Example 0: Multi-Accuracy in 6 lines of code

As a brief introduction we show how to use mcboost in only 6 lines of code. For our example, we use the data from the sonar binary classification task. We instantiate a MCBoost instance by specifying a auditor_fitter. This auditor_fitter defines the splits into groups in each boosting iteration based on the obtained residuals. In this example, we choose a Tree based model. Afterwards, we run the $multicalibrate() method on our data to start multi-calibration. We only use the first 200 samples of the sonar data set to train our multi-calibrated model. tsk = tsk("sonar") d = tsk$data(cols = tsk$feature_names) l = tsk$data(cols = tsk$target_names)[] mc = MCBoost$new(auditor_fitter = "TreeAuditorFitter")
mc$multicalibrate(d[1:200,], l[1:200]) After the calibration, we use the model to predict on the left-out data (8 observations). mc$predict_probs(d[201:208,])
#>  0.9077953 0.9077953 1.0000000 1.0000000 0.3387929 1.0000000 0.3387929
#>  0.2116734

What does mcboost do?

Internally mcboost runs the following procedure max_iter times:

1. Predict on X using the model from the previous iteration, init_predictor in the first iteration.
2. Compute the residuals res = y - y_hat
3. Split predictions into num_buckets according to y_hat.
4. Fit the auditor (auditor_fitter) (here calledc(x)) on the data in each bucket with target variable r.
5. Compute misscal = mean(c(x) * res(x))
6. if misscal > alpha: For the bucket with highest misscal, update the model using the prediction c(x). else: Stop the procedure

A lot more details can be found either in the code, or in the corresponding publications.

Example 1: Multi-Accuracy Boosting on the Adult Dataset

library(data.table)
stringsAsFactors = TRUE
)
adult_train$Country = NULL adult_train$fnlwgt = NULL
train_tsk = TaskClassif$new("adult_train", adult_train, target = "Target") We removed the features Country and fnlwgt since we expect them to have no predictive power. fnlwgt means final weight and aims to allocate similar weights to people with similar demographic characteristics, while Country has 42 distinct levels but 89 % of the observations are from the United States. 1.1 Preprocessing Then we do basic preprocessing: • Collapse rarest factors according to their prevalence • Drop missing factor levels • One-hot encode categorical variables • Impute NA’s using a histogram approach library(mlr3pipelines) pipe = po("collapsefactors", no_collapse_above_prevalence = 0.0006) %>>% po("fixfactors") %>>% po("encode") %>>% po("imputehist") prep_task = pipe$train(train_tsk)[]

In order to simulate settings where a sensitive feature is not available, we remove the (dummy encoded) feature Race from the training task.

prep_task$set_col_roles(c("Race.Amer.Indian.Eskimo", "Race.Asian.Pac.Islander", "Race.Black", "Race.Other", "Race.White"), remove_from = "feature") Now we fit a random forest. library(mlr3learners) l = lrn("classif.ranger", num.trees = 10L, predict_type = "prob") l$train(prep_task)

1.2 MCBoost

A simple way to use the predictions from any model in mcboost is to wrap the predict function and provide it as an initial predictor. This can be done from any model / any library. Note, that we have to make sure, that our init_predictor returns a numeric vector of predictions.

init_predictor = function(data) {
l$predict_newdata(data)$prob[, 2]
}

As mcboost requires the data to be provided in X, y format (a data.table or data.frame of features and a vector of labels), we create those two objects.

data = prep_task$data(cols = prep_task$feature_names)
labels = 1 - one_hot(prep_task$data(cols = prep_task$target_names)[])

We use a ridge regularized linear regression model as the auditor.

mc = MCBoost$new(auditor_fitter = "RidgeAuditorFitter", init_predictor = init_predictor) mc$multicalibrate(data, labels)

