The margins and prediction packages are a combined effort to port the functionality of Stata’s (closed source) margins command to (open source) R. These tools provide ways of obtaining common quantities of interest from regression-type models. margins provides “marginal effects” summaries of models and prediction provides unit-specific and sample average predictions from models. Marginal effects are partial derivatives of the regression equation with respect to each variable in the model for each unit in the data; average marginal effects are simply the mean of these unit-specific partial derivatives over some sample. In ordinary least squares regression with no interactions or higher-order term, the estimated slope coefficients are marginal effects. In other cases and for generalized linear models, the coefficients are not marginal effects at least not on the scale of the response variable. margins therefore provides ways of calculating the marginal effects of variables to make these models more interpretable.

The major functionality of Stata’s margins command - namely the estimation of marginal (or partial) effects - is provided here through a single function, margins(). This is an S3 generic method for calculating the marginal effects of covariates included in model objects (like those of classes “lm” and “glm”). Users interested in generating predicted (fitted) values, such as the “predictive margins” generated by Stata’s margins command, should consider using prediction() from the sibling project, prediction.


Stata’s margins command is very simple and intuitive to use:

. 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


With margins in R, replicating Stata’s results is incredibly simple using just the margins() method to obtain average marginal effects and its summary() method to obtain Stata-like output:

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
##  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. A slightly more concise expression relies on the syntactic sugar provided by margins_summary():

##  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

Using the plot() method also yields an aesthetically similar result to Stata’s marginsplot:


Using Optional Arguments in margins()

margins is intended as a port of (some of) the features of Stata’s margins command, which includes numerous options for calculating marginal effects at the mean values of a dataset (i.e., the marginal effects at the mean), an average of the marginal effects at each value of a dataset (i.e., the average marginal effect), marginal effects at representative values, and any of those operations on various subsets of a dataset. (The functionality of Stata’s command to produce predictive margins is not ported, as this is easily obtained from the prediction package.) In particular, Stata provides the following options:

  • at: calculate marginal effects at (potentially representative) specified values (i.e., replacing observed values with specified replacement values before calculating marginal effects)
  • atmeans: calculate marginal effects at the mean (MEMs) of a dataset rather than the default behavior of calculating average marginal effects (AMEs)
  • over: calculate marginal effects (including MEMs and/or AMEs at observed or specified values) on subsets of the original data (e.g., the marginal effect of a treatment separately for men and women)

The at argument has been translated into margins() in a very similar manner. It can be used by specifying a list of variable names and specified values for those variables at which to calculate marginal effects, such as margins(x, at = list(hp=150)). When using at, margins() constructs modified datasets - using build_datalist() - containing the specified values and calculates marginal effects on each modified dataset, rbind-ing them back into a single “margins” object.

Stata’s atmeans argument is not implemented in margins() for various reasons, including because it is possible to achieve the effect manually through an operation like data$var <- mean(data$var, na.rm = TRUE) and passing the modified data frame to margins(x, data = data).

At present, margins() does not implement the over option. The reason for this is also simple: R already makes data subsetting operations quite simple using simple [ extraction. If, for example, one wanted to calculate marginal effects on subsets of a data frame, those subsets can be passed directly to margins() via the data argument (as in a call to prediction()).

The rest of this vignette shows how to use at and data to obtain various kinds of marginal effects, and how to use plotting functions to visualize those inferences.

Average Marginal Effects and Average Partial Effects

We can start by loading the margins package:


We’ll use a simple example regression model based on the built-in mtcars dataset:

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

To obtain average marginal effects (AMEs), we simply call margins() on the model object created by lm():

## Average marginal effects
## lm(formula = mpg ~ cyl + hp * wt, data = mtcars)
##      cyl       hp     wt
##  -0.3652 -0.02527 -3.838

The result is a data frame with special class "margins". "margins" objects are printed in a tidy summary format, by default, as you can see above. The only difference between a "margins" object and a regular data frame are some additional data frame-level attributes that dictate how the object is printed.

