This method uses a generalized linear model to estimate the effect of each level of a factor predictor on the outcome. These values are retained to serve as the new encodings for the factor levels. This is sometimes referred to as likelihood encodings. embed has two estimation methods for accomplishing this: with and without pooling.

The example used here is the Grant data from Kuhn and Johnson (2013), these data are used to predict whether a grant application was accepted. One predictor, the sponsor, is represented by the factor variable sponsor_code. The frequencies of sponsors in the data set used here vary between 0 person and 1874 per code. There are 298 sponsor codes in the data. Rather than producing 297 indicator variables for a model, a single numeric variable can be used to represent the effect or impact of the factor level on the outcome. In this case, where a factor outcome is being predicted (accepted or not), the effects are quantified by the log-odds of the sponsor code for being accepted.

We first calculate the raw log-odds for the data (independent of any model):

library(tidymodels)
library(embed)

tidymodels_prefer()
theme_set(theme_bw() + theme(legend.position = "top"))

data(grants)

props <-
grants_other %>%
summarise(
prop = mean(class == "successful"),
log_odds  = log(prop/(1-prop)),
n = length(class)
) %>%
mutate(label = paste0(gsub("_", " ", sponsor_code), " (n=", n, ")"))

props %>%
select(-label)
## # A tibble: 291 × 4
##    <fct>        <dbl>    <dbl> <int>
##  1 100D         0.6      0.405    10
##  2 101A         0.125   -1.95     16
##  3 103C         0.111   -2.08      9
##  4 105A         0.167   -1.61      6
##  5 107C         1      Inf         2
##  6 10B          1      Inf         1
##  7 111C         0.167   -1.61      6
##  8 112D         0.667    0.693    12
##  9 113A         0.5      0        10
## 10 118B         0.5      0         2
## # … with 281 more rows
# later, for plotting
rng <- extendrange(props$log_odds[is.finite(props$log_odds)], f = 0.1)

In subsequent sections, a logistic regression model is used. When the outcome variable is numeric, the steps automatically use linear regression models to estimate effects.

## No Pooling

In this case, the effect of each sponsor code can be estimated separately for each factor level. One method for conducting this estimation step is to fit a logistic regression with the acceptance classification as the outcome and the sponsor code as the predictor. From this, the log-odds are naturally estimated by logistic regression.

For these data, a recipe is created and step_lencode_glm is used:

grants_glm <-
recipe(class ~ ., data = grants_other) %>%
# specify the variable being encoded and the outcome
# estimate the effects
prep(training = grants_other)

The tidy method can be used to extract the encodings and are merged with the raw estimates:

glm_estimates <-
tidy(grants_glm, number = 1) %>%
dplyr::select(-terms, -id)
glm_estimates
## # A tibble: 292 × 2
##    level  value
##    <chr>  <dbl>
##  1 100D   0.405
##  2 101A  -1.95
##  3 103C  -2.08
##  4 105A  -1.61
##  5 107C  16.6
##  6 10B   16.6
##  7 111C  -1.61
##  8 112D   0.693
##  9 113A   0
## 10 118B   0
## # … with 282 more rows
glm_estimates <-
glm_estimates%>%
inner_join(props, by = "sponsor_code") 

For the sponsor codes with n > 1, the estimates are effectively the same:

glm_estimates %>%
dplyr::filter(is.finite(log_odds)) %>%
mutate(difference = log_odds-glm) %>%
dplyr::select(difference) %>%
summary()
##    difference
##  Min.   :-1.155e-14
##  1st Qu.:-3.122e-16
##  Median : 0.000e+00
##  Mean   :-7.453e-17
##  3rd Qu.: 2.220e-16
##  Max.   : 1.066e-14

Note that there is also a effect that is used for a novel sponsor code for future data sets that is the average effect:

tidy(grants_glm, number = 1) %>%
dplyr::filter(level == "..new") %>%
select(-id)
## # A tibble: 1 × 3
##   level value terms
##   <chr> <dbl> <chr>
## 1 ..new -2.88 sponsor_code

## Partial Pooling

This method estimates the effects by using all of the sponsor codes at once using a hierarchical Bayesian generalized linear model. The sponsor codes are treated as a random set that contributes a random intercept to the previously used logistic regression.

Partial pooling estimates each effect as a combination of the separate empirical estimates of the log-odds and the prior distribution. For sponsor codes with small sample sizes, the final estimate is shrunken towards the overall mean of the log-odds. This makes sense since we have poor information for estimating these sponsor codes. For sponsor codes with many data points, the estimates reply more on the empirical estimates. This page has a good discussion of pooling using Bayesian models.