The print method additionally lists the average auditor values in the different buckets in each iteration:

mc
#> <MCBoost>
#>   Public:
#>     alpha: 1e-04
#>     auditor_effect: function (x, aggregate = TRUE, t = Inf, predictor_args = NULL,
#>     auditor_effects: list
#>     auditor_fitter: RidgeAuditorFitter, LearnerAuditorFitter, AuditorFitter, R6
#>     bucket_strategies: simple
#>     bucket_strategy: simple
#>     clone: function (deep = FALSE)
#>     eta: 1
#>     eval_fulldata: FALSE
#>     initialize: function (max_iter = 5, alpha = 1e-04, eta = 1, num_buckets = 2,
#>     iter_corr: list
#>     iter_models: list
#>     iter_partitions: list
#>     iter_sampling: none
#>     max_iter: 5
#>     multicalibrate: function (data, labels, predictor_args = NULL, audit = FALSE,
#>     multiplicative: TRUE
#>     num_buckets: 2
#>     partition: TRUE
#>     predict_probs: function (x, t = Inf, predictor_args = NULL, audit = FALSE, ...)
#>     predictor: function (data)
#>     print: function (...)
#>     rebucket: FALSE
#>   Private:
#>     assert_labels: function (labels, ...)
#>     assert_prob: function (prob, data, ...)
#>     calculate_corr: function (auditor, data, resid, idx)
#>     compute_residuals: function (prediction, labels)
#>     create_buckets: function (pred_probs)
#>     get_auditor_fitter: function (subpops, auditor_fitter)
#>     get_masked: function (data, resid, idx, probs, bucket)
#>     get_predictor: function (init_predictor, default_model_class)
#>     get_probs: function (pred_probs, new_probs)
#>     update_probs: function (orig_preds, model, x, mask = NULL, audit = FALSE, update_sign = 1,
#> Fitted Multi-calibration model (5 iters)
#> Correlations per iteration:
#>    iter     Bucket_1    Bucket_2     Bucket_3
#> 1:    1 0.0002345878 0.001607495 0.0111570208
#> 2:    2 0.0003028562 0.001607421 0.0018036510
#> 3:    3 0.0004304191 0.001607421 0.0005702663
#> 4:    4 0.0005746641 0.002004285 0.0005702663
#> 5:    5 0.0007652389 0.002490407 0.0005702663

1.3 Evaluation on Test Data

stringsAsFactors = TRUE
)
adult_test$Country = NULL adult_test$fnlwgt = NULL

# The first row seems to have an error
adult_test$Target = droplevels(adult_test$Target)

# Note, that we have to convert columns from numeric to integer here:
sdc = train_tsk$feature_types[type == "integer", id] adult_test[, (sdc) := lapply(.SD, as.integer), .SDcols = sdc] test_tsk = TaskClassif$new("adult_test", adult_test, target = "Target")
prep_test = pipe$predict(test_tsk)[] Now, we can again extract X, y. test_data = prep_test$data(cols = prep_test$feature_names) test_labels = 1 - one_hot(prep_test$data(cols = prep_test$target_names)[]) and predict. prs = mc$predict_probs(test_data)

The accuracy of the multi-calibrated model

mean(round(prs) == test_labels)
#>  0.8396904

is similar to the non-calibrated model.

mean(round(init_predictor(test_data)) == test_labels)
#>  0.8379706

But if we have a look at the bias for the different subpopulations of feature Race, we can see that the predictions got more calibrated. Note that we did not explicitly give neither the initial model nor the auditor access to the feature Race.

# Get bias per subgroup for multi-calibrated predictor
adult_test$biasmc = (prs - test_labels) adult_test[, .(abs(mean(biasmc)), .N), by = .(Race)] #> Race V1 N #> 1: Black 0.008414768 1561 #> 2: White 0.053498900 13946 #> 3: Asian-Pac-Islander 0.064492304 480 #> 4: Other 0.068950996 135 #> 5: Amer-Indian-Eskimo 0.002438774 159 # Get bias per subgroup for initial predictor adult_test$biasinit = (init_predictor(test_data) - test_labels)
adult_test[, .(abs(mean(biasinit)), .N), by = .(Race)]
#>                  Race          V1     N
#> 1:              Black 0.017961316  1561
#> 2:              White 0.066077755 13946
#> 3: Asian-Pac-Islander 0.074912961   480
#> 4:              Other 0.077236911   135
#> 5: Amer-Indian-Eskimo 0.009339302   159

1.4 The Auditor Effect

We can also obtain the auditor effect after multicalibration. This indicates “how much” each observation has been affected by multi-calibration (on average across iterations).

