Bayesian Analysis with brms

The marginaleffects package offers convenience functions to compute and display predictions, contrasts, and marginal effects from bayesian models estimated by the brms package. To compute these quantities, marginaleffects relies on workhorse functions from the brms package to draw from the posterior distribution. The type of draws used is controlled by using the type argument of the predictions or slopes functions:

type = "response": Compute posterior draws of the expected value using the brms::posterior_epred function.
type = "link": Compute posterior draws of the linear predictor using the brms::posterior_linpred function.
type = "prediction": Compute posterior draws of the posterior predictive distribution using the brms::posterior_predict function.

The predictions and slopes functions can also pass additional arguments to the brms prediction functions via the ... ellipsis. For example, if mod is a mixed-effects model, then this command will compute 10 draws from the posterior predictive distribution, while ignoring all group-level effects:

predictions(mod, type = "prediction", ndraws = 10, re_formula = NA)

See the brms documentation for a list of available arguments:

?brms::posterior_epred
?brms::posterior_linpred
?brms::posterior_predict

Logistic regression with multiplicative interactions

Load libraries and download data on passengers of the Titanic from the Rdatasets archive:

library(marginaleffects)
library(brms)
library(ggplot2)
library(ggdist)

dat <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/carData/TitanicSurvival.csv")
dat$survived <- ifelse(dat$survived == "yes", 1, 0)
dat$woman <- ifelse(dat$sex == "female", 1, 0)

Fit a logit model with a multiplicative interaction:

mod <- brm(survived ~ woman * age + passengerClass,
           family = bernoulli(link = "logit"),
           data = dat)

Adjusted predictions

We can compute adjusted predicted values of the outcome variable (i.e., probability of survival aboard the Titanic) using the predictions function. By default, this function calculates predictions for each row of the dataset:

predictions(mod)
#> 
#>  Estimate  2.5 % 97.5 %
#>    0.9367 0.9070 0.9590
#>    0.8493 0.7453 0.9187
#>    0.9433 0.8949 0.9704
#>    0.5131 0.4302 0.6000
#>    0.9375 0.9080 0.9601
#> --- 1036 rows omitted. See ?avg_predictions and ?print.marginaleffects --- 
#>    0.0376 0.0235 0.0581
#>    0.5859 0.5017 0.6663
#>    0.1043 0.0801 0.1337
#>    0.1017 0.0779 0.1307
#>    0.0916 0.0691 0.1189
#> Columns: rowid, estimate, conf.low, conf.high, survived, woman, age, passengerClass

To visualize the relationship between the outcome and one of the regressors, we can plot conditional adjusted predictions with the plot_predictions function:

plot_predictions(mod, condition = "age")

Compute adjusted predictions for some user-specified values of the regressors, using the newdata argument and the datagrid function:

pred <- predictions(mod,
                    newdata = datagrid(woman = 0:1,
                                       passengerClass = c("1st", "2nd", "3rd")))
pred
#> 
#>  Estimate  2.5 % 97.5 %  age woman passengerClass
#>    0.5149 0.4319  0.602 29.9     0            1st
#>    0.2013 0.1536  0.261 29.9     0            2nd
#>    0.0875 0.0656  0.114 29.9     0            3rd
#>    0.9364 0.9066  0.959 29.9     1            1st
#>    0.7783 0.7090  0.835 29.9     1            2nd
#>    0.5701 0.4938  0.644 29.9     1            3rd
#> 
#> Columns: rowid, estimate, conf.low, conf.high, survived, age, woman, passengerClass

The posterior_draws function samples from the posterior distribution of the model, and produces a data frame with drawid and draw columns.

pred <- posterior_draws(pred)
head(pred)
#>   drawid       draw rowid   estimate   conf.low conf.high  survived      age woman passengerClass
#> 1      1 0.46566713     1 0.51492993 0.43192231 0.6018749 0.4082218 29.88113     0            1st
#> 2      1 0.16658900     2 0.20128833 0.15362308 0.2613351 0.4082218 29.88113     0            2nd
#> 3      1 0.08750961     3 0.08750369 0.06555724 0.1141134 0.4082218 29.88113     0            3rd
#> 4      1 0.93735755     4 0.93641346 0.90660921 0.9587589 0.4082218 29.88113     1            1st
#> 5      1 0.77437334     5 0.77829290 0.70896643 0.8346419 0.4082218 29.88113     1            2nd
#> 6      1 0.62216334     6 0.57010265 0.49377997 0.6441967 0.4082218 29.88113     1            3rd

This “long” format makes it easy to plots results:

ggplot(pred, aes(x = draw, fill = factor(woman))) +
    geom_density() +
    facet_grid(~ passengerClass, labeller = label_both) +
    labs(x = "Predicted probability of survival", y = "", fill = "Woman")

