Compute and plot adjusted predictions, contrasts, marginal effects, and marginal means for 69 classes of statistical models in R. Conduct linear and non-linear hypothesis tests using the delta method.

## Definitions

The marginaleffects package allows R users to compute and plot four principal quantities of interest for 69 different classes of models:

### What does “marginal” mean?

One confusing aspect of the definitions above is that they use the word “marginal” in two different and opposite ways:

1. In “marginal effects,” we refer to the effect of a tiny (marginal) change in the regressor on the outcome. This is a slope, or derivative.
2. In “marginal means,” we refer to the process of marginalizing across rows of a prediction grid. This is an average, or integral.

Another potential confusion arises when some analysts use “marginal” to distinguish some estimates from “conditional” ones. As noted in the marginal effects and the contrasts vignettes, slopes and contrasts often vary from individual to individual, based on the values of all the regressors in the model. When we estimate a slope or a contrast for a specific combination of predictors – for one (possibly representative) individual – some people will call this a “conditional” estimate. When we compute the average of several individual-level estimates, some people will call this a “marginal” estimate.

On this website and in this package, we will reserve the expression “marginal effect” to mean a “slope” or “derivative”. When we take the average unit-level estimates, we will call this an “average marginal effect.”

This is all very confusing, but the terminology is so widespread and inconsistent that we must press on…

## Motivation

To calculate marginal effects we need to take derivatives of the regression equation. This can be challenging to do manually, especially when our models are non-linear, or when regressors are transformed or interacted. Computing the variance of a marginal effect is even more difficult.

The marginaleffects package hopes to do most of this hard work for you.

Many R packages advertise their ability to compute “marginal effects.” However, most of them do not actually compute marginal effects as defined above. Instead, they compute “adjusted predictions” for different regressor values, or differences in adjusted predictions (i.e., “contrasts”). The rare packages that actually compute marginal effects are typically limited in the model types they support, and in the range of transformations they allow (interactions, polynomials, etc.).

The main packages in the R ecosystem to compute marginal effects are the trailblazing and powerful margins by Thomas J. Leeper, and emmeans by Russell V. Lenth and contributors. The marginaleffects package is essentially a clone of margins, with some additional features from emmeans.

So why did I write a clone?

• Powerful: Marginal effects and contrasts can be computed for 69 different classes of models. Adjusted predictions and marginal means can be computed for about 100 model types.
• Extensible: Adding support for new models is very easy, often requiring less than 10 lines of new code. Please submit feature requests on Github.
• Fast: Computing unit-level standard errors can be orders of magnitude faster than margins in large datasets.
• Efficient: Much smaller memory footprint.
• Valid: When possible, numerical results are checked against alternative software like Stata, or other R packages.
• Beautiful: ggplot2 support for plotting (conditional) marginal effects and adjusted predictions.
• Tidy: The results produced by marginaleffects follow “tidy” principles. They are easy to program with and feed to other packages like modelsummary.
• Simple: All functions share a simple, unified, and well-documented interface.
• Thin: The package requires relatively few dependencies.
• Safe: User input is checked extensively before computation. When needed, functions fail gracefully with informative error messages.
• Active development

Downsides of marginaleffects include:

• No multiplicity adjustments. (Use p.adjust() instead.)
• Marginal means are often slower to compute than with emmeans.
• No omnibus test

## Getting started

#### Installation

You can install the released version of marginaleffects from CRAN:

install.packages("marginaleffects")

You can install the development version of marginaleffects from Github:

remotes::install_github("vincentarelbundock/marginaleffects")

First, we estimate a linear regression model with multiplicative interactions:

library(marginaleffects)

mod <- lm(mpg ~ hp * wt * am, data = mtcars)

An “adjusted prediction” is the outcome predicted by a model for some combination of the regressors’ values, such as their observed values, their means, or factor levels (a.k.a. “reference grid”).

By default, the predictions() function returns adjusted predictions for every value in original dataset:

predictions(mod) |> head()
#>   rowid     type predicted std.error statistic       p.value conf.low conf.high
#> 1     1 response  22.48857 0.8841487  25.43528 1.027254e-142 20.66378  24.31336
#> 2     2 response  20.80186 1.1942050  17.41900  5.920119e-68 18.33714  23.26658
#> 3     3 response  25.26465 0.7085307  35.65781 1.783452e-278 23.80232  26.72699
#> 4     4 response  20.25549 0.7044641  28.75305 8.296026e-182 18.80155  21.70943
#> 5     5 response  16.99782 0.7118658  23.87784 5.205109e-126 15.52860  18.46704
#> 6     6 response  19.66353 0.8753226  22.46433 9.270636e-112 17.85696  21.47011
#>    mpg  hp    wt am
#> 1 21.0 110 2.620  1
#> 2 21.0 110 2.875  1
#> 3 22.8  93 2.320  1
#> 4 21.4 110 3.215  0
#> 5 18.7 175 3.440  0
#> 6 18.1 105 3.460  0

