Marginal means are adjusted predictions, averaged across a grid of categorical predictors, holding other numeric predictors at their means. To learn more, read the marginal means vignette, visit the package website, or scroll down this page for a full list of vignettes:

## Usage

marginal_means(
model,
variables = NULL,
newdata = NULL,
vcov = TRUE,
conf_level = 0.95,
type = NULL,
transform = NULL,
cross = FALSE,
hypothesis = NULL,
equivalence = NULL,
df = Inf,
wts = "equal",
by = NULL,
...
)

## Arguments

model

Model object

variables

Focal variables

• Character vector of variable names: compute marginal means for each category of the listed variables.

• NULL: calculate marginal means for all logical, character, or factor variables in the dataset used to fit model. Hint: Set cross=TRUE to compute marginal means for combinations of focal variables.

newdata

Grid of predictor values over which we marginalize.

• NULL create a grid with all combinations of all categorical predictors in the model. Warning: can be expensive.

• Character vector: subset of categorical variables to use when building the balanced grid of predictors. Other variables are held to their mean or mode.

• Data frame: A data frame which includes all the predictors in the original model. The full dataset is replicated once for every combination of the focal variables in the variables argument, using the datagridcf() function.

vcov

Type of uncertainty estimates to report (e.g., for robust standard errors). Acceptable values:

• FALSE: Do not compute standard errors. This can speed up computation considerably.

• TRUE: Unit-level standard errors using the default vcov(model) variance-covariance matrix.

• String which indicates the kind of uncertainty estimates to return.

• Heteroskedasticity-consistent: "HC", "HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", "HC5". See ?sandwich::vcovHC

• Heteroskedasticity and autocorrelation consistent: "HAC"

• Mixed-Models degrees of freedom: "satterthwaite", "kenward-roger"

• Other: "NeweyWest", "KernHAC", "OPG". See the sandwich package documentation.

• One-sided formula which indicates the name of cluster variables (e.g., ~unit_id). This formula is passed to the cluster argument of the sandwich::vcovCL function.

• Square covariance matrix

• Function which returns a covariance matrix (e.g., stats::vcov(model))

conf_level

numeric value between 0 and 1. Confidence level to use to build a confidence interval.

type

string indicates the type (scale) of the predictions used to compute marginal effects or contrasts. This can differ based on the model type, but will typically be a string such as: "response", "link", "probs", or "zero". When an unsupported string is entered, the model-specific list of acceptable values is returned in an error message. When type is NULL, the default value is used. This default is the first model-related row in the marginaleffects:::type_dictionary dataframe. If type is NULL and the default value is "response", the function tries to compute marginal means on the link scale before backtransforming them using the inverse link function.

transform

A function applied to unit-level adjusted predictions and confidence intervals just before the function returns results. For bayesian models, this function is applied to individual draws from the posterior distribution, before computing summaries.

cross

TRUE or FALSE

• FALSE (default): Marginal means are computed for each predictor individually.

• TRUE: Marginal means are computed for each combination of predictors specified in the variables argument.

hypothesis

specify a hypothesis test or custom contrast using a numeric value, vector, or matrix, a string, or a string formula.

• Numeric:

• Single value: the null hypothesis used in the computation of Z and p (before applying transform).

• Vector: Weights to compute a linear combination of (custom contrast between) estimates. Length equal to the number of rows generated by the same function call, but without the hypothesis argument.

• Matrix: Each column is a vector of weights, as describe above, used to compute a distinct linear combination of (contrast between) estimates. The column names of the matrix are used as labels in the output.

• String formula to specify linear or non-linear hypothesis tests. If the term column uniquely identifies rows, terms can be used in the formula. Otherwise, use b1, b2, etc. to identify the position of each parameter. Examples:

• hp = drat

• hp + drat = 12

• b1 + b2 + b3 = 0

• String:

• "pairwise": pairwise differences between estimates in each row.

• "reference": differences between the estimates in each row and the estimate in the first row.

• "sequential": difference between an estimate and the estimate in the next row.

• "revpairwise", "revreference", "revsequential": inverse of the corresponding hypotheses, as described above.

