Introverts are special people. Trust me. I know. I am one. Now, not all introverts are the same. However, one thing that nearly all introverts have in common is the desire to avoid, or at least minimize, interactions. We can only take so much.
I was thinking about interactions this week. Not social ones, but rather econometric ones. Not only did I happen to cover this in class this week, but then I came across this
tweet by @StatModeling. It led to a
blog post mentioning a recent
publication on Covid-19 vaccine effectiveness. I am not interested in this article
per se for the purposes of this post. However, one of the study's authors posted this as an introduction to the blog post
Logistic regression can mean different things. It can mean a regression of the log odds ratio on covariates using Ordinary Least Squares (OLS). It can also, perhaps, refer to a logit model when the outcome is binary (since the odds ratio is undefined in this case). Since the study above refers to vaccine efficacy, I took it to mean a logit binary choice model for getting Covid or not. And, that's when I took note of what is written above.
Regardless of what is meant in the quote above (this is not a post about the authors or their study), I decided to return from my blog sabbatical and speak about interactions. In particular, the fact that in a logit binary choice model interactions are always present.
As such, it is simply false to assert that a logit model does not allow for interactions. This is why the logit model -- and, indeed, all nonlinear models -- discriminate against introverts. They force you into situations you may not like. Or, they may know what you need better than you do.
Take the logit and probit models. The data-generating process (DGP) is
Pr(y=1 | x) = F(xb)
where x is a vector of covariates, including an intercept, and b is the corresponding parameter vector. If F() is the logistic cumulative distribution function (CDF), then this is the logit model. If it is the standard normal CDF, then this is the probit model.
Now, because of the nonlinearity of the F() function, b is hard to directly interpret. When x changes, it changes the value of the index xb, which changes Pr(y=1|x). But, how much this probability changes for a given change in xb depends on the nonlinear function F().
Because of this, we discuss marginal effects, average marginal effects, and average partial effects.
The marginal effect is the change in Pr(y|x) when a particular covariate, say x_k, increases by one unit. This is given by
marginal effect of x_k = f(xb)*b_k
where f() is the derivative of F(), which is the logistic or standard normal probability density function (PDF). Notice, f() is evaluated at the index value, xb. This includes the entire x vector. This means that the marginal effect of x_k varies with every covariate in the model with a non-zero coefficient. Specifically, the derivative of the marginal effect of x_k with respect to x_j is
∂ (marginal effect of x_k) / ∂ x_j = f'(xb)*b_k*b_j
where f'() is the derivative of f(). This is non-zero if b_j, b_k ≠ 0.
Of course, one can include interactions in the index, xb, as well. This just makes the cross-derivative more flexible. For example, if the interaction x_k * x_j is included in the model, the cross-derivative becomes
∂ (marginal effect of x_k) / ∂ x_j = f'(xb)*(b_k + b_ix*x_j)*b_j + f(xb)*b_ix
where
b_ix is the coefficient on the interaction term,
x_k *
x_j. The added flexibility, but not the presence of interactions in the marginal effects themselves, is illustrated in
Berry et al. (2010).
Makes you realize quite quickly how much more complex the nonlinearity in nonlinear models makes things. Not just econometrically, but also in terms of fitting complex behaviors. But, if you are an introvert, you probably won't be discussing this with anyone else over coffee any time soon.