Marginal effects

Use slopes() to compute marginal effects (slopes of the regression equation) for each row of the dataset, and use ) to compute “Average Marginal Effects”, that is, the average of all observation-level marginal effects:

mfx <- slopes(mod)
mfx
#> 
#>            Term  Contrast Estimate   2.5 % 97.5 %
#>  woman          1 - 0       0.4076  0.3339  0.478
#>  woman          1 - 0       0.0919  0.0301  0.179
#>  woman          1 - 0       0.0997  0.0371  0.187
#>  woman          1 - 0       0.4220  0.3487  0.492
#>  woman          1 - 0       0.3517  0.2762  0.426
#> --- 4174 rows omitted. See ?avg_slopes and ?print.marginaleffects --- 
#>  passengerClass 3rd - 1st  -0.2643 -0.3397 -0.199
#>  passengerClass 3rd - 1st  -0.3527 -0.4263 -0.282
#>  passengerClass 3rd - 1st  -0.4593 -0.5507 -0.370
#>  passengerClass 3rd - 1st  -0.4550 -0.5463 -0.365
#>  passengerClass 3rd - 1st  -0.4365 -0.5261 -0.350
#> Columns: rowid, term, contrast, estimate, conf.low, conf.high, predicted, predicted_hi, predicted_lo, tmp_idx, survived, woman, age, passengerClass

Compute marginal effects with some regressors fixed at user-specified values, and other regressors held at their means:

slopes(
    mod,
    newdata = datagrid(
        woman = 1,
        passengerClass = "1st"))
#> 
#>            Term  Contrast  Estimate    2.5 %    97.5 %  age woman passengerClass
#>  woman          1 - 0      0.420368  0.34697  0.490373 29.9     1            1st
#>  age            dY/dX     -0.000238 -0.00136  0.000871 29.9     1            1st
#>  passengerClass 2nd - 1st -0.157442 -0.22327 -0.102890 29.9     1            1st
#>  passengerClass 3rd - 1st -0.365376 -0.43832 -0.294769 29.9     1            1st
#> 
#> Columns: rowid, term, contrast, estimate, conf.low, conf.high, predicted, predicted_hi, predicted_lo, tmp_idx, survived, age, woman, passengerClass

Compute and plot conditional marginal effects:

plot_slopes(mod, variables = "woman", condition = "age")

The posterior_draws produces a dataset with drawid and draw columns:

draws <- posterior_draws(mfx)

dim(draws)
#> [1] 16736000       16

head(draws)
#>   drawid       draw rowid  term contrast   estimate   conf.low conf.high predicted predicted_hi predicted_lo tmp_idx survived woman     age passengerClass
#> 1      1 0.45768070     1 woman    1 - 0 0.40758806 0.33394178 0.4783255 0.9366604    0.9366604    0.5276344       1        1     1 29.0000            1st
#> 2      1 0.09282294     2 woman    1 - 0 0.09190500 0.03006249 0.1788661 0.8493050    0.9436352    0.8493050       2        1     0  0.9167            1st
#> 3      1 0.10179526     3 woman    1 - 0 0.09973464 0.03710637 0.1873029 0.9433293    0.9433293    0.8409878       3        0     1  2.0000            1st
#> 4      1 0.47357659     4 woman    1 - 0 0.42198871 0.34867425 0.4921402 0.5131011    0.9363704    0.5131011       4        0     0 30.0000            1st
#> 5      1 0.39400824     5 woman    1 - 0 0.35167800 0.27617966 0.4262480 0.9374937    0.9374937    0.5851849       5        0     1 25.0000            1st
#> 6      1 0.72093817     6 woman    1 - 0 0.65665127 0.57622574 0.7297775 0.2730542    0.9320972    0.2730542       6        1     0 48.0000            1st

We can use this dataset to plot our results. For example, to plot the posterior density of the marginal effect of age when the woman variable is equal to 0 or 1:

mfx <- slopes(mod,
    variables = "age",
    newdata = datagrid(woman = 0:1)) |>
    posterior_draws()

ggplot(mfx, aes(x = draw, fill = factor(woman))) +
    stat_halfeye(slab_alpha = .5) +
    labs(x = "Marginal Effect of Age on Survival",
         y = "Posterior density",
         fill = "Woman")

Random effects model

This section replicates some of the analyses of a random effects model published in Andrew Heiss’ blog post: “A guide to correctly calculating posterior predictions and average marginal effects with multilevel Bayesian models.” The objective is mainly to illustrate the use of marginaleffects. Please refer to the original post for a detailed discussion of the quantities computed below.

