Using Generalized Linear Models • embed

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 %>%
  group_by(sponsor_code) %>%
  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
##    sponsor_code  prop log_odds     n
##    <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
## # ℹ 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
  step_lencode_glm(sponsor_code, outcome = vars(class)) %>%
  # 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    
## # ℹ 282 more rows

glm_estimates <-
  glm_estimates %>%
  set_names(c("sponsor_code", "glm")) %>%
  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.   :-1.332e-15  
##  1st Qu.:-2.220e-16  
##  Median : 0.000e+00  
##  Mean   :-5.139e-17  
##  3rd Qu.: 1.604e-16  
##  Max.   : 8.882e-16

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.

# due to Matrix problems
knitr::knit_exit()