Example 2: MCBoost with non-mlr3 models: GLM

mcboost does not require your model to be a mlr3 model. As an input, mcboost expects a function init_predictor that takes as input data and returns a prediction.

tsk = tsk("sonar")
data = tsk$data()[, Class := as.integer(Class) - 1L] mod = glm(data = data, formula = Class ~ .) The init_predictor could then use the glm model: init_predictor = function(data) { predict(mod, data) } … and we can calibrate this predictor. d = data[, -1] l = data$Class
mc = MCBoost$new(init_predictor = init_predictor) mc$multicalibrate(d[1:200,], l[1:200])
mc$predict_probs(d[201:208,]) #> 1 2 3 4 5 6 #> 0.0001870999 0.0178598234 0.4486733199 0.0001331085 0.1418422155 0.2885341698 #> 7 8 #> 0.2571026429 0.7356629651 Example 3: Avoiding Overfitting in MCBoost Very often MCBoost’s calibration is very aggressive and tends to overfit. This section tries to introduce a method to regularize against this overfitting. 3.1 CVLearner In this section we use a Cross-Validated learner that predicts on held-out data during the training phase. This idea is based on Wolpert (1992)’s Stacked Generalization. Other, simpler methods include choosing a smaller step size eta or reducing the number of iters. tsk = tsk("sonar") As an init_predictor we again use a ranger model from mlr3 and construct an init predictor using the convenience function provided by mcboost. learner = lrn("classif.ranger", predict_type = "prob") learner$train(tsk)
init_predictor = mlr3_init_predictor(learner)

… and we can calibrate this predictor. This time, we use a CVTreeAuditorFitter instead of a TreeAuditorFitter. This allows us to avoid overfitting similar to a technique coined stacked generalization first described by Wolpert in 1992. Note, that this can sometimes take a little longer since each learner is cross-validated using 3 folds (default).

d = data[, -1]
l = data$Class mc = MCBoost$new(init_predictor = init_predictor, auditor_fitter=CVTreeAuditorFitter$new(), max_iter = 2L) mc$multicalibrate(d[1:200,], l[1:200])
mc$predict_probs(d[201:208,]) #>  0.6506734 0.6605665 0.6303917 0.6480143 0.6037770 0.6019906 0.5845569 #>  0.5364872 3.2 Data Splitting We can also use a fresh chunk of the validation data in each iteration. mcboost implements two strategies, "bootstrap" and "split". While "split" simply splits up the data, "bootstrap" draws a new bootstrap sample of the data in each iteration. tsk = tsk("sonar") Again, we use a ranger mlr3 model as our initial predictor: learner = lrn("classif.ranger", predict_type = "prob") learner$train(tsk)
init_predictor = mlr3_init_predictor(learner)

and we can now calibrate:

d = data[, -1]
l = data$Class mc = MCBoost$new(
init_predictor = init_predictor,
auditor_fitter= TreeAuditorFitter$new(), iter_sampling = "bootstrap" ) mc$multicalibrate(d[1:200,], l[1:200])
mc$predict_probs(d[201:208,]) #>  0.6458064 0.6708513 0.6471905 0.6676154 0.5803027 0.6248506 0.5799262 #>  0.5749539 Example 4: Adjusting the SubPop Fitter For this example, we use the sonar dataset once again: tsk = tsk("sonar") data = tsk$data(cols = tsk$feature_names) labels = tsk$data(cols = tsk$target_names)[] 4.1 LearnerAuditorFitter The Subpop-fitter can be easily adjusted by constructing it from a LearnerAuditorFitter. This allows for using any mlr3 learner. See here for a list of available learners. rf = LearnerAuditorFitter$new(lrn("regr.rpart", minsplit = 10L))
mc = MCBoost$new(auditor_fitter = rf) mc$multicalibrate(data, labels)

The TreeAuditorFitter and RidgeAuditorFitter are two instantiations of this Fitter with pre-defined learners. By providing their character strings the fitter could be automatically constructed.

4.2 SubpopAuditorFitter & SubgroupAuditorFitter

In some occasions, instead of using a Learner, we might want to use a fixed set of subgroups. Those can either be defined from the data itself or provided from the outside.