The default method calculates marginal effects for all variables included in the model (ala Stata’s , dydx(*) option). To limit calculation to only a subset of variables, use the variables argument:

summary(margins(x, variables = "hp"))
##  factor     AME     SE       z      p   lower   upper
##      hp -0.0253 0.0105 -2.4046 0.0162 -0.0459 -0.0047

In an ordinary least squares regression, there is really only one way of examining marginal effects (that is, on the scale of the outcome variable). In a generalized linear model (e.g., logit), however, it is possible to examine true “marginal effects” (i.e., the marginal contribution of each variable on the scale of the linear predictor) or “partial effects” (i.e., the contribution of each variable on the outcome scale, conditional on the other variables involved in the link function transformation of the linear predictor). The latter are the default in margins(), which implicitly sets the argument margins(x, type = "response") and passes that through to prediction() methods. To obtain the former, simply set margins(x, type = "link"). There’s some debate about which of these is preferred and even what to call the two different quantities of interest. Regardless of all of that, here’s how you obtain either:

x <- glm(am ~ cyl + hp * wt, data = mtcars, family = binomial)
## Warning: fitted probabilities numerically 0 or 1 occurred
margins(x, type = "response") # the default
## Average marginal effects
## glm(formula = am ~ cyl + hp * wt, family = binomial, data = mtcars)
##      cyl       hp      wt
##  0.02156 0.002667 -0.5158
margins(x, type = "link")
## Average marginal effects
## glm(formula = am ~ cyl + hp * wt, family = binomial, data = mtcars)
##     cyl      hp     wt
##  0.5156 0.05151 -12.24

Note that some other packages available for R, as well as Stata’s margins and mfx packages enable calculation of so-called “marginal effects at means” (i.e., the marginal effect for a single observation that has covariate values equal to the means of the sample as a whole). The substantive interpretation of these is fairly ambiguous. While it was once common practice to estimate MEMs - rather than AMEs or MERs - this is now considered somewhat inappropriate because it focuses on cases that may not exist (e.g., the average of a 0/1 variable is not going to reflect a case that can actually exist in reality) and we are often interested in the effect of a variable at multiple possible values of covariates, rather than an arbitrarily selected case that is deemed “typical” in this way. As such, margins() defaults to reporting AMEs, unless modified by the at argument to calculate average “marginal effects for representative cases” (MERs). MEMs could be obtained by manually specifying at for every variable in a way that respects the variables classes and inherent meaning of the data, but that functionality is not demonstrated here.

Using the at Argument

The at argument allows you to calculate marginal effects at representative cases (sometimes “MERs”) or marginal effects at means - or any other statistic - (sometimes “MEMs”), which are marginal effects for particularly interesting (sets of) observations in a dataset. This differs from marginal effects on subsets of the original data (see the next section for a demonstration of that) in that it operates on a modified set of the full dataset wherein particular variables have been replaced by specified values. This is helpful because it allows for calculation of marginal effects for counterfactual datasets (e.g., what if all women were instead men? what if all democracies were instead autocracies? what if all foreign cars were instead domestic?).

As an example, if we wanted to know if the marginal effect of horsepower (hp) on fuel economy differed across different types of automobile transmissions, we could simply use at to obtain separate marginal effect estimates for our data as if every car observation were a manual versus if every car observation were an automatic. The output of margins() is a simplified summary of the estimated marginal effects across the requested variable levels/combinations specified in at:

x <- lm(mpg ~ cyl + wt + hp * am, data = mtcars)
margins(x, at = list(am = 0:1))
## Average marginal effects at specified values
## lm(formula = mpg ~ cyl + wt + hp * am, data = mtcars)
##  at(am)     cyl     wt        hp    am
##       0 -0.9339 -2.812 -0.008945 1.034
##       1 -0.9339 -2.812 -0.026392 1.034

