# Where does probably fit in?

## Introduction

An obvious question regarding probably might be: where does this fit in with the rest of the tidymodels ecosystem? Like the other pieces of the ecosystem, probably is designed to be modular, but plays well with other tidymodels packages.

Regarding placement in the modeling workflow, probably best fits in as a post processing step after the model has been fit, but before the model performance has been calculated.

## Example

As an example, we’ll use parsnip to fit a logistic regression on some Lending Club loan data, and then use probably to investigate what happens to performance when you vary the threshold of what a “good” loan is.

library(parsnip)
library(probably)
library(dplyr)
library(rsample)
library(modeldata)
data("lending_club")

# I think it makes more sense to have "good" as the first level
# By default it comes as the second level
lending_club <- lending_club %>%
mutate(Class = relevel(Class, "good"))

# There are a number of columns in this data set, but we will only use a few
# for this example
lending_club <- select(lending_club, Class, annual_inc, verification_status, sub_grade)

lending_club
#> # A tibble: 9,857 x 4
#>    <fct>      <dbl> <fct>               <fct>
#>  1 good       35000 Not_Verified        C4
#>  2 good       72000 Verified            C1
#>  3 good       72000 Source_Verified     D1
#>  4 good      101000 Verified            C3
#>  5 good       50100 Source_Verified     A4
#>  6 good       32000 Source_Verified     B5
#>  7 good       65000 Not_Verified        A1
#>  8 good      188000 Not_Verified        B2
#>  9 good       89000 Source_Verified     B3
#> 10 good       48000 Not_Verified        C2
#> # … with 9,847 more rows

Let’s split this into 75% training and 25% testing for something to predict on.

# 75% train, 25% test
set.seed(123)

split <- initial_split(lending_club, prop = 0.75)

lending_train <- training(split)
lending_test  <- testing(split)

Before we do anything, let’s look at the counts of what we are going to be predicting, the Class of the loan.

count(lending_train, Class)
#> # A tibble: 2 x 2
#>   Class     n
#>   <fct> <int>
#> 1 good   7003
#> 2 bad     390

Clearly there is a large imbalance here with the number of good and bad loans. This is probably a good thing for the bank, but poses an interesting issue for us because we might want to ensure we are sensitive to the bad loans and are not overwhelmed by the number of good ones. One thing that we might do is downsample the number of good loans so that the total number of them is more in line with the number of bad loans. We could do this before fitting the model using recipes::step_downsample(), but for now, let’s continue with the data unchanged.

We’ll use parsnip’s logistic_reg() to create a model specification for logistic regression, set the engine to be glm and then actually fit the model using our data and the model formula.

logi_reg <- logistic_reg()
logi_reg_glm <- logi_reg %>% set_engine("glm")

# A small model specification that defines the type of model you are
# using and the engine
logi_reg_glm
#> Logistic Regression Model Specification (classification)
#>
#> Computational engine: glm

# Fit the model
logi_reg_fit <- fit(
logi_reg_glm,
formula = Class ~ annual_inc + verification_status + sub_grade,
data = lending_train
)

logi_reg_fit
#> parsnip model object
#>
#> Fit time:  258ms
#>
#> Call:  stats::glm(formula = Class ~ annual_inc + verification_status +
#>     sub_grade, family = stats::binomial, data = data)
#>
#> Coefficients:
#>                        (Intercept)                          annual_inc
#>                         -5.651e+00                           1.583e-06
#> verification_statusSource_Verified         verification_statusVerified
#>                          2.655e-02                           1.838e-01
#>                          1.725e-01                           1.192e+00
#>                         -1.211e+01                           1.221e+00
#>                          1.743e+00                           1.015e+00
#>                          1.934e+00                           1.814e+00
#>                          2.119e+00                           2.341e+00
#>                          2.311e+00                           2.392e+00
#>                          2.916e+00                           3.364e+00
#>                          3.211e+00                           3.171e+00
#>                          3.381e+00                           3.179e+00
#>                          3.289e+00                           3.401e+00
#>                          3.413e+00                           3.490e+00
#>                          3.537e+00                           3.756e+00
#>                          3.226e+00                           4.172e+00
#>                          4.161e+00                           4.369e+00
#>                          3.890e+00                           4.123e+00
#>                          4.225e+00                           3.234e+00
#>                          5.207e+00                           3.483e+00
#>
#> Degrees of Freedom: 7392 Total (i.e. Null);  7355 Residual
#> Null Deviance:       3054
#> Residual Deviance: 2735  AIC: 2811

The output of the parsnip fit() call is a parsnip model_fit object, but the underlying print method for the glm fit is used.

