RE: st: probit with interaction dummies (significance and marginal effects)

 From "Stephen P. Jenkins" To Subject RE: st: probit with interaction dummies (significance and marginal effects) Date Tue, 29 Jul 2008 10:36:49 +0100

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Date: Mon, 28 Jul 2008 07:56:05 -0400
From: "Erasmo Giambona" <e.giambona@gmail.com>
Subject: Re: st: probit with interaction dummies (significance and
marginal effects)

This is a tremendous help. You are right. I was trying to have a one
number quick fix. This is because I have a panel of firms and
therefore I am using clogit to control for firm fixed effects. As far
as I understand, marginal effects are problematic with conditional
logit mainly because I would not be able to invoke the invlogit that
you suggest below to calculate predicted probabilities for x_1 and
x_2. I have noticed that people suggest using the sample average
predicted probability obtained with pc1 in this case. But then would
one be able to really calculate the change in the probability as x_1
changes to x_2?
Erasmo
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Yes, prediction after -clogit- is problematic for the reasons already
rehearsed on the list.

Your penultimate sentence appears to refer to the so-called "average
partial effect" rather than the "marginal effect" associated with a
particular covariate.  Read the big black book by Wooldridge
(Econometric Analysis of Cross Section and Panel Data) for more
details.

Returning to the probit discussed yesterday:
Pr(y=1|x, z, ...) = Normal(a + b*x + c*xz + ....)

For the APE associated with x, you would calculate /for every obs in
the sample/ the predicted probability implied by having x = x_1 and
leaving all other covariates at their sample observed values. Then
average these predictions.
And then repeat the exercise with, instead, x = x_2.
And then look at the difference between these averages. (Deriving
standard errors for the APE is harder.)

The same idea can of course be implemented with a logit model rather
than probit. -predictnl- is likely to be your friend in implementing
these calculations.

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Date: Mon, 28 Jul 2008 09:21:59 -0700
From: "Lachenbruch, Peter" <Peter.Lachenbruch@oregonstate.edu>
Subject: RE: st: probit with interaction dummies (significance and
marginal effects)

I think what we all were saying is that interaction represents the
difference in the Odds Ratios at different levels of the predictor
variables.  This is certainly true for logistic regression and true to
a very good approximation for probit regression.

The term 'effect modification' is common in epidemiology, probably
less so in sociology and economics.
===================================================

I can confirm that in my corner of economics  'effect modification' is
a term rarely heard. I think Peter's message set out very clearly what
it meant in the context of the logit model and odds ratio.

However, Peter's message gives me the opportunity to sound off against
the use of odds ratios in the context of interaction effects, as we
have been discussing.

In the ME discussion we talked about deriving 2 predicted
probabilities, call them P1 and P2, with the ME = P2 - P1.  One could
of course instead or also look at P2/P1.  (I think this is what some
call a "relative risk" or "risk ratio".)    I just don't think that
the "odds ratio" methodology really works at all well when we've got
interactions.   Actually, I don't think they are much use without them
either (as Norton, Wang, Ai, put it in SJ 4-2, p. 159, "it is the
ratio of a ratio, and honestly, who understands that?".

Predicted probabilities, and related concepts such as MEs or APEs, are
straightforwardly interpretable, and generalize to the interaction
context. IMHO odds ratios don't work as well. If probabilities are
small, then they approximate risk ratios, but in that case, one might
as well calculate risk ratios from the very start.

NB My equations yesterday were intended to help /explain/ the concept
of the ME in terms of predicted probabilities. I should have said that
for the ME of a continuous RHS vble, there are analytical formulae for
the ME in a probit model (and in a logit model).  Norton, Wang and
Ai's article in the SJ uses those formulae, and the associated
-inteff- code implements the calculations based on these formulae.
(But you could derive them yourself; hence reference to -nlcom-.)

Stephen
-------------------------------------------------------------
Professor Stephen P. Jenkins <stephenj@essex.ac.uk>
Director, Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.
Tel: +44 1206 873374.  Fax: +44 1206 873151.
http://www.iser.essex.ac.uk
Survival Analysis using Stata:
http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/

Learn about the UK's new household panel survey, the United Kingdom
Household Longitudinal Study: http://www.iser.essex.ac.uk/ukhls/

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