Load libraries and download data:

library(brms)
library(ggdist)
library(patchwork)
library(marginaleffects)

vdem_2015 <- read.csv("https://github.com/vincentarelbundock/marginaleffects/raw/main/data-raw/vdem_2015.csv")

head(vdem_2015)
#>   country_name country_text_id year                           region media_index party_autonomy_ord polyarchy civil_liberties party_autonomy
#> 1       Mexico             MEX 2015  Latin America and the Caribbean       0.837                  3     0.631           0.704           TRUE
#> 2     Suriname             SUR 2015  Latin America and the Caribbean       0.883                  4     0.777           0.887           TRUE
#> 3       Sweden             SWE 2015 Western Europe and North America       0.956                  4     0.915           0.968           TRUE
#> 4  Switzerland             CHE 2015 Western Europe and North America       0.939                  4     0.901           0.960           TRUE
#> 5        Ghana             GHA 2015               Sub-Saharan Africa       0.858                  4     0.724           0.921           TRUE
#> 6 South Africa             ZAF 2015               Sub-Saharan Africa       0.898                  4     0.752           0.869           TRUE

Fit a basic model:

mod <- brm(
  bf(media_index ~ party_autonomy + civil_liberties + (1 | region),
     phi ~ (1 | region)),
  data = vdem_2015,
  family = Beta(),
  control = list(adapt_delta = 0.9))

Posterior predictions

To compute posterior predictions for specific values of the regressors, we use the newdata argument and the datagrid function. We also use the type argument to compute two types of predictions: accounting for residual (observation-level) residual variance (prediction) or ignoring it (response).

nd = datagrid(model = mod,
              party_autonomy = c(TRUE, FALSE),
              civil_liberties = .5,
              region = "Middle East and North Africa")
p1 <- predictions(mod, type = "response", newdata = nd) |>
    posterior_draws() |>
    transform(type = "Response")
p2 <- predictions(mod, type = "prediction", newdata = nd) |>
    posterior_draws() |>
    transform(type = "Prediction")
pred <- rbind(p1, p2)

Extract posterior draws and plot them:

ggplot(pred, aes(x = draw, fill = party_autonomy)) +
    stat_halfeye(alpha = .5) +
    facet_wrap(~ type) +
    labs(x = "Media index (predicted)", 
         y = "Posterior density",
         fill = "Party autonomy")

Marginal effects and contrasts

As noted in the Marginal Effects vignette, there should be one distinct marginal effect for each combination of regressor values. Here, we consider only one combination of regressor values, where region is “Middle East and North Africa”, and civil_liberties is 0.5. Then, we calculate the mean of the posterior distribution of marginal effects:

mfx <- slopes(mod,
                       newdata = datagrid(civil_liberties = .5,
                                          region = "Middle East and North Africa"))
mfx
#> 
#>             Term     Contrast Estimate 2.5 % 97.5 % party_autonomy civil_liberties                       region
#>  party_autonomy  TRUE - FALSE    0.252 0.166  0.336           TRUE             0.5 Middle East and North Africa
#>  civil_liberties dY/dX           0.816 0.621  1.007           TRUE             0.5 Middle East and North Africa
#> 
#> Columns: rowid, term, contrast, estimate, conf.low, conf.high, predicted, predicted_hi, predicted_lo, tmp_idx, media_index, party_autonomy, civil_liberties, region

Use the posterior_draws() to extract draws from the posterior distribution of marginal effects, and plot them:

mfx <- posterior_draws(mfx)

ggplot(mfx, aes(x = draw, y = term)) +
  stat_halfeye() +
  labs(x = "Marginal effect", y = "")

Plot marginal effects, conditional on a regressor:

plot_slopes(mod,
         variables = "civil_liberties",
         condition = "party_autonomy")

Continuous predictors

pred <- predictions(mod,
                    newdata = datagrid(party_autonomy = FALSE,
                                       region = "Middle East and North Africa",
                                       civil_liberties = seq(0, 1, by = 0.05))) |>
        posterior_draws()

ggplot(pred, aes(x = civil_liberties, y = draw)) +
    stat_lineribbon() +
    scale_fill_brewer(palette = "Reds") +
    labs(x = "Civil liberties",
         y = "Media index (predicted)",
         fill = "")

The slope of this line for different values of civil liberties can be obtained with:

mfx <- slopes(mod,
    newdata = datagrid(
        civil_liberties = c(.2, .5, .8),
        party_autonomy = FALSE,
        region = "Middle East and North Africa"),
    variables = "civil_liberties")
mfx
#> 
#>             Term Estimate 2.5 % 97.5 % civil_liberties party_autonomy                       region
#>  civil_liberties    0.490 0.361  0.639             0.2          FALSE Middle East and North Africa
#>  civil_liberties    0.807 0.612  0.993             0.5          FALSE Middle East and North Africa
#>  civil_liberties    0.807 0.674  0.934             0.8          FALSE Middle East and North Africa
#> 
#> Columns: rowid, term, estimate, conf.low, conf.high, predicted, predicted_hi, predicted_lo, tmp_idx, media_index, civil_liberties, party_autonomy, region

