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## Marginal means, adjusted predictions, and marginal effects

Stata 11 does margins. Does estimated marginal means. Does least-squares means. Does average and conditional marginal/partial effects, as derivatives or elasticities. Does average and conditional adjusted predictions. Does predictive margins. Does more. Margins are statistics calculated from predictions of a previously fit model at fixed values of some covariates and averaging or otherwise integrating over the remaining covariates.

If that sounds overly technical, try this. **margins** answers the
question, “What does my model have to say about such-and-such a group or
such-and-such a person”, where such-and-such might be:

- my estimation sample or another sample
- a sample with the values of some covariates fixed
- a sample evaluated at each level of a treatment
- a population represented by a complex survey sample
- someone who looks like the fifth person in my sample
- someone who looks like the mean of the covariates in my sample
- someone who looks like the median of the covariates in my sample
- someone who looks like the 25th percentile of the covariates in my sample
- someone who looks like some other statistics of the covariates in my sample
- a standardized population
- a balanced experimental design
- any combination of the above
- any comparison of the above

It answers these questions either conditionally—based on fixed values of covariates—or averaged over the observations in a sample. Any sample.

It answers these questions about any prediction or any other response you can calculate as a function of your estimated parameters—linear responses, probabilities, hazards, survival times, odds ratios, risk differences, etc.

It answers these questions in terms of the response given covariate levels, or in terms of the change in the response for a change in levels, a.k.a. marginal effects.

It answers these questions providing standard errors, test statistics, and confidence intervals and those statistics can take the covariates as given or adjust for sampling, a.k.a predictive margins and survey statistics.

Say that we are interested in the outcome **y** based on a person’s gender and
packages of cigarettes smoked per day. Using Stata’s new factor-variable
notation, we can fit a logistic regression by typing

The interaction between **sex** and **smokes** makes interpretation difficult.
We can use **margins** to decipher their effects:

We obtain predictive margins. If the distribution of the cigarettes smoked remains the same in the population, but everyone were male, we would expect about 38% to have a positive outcome for y. If everyone were female; 54%. If instead the distribution of males and females were as observed but no one smoked, we would expect about 41% to have a positive outcome.

Is there a significant difference in the probability of a positive outcome
between males and females? We can run tests after **margins** to find out:

We find evidence that the predicted margins for males and females differ.

Let's see an example of marginal effects. Because of Stata 11’s new factor-variable features, we can get average partial and marginal effects for age even when age enters as a polynomial:

We are using different data than before. The probability that a person is in
a union increases by 0.0015 as age increases by one year. By default,
**margins** reports average marginal (partial) effects, which means effects
are calculated for each observation in the data and then averaged.

Alternatively, if we wanted effects at the average of the covariates, we could type

. margins, dydx(age) atmeans

Stata 11’s **margins** command includes options to control whether the
standard errors reflect just the sampling variation of the estimated
coefficients or whether they also reflect the sampling variation of the
estimation sample. In the latter case, **margins** can account for complex
survey sampling including weights, sampling units, pre- and
poststratification, and subpopulations.

**margins** works after EVERY Stata estimation command except exact logistic
and exact Poisson; alternative-specific conditional logistic,
alternative-specific multinomial probit, and alternative-specific rank-ordered
probit; nested logit; generalized method of moments; and structural vector
autoregressive models.