Because of the hp * am interaction in the regression, the marginal effect of horsepower differs between the two sets of results. We can also specify more than one variable to at, creating a potentially long list of marginal effects results. For example, we can produce marginal effects at both levels of am and the values from the five-number summary (minimum, Q1, median, Q3, and maximum) of observed values of hp. This produces 2 * 5 = 10 sets of marginal effects estimates:

margins(x, at = list(am = 0:1, hp = fivenum(mtcars$hp)))
## Average marginal effects at specified values
## lm(formula = mpg ~ cyl + wt + hp * am, data = mtcars)
##  at(am) at(hp)     cyl     wt        hp      am
##       0     52 -0.9339 -2.812 -0.008945  2.6864
##       1     52 -0.9339 -2.812 -0.026392  2.6864
##       0     96 -0.9339 -2.812 -0.008945  1.9188
##       1     96 -0.9339 -2.812 -0.026392  1.9188
##       0    123 -0.9339 -2.812 -0.008945  1.4477
##       1    123 -0.9339 -2.812 -0.026392  1.4477
##       0    180 -0.9339 -2.812 -0.008945  0.4533
##       1    180 -0.9339 -2.812 -0.026392  0.4533
##       0    335 -0.9339 -2.812 -0.008945 -2.2509
##       1    335 -0.9339 -2.812 -0.026392 -2.2509

Because this is a linear model, the marginal effects of cyl and wt do not vary across levels of am or hp. The minimum and Q1 value of hp are also the same, so the marginal effects of am are the same in the first two results. As you can see, however, the marginal effect of hp differs when am == 0 versus am == 1 (first and second rows) and the marginal effect of am differs across levels of hp (e.g., between the first and third rows). As should be clear, the at argument is incredibly useful for getting a better grasp of the marginal effects of different covariates.

This becomes especially apparent when a model includes power-terms (or any other alternative functional form of a covariate). Consider, for example, the simple model of fuel economy as a function of weight, with weight included as both a first- and second-order term:

x <- lm(mpg ~ wt + I(wt^2), data = mtcars)
## Call:
## lm(formula = mpg ~ wt + I(wt^2), data = mtcars)
## Residuals:
##    Min     1Q Median     3Q    Max 
## -3.483 -1.998 -0.773  1.462  6.238 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  49.9308     4.2113  11.856 1.21e-12 ***
## wt          -13.3803     2.5140  -5.322 1.04e-05 ***
## I(wt^2)       1.1711     0.3594   3.258  0.00286 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Residual standard error: 2.651 on 29 degrees of freedom
## Multiple R-squared:  0.8191, Adjusted R-squared:  0.8066 
## F-statistic: 65.64 on 2 and 29 DF,  p-value: 1.715e-11

Looking only at the regression results table, it is actually quite difficult to understand the effect of wt on fuel economy because it requires performing mental multiplication and addition on all possible values of wt. Using the at option to margins, you could quickly obtain a sense of the average marginal effect of wt at a range of plausible values:

margins(x, at = list(wt = fivenum(mtcars$wt)))
## Average marginal effects at specified values
## lm(formula = mpg ~ wt + I(wt^2), data = mtcars)
##  at(wt)      wt
##   1.513 -9.8366
##   2.542 -7.4254
##   3.325 -5.5926
##   3.650 -4.8314
##   5.424 -0.6764

The marginal effects in the first column of results reveal that the average marginal effect of wt is large and negative except when wt is very large, in which case it has an effect not distinguishable from zero. We can easily plot these results using the cplot() function to see the effect visually in terms of either predicted fuel economy or the marginal effect of wt:

cplot(x, "wt", what = "prediction", main = "Predicted Fuel Economy, Given Weight")

cplot(x, "wt", what = "effect", main = "Average Marginal Effect of Weight")

A really nice feature of Stata’s margins command is that it handles factor variables gracefully. This functionality is difficult to emulate in R, but the margins() function does its best. Here we see the marginal effects of a simple regression that includes a factor variable:

x <- lm(mpg ~ factor(cyl) * hp + wt, data = mtcars)
## Average marginal effects
## lm(formula = mpg ~ factor(cyl) * hp + wt, data = mtcars)
##        hp    wt  cyl6   cyl8
##  -0.04475 -3.06 1.473 0.8909

margins() recognizes the factor and displays the marginal effect for each level of the factor separately. (Caveat: this may not work with certain at specifications, yet.)