And plotted:

mfx <- posterior_draws(mfx)

ggplot(mfx, aes(x = draw, fill = factor(civil_liberties))) +
    stat_halfeye(slab_alpha = .5) +
    labs(x = "Marginal effect of Civil Liberties on Media Index",
         y = "Posterior density",
         fill = "Civil liberties")

The slopes function can use the ellipsis (...) to push any argument forward to the posterior_predict function. This can alter the types of predictions returned. For example, the re_formula=NA argument of the posterior_predict.brmsfit method will compute marginaleffects without including any group-level effects:

mfx <- slopes(
    mod,
    newdata = datagrid(
        civil_liberties = c(.2, .5, .8),
        party_autonomy = FALSE,
        region = "Middle East and North Africa"),
    variables = "civil_liberties",
    re_formula = NA) |>
    posterior_draws()

ggplot(mfx, aes(x = draw, fill = factor(civil_liberties))) +
    stat_halfeye(slab_alpha = .5) +
    labs(x = "Marginal effect of Civil Liberties on Media Index",
         y = "Posterior density",
         fill = "Civil liberties")

Global grand mean

pred <- predictions(
    mod,
    re_formula = NA,
    newdata = datagrid(party_autonomy = c(TRUE, FALSE))) |>
    posterior_draws()

mfx <- slopes(
    mod,
    re_formula = NA,
    variables = "party_autonomy") |>
    posterior_draws()

plot1 <- ggplot(pred, aes(x = draw, fill = party_autonomy)) +
         stat_halfeye(slab_alpha = .5) +
         labs(x = "Media index (Predicted)",
              y = "Posterior density",
              fill = "Party autonomy")

plot2 <- ggplot(mfx, aes(x = draw)) +
         stat_halfeye(slab_alpha = .5)  +
         labs(x = "Contrast: Party autonomy TRUE - FALSE",
              y = "",
              fill = "Party autonomy")

# combine plots using the `patchwork` package
plot1 + plot2

Region-specific predictions and contrasts

Predicted media index by region and level of civil liberties:

pred <- predictions(mod,
                    newdata = datagrid(region = vdem_2015$region,
                                       party_autonomy = FALSE, 
                                       civil_liberties = seq(0, 1, length.out = 100))) |> 
        posterior_draws()

ggplot(pred, aes(x = civil_liberties, y = draw)) +
    stat_lineribbon() +
    scale_fill_brewer(palette = "Reds") +
    facet_wrap(~ region) +
    labs(x = "Civil liberties",
         y = "Media index (predicted)",
         fill = "")

Predicted media index by region and level of civil liberties:

pred <- predictions(mod,
                    newdata = datagrid(region = vdem_2015$region,
                                       civil_liberties = c(.2, .8),
                                      party_autonomy = FALSE)) |>
        posterior_draws()

ggplot(pred, aes(x = draw, fill = factor(civil_liberties))) +
    stat_halfeye(slab_alpha = .5) +
    facet_wrap(~ region) +
    labs(x = "Media index (predicted)",
         y = "Posterior density",
         fill = "Civil liberties")

Predicted media index by region and party autonomy:

pred <- predictions(mod,
                    newdata = datagrid(region = vdem_2015$region,
                                       party_autonomy = c(TRUE, FALSE),
                                       civil_liberties = .5)) |>
        posterior_draws()

ggplot(pred, aes(x = draw, y = region , fill = party_autonomy)) +
    stat_halfeye(slab_alpha = .5) +
    labs(x = "Media index (predicted)",
         y = "",
         fill = "Party autonomy")

TRUE/FALSE contrasts (marginal effects) of party autonomy by region:

mfx <- slopes(
    mod,
    variables = "party_autonomy",
    newdata = datagrid(
        region = vdem_2015$region,
        civil_liberties = .5)) |>
    posterior_draws()

ggplot(mfx, aes(x = draw, y = region , fill = party_autonomy)) +
    stat_halfeye(slab_alpha = .5) +
    labs(x = "Media index (predicted)",
         y = "",
         fill = "Party autonomy")

Hypothetical groups

We can also obtain predictions or marginal effects for a hypothetical group instead of one of the observed regions. To achieve this, we create a dataset with NA in the region column. Then we call the marginaleffects or predictions functions with the allow_new_levels argument. This argument is pushed through via the ellipsis (...) to the posterior_epred function of the brms package:

dat <- data.frame(civil_liberties = .5,
                  party_autonomy = FALSE,
                  region = "New Region")

mfx <- slopes(
    mod,
    variables = "party_autonomy",
    allow_new_levels = TRUE,
    newdata = dat)

draws <- posterior_draws(mfx)

ggplot(draws, aes(x = draw)) +
     stat_halfeye() +
     labs(x = "Marginal effect of party autonomy in a generic world region", y = "")

