Re: st: Testing the extent of difference between two coefficients in the same model

[David Hoaglin <dchoaglin@gmail.com>]

In the modified model, the predictors that we're focusing on are "old"
and "new MINUS old."  The coefficient of "old" in this model tells us
about the contribution of "old" to the log-odds when we are also
taking into account the contribution of the difference between "new"
and "old" (via the predictor "new MINUS old").  And the coefficient of
"new MINUS old" tells us about the contribution of that difference to
the log-odds when we are also taking into account the contribution of
"old."  We have simply rearranged the information supplied by "old"
and "new."  (Technically, "old" and "new MINUS old" span the same
subspace of predictor space as "old" and "new.")  Thus, "old" makes a
significant contribution to the log-odds, and "new MINUS old" makes a
significant additional contribution.  You could turn the modification
around and use "new" and "old MINUS new" as the predictors, to see
whether, with "new" in the model, "old" makes a significant additional
contribution.  The choice among models reflects the question that you
are asking.

Since this is a regression model, it is an oversimplification
(generally not a good idea) to interpret the coefficient of one
predictor as telling you about the effect of a change of one unit in
that predictor when the other predictors are held fixed.  In general,
that coefficient tells us about the effect of changing that predictor,
adjusting for simultaneous change in the other predictors (in the data
at hand).

David Hoaglin

> I also tried David's suggestion and changed the model. When adding the "new MINUS old" covariate and the "old" covariate in the same model, I get a positive significant coefficient for the "new MINUS old" covariate. However, in this model the "old" coefficient also turns positive and significant. David says that my covariate of interest is the "new MINUS old". This covariate operates as expected (postive and significant). But what to make of the "old" coefficient that is now significant? How to interpret this when "old" has also been subtracted from my main covariate of interest?

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