Marginal Effects for Model Objects

margins is an effort to port the marginal effect estimation functionality of Stata's (closed source) margins command to R. This is done through a single function, margins(), which is an S3 generic method for calculating the marginal effects (or "partial effects") of covariates included in model objects (like those of classes "lm" and "glm").

Underlying functions marginal_effects() and dydx() provide low-level interfaces to the numerical differentiation, and the package implements several useful additional features for summarizing model objects, including:

• A plot() method for the new "margins" class that ports Stata's marginsplot command.
• A persp() method for "lm", "glm", and "loess" objects to provide three-dimensional representations of response surfaces.
• An image() method for the same that produces flat, two-dimensional heatmap-style representations of response surfaces.
• A plotting function cplot() to provide the commonly needed visual summaries of predictions or marginal effects conditional on a second variable.

Users interested in generating predicted (fitted) values, such as the "predictive margins" generated by Stata's margins command, should consider using prediction() from the prediction package.

Motivation

With the introduction of Stata's margins command, it has become incredibly simple to estimate average marginal effects (i.e., "average partial effects") and marginal effects at representative cases. Indeed, in just a few lines of Stata code, regression results for almost any kind model can be transformed into meaningful quantities of interest and related plots:

. import delimited mtcars.csv
. quietly reg mpg c.cyl##c.hp wt
. margins, dydx(*)
------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         cyl |   .0381376   .5998897     0.06   0.950    -1.192735     1.26901
          hp |  -.0463187    .014516    -3.19   0.004     -.076103   -.0165343
          wt |  -3.119815    .661322    -4.72   0.000    -4.476736   -1.762894
------------------------------------------------------------------------------
. marginsplot

Stata's margins command is incredibly robust. It works with nearly any kind of statistical model and estimation procedure, including OLS, generalized linear models, panel regression models, and so forth. It also represents a significant improvement over Stata's previous marginal effects command - mfx - which was subject to various well-known bugs. While other Stata modules have provided functionality for deriving quantities of interest from regression estimates (e.g., Clarify), none has done so with the simplicity and genearlity of margins.

By comparison, R has no robust functionality in the base tools for drawing out marginal effects from model estimates (though the S3 predict() methods implement some of the functionality for computing fitted/predicted values). Nor do any add-on packages implement appropriate marginal effect estimates. Notably, several packages provide estimates of marginal effects for different types of models. Among these are car, alr3, mfx, erer, among others. Unfortunately, none of these packages implement marginal effects correctly (i.e., correctly account for interrelated variables such as interaction terms (e.g., a:b) or power terms (e.g., I(a^2)) and the packages all implement quite different interfaces for different types of models. interplot and plotMElm provide functionality simply for plotting quantities of interest from multiplicative interaction terms in models but do not appear to support general marginal effects displays (in either tabular or graphical form), while visreg provides a more general plotting function but no tabular output. interactionTest provides some additional useful functionality for controlling the false discovery rate when making such plots and interpretations, but is again not a general tool for marginal effect estimation.

Given the challenges of interpreting the contribution of a given regressor in any model that includes quadratic terms, multiplicative interactions, a non-linear transformation, or other complexities, there is a clear need for a simple, consistent way to estimate marginal effects for popular statistical models. This package aims to correctly calculate marginal effects that include complex terms and provide a uniform interface for doing those calculations. Thus, the package implements a single S3 generic method (margins()) that can be easily generalized for any type of model implemented in R.

Some technical details of the package are worth briefly noting. The estimation of marginal effects relies on numerical approximations of derivatives produced using predict() (actually, a wrapper around predict() called prediction() that is type-safe). Variance estimation, by default is provided using the delta method a numerical approximation of the Jacobian matrix. While symbolic differentiation of some models (e.g., basic linear models) is possible using D() and deriv(), R's modelling language (the "formula" class) is sufficiently general to enable the construction of model formulae that contain terms that fall outside of R's symbolic differentiation rule table (e.g., y ~ factor(x) or y ~ I(FUN(x)) for any arbitrary FUN()). By relying on numeric differentiation, margins() supports any model that can be expressed in R formula syntax. Even Stata's margins command is limited in its ability to handle variable transformations (e.g., including x and log(x) as predictors) and quadratic terms (e.g., x^3); these scenarios are easily expressed in an R formula and easily handled, correctly, by margins().