• See the Examples section below and the vignette: https://vincentarelbundock.github.io/marginaleffects/articles/hypothesis.html

equivalence

Numeric vector of length 2: bounds used for the two-one-sided test (TOST) of equivalence, and for the non-inferiority and non-superiority tests. See Details section below.

Adjust p-values for multiple comparisons: "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", or "fdr". See stats::p.adjust

df

Degrees of freedom used to compute p values and confidence intervals. A single numeric value between 1 and Inf. When df is Inf, the normal distribution is used. When df is finite, the t distribution is used. See insight::get_df for a convenient function to extract degrees of freedom. Ex: slopes(model, df = insight::get_df(model))

wts

character value. Weights to use in the averaging.

• "equal": each combination of variables in newdata gets equal weight.

• "cells": each combination of values for the variables in the newdata gets a weight proportional to its frequency in the original data.

• "proportional": each combination of values for the variables in newdata -- except for those in the variables argument -- gets a weight proportional to its frequency in the original data.

by

Collapse marginal means into categories. Data frame with a by column of group labels, and merging columns shared by newdata or the data frame produced by calling the same function without the by argument.

...

Additional arguments are passed to the predict() method supplied by the modeling package.These arguments are particularly useful for mixed-effects or bayesian models (see the online vignettes on the marginaleffects website). Available arguments can vary from model to model, depending on the range of supported arguments by each modeling package. See the "Model-Specific Arguments" section of the ?marginaleffects documentation for a non-exhaustive list of available arguments.

## Value

Data frame of marginal means with one row per variable-value combination.

## Details

This function begins by calling the predictions function to obtain a grid of predictors, and adjusted predictions for each cell. The grid includes all combinations of the categorical variables listed in the variables and newdata arguments, or all combinations of the categorical variables used to fit the model if newdata is NULL. In the prediction grid, numeric variables are held at their means.

After constructing the grid and filling the grid with adjusted predictions, marginal_means computes marginal means for the variables listed in the variables argument, by average across all categories in the grid.

marginal_means can only compute standard errors for linear models, or for predictions on the link scale, that is, with the type argument set to "link".

The marginaleffects website compares the output of this function to the popular emmeans package, which provides similar but more advanced functionality: https://vincentarelbundock.github.io/marginaleffects/

## Standard errors using the delta method

Standard errors for all quantities estimated by marginaleffects can be obtained via the delta method. This requires differentiating a function with respect to the coefficients in the model using a finite difference approach. In some models, the delta method standard errors can be sensitive to various aspects of the numeric differentiation strategy, including the step size. By default, the step size is set to 1e-8, or to 1e-4 times the smallest absolute model coefficient, whichever is largest.

marginaleffects can delegate numeric differentiation to the numDeriv package, which allows more flexibility. To do this, users can pass arguments to the numDeriv::jacobian function through a global option. For example:

• options(marginaleffects_numDeriv = list(method = "simple", method.args = list(eps = 1e-6)))

• options(marginaleffects_numDeriv = list(method = "Richardson", method.args = list(eps = 1e-5)))

• options(marginaleffects_numDeriv = NULL)

See the "Standard Errors and Confidence Intervals" vignette on the marginaleffects website for more details on the computation of standard errors:

https://vincentarelbundock.github.io/marginaleffects/articles/uncertainty.html

Note that the inferences() function can be used to compute uncertainty estimates using a bootstrap or simulation-based inference. See the vignette:

https://vincentarelbundock.github.io/marginaleffects/articles/bootstrap.html

## Model-Specific Arguments

Some model types allow model-specific arguments to modify the nature of marginal effects, predictions, marginal means, and contrasts. Please report other package-specific predict() arguments on Github so we can add them to the table below.

https://github.com/vincentarelbundock/marginaleffects/issues

 Package Class Argument Documentation brms brmsfit ndraws brms::posterior_predict re_formula brms::posterior_predict lme4 merMod re.form lme4::predict.merMod allow.new.levels lme4::predict.merMod glmmTMB glmmTMB re.form glmmTMB::predict.glmmTMB allow.new.levels glmmTMB::predict.glmmTMB zitype glmmTMB::predict.glmmTMB mgcv bam exclude mgcv::predict.bam robustlmm rlmerMod re.form robustlmm::predict.rlmerMod allow.new.levels robustlmm::predict.rlmerMod MCMCglmm MCMCglmm ndraws

