We are about to tell you that margins can make meaningful predictions in the presence of random effects, random coefficients, and latent variables. We are about to tell you that margins and Stata's predict now integrate over the unobserved effects. This is exciting. Here's why.
Making meaningful predictions can be difficult even in the absence of random effects or random coefficients. For instance, consider a model as simple as logistic regression. Assume you have a special interest in variable's x's effect on outcome. You fit a model. Based on that, you can now say the effect of a 1-unit increase in x is to decrease the log odds by 0.22. Even those of us familiar with logistic regression would probably respond, "Yes, and that means what?"
"That's easy", you reply, exponentiating -0.22 in your head. "If x increases by 1, the odds fall to 80% of what it otherwise would be".
To which a reasonable person could reasonably reply, "And what would the odds otherwise be?"
"Well, that depends on the patient's (worker's, student's, etc.) other characteristics".
"That's rather obvious. Perhaps you could tell me the increase in terms of probabilities?"
"Well, that's rather difficult. You see, probabilities are a nonlinear function of odds ratios, so the baseline probability enters into it, and really, it's very similar to the previous problem, with an additional complication".
Irritation shows on the face of our reasonable person. "Surely you can tell me something interpretable. How about you tell me the number of lives that would be saved in the data if we increased x by one over what it otherwise would be. Surely you can do that. Perhaps with a 95% confidence interval?"
You realize that you can do that, but that's not a calculation you are going to make in your head. Fortunately for you, margins can help; it will make the calculation, with standard errors, and with confidence intervals. So you agree to make the calculation to return later.
"We have 2,500 people in our data", you report. "If we increased x by 1, we would expect to save 102 lives, say, 70 to 140".
You just got the reasonable person's attention. Of course, you would have lost it if you had said that the expected value was around 0.5, somewhere between 0.2 and 0.7. That's why we make such calculations.
Now, let's complicate our model. Let's assume that the model does not include just x, it includes an interaction of x with age, and that while x is estimated to have a whopping effect, x multiplied by age has a negative effect.
We do not need to repeat the dialog, clearly estimating the effect of x on lives saved is going to be more difficult, and more important. Reporting just the coefficients or the odds ratios does not even reveal whether increasing x results in lives saved or lives lost.
margins can do that calculation, too, and in fact, it is no more difficult for margins than the first calculation. The statement that you would save 70 to 140 lives is now even more impressive.
Now, let's complicate our model one last time. Let's add random effects. Let's assume that our outcomes come from patients who have doctors who work in hospitals and that we have introduced random effects for those doctors and hospitals. And perhaps we have introduced a random coefficient as well, either on x or on some other variable.
Now, calculating the prediction is truly difficult. We want to predict an average effect, an effect as if patients had been assigned randomly to hospitals and doctors, and we have a nonlinear model, and that means we cannot ignore the variance of the random effects. Instead we will need to dust off our calculus tools and, for each patient, integrate over all unobserved variables.
Now, margins can do that, too.
And margins can do it not just for logistic regression and multilevel logistic regression but for
In addition, Stata's predict will also integrate over unobserved variables.
We have fictional hospital data on 2,500 patients treated in 20 hospitals by roughly 600 different doctors. We fit the following model,
. melogit outcome i.x c.age i.x#c.age other || hosp: || doctor: (output omitted)
which is how you specify in Stata that you want a multilevel logistic regression to be fit containing (indicator variable) x, (continuous variable) age, x*age, and other and that you want random effects for each hospital and for each doctor within hospital.
The omitted output reports that
The coefficient on x is outlandish, but we remind you that that sometimes happens when you include a variable and its interaction with age. The sum of effects in this case is reasonable.
The omitted output also reported large estimated variances of the random effects, namely, 8.33 and 11.64. We could tell a story about how we expected those large variances in the case of this treatment, but because the story is fictional, we will not bother.
In fact, in these fictional data, we intentionally made the variances large just so they would have a larger effect on the predicted number of lives saved.
Based on the above results, does treatment x save lives or lose them? We type
. margins r.x Contrasts of predictive margins Model VCE : OIM Expression : Marginal predicted mean, predict()
|df chi2 P>chi2|
|x||1 52.05 0.0000|
|Contrast Std. Err. [95% Conf. Interval]|
|(1 vs 0)||-.1745077 .0241884 -.2219161 -.1270992|
The output reports the difference in probabilities is -0.175; patients die less with x=1 instead of x=0, and the output reports the corresponding 95% confidence interval.
We have 2,500 patients in these data, so multiplying the difference in probabilities and changing the sign results in lives saved. We obtain
436 fewer deaths if all had x=1
with a 95% confidence interval of [318, 555].
The 436 results take into account the estimated coefficients for x and x*age along with the values of age and other in our data, along with the estimated random effects for doctor and hospital.
The confidence interval takes into account all the above plus the uncertainty because some of the ingredients were estimated rather than known.