Marginal Means

In the context of this package, “marginal means” refer to the values obtained by this three step process:

  1. Construct a “grid” of predictor values with all combinations of categorical variables, and where numeric variables are held at their means.
  2. Calculate adjusted predictions for each cell in that grid.
  3. Take the average of those adjusted predictions across one dimension of the grid to obtain the marginal means.

For example, consider a model with a numeric, a factor, and a logical predictor:

library(marginaleffects)

dat <- mtcars
dat$cyl <- as.factor(dat$cyl)
dat$am <- as.logical(dat$am)
mod <- lm(mpg ~ hp + cyl + am, data = dat)

Using the predictions function, we set the hp variable at its mean and compute predictions for all combinations for am and cyl:

predictions(mod, variables = c("am", "cyl"))
#>   rowid     type predicted std.error conf.low conf.high      mpg       hp    am
#> 1     1 response  21.03914  1.213043 18.55019  23.52810 20.09062 146.6875  TRUE
#> 2     2 response  24.96372  1.176830 22.54907  27.37838 20.09062 146.6875  TRUE
#> 3     3 response  21.43031  1.826126 17.68341  25.17721 20.09062 146.6875  TRUE
#> 4     4 response  16.88129  1.272938 14.26944  19.49314 20.09062 146.6875 FALSE
#> 5     5 response  20.80587  1.756564 17.20169  24.41004 20.09062 146.6875 FALSE
#> 6     6 response  17.27245  1.116885 14.98079  19.56411 20.09062 146.6875 FALSE
#>   cyl
#> 1   6
#> 2   4
#> 3   8
#> 4   6
#> 5   4
#> 6   8

For illustration purposes, it is useful to reshape the above results:

am
cyl TRUE FALSE Marginal mean of cyl
6 21.0 16.9 19.0
4 25.0 20.8 22.9
8 21.4 17.3 19.4
Marginal means of am 22.5 18.3

The marginal means of am and cyl are obtained by taking the mean of the adjusted predictions across cells. The marginalmeans function gives us the same results easily:

marginalmeans(mod)
#>   term value marginalmean std.error conf.low conf.high
#> 1   am FALSE     18.31987 0.7853925 16.78053  19.85921
#> 2   am  TRUE     22.47772 0.8343346 20.84246  24.11299
#> 3  cyl     4     22.88479 1.3566479 20.22581  25.54378
#> 4  cyl     6     18.96022 1.0729360 16.85730  21.06313
#> 5  cyl     8     19.35138 1.3770817 16.65235  22.05041

The same results can be obtained using the very powerful emmeans package:

library(emmeans)
emmeans(mod, specs = "cyl")
#>  cyl emmean   SE df lower.CL upper.CL
#>  4     22.9 1.36 27     20.1     25.7
#>  6     19.0 1.07 27     16.8     21.2
#>  8     19.4 1.38 27     16.5     22.2
#> 
#> Results are averaged over the levels of: am 
#> Confidence level used: 0.95
emmeans(mod, specs = "am")
#>  am    emmean    SE df lower.CL upper.CL
#>  FALSE   18.3 0.785 27     16.7     19.9
#>   TRUE   22.5 0.834 27     20.8     24.2
#> 
#> Results are averaged over the levels of: cyl 
#> Confidence level used: 0.95

Interactions

By default, the marginalmeans() function calculates marginal means for each categorical predictor one after the other. We can also compute marginal means for combinations of categories by setting interaction=TRUE:

library(glmmTMB)

dat <- "https://vincentarelbundock.github.io/Rdatasets/csv/Stat2Data/Titanic.csv"
dat <- read.csv(dat)
titanic <- glmmTMB(
    Survived ~ Sex * PClass + Age + (1 | PClass),
    family = binomial,
    data = dat)

Regardless of the scale of the predictions (type argument), marginalmeans() always computes standard errors using the Delta Method:

marginalmeans(titanic,
    type = "response",
    variables = c("Sex", "PClass"))
#>      Sex PClass marginalmean  std.error
#> 1 female    1st    0.9701724 0.01392585
#> 2 female    2nd    0.8803769 0.03608862
#> 3 female    3rd    0.3644761 0.05020662
#> 4   male    1st    0.4450399 0.05150155
#> 5   male    2nd    0.1422606 0.03045734
#> 6   male    3rd    0.1189557 0.02124176

When the model is linear or on the link scale, it also produces confidence intervals:

marginalmeans(
    titanic,
    type = "link",
    variables = c("Sex", "PClass"))
#>      Sex PClass marginalmean std.error   conf.low  conf.high
#> 1 female    1st    3.4820408 0.4811625  2.5389797  4.4251019
#> 2 female    2nd    1.9960037 0.3426777  1.3243678  2.6676396
#> 3 female    3rd   -0.5559886 0.2167161 -0.9807443 -0.1312329
#> 4   male    1st   -0.2207321 0.2085377 -0.6294584  0.1879942
#> 5   male    2nd   -1.7966396 0.2495423 -2.2857336 -1.3075456
#> 6   male    3rd   -2.0023565 0.2025903 -2.3994262 -1.6052869