Averaging, marginalizing, integrating random effects

Consider a logistic regression model with random effects:

dat <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/plm/EmplUK.csv")
dat$x <- as.numeric(dat$output > median(dat$output))
dat$y <- as.numeric(dat$emp > median(dat$emp))
mod <- brm(y ~ x + (1 | firm), data = dat, backend = "cmdstanr", family = "bernoulli")

We can compute adjusted predictions for a given value of x and for each firm (random effects) as follows:

p <- predictions(mod, newdata = datagrid(x = 0, firm = unique))
head(p)
#> 
#>  Estimate    2.5 % 97.5 % x firm
#>   1.0e+00 9.01e-01 1.0000 0    1
#>   1.0e+00 8.95e-01 1.0000 0    2
#>   1.0e+00 9.12e-01 1.0000 0    3
#>   1.0e+00 7.97e-01 1.0000 0    4
#>   1.0e+00 9.09e-01 1.0000 0    5
#>   4.9e-08 8.42e-21 0.0019 0    6
#> 
#> Columns: rowid, estimate, conf.low, conf.high, y, x, firm

We can average/marginalize/integrate across random effects with the avg_predictions() function or the by argument:

avg_predictions(mod, newdata = datagrid(x = 0, firm = unique))
#> 
#>  Estimate 2.5 % 97.5 %
#>     0.454  0.44  0.468
#> 
#> Columns: estimate, conf.low, conf.high

predictions(mod, newdata = datagrid(x = 0:1, firm = unique), by = "x")
#> 
#>  x Estimate 2.5 % 97.5 %
#>  0    0.454 0.440  0.468
#>  1    0.557 0.546  0.570
#> 
#> Columns: x, estimate, conf.low, conf.high

We can also draw from the (assumed gaussian) population distribution of random effects, by asking predictions() to make predictions for new “levels” of the random effects. If we then take an average of predictions using avg_predictions() or the by argument, we will have “integrated out the random effects”, as described in the brmsmargins package vignette. In the code below, we make predictions for 100 firm identifiers which were not in the original dataset. We also ask predictions() to push forward the allow_new_levels and sample_new_levels arguments to the brms::posterior_epred function:

predictions(
    mod,
    newdata = datagrid(x = 0:1, firm = -1:-100),
    allow_new_levels = TRUE,
    sample_new_levels = "gaussian",
    by = "x")
#> 
#>  x Estimate 2.5 % 97.5 %
#>  0    0.452 0.342  0.566
#>  1    0.551 0.437  0.663
#> 
#> Columns: x, estimate, conf.low, conf.high

We can “integrate out” random effects in the other slopes functions too. For instance,

avg_comparisons(
    mod,
    newdata = datagrid(firm = -1:-100),
    allow_new_levels = TRUE,
    sample_new_levels = "gaussian")
#> 
#>  Term Contrast Estimate  2.5 % 97.5 %
#>     x    1 - 0   0.0978 0.0468  0.162
#> 
#> Columns: term, contrast, estimate, conf.low, conf.high

This is nearly equivalent the brmsmargins command output (with slight variations due to different random seeds):

library(brmsmargins)
bm <- brmsmargins(
  k = 100,
  object = mod,
  at = data.frame(x = c(0, 1)),
  CI = .95,
  CIType = "ETI",
  contrasts = cbind("AME x" = c(-1, 1)),
  effects = "integrateoutRE")
bm$ContrastSummary |> data.frame()
#>            M        Mdn        LL        UL PercentROPE PercentMID   CI CIType ROPE  MID Label
#> 1 0.09901885 0.09702168 0.0482447 0.1630884          NA         NA 0.95    ETI <NA> <NA> AME x

See the alternative software vignette for more information on brmsmargins.

Multinomial logit

Fit a model with categorical outcome (heating system choice in California houses) and logit link:

dat <- "https://vincentarelbundock.github.io/Rdatasets/csv/Ecdat/Heating.csv"
dat <- read.csv(dat)
mod <- brm(depvar ~ ic.gc + oc.gc,
           data = dat,
           family = categorical(link = "logit"))

Adjusted predictions

Compute predicted probabilities for each level of the outcome variable:

pred <- predictions(mod)

head(pred)
#> 
#>  Group Estimate  2.5 % 97.5 %
#>     ec   0.0663 0.0447 0.0930
#>     ec   0.0768 0.0590 0.0974
#>     ec   0.1030 0.0618 0.1585
#>     ec   0.0634 0.0459 0.0838
#>     ec   0.0745 0.0574 0.0947
#>     ec   0.0709 0.0455 0.1036
#> 
#> Columns: rowid, group, estimate, conf.low, conf.high, depvar, ic.gc, oc.gc

