I have found a signficant 3 way interaction (X by Y by Z) using MANOVA, but do not have any significant 2 way interactions (X by Z -or- Y by Z) involving relevant factors; is this possible? If so, how?
Re: Can you have a significant 3 way interaction without a corresponding 2 way intera
For an ANOVA, it is possible to have a significant 3-way interaction without any significant 2-way interactions, certainly for a balanced (orthogonal) design, and possibly for a nonorthogonal design as well (I haven't confirmed the latter though). Regarding how this is possible, just think of the linear model describing the relationship between dependent and independent variables. The coefficients for the 2-way interactions are zero (or very small) whereas the 3-way interaction is large. You could simulate data from such a model and graph it to get a feel for what this relationship might look like. I think the above should generalise to MANOVAs as well.
Trying to interpret a 3-way MANOVA interaction is certainly a challenge, and perhaps impossible without graphing the data. Univariate graphs for each dependent variable vs. the three independent variables might a be a good place to start, along with scatterplots of the dependent variables.