Interpretation of elasticity with logit model

[sa sa]

Hi everyone,

I run a logit model to estimate the impact of hospital characteristics on the probability of having a cesarean delivery.
As covariates, i have the number of physicians per hospital that i used as log to estimate the elasticity but i have difficulty to interpret.
My results are:
Coef. Std. Err. z P>|z| [95% Conf. Interval]
log physicians -0.32 0.11 -2.94 0.003 -0.53 -0.11

May i have some help to interpret this result ?
Thanks in advance.

[Clyde Schechter]

That's the hard way to get elasticity. Re-run your model as -logit cesarean physicians ...- and then run -margins, eyex(physicians)-, which will give you the average elasticity. The key is to use physicians itself, not the log, and let Stata worry about the logarithms involved.

[John Mullahy]

Sa Sa: I suggest that you look at -help margins-. For example, if you were looking to compute the elasticity of Pr(cesarian) with respect to the number of physicians, you could either estimate your logit model followed by

margins, eydx(logphys)

when physicians is entered in log form, or followed by

margins, eyex(phys)

if physicians in entered untransformed in natural units.

A larger issue to consider is whether you want an elasticity — percent change in probability due to a percent change in physicians — or a semi-elasticity — change in probability due to a percent change in physicians. This is up to you to decide, but will determine whether to use dyex, eyex, or eydx in -margins-, depending on how you specify your physicians variable.

[John Mullahy]

Sa Sa: Clyde's explanation is a better-articulated version of what I was suggesting.

[sa sa]

Thanks a lot.
With margins, eyex(physicians), i have a similar finding.
And how i can interpret my result ?
a 10% increase in physicians staff could reduce the rate of cesarean deliveries by 3 (or 0.3 ?) percentage points.

[Clyde Schechter]

How do you suppose someone can validate your interpretation of results that you do not show? Show the output and then somebody can tell you whether your interpretation is correct.

[sa sa]

Coef of the log physicians -0.32 with p-value 0.003.
Thanks!

[Clyde Schechter]

This looks like a replay of the output in #1, which is essentially impossible to interpret as an elasticity. It almost certainly does not mean what you propose in #5--if that turns out to be correct it is purely a coincidence. Did you try what I suggested in #2? If so, show the exact output from that. (Meaning copy/paste directly from the Results window or your log file and paste it here in the Forum editor between code delimiters.)

[sa sa]

I was pretty sure that my first result from logit cesarean log(physicians) can be intrepreted as an elasticity, the coef of the log physicians is the elasticity for the odds of a cesarean with respect to the number of physicians. I don't understand why it is not possible ? Can u help me to understand please...

[Clyde Schechter]

Yes, it can be interpreted as the elasticity of the odds. But ordinarily one would be interested in the elasticity of the probability of the outcome, and for that purpose the coefficient is not useful.

If the elasticity of the odds is what you are interested in, then, we can reason as follows. I assume that you used natural logarithms to calculate log physicians. If the number of physicians increase by 1%, then log physicians will increase by log(1.01) which is, to a reasonable approximation 0.01. Then the log odds of the outcome will decrease by 0.32*.01 = 0.0032. This then corresponds to a decrease in outcome odds by a factor of exp(.0032) = 0.9968. So there is a roughly 0.32% decrease in the odds of the outcome.

You have asked about a 10% increase in number of physicians. In that case, log physicians goes up by 0.095 ( = log(1.1)). So log odds goes up by 0.32 * 0.095 = 0.0304. This then reduces the odds by a factor of 0.97, or a 3.0% reduction in the outcome odds.

[sa sa]

Thanks a lot Clyde for all these explanations. I better understand.
However, i have a little problem when i try margins, eyex(physicians), i get the error message:
could not calculate numerical derivatives -- discontinuous region with missing values
encountered
r(459);
In fact, i have previously run margins, dyex(physicians) and not margins, eyex(physicians).

. margins, dyex(physicians)

Average marginal effects Number of obs = 99728
Model VCE : Robust

Expression : Linear prediction, predict()
dy/ex w.r.t. : physicians

-------------------------------------------------------------------------------
| Delta-method
| dy/ex Std. Err. z P>|z| [95% Conf. Interval]
--------------+----------------------------------------------------------------
physicians | -.3204434 .0443224 -5.65 0.000 -.3373137 -.1635732
-------------------------------------------------------------------------------

Since the dependent variable is in log, i wonder if we should use margins eyex ? dyex might be sufficient to estimate the elasticity (with logit models, we always estimate a semielasticity which we can interpret as elasticity) ?

[sa sa]

What de you think Clyde? thanks a lot...

[Clyde Schechter]

The semi-elasticity you calculated with -margins, dyex(physicians)- is the same as the regression coefficient (at least to several decimal places) here. (If there were a lot of other variables in the model, depending on their distributions, they might not be.) The outcome variable in a logistic regression is, of course, a log transform of the odds. So you can interpret that as elasticity of the odds of the outcome, and if that is what you are interested in, then this way of doing it is fine. In fact, it is better than doing it from the regression coefficient if there are multiple predictor variables in the logistic model.

I do not know why you encountered that difficulty with -margins, eyex()-. It must be something in your data that causes numerical problems with the calculation, and I don't know what the problem might be. In general, this approach works and it gives the elasticity of the outcome probability.

[sa sa]

I need also the elasticity of the probability in order to estimate the resulting change in the cesarean rate, so i'm also interested in margins, eyex().
I check my data, i have no missing value.
Any idea...

[Clyde Schechter]

I'm afraid not. Perhaps somebody else knows and will chime in.

Just one last minute thought. Try running the logit regression with log_physicians as the predictor and then do -margins, eydx(log_physicians)-. That is, conceptually, equivalent to eyex() with physicians as the predictor. My instinct tells me that you will run into the same inability to calculate numerical derivatives, but you might get lucky.