Marginal effects: interpret as percentage or percentage points.

[Elin Vimefall]

Dear all

I do have a question about how to interpret the marginal effects in a model with one (or several) binary dependent variable (For example I'm using a probit and a multinominal logit).

If the marigal effect is 0.07, do I write this: the probability to be in school (for example) increase with 7 percentage, or is it more correct to write 7 percenatge points?


Best regards /ElinVimefall

[Maarten Buis]

Marginal effects are additive approximations of effects in non-additive models, so a marginal effect of 0.07 corresponds to a 7 percentage point increase.

Odds ratios, Relative Risk Ratios, Incidence Rate Ratios, Hazard Ratios are multiplicative effects, so a * ratio of 1.07 corresponds to an increase of 7 percent.

[Sebastian Kripfganz]

The marginal effect tells you by how many units the probability to be in school changes if the explanatory variable changes by one unit. A probability is measured in percent (here divided by 100) and its units are percentage points. Thus, your second suggestion is correct.

[Elin Vimefall]

Thanks a lot!

[Elin Vimefall]

So is the correct way to write it that they are about 7 percentage points less likely to be in school?

[Maarten Buis]

The marginal effect is positive, so they are more likely. If the marginal effect was -0.07 then it would be less likely.

[Elin Vimefall]

Off course! Sorry for my sloppy mistake and thanks again!

[Yaara Nussi]

Hi,
I am dealing with a similar issue and found this post very helpful.
I would like to present a followup for discussion.

I ran an IV Probit regression using CMP for a binary y. I am using Stata 13.0 MP.

I used . margins to calculate the marginal effect of the constant:

Delta-method
Margin Std. Err. z P>z [95% Conf. Interval]

_cons .6531714 .2749499 2.38 0.018 .1142796 1.192063

Would the correct interpretation of the constant coefficient be that the probability of y=1 increases by 65 percentage points?

I also used . margins dydx(*) predict(pr) to calculate the marginal effect of my explanatory variables:

		Delta-method
	dy/dx	Std. Err.	z	P>z	[95% Conf.	Interval]
age	0.0108648	0.002687	4.04	0	0.005599	0.016131
working	-0.1372241	0.070705	-1.94	0.052	-0.2758	0.001356

Would the correct interpretation of the age coefficient be that the probability of y=1 increases by 1 percentage point for each year of age?
Would the correct interpretation of the working coefficient be that the probability of y=1 decreases by 13 percentage points when working?

I tried referring to the Stata help section for Margins (http://www.stata.com/manuals13/rmargins.pdf) and was able to determine this calculates the Average marginal effects (average probabilities of outcome) and not the predicted probabilities evaluated at the mean of the covariates as would be had I used the -atmens- option, but was unable to incorporate this into reporting my results.

Any insight is appreciated.
Many thanks,
Yaara

[Clyde Schechter]

Would the correct interpretation of the age coefficient be that the probability of y=1 increases by 1 percentage point for each year of age?
Would the correct interpretation of the working coefficient be that the probability of y=1 decreases by 13 percentage points when working?

Pretty much yes. I would rephrase these in terms that make it clear that there is no implication of causality. But otherwise, this is correct.

[Yaara Nussi]

Could you suggest such an alternative phrasing?

Would the AME in fact be the following difference in probabilities: Pr( y=1|x=1)-Pr(y=1|x=0)?
Thinking about it this way made the interpretation as percentage points more intuitive for working (as well as any other binary regressors).

If this reasoning is correct, would the same principle apply to continuous variables such as age?
In this case would the AME calculated by margins be the average of all the differences in probability of y=1 for the different ages?
For ages ranging from 25 to 75 AME=(Avg {Pr(y=1|x=26)-Pr(y=1|x=25),...., Pr(y=1|x=75)-Pr(y=1|x=74)}

Would the constant coefficient have any meaning in this case? If all the variables were binary I understand the interpretation would be the probability of y=1 when all Xs are at the baseline level of 0 (not working, female and so on), but when I incorporate age into the mix age=0 has no meaning (especially since the minimum age is 25)

Thank you again,
Yaara

[Clyde Schechter]

Could you suggest such an alternative phrasing?