## Bayesian posterior summaries

By default, credible intervals in bayesian models are built as equal-tailed intervals. This can be changed to a highest density interval by setting a global option:

options("marginaleffects_posterior_interval" = "eti")

options("marginaleffects_posterior_interval" = "hdi")

By default, the center of the posterior distribution in bayesian models is identified by the median. Users can use a different summary function by setting a global option:

options("marginaleffects_posterior_center" = "mean")

options("marginaleffects_posterior_center" = "median")

When estimates are averaged using the by argument, the tidy() function, or the summary() function, the posterior distribution is marginalized twice over. First, we take the average across units but within each iteration of the MCMC chain, according to what the user requested in by argument or tidy()/summary() functions. Then, we identify the center of the resulting posterior using the function supplied to the "marginaleffects_posterior_center" option (the median by default).

## Equivalence, Inferiority, Superiority

$$\theta$$ is an estimate, $$\sigma_\theta$$ its estimated standard error, and $$[a, b]$$ are the bounds of the interval supplied to the equivalence argument.

Non-inferiority:

• $$H_0$$: $$\theta \leq a$$

• $$H_1$$: $$\theta > a$$

• $$t=(\theta - a)/\sigma_\theta$$

• p: Upper-tail probability

Non-superiority:

• $$H_0$$: $$\theta \geq b$$

• $$H_1$$: $$\theta < b$$

• $$t=(\theta - b)/\sigma_\theta$$

• p: Lower-tail probability

Equivalence: Two One-Sided Tests (TOST)

• p: Maximum of the non-inferiority and non-superiority p values.

Thanks to Russell V. Lenth for the excellent emmeans package and documentation which inspired this feature.

## Examples

library(marginaleffects)

# simple marginal means for each level of cyl
dat <- mtcars
dat$carb <- factor(dat$carb)
dat$cyl <- factor(dat$cyl)
dat$am <- as.logical(dat$am)
mod <- lm(mpg ~ carb + cyl + am, dat)

marginal_means(
mod,
variables = "cyl")
#>
#>  Term Value Mean Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>   cyl     4 23.1       1.66 13.9   <0.001  19.9   26.4
#>   cyl     6 20.4       1.34 15.2   <0.001  17.8   23.0
#>   cyl     8 16.2       1.07 15.1   <0.001  14.1   18.3
#>
#> Results averaged over levels of: carb, am, cyl
#> Columns: term, value, cyl, estimate, std.error, statistic, p.value, conf.low, conf.high
#>

# collapse levels of cyl by averaging
by <- data.frame(
cyl = c(4, 6, 8),
by = c("4 & 6", "4 & 6", "8"))
marginal_means(mod,
variables = "cyl",
by = by)
#>
#>     By Mean Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>  4 & 6 21.7       1.13 19.2   <0.001  19.5   24.0
#>  8     16.2       1.07 15.1   <0.001  14.1   18.3
#>
#> Results averaged over levels of: carb, am, cyl
#> Columns: by, estimate, std.error, statistic, p.value, conf.low, conf.high
#>

# pairwise differences between collapsed levels
marginal_means(mod,
variables = "cyl",
by = by,
hypothesis = "pairwise")
#>
#>       Term Mean Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>  4 & 6 - 8 5.54       1.51 3.66   <0.001  2.57    8.5
#>
#> Results averaged over levels of: carb, am, cyl
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high
#>