Stata’s margins command includes an over() option, which allows you to very easily calculate marginal effects on subsets of the data (e.g., separately for men and women). This is useful in Stata because the program only allows one dataset in memory. Because R does not impose this restriction and further makes subsetting expressions very simple, that feature is not really useful and can be achieved using standard subsetting notation in R:

x <- lm(mpg ~ factor(cyl) * am + hp + wt, data = mtcars)
# automatic vehicles
margins(x, data = mtcars[mtcars$am == 0, ])
## Average marginal effects
## lm(formula = mpg ~ factor(cyl) * am + hp + wt, data = mtcars)
##     am       hp     wt   cyl6   cyl8
##  1.706 -0.03115 -2.441 -1.715 -1.586
# manual vehicles
margins(x, data = mtcars[mtcars$am == 1, ])
## Average marginal effects
## lm(formula = mpg ~ factor(cyl) * am + hp + wt, data = mtcars)
##     am       hp     wt   cyl6   cyl8
##  2.167 -0.03115 -2.441 -4.218 -2.656

Because a "margins" object is just a data frame, it is also possible to obtain the same result by subsetting the output of margins():

m <- margins(x)
split(m, m$am)
## $`0`
## Average marginal effects
## lm(formula = mpg ~ factor(cyl) * am + hp + wt, data = mtcars)
##     am       hp     wt   cyl6   cyl8
##  1.706 -0.03115 -2.441 -1.715 -1.586
## $`1`
## Average marginal effects
## lm(formula = mpg ~ factor(cyl) * am + hp + wt, data = mtcars)
##     am       hp     wt   cyl6   cyl8
##  2.167 -0.03115 -2.441 -4.218 -2.656

Marginal Effects Plots

Using margins() to calculate marginal effects enables several kinds of plotting. The built-in plot() method for objects of class "margins" creates simple diagnostic plots for examining the output of margins() in visual rather than tabular format. It is also possible to use the output of margins() to produce more typical marginal effects plots that show the marginal effect of one variable across levels of another variable. This section walks through the plot() method and then shows how to produce marginal effects plots using base graphics.

The plot() method for “margins” objects

The margins package implements a plot() method for objects of class "margins" (seen above). This produces a plot similar (in spirit) to the output of Stata’s marginsplot. It is highly customizable, but is meant primarily as a diagnostic tool to examine the results of margins(). It simply produces, by default, a plot of marginal effects along with 95% confidence intervals for those effects. The confidence level can be modified using the levels argument, which is vectorized to allow multiple levels to be specified simultaneously.

More advanced plotting

There are two common ways of visually representing the substantive results of a regression model: (1) fitted values plots, which display the fitted conditional mean outcome across levels of a covariate, and (2) marginal effects plots, which display the estimated marginal effect of a variable across levels of a covariate. This section discusses both approaches.

Fitted value plots can be created using cplot() (to provide conditional predicted value plots or conditional effect plots) and both the persp() method and image() method for "lm" objects, which display the same type of relationships in three-dimensions (i.e., across two conditioning covariates).

For example, we can use cplot() to quickly display the predicted fuel economy of a vehicle from a model:

x <- lm(mpg ~ cyl + wt * am, data = mtcars)
cplot(x, "cyl")

cplot(x, "wt")

The slopes of the predicted value lines are the marginal effect of wt when am == 0 and am == 1. We can obtain these slopes using margins() and specifying the at argument:

margins(x, at = list(am = 0:1))
## Average marginal effects at specified values
## lm(formula = mpg ~ cyl + wt * am, data = mtcars)
##  at(am)    cyl     wt     am
##       0 -1.181 -2.369 -1.566
##       1 -1.181 -6.566 -1.566

Another plotting function - the persp() method for “lm” objects - gives even more functionality:

persp(x, "cyl", "wt")

This three-dimensional representation can be easily rotated using the theta and phi arguments. If multiple values are specified for each, facetting is automatically provided:

persp(x, "cyl", "wt", theta = c(0, 90))