Average Marginal Effects Versus Effects at the Means

Two Different Summaries of a Nonlinear Effect

A marginal effect in a binary choice model depends on the covariate values at which it is evaluated. A logit or probit coefficient therefore does not, by itself, give one constant change in outcome probability for a one-unit change in a covariate. Once pointwise marginal effects are available, a natural question is how to summarize them across individuals. Two common answers are to average the individual effects or to evaluate the effect at the average covariate profile. These calculations sound similar, but generally answer different questions.

The distinction matters when the covariates vary substantially across the target population or when the probability model is strongly nonlinear. An average marginal effect describes the average model-implied local effect across the covariate profiles being considered. A marginal effect at the mean describes the local effect at one constructed profile whose covariates equal their means. That profile need not correspond to any actual individual.

Definitions and Model Setup

Let \(Y_i\) be a binary outcome and let \(x_i\) be an individual’s covariate vector. In a logit or probit model, the conditional probability of the outcome is specified as

where \(\beta\) is a parameter vector and \(G\) is the logistic cumulative distribution function for logit or the standard normal cumulative distribution function \(\Phi\) for probit. For a continuously varying covariate \(x_{ij}\), the pointwise marginal effect is the derivative of this conditional probability with respect to that covariate, holding the other covariates fixed. Writing \(g=G'\), the derivative is

This derivative has units of probability change per one-unit change in \(x_{ij}\). It is local: it describes the slope at the specified covariate profile, not necessarily a finite change over a full unit for a large change in \(x_{ij}\). For logit, \(g(\eta)=\Lambda(\eta)[1-\Lambda(\eta)]\), where \(\Lambda\) is the logistic cumulative distribution function. For probit, \(g(\eta)=\phi(\eta)\), the standard normal density.

These definitions use estimates \(\hat{\beta}\) from a fitted model. The same distinction applies to population quantities: an AME averages effects over the target population’s covariate distribution, whereas an effect at the population mean evaluates the derivative at that distribution’s mean covariate vector. The sample formulas use each observed profile with equal weight; other target populations or sampling designs may call for different weights.

The model setup assumes a correctly specified conditional probability of the stated form and a differentiable link function. A derivative-based effect also requires the covariate to be meaningfully treated as continuous over the range being interpreted. These are model-implied associations unless a separate causal design and its assumptions justify a causal interpretation. The calculations below illustrate model-based probability slopes; they do not establish causal effects.

Worked Calculations

Why Averaging and Evaluating Do Not Usually Commute

The linear index has a useful property: with an intercept and the same observations and weights, the index at the mean covariate vector equals the mean of the individual indices. In symbols, \(\bar{x}'\hat{\beta}\) equals the average of \(x_i'\hat{\beta}\). But a marginal effect also contains \(g(x_i'\hat{\beta})\), and \(g\) is nonlinear in logit and probit models. In general, applying a nonlinear function to a mean does not give the mean of the transformed values.

A short Taylor expansion makes the source of the difference precise. Let \(\eta_i=x_i'\hat{\beta}\), let \(\bar{\eta}\) be their average, and expand \(g(\eta_i)\) around \(\bar{\eta}\). To second order,

Averaging this approximation over \(i\), the first-order term vanishes because the deviations from the mean sum to zero. Thus, when the approximation is adequate,

The second term depends on both curvature and the spread of the indices. This explains why differences can be small when indices cluster tightly and more noticeable when profiles span regions with different slopes or curvature. It does not give a universal direction for the difference: the curvature can change sign, and the effects also include the coefficient \(\hat{\beta}_j\). The examples show that one cannot assume either the AME or the MEM must be larger.

Choosing and Interpreting the Summary

The AME is usually the relevant summary when the question concerns the average model-implied local effect across a specified population or sample. Its value depends on which covariate profiles are included and how they are weighted. The MEM instead answers a narrower question: what is the local marginal effect at the profile formed by setting each covariate to its mean? It can be useful as a concise description of the slope at that profile, but should not be described as an average effect.

Both summaries are model-based and are expressed in outcome-probability units per unit of the continuous covariate. A value of \(0.05\), for example, means a local change of about 0.05 in predicted probability per one-unit increase in the covariate at the profile or profiles being summarized. It does not mean a five-percentage-point change for every individual over any arbitrary finite increase. Nor does either summary by itself establish a causal effect.

A derivative is generally not the appropriate way to summarize a binary regressor: there is no meaningful infinitesimal change from zero to one. Its probability effect is instead described by comparing predicted probabilities at the two regressor values. The interpretation must match the covariate’s scale and support.

Common Mistake