The expected difference in probability of y=1 associated with working is a 13 percentage point decrease.
The average marginal effect on probability of y = 1 associated with a one year difference in age is a 1% increase.


For working, which I take it is a dichotomous variable, the marginal effect is calculated by predicting the outcome probability for each observation substituting working = 1, and then again for each observation substituting 0. The sample average of the difference between those outcome probabilities is then shown as the average marginal effect.

For age, which I take it is a continuous variable, it is somewhat different. Using the form of the regression equation, Stata calculates the partial derivative of the probability of the outcome with respect to age for each observation in the data set, using the observed values of all variables. The sample average of that is then reported as the average marginal effect.

Your interpretation of the constant term is correct. If you want a meaningful constant term in your model, you could redo the analysis after centering age at some interesting value that is within the range of observed values.

By the way, everything said here presumes that the model does not contain any interaction terms involving age or working. If that is wrong, then everything changes.

[Yaara Nussi]

Hi,
Would the change in phrasing (and subsequent interpretation as PP or percent) stem from the binary vs. continuous nature of the variable used?
The model has no interaction terms at this point.
Many thanks,
Yaara

[Clyde Schechter]

OK, I see how this is confusing because there are several things going on at once, and I also made an error in one place.

The expected difference in probability of y=1 associated with working is a 13 percentage point decrease.
The average marginal effect on probability of y = 1 associated with a one year difference in age is a 1 percentage point increase.

I have corrected the error in the above: the change is marked in red. My apologies for the mistake.

Principles and explanations:

1. A difference between two percentages is always measured in percentage points. A ratio of percentages may, if multiplied by 100, be measured as a percentage.
For example, if I am comparing rate (of something) of 3% with a rate of 2%, the difference is 1 percentage point, and the ratio is 1.5, or 150 percent. This is true regardless of the metric properties or meaning of the variable: it applies without exception.

2. When calculating marginal effects for discrete variables, -margins- calculates the actual difference between the predicted value at n+1 and the predicted value at n. It is important to remember that -margins- will only know that the variable is discrete if it is prefixed with an i. in the regression equation (or, confusingly, if it appears with no prefix in an interaction term). If the variable bears no prefix and is not part of an interaction term, Stata assumes it is a continuous variable--Stata does not examine the actual values of the variable to make this distinction. For variables Stata interprets as continuous, marginal effects are calculated as the partial derivative of the predicted outcome with respect to that variable.

3. The phrase "associated with" is a non-causal way of saying that "when we observe this, we expect to observe that."

[natalia malancu]

hi guys!

i've been confused for a while about the following, and although this thread is not an exact match, i'll ask my questions here instead of starting yet another thread.
in a mincerian equation, same as in any ols, coefficients are marginal effects ( is is average marginal or at the average? i always thought average marginal). often times the coefficient is interpreted as a per cent change in average wages. From here my confusion... shouldnt it be a percentage point change? what am I missing?

I even went back to wooldridge's econometrics intro, where in part I, regression analysis with cross-section data, he give the following example:

log(price)= 9.23-0.718 log(nox)+0.306rooms

followed by the interpretation:

when nox increases by 1% price fall by 0.718%, holding rooms fix.
when rooms increases by 1 price increases by approximately 100(0.306) =30.6%

best,
natalia

[lucho sanchez]

The expected difference in probability of y=1 associated with working is a 13 percentage point decrease.
The average marginal effect on probability of y = 1 associated with a one year difference in age is a 1% increase.


For working, which I take it is a dichotomous variable, the marginal effect is calculated by predicting the outcome probability for each observation substituting working = 1, and then again for each observation substituting 0. The sample average of the difference between those outcome probabilities is then shown as the average marginal effect.

For age, which I take it is a continuous variable, it is somewhat different. Using the form of the regression equation, Stata calculates the partial derivative of the probability of the outcome with respect to age for each observation in the data set, using the observed values of all variables. The sample average of that is then reported as the average marginal effect.

Your interpretation of the constant term is correct. If you want a meaningful constant term in your model, you could redo the analysis after centering age at some interesting value that is within the range of observed values.

By the way, everything said here presumes that the model does not contain any interaction terms involving age or working. If that is wrong, then everything changes.

In most books (for example Gujarati) they do not explain the unit change of the X ones. Why is this? Thanks a lot.