Statistical Group Fairness Metrics

One way to assess the fairness of a model is to find a definition of fairness that is relevant to a problem at hand. We might for example define a model to be fair if the chance to be accepted for a job given you are qualified is independent of a protected attribute e.g. gender. This can e.g. be measured using the true positive rate(TPR): in mlr3 this metric is called "classif.tpr". In this case we measure discrepancies between groups by computing differences (-) but we could also compute quotients. In practice, we often compute absolute differences.

\[ \Delta_{TPR} = TPR_{Group 1} - TPR_{Group 2} \]

We will use metrics like the one defined above throughout the remainder of this vignette. Some predefined measures are readily available in mlr3fairness, but we will also showcase how custom measures can be constructed below.

In general, fairness measures have a fairness. prefix followed by the metric that is compared across groups. We will thus e.g. call the difference in accuracies across groups fairness.acc. A full list can be found below.

Assessing for Bias: A first look

This vignette assumes that you are familiar with the basics of mlr3 and it’s core objects. The mlr3 book can be a great resource in case you want to learn more about mlr3’s internals.

We first start by training a model for which we want to conduct an audit. For this example, we use the adult_train dataset. Keep in mind all the datasets from mlr3fairness package already set protected attribute via the col_role “pta”, here the “sex” column. To speed things up, we only use the first 1000 rows.

library(mlr3fairness)
library(mlr3learners)

t = tsk("adult_train")$filter(1:1000)
t$col_roles$pta
#> [1] "sex"

Our model is a random forest model fitted on the dataset:

l = lrn("classif.ranger")
l$train(t)

We can now predict on a new dataset and use those predictions to assess for bias:

test = tsk("adult_test")
prd = l$predict(test)

Using the $score method and a measure we can e.g. compute the absolute differences in true positive rates.

prd$score(msr("fairness.tpr"), task = test)
#> fairness.tpr 
#>   0.08266254

The exact measure to choose is often data-set and situation dependent. The Aequitas Fairness Tree can be a great ressource to get started.

We can furthermore simply look at the per-group measures:

meas = groupwise_metrics(msr("classif.tpr"), test)
prd$score(meas, task = test)
#>   subgroup.tpr_Male subgroup.tpr_Female 
#>           0.8920095           0.9746721

Fairness Measures - An in-depth look

Before, we have used msr("fairness.tpr") to assess differences in false positive rates across groups. But what happens internally?

The msr() function is used to obtain a Measure from a dictionary of pre-defined Measures. We can use msr() without any arguments in order to print a list of available measures. In the following example, we will build a Measure that computes differences in False Positive Rates making use of the "classif.fpr" measure readily implemented in mlr3.

# Binary Class false positive rates
msr("classif.fpr")
#> <MeasureBinarySimple:classif.fpr>: False Positive Rate
#> * Packages: mlr3, mlr3measures
#> * Range: [0, 1]
#> * Minimize: TRUE
#> * Average: macro
#> * Parameters: list()
#> * Properties: -
#> * Predict type: response

The core Measure in mlr3fairness is a MeasureFairness. It can be used to construct arbitrary measures that compute a difference between a specific metric across groups. We can therefore build a new metric as follows:

m1 = MeasureFairness$new(base_measure = msr("classif.fpr"), operation = function(x) {abs(x[1] - x[2])})
m1
#> <MeasureFairness:fairness.fpr>
#> * Packages: mlr3, mlr3fairness
#> * Range: [-Inf, Inf]
#> * Minimize: TRUE
#> * Average: macro
#> * Parameters: list()
#> * Properties: requires_task
#> * Predict type: response

This measure does the following steps: - Compute the metric supplied as base_measure in each group defined by the "pta" column. - Compute operation (here abs(x[1] - x[2])) and return the result.

In some cases, we might also want to replace the operation with a different operation, e.g. x[1] / x[2] in order to compute a different perspective.

mlr3fairness comes with two built-in functions that can be used to compute fairness metrics also across protected attributes that have more than two classes.

• groupdiff_absdiff: maximum absolute difference between all classes (the default for all metrics)
• groupdiff_tau: minimum quotient between all classes

Note: Depending on the operation we need to set a different minimize flag for the measure, so subsequent operations based on the measure automatically know if the measure is to be minimized or maximized e.g. during tuning.

We can also use those operations to construct a measure using msr(), since MeasureFairness (key: msr("fairness")) can be constructed from the dictionary with additional arguments.

m2 = msr("fairness", operation = groupdiff_absdiff, base_measure = msr("classif.tpr"))

This allows us to construct pretty flexible metrics e.g. for regression settings:

m2 = msr("fairness", operation = groupdiff_absdiff, base_measure = msr("regr.mse"))

Composite Fairness Measures

Some fairness measures also require a combination of multiple Fairness Metrics. In the following example we show how to compute the mean of two fairness metrics, here false negative and true negative rates and create a new Measure that computes the mean (see aggfun) of those metrics:

ms = list(msr("fairness.fnr"), msr("fairness.tnr"))
ms
#> [[1]]
#> <MeasureFairness:fairness.fnr>
#> * Packages: mlr3, mlr3fairness
#> * Range: [0, 1]
#> * Minimize: TRUE
#> * Average: macro
#> * Parameters: list()
#> * Properties: requires_task
#> * Predict type: response
#> 
#> [[2]]
#> <MeasureFairness:fairness.tnr>
#> * Packages: mlr3, mlr3fairness
#> * Range: [0, 1]
#> * Minimize: TRUE
#> * Average: macro
#> * Parameters: list()
#> * Properties: requires_task
#> * Predict type: response

m = MeasureFairnessComposite$new(measures = ms, aggfun = mean)