It is easy to transform those link-scale marginal means with arbitrary functions using the transform_post argument:

marginalmeans(titanic,
    type = "link",
    transform_post = insight::link_inverse(titanic),
    variables = c("Sex", "PClass"))
#>      Sex PClass marginalmean   conf.low conf.high
#> 1 female    1st    0.9701724 0.92682967 0.9881687
#> 2 female    2nd    0.8803769 0.78990748 0.9350899
#> 3 female    3rd    0.3644761 0.27274413 0.4672388
#> 4   male    1st    0.4450399 0.34763335 0.5468606
#> 5   male    2nd    0.1422606 0.09231141 0.2128979
#> 6   male    3rd    0.1189557 0.08321647 0.1672440

When a model does not include interactions, marginalmeans() defaults to reporting EMMs for each category individually, without interactions:

titanic2 <- glmmTMB(
    Survived ~ Sex + PClass + Age + (1 | PClass),
    family = binomial,
    data = dat)
marginalmeans(
    titanic2,
    variables = c("Sex", "PClass"))
#>     term  value marginalmean  std.error
#> 1 PClass    1st    0.7065907 0.02889053
#> 2 PClass    2nd    0.4935160 0.02871382
#> 3 PClass    3rd    0.2910045 0.02680176
#> 4    Sex female    0.7408546 0.02402629
#> 5    Sex   male    0.2532196 0.02031785

We can force the interactions:

marginalmeans(
    titanic2,
    interaction = TRUE,
    variables = c("Sex", "PClass"))
#>      Sex PClass marginalmean  std.error
#> 1 female    1st   0.92882414 0.01610223
#> 2 female    2nd   0.78190513 0.03564791
#> 3 female    3rd   0.51183442 0.04583722
#> 4   male    1st   0.48435732 0.04680392
#> 5   male    2nd   0.20512692 0.03080665
#> 6   male    3rd   0.07017461 0.01354387

Tidy summaries

The summary, tidy, and glance functions are also available to summarize and manipulate the results:

mm <- marginalmeans(mod)

tidy(mm)
#>   term value estimate std.error statistic p.value conf.low conf.high
#> 1   am FALSE 18.31987 0.7853925  23.32575       0 16.78053  19.85921
#> 2   am  TRUE 22.47772 0.8343346  26.94090       0 20.84246  24.11299
#> 3  cyl     4 22.88479 1.3566479  16.86863       0 20.22581  25.54378
#> 4  cyl     6 18.96022 1.0729360  17.67134       0 16.85730  21.06313
#> 5  cyl     8 19.35138 1.3770817  14.05246       0 16.65235  22.05041

glance(mm)
#>        aic      bic r.squared adj.r.squared     rmse nobs
#> 1 161.0033 169.7978  0.824875     0.7989306 2.482432   32

summary(mm)
#> Estimated marginal means 
#>   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

Thanks to those tidiers, we can also present the results in the style of a regression table using the modelsummary package. For examples, see the tables and plots vignette.

Case study: Multinomial Logit

This example requires version 0.2.0 of the marginaleffects package.

To begin, we generate data and estimate a large model:

library(nnet)
library(marginaleffects)

set.seed(1839)
n <- 1200
x <- factor(sample(letters[1:3], n, TRUE))
y <- vector(length = n)
y[x == "a"] <- sample(letters[4:6], sum(x == "a"), TRUE)
y[x == "b"] <- sample(letters[4:6], sum(x == "b"), TRUE, c(1 / 4, 2 / 4, 1 / 4))
y[x == "c"] <- sample(letters[4:6], sum(x == "c"), TRUE, c(1 / 5, 3 / 5, 2 / 5))

dat <- data.frame(x = x, y = factor(y))
tmp <- as.data.frame(replicate(20, factor(sample(letters[7:9], n, TRUE))))
dat <- cbind(dat, tmp)
void <- capture.output({
    mod <- multinom(y ~ ., dat)
})

Try to compute marginal means, but realize that your grid won’t fit in memory:

marginalmeans(mod, type = "probs")
#> Error: You are trying to create a prediction grid with more than 1 billion rows, which is likely to exceed the memory and computational power available on your local machine. Presumably this is because you are considering many variables with many levels. All of the functions in the `marginaleffects` package include arguments to specify a restricted list of variables over which to create a prediction grid.

Use the variables and variables_grid arguments to compute marginal means over a more reasonably sized grid:

marginalmeans(mod,
              type = "probs",
              variables = c("x", "V1"),
              variables_grid = paste0("V", 2:3))