The datagrid function gives us a powerful way to define a grid of predictors. All the variables not mentioned explicitly in datagrid() are fixed to their mean or mode:

predictions(mod, newdata = datagrid(am = 0, wt = seq(2, 3, .2)))
#>   rowid     type predicted std.error statistic       p.value conf.low conf.high
#> 1     1 response  21.95621 2.0386301  10.77008  4.765935e-27 17.74868  26.16373
#> 2     2 response  21.42097 1.7699036  12.10290  1.019401e-33 17.76807  25.07388
#> 3     3 response  20.88574 1.5067373  13.86157  1.082834e-43 17.77599  23.99549
#> 4     4 response  20.35051 1.2526403  16.24609  2.380723e-59 17.76518  22.93583
#> 5     5 response  19.81527 1.0144509  19.53301  5.755097e-85 17.72155  21.90900
#> 6     6 response  19.28004 0.8063905  23.90906 2.465206e-126 17.61573  20.94435
#>         hp am  wt mpg
#> 1 146.6875  0 2.0  21
#> 2 146.6875  0 2.2  21
#> 3 146.6875  0 2.4  21
#> 4 146.6875  0 2.6  21
#> 5 146.6875  0 2.8  21
#> 6 146.6875  0 3.0  21

We can plot how predictions change for different values of one or more variables – Conditional Adjusted Predictions – using the plot_cap function:

plot_cap(mod, condition = c("hp", "wt"))

mod2 <- lm(mpg ~ factor(cyl), data = mtcars)
plot_cap(mod2, condition = "cyl")

The Adjusted Predictions vignette shows how to use the predictions() and plot_cap() functions to compute a wide variety of quantities of interest:

• Adjusted Predictions at User-Specified Values (aka Predictions at Representative Values)
• Adjusted Predictions at the Mean
• Average Predictions at the Mean
• Conditional Predictions

#### Contrasts

A contrast is the difference between two adjusted predictions, calculated for meaningfully different predictor values (e.g., College graduates vs. Others).

What happens to the predicted outcome when a numeric predictor increases by one unit, and logical variable flips from FALSE to TRUE, and a factor variable shifts from baseline?

titanic <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/Stat2Data/Titanic.csv")
titanic$Woman <- titanic$Sex == "female"
mod3 <- glm(Survived ~ Woman + Age * PClass, data = titanic, family = binomial)

cmp <- comparisons(mod3)
summary(cmp)
#>     Term     Contrast   Effect Std. Error z value   Pr(>|z|)     2.5 %
#> 1  Woman TRUE - FALSE  0.50329   0.031654  15.899 < 2.22e-16  0.441244
#> 2    Age           +1 -0.00558   0.001084  -5.147 2.6471e-07 -0.007705
#> 3 PClass    2nd - 1st -0.22603   0.043546  -5.191 2.0950e-07 -0.311383
#> 4 PClass    3rd - 1st -0.38397   0.041845  -9.176 < 2.22e-16 -0.465985
#>      97.5 %
#> 1  0.565327
#> 2 -0.003455
#> 3 -0.140686
#> 4 -0.301957
#>
#> Model type:  glm
#> Prediction type:  response

The contrast above used a simple difference between adjusted predictions. We can also used different functions to combine and contrast predictions in different ways. For instance, researchers often compute Adjusted Risk Ratios, which are ratios of predicted probabilities. We can compute such ratios by applying a transformation using the transform_pre argument. We can also present the results of “interactions” between contrasts. What happens to the ratio of predicted probabilities for survival when PClass changes between each pair of factor levels (“pairwise”) and Age changes by 2 standard deviations simultaneously:

cmp <- comparisons(
mod3,
transform_pre = "ratio",
variables = list(Age = "2sd", PClass = "pairwise"))
summary(cmp)
#>                   Age    PClass Effect Std. Error z value   Pr(>|z|)  2.5 %
#> 1 (x - sd) / (x + sd) 1st / 1st 0.7043    0.05946  11.846 < 2.22e-16 0.5878
#> 2 (x - sd) / (x + sd) 2nd / 1st 0.3185    0.05566   5.723 1.0442e-08 0.2095
#> 3 (x - sd) / (x + sd) 3rd / 1st 0.2604    0.05308   4.907 9.2681e-07 0.1564
#> 4 (x - sd) / (x + sd) 2nd / 2nd 0.3926    0.08101   4.846 1.2588e-06 0.2338
#> 5 (x - sd) / (x + sd) 3rd / 2nd 0.3162    0.07023   4.503 6.7096e-06 0.1786
#> 6 (x - sd) / (x + sd) 3rd / 3rd 0.7053    0.20273   3.479 0.00050342 0.3079
#>   97.5 %
#> 1 0.8209
#> 2 0.4276
#> 3 0.3645
#> 4 0.5514
#> 5 0.4539
#> 6 1.1026
#>
#> Model type:  glm
#> Prediction type:  response

The code above is explained in detail in the vignette on Transformations and Custom Contrasts.

