Re: st: Interpretation of interaction term in log linear (non linear) model

[David Hoaglin <dchoaglin@gmail.com>]

Dear Suryadipta,

It may be helpful to focus, initially, on the fitted model in the log scale.

The definition of the coefficient of "dummy" includes the list of
other predictors in the model (constant, x1, and dummy*x1).  Also,
when you interpret the coefficient of "dummy", you should mention that
it summarizes the effect of "dummy" on log(Trade) after adjusting for
simultaneous linear change in x1 and dummy*x1.  If the interpretation
of the coefficient of "dummy" does not mention those adjustments, it
gives the impression that the coefficient summarizes the change in
log(Trade) corresponding to an increase of 1 unit (i.e., 1 SD) in x1
when the other predictors are held constant.  In your model that
oversimplified interpretation is misleading, because one cannot change
"dummy" and hold dummy*x1 constant.  More generally, the "held
constant" interpretation does not reflect the way multiple regression
works.

The presence of the interaction term implies that the model makes
separate adjustments for the contribution of x1 in the two groups
defined by "dummy".  It also implies (as you mentioned) that the
effect of "dummy" depends on the value of x1.  It is easiest to
calculate that effect when x1 = 0.  That may be an appropriate
starting point, but you should also show the mean of x1 when "dummy" =
0 and the mean of x1 when "dummy" = 1 (and look at the relation
between the ranges of x1 in the two groups).  Centering the variable
underlying x1 is likely to be a good idea, but the case for dividing
by its standard deviation is less clear.

This discussion should clarify the interpretation and provide a basis
for translating it to the original scale of the data.  It applies also
if you use quasi-likelihood, conveniently available in the glm/poisson
framework.  If you want to work in terms of elasticities, please check
that any derivatives involved do not (inappropriately) assume that the
other predictors can be held constant.

David Hoaglin

On Sat, Jun 8, 2013 at 12:12 PM, Suryadipta Roy <sroy2138@gmail.com> wrote:
> Dear Statalisters,
>
> I was wondering if some one would be kind enough to clarify if I am on
> the right track in clarifying the coefficient of the interaction term
> when the dependent variable is in logarithm. The estimated model is of
> the form: log(Trade) = constant + 0.15dummy - 0.15x1 + 0.12dummy*x1,
> where dummy is (0,1) categorical variable, x1 is a continuous variable
> (standardized 0 - 1), and dummy*x1 is the interaction term. The result
> has been obtained from a fixed effects panel regression using -areg-
> with robust standard error option, and all the variables are
> statistically significant. Based on readings of Maarten's Stata tip
> 87: Interpretation of interactions in non-linear model, several
> Statalist postings, and the following link
> http://www.stanford.edu/~mrosenfe/soc_388_notes/soc_388_2002/Interpreting%20the%20coefficients%20of%20loglinear%20models.pdf
> , I wanted to make sure if any of the following interpretation of the
> above result is correct:
>
> 1. The coefficient of "dummy" indicates that this category (dummy
> variable = 1) has 16% (= exp(0.15) - 1) more of "Trade" compared to
> the base category (dummy variable = 0).
> 2. The effect of being in this category on "Trade" increases when the
> value of x1 increases. For every standard deviation increase in x1,
> the effect of "dummy" increases by about 13% (exp(0.12) - 1), OR there
> is a statistically significant 13% increase in "Trade" to countries
> having more of x1 relative to countries that have one standard
> deviation lower value of x1, OR the effect of being in the "dummy = 1"
> category in a country with one standard deviation more of x1 than
> average is exp(0.12)*exp(0.15) = 1.31, which means that "dummy=1"
> category has about 31% more "Trade" than "dummy=0" category.
>
> Following suggestions elsewhere in the Statalist, I have pursued other
> non-linear estimation strategies (and have asked questions to that
> effect earlier), but there is a tradition in this literature to use
> log-linear models. Any suggestion is greatly appreciated.
>
> Sincerely,
> Suryadipta Roy.
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