Extract posterior draws and plot them:

draws <- posterior_draws(pred)

ggplot(draws, aes(x = draw, fill = group)) +
    geom_density(alpha = .2, color = "white") +
    labs(x = "Predicted probability",
         y = "Density",
         fill = "Heating system")

Use the plot_predictions function to plot conditional adjusted predictions for each level of the outcome variable gear, conditional on the value of the mpg regressor:

plot_predictions(mod, condition = "oc.gc") +
    facet_wrap(~ group) +
    labs(y = "Predicted probability")

Marginal effects

avg_slopes(mod)
#> 
#>  Group  Term  Estimate     2.5 %   97.5 %
#>     ec ic.gc -1.77e-04 -3.96e-04 2.37e-05
#>     er ic.gc  1.65e-05 -2.26e-04 2.51e-04
#>     gc ic.gc  1.38e-05 -3.72e-04 4.00e-04
#>     gr ic.gc  4.24e-05 -2.37e-04 3.30e-04
#>     hp ic.gc  1.07e-04 -7.73e-05 2.97e-04
#>     ec oc.gc  4.88e-04 -4.04e-04 1.45e-03
#>     er oc.gc -1.02e-03 -2.07e-03 2.98e-05
#>     gc oc.gc  1.04e-03 -7.39e-04 2.78e-03
#>     gr oc.gc  9.46e-05 -1.19e-03 1.34e-03
#>     hp oc.gc -5.85e-04 -1.45e-03 2.30e-04
#> 
#> Columns: group, term, estimate, conf.low, conf.high

Hurdle models

This section replicates some analyses from yet another amazing blog post by Andrew Heiss.

To begin, we estimate a hurdle model in brms with random effects, using data from the gapminder package: 704G

library(gapminder)
library(brms)
library(dplyr)
library(ggplot2)
library(ggdist)
library(cmdstanr)
library(patchwork)
library(marginaleffects)

set.seed(1024)

CHAINS <- 4
ITER <- 2000
WARMUP <- 1000
BAYES_SEED <- 1234

gapminder <- gapminder::gapminder |> 
  filter(continent != "Oceania") |> 
  # Make a bunch of GDP values 0
  mutate(prob_zero = ifelse(lifeExp < 50, 0.3, 0.02),
         will_be_zero = rbinom(n(), 1, prob = prob_zero),
         gdpPercap = ifelse(will_be_zero, 0, gdpPercap)) |> 
  select(-prob_zero, -will_be_zero) |> 
  # Make a logged version of GDP per capita
  mutate(log_gdpPercap = log1p(gdpPercap)) |> 
  mutate(is_zero = gdpPercap == 0)

mod <- brm(
  bf(gdpPercap ~ lifeExp + year + (1 + lifeExp + year | continent),
     hu ~ lifeExp),
  data = gapminder,
  backend = "cmdstanr",
  family = hurdle_lognormal(),
  cores = 2,
  chains = CHAINS, iter = ITER, warmup = WARMUP, seed = BAYES_SEED,
  silent = 2)

Adjusted predictions

Adjusted predictions for every observation in the original data:

predictions(mod) |> head()
#> 
#>  Estimate 2.5 % 97.5 %
#>       143   103    219
#>       168   125    256
#>       202   153    304
#>       251   197    373
#>       312   250    454
#>       398   325    567
#> 
#> Columns: rowid, estimate, conf.low, conf.high, gdpPercap, lifeExp, year, continent

Adjusted predictions for the hu parameter:

predictions(mod, dpar = "hu") |> head()
#> 
#>  Estimate 2.5 % 97.5 %
#>     0.574 0.475  0.652
#>     0.537 0.442  0.611
#>     0.496 0.407  0.566
#>     0.446 0.366  0.511
#>     0.396 0.325  0.454
#>     0.341 0.282  0.391
#> 
#> Columns: rowid, estimate, conf.low, conf.high, gdpPercap, lifeExp, year, continent

Predictions on a different scale:

predictions(mod, type = "link", dpar = "hu") |> head()
#> 
#>  Estimate  2.5 %  97.5 %
#>    0.2980 -0.101  0.6259
#>    0.1463 -0.235  0.4527
#>   -0.0178 -0.377  0.2673
#>   -0.2189 -0.551  0.0424
#>   -0.4234 -0.730 -0.1857
#>   -0.6573 -0.933 -0.4443
#> 
#> Columns: rowid, estimate, conf.low, conf.high, gdpPercap, lifeExp, year, continent