Splitting via the dataset

In order to split the data into groups according to a set of columns, we use a SubpopAuditorFitter together with a list of subpops. Those define the group splits to multi-calibrate on. These splits can be either a character string, referencing a binary variable in the data or a function that, when evaluated on the data, returns a binary vector.

In order to showcase both options, we add a binary variable to our data:

data[, Bin := sample(c(1, 0), nrow(data), replace = TRUE)]
rf = SubpopAuditorFitter$new(list( "Bin", function(data) {data[["V1"]] > 0.2}, function(data) {data[["V1"]] > 0.2 | data[["V3"]] < 0.29} )) mc = MCBoost$new(auditor_fitter = rf)
mc$multicalibrate(data, labels) And we can again apply it to predict on new data: mc$predict_probs(data)
#>    0.500000 0.500000 0.850072 0.500000 0.500000 0.500000 0.500000 0.500000
#>    0.500000 0.500000 0.850072 0.850072 0.500000 0.850072 0.500000 0.500000
#>   0.500000 0.500000 0.500000 0.500000 0.850072 0.850072 0.500000 0.500000
#>   0.850072 0.850072 0.850072 0.500000 0.500000 0.850072 0.500000 0.850072
#>   0.850072 0.850072 0.500000 0.500000 0.500000 0.500000 0.850072 0.500000
#>   0.850072 0.850072 0.850072 0.850072 0.500000 0.850072 0.850072 0.850072
#>   0.850072 0.500000 0.500000 0.500000 0.850072 0.500000 0.500000 0.500000
#>   0.500000 0.500000 0.500000 0.500000 0.850072 0.850072 0.500000 0.500000
#>   0.500000 0.500000 0.850072 0.500000 0.850072 0.850072 0.850072 0.500000
#>   0.850072 0.500000 0.500000 0.850072 0.850072 0.500000 0.500000 0.500000
#>   0.500000 0.850072 0.500000 0.850072 0.850072 0.850072 0.850072 0.500000
#>   0.500000 0.500000 0.500000 0.850072 0.850072 0.500000 0.500000 0.500000
#>   0.850072 0.500000 0.500000 0.850072 0.850072 0.850072 0.850072 0.500000
#>  0.850072 0.850072 0.850072 0.850072 0.850072 0.500000 0.850072 0.850072
#>  0.850072 0.850072 0.850072 0.850072 0.500000 0.850072 0.850072 0.850072
#>  0.850072 0.500000 0.500000 0.500000 0.850072 0.850072 0.500000 0.500000
#>  0.850072 0.850072 0.500000 0.850072 0.500000 0.500000 0.500000 0.850072
#>  0.850072 0.850072 0.850072 0.850072 0.850072 0.850072 0.500000 0.850072
#>  0.500000 0.500000 0.850072 0.850072 0.850072 0.850072 0.850072 0.850072
#>  0.500000 0.850072 0.500000 0.850072 0.500000 0.500000 0.850072 0.500000
#>  0.850072 0.850072 0.500000 0.500000 0.850072 0.850072 0.500000 0.500000
#>  0.500000 0.850072 0.850072 0.850072 0.500000 0.850072 0.850072 0.850072
#>  0.500000 0.850072 0.500000 0.850072 0.500000 0.850072 0.850072 0.500000
#>  0.850072 0.850072 0.500000 0.850072 0.850072 0.500000 0.850072 0.500000
#>  0.850072 0.500000 0.500000 0.500000 0.500000 0.850072 0.850072 0.850072
#>  0.850072 0.850072 0.500000 0.500000 0.500000 0.850072 0.850072 0.500000

If we want to add the splitting from the outside, by supplying binary masks for the rows of the data, we can provide manually defined masks. Note, that the masks have to correspond with the number of rows in the dataset.