The Contrasts vignette shows how to use the comparisons() function to compute a wide variety of quantities of interest:

• Custom comparisons for:
• Numeric variables (e.g., 1 standard deviation, interquartile range, custom values)
• Factor or character
• Logical
• Contrast interactions
• Unit-level Contrasts
• Average Contrasts
• Group-Average Contrasts
• Contrasts at the Mean
• Contrasts Between Marginal Means

#### Marginal effects

A “marginal effect” is a partial derivative (slope) of the regression equation with respect to a regressor of interest. It is unit-specific measure of association between a change in a regressor and a change in the regressand. The marginaleffects() function uses numerical derivatives to estimate the slope of the regression equation with respect to each of the variables in the model (or contrasts for categorical variables).

By default, marginaleffects() estimates the slope for each row of the original dataset that was used to fit the model:

mfx <- marginaleffects(mod)

#>   rowid     type term        dydx  std.error statistic     p.value    conf.low
#> 1     1 response   hp -0.03690556 0.01850172 -1.994710 0.046074551 -0.07316825
#> 2     2 response   hp -0.02868936 0.01562861 -1.835695 0.066402771 -0.05932087
#> 3     3 response   hp -0.04657166 0.02258715 -2.061866 0.039220507 -0.09084166
#> 4     4 response   hp -0.04227128 0.01328278 -3.182412 0.001460541 -0.06830506
#>       conf.high predicted predicted_hi predicted_lo  mpg  hp    wt am    eps
#> 1 -0.0006428553  22.48857     22.48752     22.48857 21.0 110 2.620  1 0.0283
#> 2  0.0019421508  20.80186     20.80105     20.80186 21.0 110 2.875  1 0.0283
#> 3 -0.0023016728  25.26465     25.26333     25.26465 22.8  93 2.320  1 0.0283
#> 4 -0.0162375066  20.25549     20.25430     20.25549 21.4 110 3.215  0 0.0283

The function summary calculates the “Average Marginal Effect,” that is, the average of all unit-specific marginal effects:

summary(mfx)
#>   Term   Effect Std. Error  z value   Pr(>|z|)    2.5 %   97.5 %
#> 1   hp -0.03807    0.01279 -2.97725 0.00290848 -0.06314 -0.01301
#> 2   wt -3.93909    1.08596 -3.62728 0.00028642 -6.06754 -1.81065
#> 3   am -0.04811    1.85260 -0.02597 0.97928234 -3.67913  3.58292
#>
#> Model type:  lm
#> Prediction type:  response

The plot_cme plots “Conditional Marginal Effects,” that is, the marginal effects estimated at different values of a regressor (often an interaction):

plot_cme(mod, effect = "hp", condition = c("wt", "am"))

The Marginal Effects vignette shows how to use the marginaleffects() function to compute a wide variety of quantities of interest:

• Unit-level Marginal Effects
• Average Marginal Effects
• Group-Average Marginal Effects
• Marginal Effects at the Mean
• Marginal Effects Between Marginal Means
• Conditional Marginal Effects
• Tables and Plots

#### Marginal means

Marginal Means are the adjusted predictions of a model, averaged across a “reference grid” of categorical predictors. To compute marginal means, we first need to make sure that the categorical variables of our model are coded as such in the dataset:

dat <- mtcars
dat$am <- as.logical(dat$am)
dat$cyl <- as.factor(dat$cyl)

Then, we estimate the model and call the marginalmeans function:

mod <- lm(mpg ~ am + cyl + hp, data = dat)
mm <- marginalmeans(mod)
summary(mm)
#>   Term Value  Mean Std. Error z value   Pr(>|z|) 2.5 % 97.5 %
#> 1   am FALSE 18.32     0.7854   23.33 < 2.22e-16 16.78  19.86
#> 2   am  TRUE 22.48     0.8343   26.94 < 2.22e-16 20.84  24.11
#> 3  cyl     4 22.88     1.3566   16.87 < 2.22e-16 20.23  25.54
#> 4  cyl     6 18.96     1.0729   17.67 < 2.22e-16 16.86  21.06
#> 5  cyl     8 19.35     1.3771   14.05 < 2.22e-16 16.65  22.05
#>
#> Model type:  lm
#> Prediction type:  response
#> Results averaged over levels of: am, cyl

The Marginal Means vignette offers more detail.

#### More

There is much more you can do with marginaleffects. Return to the Table of Contents to read the vignettes, learn how to report marginal effects and means in nice tables with the modelsummary package, how to define your own prediction “grid”, and much more.