despite the infinite mathematical solutions for the interaction effects (and also for the main effects), most of these solutions cannot be treated as a possible solutions because, by definition, the interaction effects must be independent from the main effects of each of the two variables, and the only pattern for the interaction effects that fits this statistical demand is one in which the two effects on the main diagonal have the same value, and the two other effects, on the minor diagonal, have the same value. only patterns of this kind can fulfill the demand for zero correlations between the interaction and main effects, and can be consider as a possible solutions. So if we take, for example, the following four means: 14,17,19,28, we can say that a possible population's parameters for them can be: 16 for the constant, -3,-3,5,5 for A, -2,4,-2,4 for B, and 3,0,0,3 for the interaction. for this reasons, the estimates we gets from sas (which use the "set to zero" constraints): 28 for the constant, -11 -11 0 0 for A, -9 0 -9 0 for B, and 6 0 0 0 for the interaction, cannot be consider a possible solution, because the interaction effects in it correlates with the main effects (r=-.577 with both main effects). If the plausible patterns for the interaction parameters are as I mentioned above (same for main diagonal, same for minor diagonal) they cancel each other if we are using the estimable function comparing the two marginal means for either one of the variables (A or B), in a way that makes this marginal means' comparison an estimate of the two main effects' comparison.

Isn't because of all the above that we can interpret the marginal means' comparison as a main effects' comparison?