Now let’s predict on our testing set, and use type = "prob" to get class probabilities back rather than hard predictions. We will use these with probably to investigate performance.

predictions <- logi_reg_fit %>%
predict(new_data = lending_test, type = "prob")

#> # A tibble: 2 x 2
#>        <dbl>     <dbl>
#> 1      0.936    0.0642
#> 2      0.972    0.0283

lending_test_pred <- bind_cols(predictions, lending_test)

lending_test_pred
#> # A tibble: 2,464 x 6
#>         <dbl>     <dbl> <fct>      <dbl> <fct>               <fct>
#>  1      0.936   0.0642  good      35000  Not_Verified        C4
#>  2      0.972   0.0283  good      75000  Verified            B4
#>  3      0.967   0.0334  good      89000  Source_Verified     B5
#>  4      0.976   0.0239  good     108000  Source_Verified     B1
#>  5      0.932   0.0685  good      62000  Source_Verified     C4
#>  6      0.957   0.0429  good      80662. Source_Verified     C3
#>  7      0.996   0.00444 good      42000  Not_Verified        A2
#>  8      0.889   0.111   good      81000  Source_Verified     E2
#>  9      0.960   0.0403  good      56000  Not_Verified        C3
#> 10      0.989   0.0111  good      74880  Source_Verified     B2
#> # … with 2,454 more rows

With our class probabilities in hand, we can use make_two_class_pred() to convert these probabilities into hard predictions using a threshold. A threshold of 0.5 just says that if the predicted probability is above 0.5, then classify this prediction as a “good” loan, otherwise, bad.

hard_pred_0.5 <- lending_test_pred %>%
mutate(.pred = make_two_class_pred(.pred_good, levels(Class), threshold = .5)) %>%
select(Class, contains(".pred"))

hard_pred_0.5 %>%
count(.truth = Class, .pred)
#> # A tibble: 2 x 3
#>   .truth      .pred     n
#>   <fct>  <clss_prd> <int>
#> 1 good         good  2337
#> 2 bad          good   127

Hmm, with a 0.5 threshold, almost all of the loans were predicted as “good”. Perhaps this has something to do with the large class imbalance. On the other hand, the bank might want to be more stringent with what is classified as a “good” loan, and might require a probability of 0.75 as the threshold.

hard_pred_0.75 <- lending_test_pred %>%
mutate(.pred = make_two_class_pred(.pred_good, levels(Class), threshold = .75)) %>%
select(Class, contains(".pred"))

hard_pred_0.75 %>%
count(.truth = Class, .pred)
#> # A tibble: 4 x 3
#>   .truth      .pred     n
#>   <fct>  <clss_prd> <int>
#> 1 good         good  2333
#> 4 bad           bad     3

In this case, 3 of the bad loans were correctly classified as bad, but more of the good loans were also misclassified as bad now. There is a tradeoff here, which can be somewhat captured by the metrics sensitivity and specificity. Both metrics have a max value of 1.