# cross
marginal_means(mod,
variables = c("cyl", "carb"),
cross = TRUE)
#>
#>  Mean Std. Error     z Pr(>|z|) 2.5 % 97.5 %
#>  25.8       1.26 20.43   <0.001 23.34   28.3
#>  25.6       1.17 21.93   <0.001 23.30   27.9
#>  25.3       2.37 10.71   <0.001 20.70   30.0
#>  21.9       1.90 11.51   <0.001 18.15   25.6
#>  20.3       3.77  5.39   <0.001 12.91   27.7
#>  19.8       3.81  5.18   <0.001 12.29   27.2
#>  23.1       1.77 13.08   <0.001 19.63   26.5
#>  22.9       1.87 12.24   <0.001 19.20   26.5
#>  22.6       2.37  9.56   <0.001 17.98   27.2
#>  19.1       1.34 14.31   <0.001 16.53   21.8
#>  17.6       3.00  5.85   <0.001 11.68   23.5
#>  17.0       3.48  4.89   <0.001 10.21   23.9
#>  18.9       1.94  9.74   <0.001 15.11   22.7
#>  18.7       1.57 11.90   <0.001 15.61   21.8
#>  18.4       1.83 10.07   <0.001 14.85   22.0
#>  15.0       1.20 12.53   <0.001 12.63   17.3
#>  13.4       3.36  3.99   <0.001  6.81   20.0
#>  12.9       3.00  4.28   <0.001  6.98   18.8
#>
#> Results averaged over levels of: am
#> Columns: cyl, carb, estimate, std.error, statistic, p.value, conf.low, conf.high
#>

# collapsed cross
by <- expand.grid(
cyl = unique(mtcars$cyl), carb = unique(mtcars$carb))
by$by <- ifelse( by$cyl == 4,
paste("Control:", by$carb), paste("Treatment:", by$carb))

# Convert numeric variables to categorical before fitting the model
dat <- mtcars
dat$am <- as.logical(dat$am)
dat$carb <- as.factor(dat$carb)
mod <- lm(mpg ~ hp + am + carb, data = dat)

# Compute and summarize marginal means
marginal_means(mod)
#>
#>  Term Value Mean Std. Error     z Pr(>|z|) 2.5 % 97.5 %
#>  carb 1     22.0      1.345 16.35   <0.001  19.4   24.6
#>  carb 2     21.5      1.025 20.95   <0.001  19.5   23.5
#>  carb 3     20.6      1.780 11.55   <0.001  17.1   24.0
#>  carb 4     18.8      1.042 18.06   <0.001  16.8   20.9
#>  carb 6     18.5      3.019  6.12   <0.001  12.6   24.4
#>  carb 8     21.6      4.055  5.33   <0.001  13.7   29.6
#>  am   FALSE 17.9      1.244 14.37   <0.001  15.4   20.3
#>  am   TRUE  23.1      0.974 23.72   <0.001  21.2   25.0
#>
#> Results averaged over levels of: hp, am, carb
#> Columns: term, value, estimate, std.error, statistic, p.value, conf.low, conf.high
#>

# Contrast between marginal means (carb2 - carb1), or "is the 1st marginal means equal to the 2nd?"
# see the vignette on "Hypothesis Tests and Custom Contrasts" on the marginaleffects website.
lc <- c(-1, 1, 0, 0, 0, 0)
marginal_means(mod, variables = "carb", hypothesis = "b2 = b1")
#>
#>   Term   Mean Std. Error      z Pr(>|z|) 2.5 % 97.5 %
#>  b2=b1 -0.514       1.48 -0.348    0.728 -3.41   2.38
#>
#> Results averaged over levels of: am, carb
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high
#>

marginal_means(mod, variables = "carb", hypothesis = lc)
#>
#>    Term   Mean Std. Error      z Pr(>|z|) 2.5 % 97.5 %
#>  custom -0.514       1.48 -0.348    0.728 -3.41   2.38
#>
#> Results averaged over levels of: am, carb
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high
#>

# Multiple custom contrasts
lc <- matrix(c(
-2, 1, 1, 0, -1, 1,
-1, 1, 0, 0, 0, 0
),
ncol = 2,
dimnames = list(NULL, c("A", "B")))
marginal_means(mod, variables = "carb", hypothesis = lc)
#>
#>  Term   Mean Std. Error      z Pr(>|z|)  2.5 % 97.5 %
#>     A  1.199       6.15  0.195    0.845 -10.85  13.25
#>     B -0.514       1.48 -0.348    0.728  -3.41   2.38
#>
#> Results averaged over levels of: am, carb
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high
#>