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Re: st: Clarification requested about the at() option of -margins-


From   Trevor Zink <tzink@bren.ucsb.edu>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Clarification requested about the at() option of -margins-
Date   Thu, 24 Oct 2013 09:01:27 -0700

Paul,

Thanks very much for your detailed answer. If I may ask a few follow-up questions to make sure I understand properly...

1) "-margins- converts log-odds (and their slopes) to probabilities (and their slopes) for us". I have spent a lot of time reading about -margins- over the past few weeks, and I can't recall ever hearing it explained like this. Is this really what -margins- does? Simply converts from log-odds back to probability? If so, that is great news--it makes the interpretation of the output much easier. 2) "Although the slope for the log-odds is fixed; that for the probability is not. As 0 and 1 are approached, the slope tends to 0, and the possible values and SE are also constrained". So what you're saying is the fact that the slope goes to 0 at the 0 and 1 isn't because of any extrapolation like I assumed, it's simply a product of mapping onto the logic function? 3) You used both -margins, at()- and -margins, dydx() at()-. My understanding of the difference after reading your answer is that -margins, at()- gives the /probability/ of Y==1 at the specified values of X. Whereas -margins, dydx() at()- gives the /change in the probability/ of Y==1 from an infintesimal change in X at the specified values of X. Correct?

(as a side note, I wouldn't have expected weight to predict foreign vs domestic as well as it does)

Thanks again for your answer.
Trevor


On 10/24/2013 3:13 AM, Seed, Paul wrote:
Dear Statalist,
Trevor Zink asks why -margins- does not behave as he would expect following
logistic regression.

The answer is found only by going back to exactly what logistic regression
actually does; and how it compares to linear regression.

Linear regression is carried out under an assumption of constant slope, and has no problem
therefore in estimating the slope at any value of the predictors.  With a single predictor,
the estimated slope does not change. (Point 1 of Stata output).

However, it is inappropriate for a binary outcome, as it can lead to estimated proportions beyond 0 and 1.
(Point 2).

Logistic regression solves this by working with the log-odds, rather than the probability.  There are no
impossible values.  Extreme log-odds correspond to probabilities close to 0 or 1.
-margins- converts log-odds (and their slopes) to probabilities (and their slopes) for us.  (Point 3)

Although the slope for the log-odds is fixed;  that for the probability is not.  As 0 and 1 are approached, the
slope tends to 0, and the possible values and SE are also constrained. (Point 4)

Plotting the estimated values against weight reveals this quite clearly. (Point 5)

The code below uses Trevor's example (amended and expanded).

************** Begin Stata code*******************
set more off
sysuse auto, clear

gen wt_tons = weight/2240
* Change units to make results easier to understand
summarize wt_tons
* maximum weight is 2.1607 tons

regress foreign wt_tons
margins, dydx(wt_tons) at(wt_tons=(0(0.2)2 20))
* Point 1

margins, at(wt_tons=(0(0.2)2 ))
* Point 2

logit foreign wt_tons
margins, at(wt_tons=(0(0.2)2 ))
* Point 3

margins, dydx(wt_tons) at(wt_tons=(0(0.2)2 ))
* Point 4

predict Foreign if foreign
predict USA if !foreign

label var Foreign Foreign
label var USA USA
label var wt_tons "Car weight (tons)"

gr7 Foreign USA wt_tons, xlab(0 1 2) ylab(0 .5 1.0) l1title("Estimated probability of car being foreign")
* Point 5
**************** End Stata code *************

Best wishes,

Paul T Seed, Senior Lecturer in Medical Statistics,
Division of Women's Health, King's College London
Women's Health Academic Centre, King's Health Partners
(+44) (0) 20 7188 3642.



Date: Wed, 23 Oct 2013 23:21:13 -0700
From: Trevor Zink <tzink@bren.ucsb.edu>
Subject: st: Clarification requested about the at() option of -margins-

Long-time lurker, first-time post. I couldn't find a good explanation in
the archives.

I'm confused about what, specifically, -margins- is doing with the at()
option, such that it can calculate margins for values of variable that
don't exist in the data. To articulate with an example:

sysuse auto
summarize weight //maximum weight is 4840
logit foreign weight  //nonsensical, but ok for the example
margins, dydx(weight) at(weight=(0(1000)10000 100000))

Here I ask for the slope of the function at a variety of weights from 0
to 10,000 and also 100,000. The maximum weight observed in the data is 4840.

My understanding of -margins- with at() was that it calculates the slope
of the function holding the specified variables constant at the
specified levels. But if the specified level doesn't appear in the data,
how can Stata determine what the slope is at this value? Ok, it's
clearly extrapolating, but based on what information? The only other
information included in the above model is a constant. When I try the
above but specifying the nocons option to -logit- Stata returns an
error, so it must be forecasting based on the constant; but specifically
how?

What's even more strange to me is that the standard errors *shrink* as
the estimates extend beyond the observed data. If Stata is forecasting
based on only the constant this seems counter-intuitive to me.

Thanks, and sorry if this is silly.

Trevor Zink

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--
Trevor Zink, MBA, MA
Ph.D. Candidate
UC Regents Special Fellow
Bren School of Environmental Science and Management
University of California, Santa Barbara
tzink@bren.ucsb.edu <mailto:tzink@bren.ucsb.edu>
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