Re: st: Re: interpreting marginal effects of fractional logit with continuous independent variables

[David Hoaglin <dchoaglin@gmail.com>]

Sandra,

Thanks for the additional information.

For a proportion p in a sample of n, the estimated variance is p(1 -
p)/n.  I haven't looked into whether a fractional logit analysis takes
the p(1 - p) part into account.  If the n's vary substantially, the
analysis should take them into account; but you can't because you
don't have them.

My comment about the region of "predictor space" covered by the data
aims at avoiding predictions where you have little or no data.  In
some situations, setting each covariate at its mean could produce a
point that is not supported by the data.  The challenge is to
determine the regions of "predictor space" that are adequately covered
by the data.

Even if your data are not troublesome in that way, when you assess the
effect of ple, I think one of the approaches holds llti_stand at its
mean (and similarly for the effect of llti_stand).  I don't recall a
discussion of the association between ple and llti_stand.  If they
have substantial correlation, statements about the effect of each need
to acknowledge that relation.

Underlying this comment is the need to move away (as far as possible)
from interpretations of coefficients in regression and similar models
that involve the idea that the other variables are "held constant."
That interpretation does not reflect the way regression works.  The
appropriate general interpretation is that the coefficient of a
particular predictor tells how the dependent variable changes per unit
increase in that predictor after adjusting for simultaneous linear
change in the other predictors in the data at hand.  I realize that
numerous textbooks present the "held constant" interpretation, but
that does not establish its validity.

By using the combinations of values that the covariates actually have
in the data, Richard Williams's Average Marginal Effect avoids making
assumptions about holding those variables constant.

David Hoaglin

On Tue, Nov 19, 2013 at 7:10 AM, Sandra Virgo <Sandra.Virgo@lshtm.ac.uk> wrote:
>
> To David Hoaglin:
>
> Hello David - thanks so much for your help.
>
> I get what you mean about all cases not having equal weight as
> denominators and denominators might vary a lot.
>
> I think I understand what you mean about the 'region of predictor
> space' - presumably you're asking how much of the variation in ple and
> llti_stand is actually present when I hold covariates at their means?
>
> I am happy to ignore the 'all else held at means' assumption, and
> therefore I won't try to calculate the Marginal Effects at the Mean as I
> also know the limitations of this.
>
> However, I'd still be interested in calculating Average Marginal
> Effects (i.e. with covariates taking on the actual values they have in
> my data) in the way that Richard Williams describes.
>
> And I'm guessing that it would still be OK from what you say to
> calculate predicted probabilities (fitted values/adjusted predictions to
> use other terminology) for specific scenarios. I would also like to
> calculate marginal effects at representative values (i.e. with
> covariates at their existing values apart from any I choose to fix at a
> series of values in order to explore 'interactions') as well as
> exploring the effects of discrete changes in my continuous variable.
>
> Hopefully these analyses will be OK from what you say.
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