### Bayesian Methods

One appraoch to partial pooling is the function step_lencode_bayes uses the stan_glmer function in the rstanarm package. There are a number of options that can be used to control the model estimation routine, including:

opts <-
list(
## the number of chains
chains = 4,
## how many cores to use
cores = 4,
## the total number of iterations per chain (low here for time)
iter = 1000,
## set the random number seed
seed = 8779
)

Using the default priors, the model is estimated via:

grants_glmer <-
recipe(class ~ ., data = grants_other) %>%
step_lencode_bayes(
outcome = vars(class),
options = opts
) %>%
prep(training = grants_other)
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess

This took more time than the simple non-pooled model. The embeddings are extracted in the same way:

all_estimates <-
tidy(grants_glmer, number = 1) %>%
dplyr::select(-terms, -id) %>%

all_estimates %>% dplyr::select(sponsor_code, log_odds, glm, glmer)
## # A tibble: 291 × 4
##    <chr>           <dbl>  <dbl>  <dbl>
##  1 100D            0.405  0.405  0.273
##  2 101A           -1.95  -1.95  -1.67
##  3 103C           -2.08  -2.08  -1.58
##  4 105A           -1.61  -1.61  -1.21
##  5 107C          Inf     16.6    0.739
##  6 10B           Inf     16.6    0.305
##  7 111C           -1.61  -1.61  -1.24
##  8 112D            0.693  0.693  0.518
##  9 113A            0      0     -0.105
## 10 118B            0      0     -0.262
## # … with 281 more rows

Note that the n = 1 sponsor codes have estimates that are less extreme that the naive estimates. Also,

Let’s see the effect of the shrinkage induced by partial pooling by plotting the naive results versus the new results (finite data only):

pooled_plot <-
all_estimates %>%
dplyr::filter(is.finite(log_odds)) %>%
ggplot(aes(x = log_odds, y = glmer)) +
geom_abline(col = "red", alpha = .5) +
geom_point_interactive(aes(size = sqrt(n), tooltip = label),
alpha = .5) +
xlim(rng) + ylim(rng)

# Convert the plot to a format that the html file can handle
ggiraph(ggobj = pooled_plot)  

New levels are encoded as:

tidy(grants_glmer, number = 1) %>%
dplyr::filter(level == "..new") %>%
dplyr::select(-terms, -id)
## # A tibble: 1 × 2
##   level  value
##   <chr>  <dbl>
## 1 ..new -0.467

### Empirical Bayesian Methods/Mixed Models

The same generalized linear model can be fit using mixed models via a random intercept. The lme4 package can also be used to get pooled estimates via step_lencode_mixed.

grants_mixed <-
recipe(class ~ ., data = grants_other) %>%
step_lencode_mixed(
outcome = vars(class),
) %>%
prep(training = grants_other)

all_estimates <-
tidy(grants_mixed, number = 1) %>%
dplyr::select(-terms, -id) %>%

all_estimates %>%
dplyr::select(sponsor_code, log_odds, glm, glmer, mixed)
## # A tibble: 291 × 5
##    sponsor_code log_odds    glm  glmer   mixed
##    <chr>           <dbl>  <dbl>  <dbl>   <dbl>
##  1 100D            0.405  0.405  0.273  0.222
##  2 101A           -1.95  -1.95  -1.67  -1.57
##  3 103C           -2.08  -2.08  -1.58  -1.47
##  4 105A           -1.61  -1.61  -1.21  -1.14
##  5 107C          Inf     16.6    0.739  0.607
##  6 10B           Inf     16.6    0.305  0.218
##  7 111C           -1.61  -1.61  -1.24  -1.14
##  8 112D            0.693  0.693  0.518  0.470
##  9 113A            0      0     -0.105 -0.0949
## 10 118B            0      0     -0.262 -0.258
## # … with 281 more rows

Comparing the raw and mixed model estimates:

mixed_plot <-
all_estimates %>%
dplyr::filter(is.finite(log_odds)) %>%
ggplot(aes(x = log_odds, y = mixed)) +
geom_abline(col = "red", alpha = .5) +
geom_point_interactive(aes(size = sqrt(n), tooltip = label),
alpha = .5) +
xlim(rng) + ylim(rng)

ggiraph(ggobj = mixed_plot)

These values are very similar to the Bayesian estimates.