What is the actual goal here? Is this a small step in some bigger picture?
i feel like the answer to my question is a 'no' but i'll ask it anyway just to be absolutely sure.
say you have 3 variables X, Y and Z each one with some correlation , , . we know from the formula of the determinant of the correlation matrix that if, for instance, and are fixed, then must necessarily fall within the interval:
so the question now becomes... if we consider the OLS multiple regression models and , is there some way to calculate the range of values that can have when gets introduced into the model? in general, the will not be the same in the first and in the second model. i was hoping maybe some function of maybe the correlations/covariances and variances of the constituting variables could give me a range of values...
thaaanks!
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What is the actual goal here? Is this a small step in some bigger picture?
I don't have emotions and sometimes that makes me very sad.
hello. yes, it is a small part of a wider problem here (i'll try to be brief).
here in social-sciency land we have a regression-based method called 'mediation' where you have three variables (a predictor/independent variable X, a response/dependent variable Y and a mediator Z). the way it works if by first running the regression:
and you look to see if is significant
then you do other regressions (you predict Z from X, you predict Y from Z, nothing too important for this question).
what matters, however is that when then you run this regression:
you need to see see how the coefficient changes. if it becomes non-significant then you say Z "fully mediates" the relationship between X and Y (which rarely happens). if is still significant but it's reduced (the most common case) then that means Z "partially mediates" X and Y.
we reviewed these concepts in class last tuesday and i was thinking to myself "well, it seems like in the most common case of partial mediation (i.e. is still significant but smaller once Z is introduced in the regression equation) it would be useful to know the range of values could have. that made me think about Dragan's formulas for regression coefficients and the bounds that correlations impose each other to keep the correlation matrix as positive-definite. that's when i thought "what if i could find a way to provide a range of values that can have when Z is introduced verus absent in the regression equation? which prompted my question.
but the formulas that Dragan posted have too much going on within them. i'm thinking there could always be a way that if something changes any potential range of values i could generate for could be violated.
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for future reference, i was able to find someone who articulated more or less what i wanted to say in this thread. although my original question is wrong (i.e. that specific b-coefficient has no limits in its range) there ARE limits in the range, imposed by the correlation/covariance matrix's property of positive definiteness, that some of these coefficients together can have.
the source is here. it starts on page #12 of the PDF:
http://quantpsy.org/pubs/preacher_kelley_2011.pdf
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