rf = SubgroupAuditorFitter$new(list( rep(c(0, 1), 104), rep(c(1, 1, 1, 0), 52) )) mc = MCBoost$new(auditor_fitter = rf)
mc$multicalibrate(data, labels) During prediction, we now have to supply a set of masks for the prediction data. predict_masks = list( rep(c(0, 1), 52), rep(c(1, 1, 1, 0), 26) ) mc$predict_probs(data[1:104,], subgroup_masks = predict_masks)
#>    0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895 0.5244895
#>    0.5303758 0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895
#>   0.5244895 0.5303758 0.5244895 0.5244895 0.5244895 0.5303758 0.5244895
#>   0.5244895 0.5244895 0.5303758 0.5244895 0.5244895 0.5244895 0.5303758
#>   0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895 0.5244895
#>   0.5303758 0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895
#>   0.5244895 0.5303758 0.5244895 0.5244895 0.5244895 0.5303758 0.5244895
#>   0.5244895 0.5244895 0.5303758 0.5244895 0.5244895 0.5244895 0.5303758
#>   0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895 0.5244895
#>   0.5303758 0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895
#>   0.5244895 0.5303758 0.5244895 0.5244895 0.5244895 0.5303758 0.5244895
#>   0.5244895 0.5244895 0.5303758 0.5244895 0.5244895 0.5244895 0.5303758
#>   0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895 0.5244895
#>   0.5303758 0.5244895 0.5244895 0.5244895 0.5303758 0.5244895 0.5244895
#>   0.5244895 0.5303758 0.5244895 0.5244895 0.5244895 0.5303758

Example 5: Multi-Calibrating data with missing values using a pipeline

When data has missing values or other non-standard columns, we often have to pre-process data in order to be able to fit models. Those preprocessing steps can be embedded into the SubPopFitter by using a mlr3pipelines Pipeline. The following code shows a brief example:

tsk = tsk("penguins")
# first we convert to a binary task
row_ids = tsk$data(cols = c("species", "..row_id"))[species %in% c("Adelie", "Gentoo")][["..row_id"]] tsk$filter(row_ids)$droplevels() tsk #> <TaskClassif:penguins> (276 x 8) #> * Target: species #> * Properties: twoclass #> * Features (7): #> - int (3): body_mass, flipper_length, year #> - dbl (2): bill_depth, bill_length #> - fct (2): island, sex library("mlr3pipelines") library("mlr3learners") # Convert task to X,y X = tsk$data(cols = tsk$feature_names) y = tsk$data(cols = tsk$target_names) # Our inital model is a pipeline that imputes missings and encodes categoricals init_model = as_learner(po("encode") %>>% po("imputehist") %>>% lrn("classif.glmnet", predict_type = "prob")) # And we fit it on a subset of the data in order to simulate a poorly performing model. init_model$train(tsk$clone()$filter(row_ids[c(1:9, 160:170)]))
init_model$predict(tsk)$score()
#> Warning: Multiple lambdas have been fit. Lambda will be set to 0.01 (see parameter 's').
#> This happened PipeOp classif.glmnet's $predict() #> classif.ce #> 0.1594203 # We define a pipeline that imputes missings and encodes categoricals auditor = as_learner(po("encode") %>>% po("imputehist") %>>% lrn("regr.rpart")) mc = MCBoost$new(auditor_fitter = auditor, init_predictor = init_model)
mc$multicalibrate(X, y) #> Warning: Multiple lambdas have been fit. Lambda will be set to 0.01 (see parameter 's'). #> This happened PipeOp classif.glmnet's$predict()

and we can observe where it improved:

mc
#> <MCBoost>
#>   Public:
#>     alpha: 1e-04
#>     auditor_effect: function (x, aggregate = TRUE, t = Inf, predictor_args = NULL,
#>     auditor_effects: list
#>     auditor_fitter: LearnerAuditorFitter, AuditorFitter, R6
#>     bucket_strategies: simple
#>     bucket_strategy: simple
#>     clone: function (deep = FALSE)
#>     eta: 1
#>     eval_fulldata: FALSE
#>     initialize: function (max_iter = 5, alpha = 1e-04, eta = 1, num_buckets = 2,
#>     iter_corr: list
#>     iter_models: list
#>     iter_partitions: list
#>     iter_sampling: none
#>     max_iter: 5
#>     multicalibrate: function (data, labels, predictor_args = NULL, audit = FALSE,
#>     multiplicative: TRUE
#>     num_buckets: 2
#>     partition: TRUE
#>     predict_probs: function (x, t = Inf, predictor_args = NULL, audit = FALSE, ...)
#>     predictor: function (data, ...)
#>     print: function (...)
#>     rebucket: FALSE
#>   Private:
#>     assert_labels: function (labels, ...)
#>     assert_prob: function (prob, data, ...)
#>     calculate_corr: function (auditor, data, resid, idx)
#>     compute_residuals: function (prediction, labels)
#>     create_buckets: function (pred_probs)
#>     get_auditor_fitter: function (subpops, auditor_fitter)
#>     get_masked: function (data, resid, idx, probs, bucket)
#>     get_predictor: function (init_predictor, default_model_class)
#>     get_probs: function (pred_probs, new_probs)
#>     update_probs: function (orig_preds, model, x, mask = NULL, audit = FALSE, update_sign = 1,
#> Fitted Multi-calibration model (5 iters)
#> Correlations per iteration:
#>    iter   Bucket_1  Bucket_2 Bucket_3
#> 1:    1 0.15308560 0.2487837        0
#> 2:    2 0.14699794 0.2414251        0
#> 3:    3 0.13422070 0.2204713        0
#> 4:    4 0.11225541 0.1844612        0
#> 5:    5 0.06501729 0.1092000        0