Plot adjusted predictions as a function of lifeExp:

plot_predictions(
    mod,
    condition = "lifeExp") +
    labs(y = "mu") +
plot_predictions(
    mod,
    dpar = "hu",
    condition = "lifeExp") +
    labs(y = "hu")

Predictions with more than one condition and the re_formula argument from brms:

plot_predictions(
    mod,
    re_formula = NULL,
    condition = c("lifeExp", "continent"))

Extract draws with `posterior_draws()`

The posterior_draws() function extract raw samples from the posterior from objects produced by marginaleffects. This allows us to use richer geoms and summaries, such as those in the ggdist package:

predictions(
    mod,
    re_formula = NULL,
    newdata = datagrid(model = mod,
                       continent = gapminder$continent,
                       year = c(1952, 2007),
                       lifeExp = seq(30, 80, 1))) |>
    posterior_draws() |>
    ggplot(aes(lifeExp, draw, fill = continent, color = continent)) +
    stat_lineribbon(alpha = .25) +
    facet_grid(year ~ continent)

Average Contrasts

What happens to gdpPercap when lifeExp increases by one?

avg_comparisons(mod)
#> 
#>     Term Contrast Estimate 2.5 % 97.5 %
#>  lifeExp       +1    718.9 515.6  812.0
#>  year          +1    -63.8 -84.4  -41.1
#> 
#> Columns: term, contrast, estimate, conf.low, conf.high

What happens to gdpPercap when lifeExp increases by one standard deviation?

avg_comparisons(mod, variables = list(lifeExp = "sd"))
#> 
#>     Term                Contrast Estimate 2.5 % 97.5 %
#>  lifeExp (x + sd/2) - (x - sd/2)     4050  3718   4741
#> 
#> Columns: term, contrast, estimate, conf.low, conf.high

What happens to gdpPercap when lifeExp increases from 50 to 60 and year simultaneously increases its min to its max?

avg_comparisons(
    mod,
    variables = list(lifeExp = c(50, 60), year = "minmax"),
    cross = TRUE)
#> 
#>  C: lifeExp   C: year Estimate 2.5 % 97.5 %
#>     60 - 50 Max - Min      835   523   1404
#> 
#> Columns: term, contrast_lifeExp, contrast_year, estimate, conf.low, conf.high

Plot draws from the posterior distribution of average contrasts (not the same thing as draws from the posterior distribution of contrasts):

avg_comparisons(mod) |>
    posterior_draws() |>
    ggplot(aes(estimate, term)) +
    stat_dotsinterval() +
    labs(x = "Posterior distribution of average contrasts", y = "")

Marginal effects (slopes)

Average Marginal Effect of lifeExp on different scales and for different parameters:

avg_slopes(mod)
#> 
#>     Term Estimate 2.5 % 97.5 %
#>  lifeExp    718.7 515.6    812
#>  year       -63.8 -84.4    -41
#> 
#> Columns: term, estimate, conf.low, conf.high

avg_slopes(mod, type = "link")
#> 
#>     Term Estimate   2.5 %   97.5 %
#>  lifeExp  0.08249  0.0742  0.08856
#>  year    -0.00937 -0.0120 -0.00632
#> 
#> Columns: term, estimate, conf.low, conf.high

avg_slopes(mod, dpar = "hu")
#> 
#>     Term Estimate    2.5 %   97.5 %
#>  lifeExp -0.00817 -0.00937 -0.00669
#>  year     0.00000  0.00000  0.00000
#> 
#> Columns: term, estimate, conf.low, conf.high

avg_slopes(mod, dpar = "hu", type = "link")
#> 
#>     Term Estimate  2.5 %  97.5 %
#>  lifeExp  -0.0993 -0.113 -0.0838
#>  year      0.0000  0.000  0.0000
#> 
#> Columns: term, estimate, conf.low, conf.high

Plot Conditional Marginal Effects

plot_slopes(
    mod,
    variables = "lifeExp",
    condition = "lifeExp") +
    labs(y = "mu") +

plot_slopes(
    mod,
    dpar = "hu",
    variables = "lifeExp",
    condition = "lifeExp") +
    labs(y = "hu")

Or we can call slopes() or comparisons() with posterior_draws() function to have even more control:

comparisons(
    mod,
    type = "link",
    variables = "lifeExp",
    newdata = datagrid(lifeExp = c(40, 70), continent = gapminder$continent)) |>
    posterior_draws() |>
    ggplot(aes(draw, continent, fill = continent)) +
    stat_dotsinterval() +
    facet_grid(lifeExp ~ .) +
    labs(x = "Effect of a 1 unit change in Life Expectancy")

Bayesian estimates and credible intervals

For bayesian models like those produced by the brms or rstanarm packages, the marginaleffects package functions report the median of the posterior distribution as their main estimates.