• sensitivity - The proportion of predicted “good” loans out of all “good” loans
• specificity - The proportion of predicted “bad” loans out of all “bad” loans
library(yardstick)

# Currently yardstick can't deal with the class_pred objects that come from
# probably, but it will be able to soon!
hard_pred_0.5 <- mutate(hard_pred_0.5, .pred = as.factor(.pred))
hard_pred_0.75 <- mutate(hard_pred_0.75, .pred = as.factor(.pred))

sens(hard_pred_0.5, Class, .pred)
#> # A tibble: 1 x 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 sens    binary             1
spec(hard_pred_0.5, Class, .pred)
#> # A tibble: 1 x 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 spec    binary             0

sens(hard_pred_0.75, Class, .pred)
#> # A tibble: 1 x 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 sens    binary         0.998
spec(hard_pred_0.75, Class, .pred)
#> # A tibble: 1 x 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 spec    binary        0.0236

In this example, as we increased specificity (by capturing those 3 bad loans with a higher threshold), we lowered sensitivity (by incorrectly reclassifying some of the good loans as bad). It would be nice to have some combination of these metrics to represent this tradeoff. Luckily, j_index is exactly that.

$j\_index = sens + spec - 1$

j_index has a maximum value of 1 when there are no false positives and no false negatives. It can be used as justification of whether or not an increase in the threshold value is worth it. If increasing the threshold results in more of an increase in the specificity than a decrease in the sensitivity, we can see that with j_index.

j_index(hard_pred_0.5, Class, .pred)
#> # A tibble: 1 x 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 j_index binary             0
j_index(hard_pred_0.75, Class, .pred)
#> # A tibble: 1 x 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 j_index binary        0.0219

Now, this is not the only way to optimize things. If you care about low false positives, you might be more interested in keeping sensitivity high, and this wouldn’t be the best way to tackle this problem. But for now, let’s see how we can use probably to optimize the j_index.

threshold_perf() will recalculate a number of metrics across varying thresholds. One of these is j_index.

threshold_data <- lending_test_pred %>%
threshold_perf(Class, .pred_good, thresholds = seq(0.5, 1, by = 0.0025))

threshold_data %>%
filter(.threshold %in% c(0.5, 0.6, 0.7))
#> # A tibble: 12 x 4
#>    .threshold .metric  .estimator .estimate
#>         <dbl> <chr>    <chr>          <dbl>
#>  1        0.5 sens     binary        1
#>  2        0.6 sens     binary        1.00
#>  3        0.7 sens     binary        1.00
#>  4        0.5 spec     binary        0
#>  5        0.6 spec     binary        0.0157
#>  6        0.7 spec     binary        0.0157
#>  7        0.5 j_index  binary        0
#>  8        0.6 j_index  binary        0.0153
#>  9        0.7 j_index  binary        0.0153
#> 10        0.5 distance binary        1
#> 11        0.6 distance binary        0.969
#> 12        0.7 distance binary        0.969

With ggplot2, we can easily visualize this varying performance to find our optimal threshold for maximizing j_index.

library(ggplot2)

threshold_data <- threshold_data %>%
filter(.metric != "distance") %>%
mutate(group = case_when(
.metric == "sens" | .metric == "spec" ~ "1",
TRUE ~ "2"
))

max_j_index_threshold <- threshold_data %>%
filter(.metric == "j_index") %>%
filter(.estimate == max(.estimate)) %>%
pull(.threshold)

ggplot(threshold_data, aes(x = .threshold, y = .estimate, color = .metric, alpha = group)) +
geom_line() +
theme_minimal() +
scale_color_viridis_d(end = 0.9) +
scale_alpha_manual(values = c(.4, 1), guide = "none") +
geom_vline(xintercept = max_j_index_threshold, alpha = .6, color = "grey30") +
labs(
x = "'Good' Threshold\n(above this value is considered 'good')",
y = "Metric Estimate",
title = "Balancing performance by varying the threshold",
subtitle = "Sensitivity or specificity alone might not be enough!\nVertical line = Max J-Index"
)

It’s clear from this visual that the optimal threshold is very high, exactly 0.9575. This is pretty high, so again, this optimization method won’t be useful for all cases. To wrap up, here are all of the test set metrics for that threshold value.

threshold_data %>%
filter(.threshold == max_j_index_threshold)
#> # A tibble: 3 x 5
#>   .threshold .metric .estimator .estimate group
#>        <dbl> <chr>   <chr>          <dbl> <chr>
#> 1      0.958 sens    binary         0.619 1
#> 2      0.958 spec    binary         0.756 1
#> 3      0.958 j_index binary         0.375 2