Example 6: Multi-Calibration Regression

We abuse the Communities & Crime dataset in order to showcase how mcboost can be used in a regression setting.

library(data.table)
library(mlr3oml)
oml = OMLData$new(42730) data = oml$data

tsk = TaskRegr$new("communities_crime", data, target = "ViolentCrimesPerPop") Currently, mcboost only allows to work with targets between 0 and 1. Luckily, our target variable’s values are already in that range, but if they were not, we could simply scale them to [0;1] before our analysis. summary(data$ViolentCrimesPerPop)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>   0.000   0.070   0.150   0.238   0.330   1.000

We again split our task into train and test. We do this in mlr3 by simply setting some (here 500) row roles to "validation".

tsk$set_row_roles(sample(tsk$row_roles$use, 500), "validation") 6.1 Preprocessing Then we do basic preprocessing, since we do not have any categorical variables, we only impute NA’s using a histogram approach. library(mlr3pipelines) pipe = po("imputehist") prep_task = pipe$train(list(tsk))[]

prep_task$set_col_roles(c("racepctblack", "racePctWhite", "racePctAsian", "racePctHisp", "community"), remove_from = "feature") Now we fit our first Learner: A random forest. library(mlr3learners) l = lrn("regr.ranger", num.trees = 10L) l$train(prep_task)

6.2 MCBoost

A simple way to use the predictions from any Model in mcboost is to wrap the predict function and provide it as an initial predictor. This can be done from any model / any library. Note, that we have to make sure, that our init_predictor returns a numeric vector of predictions.

init_predictor = function(data) {
l$predict_newdata(data)$response
}

As mcboost requires the data to be provided in X, y format (a data.table or data.frame of features and a vector of labels), we create those two objects.

data = prep_task$data(cols = prep_task$feature_names)
labels = prep_task$data(cols = prep_task$target_names)[]
mc = MCBoost$new(auditor_fitter = "RidgeAuditorFitter", init_predictor = init_predictor, eta = 0.1) mc$multicalibrate(data, labels)

6.3 Evaluation on Test Data

We first create the test task by setting the validation rows to use, and then use our preprocessing pipe's predict function to also impute missing values for the validation data. Then we again extract features X and target y.

test_task = tsk$clone() test_task$row_roles$use = test_task$row_roles$validation test_task = pipe$predict(list(test_task))[]
test_data = test_task$data(cols = tsk$feature_names)
test_labels = test_task$data(cols = tsk$target_names)[]

and predict.

prs = mc$predict_probs(test_data) Now we can compute the MSE of the multi-calibrated model mean((prs - test_labels)^2) #>  0.01909034 and compare to the non-calibrated version: mean((init_predictor(test_data) - test_labels)^2) #>  0.01880431 But looking at sub-populations we can see that the predictions got more calibrated. Since we cannot show all subpopulations we only show the MSE for the feature racepctblack. test_data$se_mcboost = (prs - test_labels)^2
test_data\$se_init = (init_predictor(test_data) - test_labels)^2

test_data[, .(mcboost = mean(se_mcboost), initial = mean(se_init), .N), by = .(racepctblack > 0.5)]
#>    racepctblack    mcboost    initial   N
#> 1:        FALSE 0.01530715 0.01522334 447
#> 2:         TRUE 0.05099764 0.04900611  53