How to compare the fairness performance between different learners using Benchmarks

In this example, we create a BenchmarkInstance. Then by using aggregate() function they could access the fairness measures easily. The following example demonstrates the process to evaluate the fairness metrics on Benchmark Results:

design = benchmark_grid(
  tasks = tsks("adult_train"),
  learners = lrns(c("classif.ranger", "classif.rpart"),
    predict_type = "prob", predict_sets = c("train", "test")),
  resamplings = rsmps("cv", folds = 3)
)

bmr = benchmark(design)
#> INFO  [22:45:23.870] [mlr3] Running benchmark with 6 resampling iterations
#> INFO  [22:45:23.874] [mlr3] Applying learner 'classif.ranger' on task 'adult_train' (iter 1/3)
#> INFO  [22:45:31.106] [mlr3] Applying learner 'classif.ranger' on task 'adult_train' (iter 2/3)
#> INFO  [22:45:38.404] [mlr3] Applying learner 'classif.ranger' on task 'adult_train' (iter 3/3)
#> INFO  [22:45:44.961] [mlr3] Applying learner 'classif.rpart' on task 'adult_train' (iter 1/3)
#> INFO  [22:45:45.056] [mlr3] Applying learner 'classif.rpart' on task 'adult_train' (iter 2/3)
#> INFO  [22:45:45.158] [mlr3] Applying learner 'classif.rpart' on task 'adult_train' (iter 3/3)
#> INFO  [22:45:45.243] [mlr3] Finished benchmark

# Operations have been set to `groupwise_quotient()`
measures = list( msr("fairness.tpr"), msr("fairness.npv"), msr("fairness.acc"), msr("classif.acc") )

tab = bmr$aggregate(measures)
tab
#>    nr     task_id     learner_id resampling_id iters fairness.tpr fairness.npv
#> 1:  1 adult_train classif.ranger            cv     3   0.06378266   0.01906678
#> 2:  2 adult_train  classif.rpart            cv     3   0.05976163   0.03593223
#>    fairness.acc classif.acc
#> 1:    0.1051482   0.8600495
#> 2:    0.1218701   0.8407775
#> Hidden columns: resample_result

Metrics aggregation - groupdiff_*

For MeasureFairness, mlr3 computes the base_measure in each group specified by the pta column. If we now want to return those measures, we need to aggregate this to a single metric - e.g. using one of the groupdiff_* functions available with mlr3. See ?groupdiff_tau for a list. Note, that the operation below also accepts custom aggregation function, see the example below.

msr("fairness.acc", operation = groupdiff_diff)
#> <MeasureFairness:fairness.acc>
#> * Packages: mlr3, mlr3fairness
#> * Range: [0, 1]
#> * Minimize: TRUE
#> * Average: macro
#> * Parameters: list()
#> * Properties: requires_task
#> * Predict type: response

We can also report other metrics, e.g. the error in a specific group:

t = tsk("adult_train")$filter(1:1000)
mm = msr("fairness.acc", operation = function(x) {x["Female"]})
l = lrn("classif.rpart")
prds = l$train(t)$predict(t)
prds$score(mm, t)
#> fairness.acc 
#>    0.9404389

Individual Fairness

Individual fairness notions were first proposed by (Dwork et al. 2012). The core idea comes from the principle of treating similar cases similarly and different cases differently. In contrast to statistical group fairness notions, this notion allows assessing fairness at an individual level and would therefore allow determining whether an individual is treated fairly. A more in-depth treatment of individual fairness notions is given by (Binns 2020).

In order to translate this from an abstract concept into practice, we need to define two distance metrics: - A distance metric \(d(x_i, x_j)\) that measures how similar two cases \(x_i\) and \(x_j\) are - A distance metric between treatments, here the predictions of our model \(f\): \(\phi(f({x}_i), f({x}_j))\).

Intuitively, we would now want, that if \(d(x_i, x_j)\) is small, the difference in predictions \(\phi(f({x}_i), f({x}_j))\) should also be small. This essentially requires Lipschitz continuity of \(f\) with respect to \(d\). Given a Lipschitz constant \(L > 0\), we can write this as:

\[ \phi(f({x}_i), f({x}_j)) \leq L \cdot d(x_i, x_j).\]

Currently, mlr3fairness does not support individual fairness metrics, but we aim to introduce such metrics in the future.

Using metrics for non-mlr3 predictions

We can similarly employ mlr3 metrics on predictions stemming from different models. To do so, we create a data.table containing the different components.

# Get adult data as a data.table
train = tsk("adult_train")$data()
mod = rpart::rpart(target ~ ., train)

# Predict on test data
test = tsk("adult_test")$data()
yhat = predict(mod, test, type = "vector")

# Convert to a factor with the same levels
yhat = as.factor(yhat)
levels(yhat) = levels(test$target)

compute_metrics(
  data = test, 
  target = "target",
  prediction = yhat,
  protected_attribute = "sex",
  metrics = msr("fairness.acc")
)
#> fairness.acc 
#>    0.1248581