The default credible intervals are equal-tailed intervals (quantiles), and the default function to identify the center of the distribution is the median. Users can customize the type of intervals reported by setting global options. Note that both the reported estimate and the intervals change slightly:

library(insight)
library(marginaleffects)

mod <- insight::download_model("brms_1")

options(marginaleffects_posterior_interval = "hdi")
options(marginaleffects_posterior_center = mean)
avg_comparisons(mod)
#> 
#>  Term Contrast Estimate 2.5 % 97.5 %
#>   cyl       +1    -1.50 -2.38 -0.677
#>   wt        +1    -3.21 -4.70 -1.570
#> 
#> Columns: term, contrast, estimate, conf.low, conf.high

options(marginaleffects_posterior_interval = "eti")
options(marginaleffects_posterior_center = stats::median)
avg_comparisons(mod)
#> 
#>  Term Contrast Estimate 2.5 % 97.5 %
#>   cyl       +1    -1.49 -2.36 -0.636
#>   wt        +1    -3.20 -4.79 -1.645
#> 
#> Columns: term, contrast, estimate, conf.low, conf.high

Random variables: `posterior` and `ggdist`

Recent versions of the posterior, brms, and ggdist packages make it easy to draw, summarize and plot random variables. The posterior_draws() can produce objects of class rvar which make it easy to use those features by returning a data frame with a column of type rvar:

library(brms)
library(ggdist)
library(ggplot2)
library(marginaleffects)
mod <- brm(am ~ mpg + hp, data = mtcars, family = bernoulli)

avg_comparisons(mod) |>
  posterior_draws(shape = "rvar") |>
  ggplot(aes(y = term, xdist = rvar)) + 
  stat_slabinterval()

Non-linear hypothesis testing

We begin by estimating a model:

mod <- brm(am ~ mpg + hp, data = mtcars, family = bernoulli(),
           seed = 1024, silent = 2, chains = 4, iter = 1000)

Notice that we can compute average contrasts in two different ways, using the avg_comparisons() function or the comparison argument:

avg_comparisons(mod)
#> 
#>  Term Contrast Estimate   2.5 %  97.5 %
#>   hp        +1  0.00599 0.00288 0.00886
#>   mpg       +1  0.13547 0.07871 0.17472
#> 
#> Columns: term, contrast, estimate, conf.low, conf.high

comparisons(mod, comparison = "differenceavg")
#> 
#>  Term Contrast Estimate   2.5 %  97.5 %
#>   mpg mean(+1)  0.13547 0.07871 0.17472
#>   hp  mean(+1)  0.00599 0.00288 0.00886
#> 
#> Columns: term, contrast, estimate, conf.low, conf.high, predicted, predicted_hi, predicted_lo, tmp_idx

Now, we use the hypothesis argument to compare the first to the second rows of the comparisons() output:

comparisons(
    mod,
    comparison = "differenceavg",
    hypothesis = "b2 - b1 = 0.2")
#> 
#>       Term Estimate  2.5 % 97.5 %
#>  b2-b1=0.2    -0.33 -0.367 -0.275
#> 
#> Columns: term, estimate, conf.low, conf.high

The hypotheses function of the brms package can also perform non-linear hypothesis testing, and it generates some convenient statistics and summaries. This function accepts a D-by-P matrix of draws from the posterior distribution, where D is the number of draws and N is the number of parameters. We can obtain such a matrix using the posterior_draws(x, shape = "DxP"), and we can simply add a couple calls to our chain of operations:

avg_comparisons(mod, comparison = "differenceavg") |>
    posterior_draws(shape = "DxP") |>
    brms::hypothesis("b2 - b1 > .2")
#> Hypothesis Tests for class :
#>         Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
#> 1 (b2-b1)-(.2) > 0    -0.33      0.02    -0.36    -0.28          0         0     
#> ---
#> 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
#> '*': For one-sided hypotheses, the posterior probability exceeds 95%;
#> for two-sided hypotheses, the value tested against lies outside the 95%-CI.
#> Posterior probabilities of point hypotheses assume equal prior probabilities.

Logistic regression with multiplicative interactions

Adjusted predictions

Marginal effects

Random effects model

Posterior predictions

Marginal effects and contrasts

Continuous predictors

Global grand mean

Region-specific predictions and contrasts

Hypothetical groups

Averaging, marginalizing, integrating random effects

Multinomial logit

Adjusted predictions

Marginal effects

Hurdle models

Adjusted predictions

Extract draws with posterior_draws()

Average Contrasts

Marginal effects (slopes)

Bayesian estimates and credible intervals

Random variables: posterior and ggdist

Non-linear hypothesis testing

Extract draws with `posterior_draws()`

Random variables: `posterior` and `ggdist`