Do non-significant correlates of a DV, which is significant, suggest anything about the effect of that DV? Let's assume one has an analysis in which there are multiple correlated DVs (average correlation .46) being examined in separate univariate analyses (e.g. t-tests; insufficient df and frustration tolerance to restructure the data fro a MANOVA) in relation to a single IV.  All except one of these DVs is not significantly affected by an IV manipulation.
I was thinking that in general, dependence between DVs would tend to globally trend analyses to be significant or non-significant.  However, when a DV with a legitimate effect is correlated with a variable which has no effect, wouldn't that correlation represent an association between them which is a source of error variance in the DV with the legitimate effect? Thus, if anything, the presence of significance in a DV associated with non-significant DVs should be taken as a sign of a true effect on that DV?
In short:


*

*What, if anything, do the non-significant tests tells us about the one that is significant?  

*Is there a relationship between this question and the notion of suppressor variables in regression?

 A: Not sure I grasped your question. But how can you answer your questions with just univariate analyses at hand? You was thinking of MANOVA of several DVs (say, Y1, Y2) on single IV (X). Let's notice that such MANOVA is equivalent to multiple regression of X on predictors Y1, Y2: direction of prediction changes nothing in this case; the quantity you called Pillai's trace will be now called R-square. In milieu of multiple regression it is no wonder to find a picture where IVs Y1 and Y2 are moderately correlated, still only Y1 predicts X significantly. Can Y1 or Y2 be a suppressor? Yes, and one can check this. If the increment of R-square in responce to adding Y2 into the model containing the rest predictors (Y1) is greater than R-square of simple regression of X on Y2 then Y2 is a suppressor.
A: It is not completely clear to me what are you trying to achieve here, an example would probably help. Adding one of the "insignificant" dependent variables to a regression with the "significant" one and checking if the effect is still there should probably answer the question from the second paragraph. More systematically, I would estimate a seemingly unrelated regression system of three equations and look at the correlation of their residuals, including testing of cross-equation equality of the parameters.