Simple code examples

Replicating Stata's results is incredibly simple using just the margins() method to obtain average marginal effects:

library("margins")
x <- lm(mpg ~ cyl * hp + wt, data = mtcars)
(m <- margins(x))

## Average marginal effects

## lm(formula = mpg ~ cyl * hp + wt, data = mtcars)

##      cyl       hp    wt
##  0.03814 -0.04632 -3.12

summary(m)

##  factor     AME     SE       z      p   lower   upper
##     cyl  0.0381 0.5999  0.0636 0.9493 -1.1376  1.2139
##      hp -0.0463 0.0145 -3.1909 0.0014 -0.0748 -0.0179
##      wt -3.1198 0.6613 -4.7176 0.0000 -4.4160 -1.8236

With the exception of differences in rounding, the above results match identically what Stata's margins command produces. Using the plot() method also yields an aesthetically similar result to Stata's marginsplot:

plot(m)

If you are only interested in obtaining the marginal effects (without corresponding variances or the overhead of creating a "margins" object), you can call marginal_effects(x) directly. Furthermore, the dydx() function enables the calculation of the marginal effect of a single named variable:

# all marginal effects, as a data.frame
head(marginal_effects(x))

##     dydx_cyl     dydx_hp   dydx_wt
## 1 -0.6572244 -0.04987248 -3.119815
## 2 -0.6572244 -0.04987248 -3.119815
## 3 -0.9794364 -0.08777977 -3.119815
## 4 -0.6572244 -0.04987248 -3.119815
## 5  0.5747624 -0.01196519 -3.119815
## 6 -0.7519926 -0.04987248 -3.119815

# marginal effect of one variable
head(dydx(mtcars, x, "hp"))

##       dydx_hp
## 1 -0.04987248
## 2 -0.04987248
## 3 -0.08777977
## 4 -0.04987248
## 5 -0.01196519
## 6 -0.04987248

These functions may be useful, for example, for plotting, or getting a quick impression of the results.

While there is still work to be done to improve performance, margins is reasonably speedy:

library("microbenchmark")
microbenchmark(marginal_effects(x))

## Unit: milliseconds
##                 expr      min       lq     mean   median       uq      max neval
##  marginal_effects(x) 11.67015 11.98566 13.39177 12.31466 13.59553 63.25713   100

microbenchmark(margins(x))

## Unit: milliseconds
##        expr      min       lq     mean   median       uq      max neval
##  margins(x) 88.40953 91.17482 95.45244 92.99871 96.37431 183.5197   100

In addition to the estimation procedures and plot() generic, margins offers several plotting methods for model objects. First, there is a new generic cplot() that displays predictions or marginal effects (from an "lm" or "glm" model) of a variable conditional across values of third variable (or itself). For example, here is a graph of predicted probabilities from a logit model:

m <- glm(am ~ wt*drat, data = mtcars, family = binomial)
cplot(m, x = "wt", se.type = "shade")

And fitted values with a factor independent variable:

cplot(lm(Sepal.Length ~ Species, data = iris))

and a graph of the effect of drat across levels of wt:

cplot(m, x = "wt", dx = "drat", what = "effect", se.type = "shade")

## Warning in warn_for_weights(model): 'weights' used in model estimation are currently ignored!

Second, the package implements methods for "lm" and "glm" class objects for the persp() generic plotting function. This enables three-dimensional representations of predicted outcomes:

persp(x, xvar = "cyl", yvar = "hp")

and marginal effects:

persp(x, xvar = "cyl", yvar = "hp", what = "effect", nx = 10)

And if three-dimensional plots aren't your thing, there are also analogous methods for the image() generic, to produce heatmap-style representations:

image(x, xvar = "cyl", yvar = "hp", main = "Predicted Fuel Efficiency,\nby Cylinders and Horsepower")

The numerous package vignettes and help files contain extensive documentation and examples of all package functionality.

Requirements and Installation

The development version of this package can be installed directly from GitHub using ghit:

if (!require("ghit")) {
    install.packages("ghit")
    library("ghit")
}
install_github("leeper/prediction")
install_github("leeper/margins")

# building vignettes takes a moment, so for a quicker install set:
install_github("leeper/margins", build_vignettes = FALSE)