fairnessThresholder

Optimize classification threshold to include fairness

Since R2023a

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    Description

    fairnessThresholder searches for an optimal score threshold to maximize accuracy while satisfying fairness bounds. For observations in the critical region below the optimal threshold, the function adjusts the labels so that the fairness constraints hold for the reference and nonreference groups in the sensitive attribute. After you create a fairnessThresholder object, you can use the predict and loss object functions on new data to predict fairness labels and calculate the classification loss, respectively.

    Creation

    Syntax

    fairnessMdl = fairnessThresholder(Mdl,Tbl,AttributeName,ResponseVarName)
    fairnessMdl = fairnessThresholder(Mdl,X,attribute,Y)
    fairnessMdl = fairnessThresholder(Mdl,attribute,threshold)
    fairnessMdl = fairnessThresholder(___,Name=Value)

    Description

    fairnessMdl = fairnessThresholder(Mdl,Tbl,AttributeName,ResponseVarName) optimizes the score threshold for a binary classifier Mdl while satisfying fairness bounds. The function tries a vector of thresholds for classifying observations in the validation data table Tbl with the class labels in the ResponseVarName table variable. For observations in the critical region below the optimal threshold, the function adjusts the labels so that the fairness constraints hold for the reference and nonreference groups in the AttributeName sensitive attribute. For more information, see Reject Option-Based Classification.

    example

    fairnessMdl = fairnessThresholder(Mdl,X,attribute,Y) uses the numeric predictor data X, the sensitive attribute specified by attribute, and the binary response specified by Y.

    example

    fairnessMdl = fairnessThresholder(Mdl,attribute,threshold) uses the specified threshold as the score threshold instead of solving an optimization problem. Select a fixed threshold based on domain expertise.

    fairnessMdl = fairnessThresholder(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. For example, specify the bias metric by using the BiasMetric name-value argument.

    example

    Input Arguments

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    Binary classifier, specified as a full or compact classification model object or a function handle.

    Validation data set, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one variable. The table must include all predictor variables used to train Mdl, the sensitive attribute, and the response variable. The table can include additional variables, such as observation weights. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

    Data Types: table

    Sensitive attribute name, specified as the name of a variable in Tbl. You must specify AttributeName as a character vector or a string scalar. For example, if the sensitive attribute is stored as Tbl.Attribute, then specify it as "Attribute".

    The sensitive attribute must be a numeric vector, logical vector, character array, string array, cell array of character vectors, or categorical vector.

    Data Types: char | string

    Response variable name, specified as the name of a variable in Tbl. You must specify ResponseVarName as a character vector or a string scalar. For example, if the response variable is stored as Tbl.Y, then specify it as "Y".

    The response variable must be a numeric vector, logical vector, character array, string array, cell array of character vectors, or categorical vector. The data type must be the same as the data type of the response variable used to train Mdl.

    Data Types: char | string

    Validation predictor data, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one predictor variable.

    • X, attribute, and Y must have the same number of rows.

    • The columns of X must have the same order as the predictor variables used to train Mdl.

    Data Types: single | double

    Sensitive attribute, specified as a numeric column vector, logical column vector, character array, string array, cell array of character vectors, or categorical column vector.

    • X, attribute, and Y must have the same number of rows.

    • If attribute is a character array, then each row of the array must correspond to a group in the sensitive attribute.

    Data Types: single | double | logical | char | string | cell | categorical

    Class labels, specified as a numeric column vector, logical column vector, character array, string array, cell array of character vectors, or categorical column vector.

    • X, attribute, and Y must have the same number of rows.

    • If Y is a character array, then each row of the array must correspond to a class label.

    • The data type of Y must be the same as the data type of the response variable used to train Mdl.

    • If Mdl is a classification model object, then the distinct classes in Y must be a subset of the classes in Mdl.ClassNames.

    Data Types: single | double | logical | char | string | cell | categorical

    Score threshold, specified as a numeric scalar. fairnessThresholder adjusts the label for each observation whose maximum score is less than the threshold value.

    • If Mdl or its predict object function returns classification scores that are posterior probabilities, then specify a threshold value in the range [0.5, 1].

    • If the predict object function of Mdl returns classification scores in the range (–∞,∞), then specify a nonnegative threshold value.

    Data Types: single | double

    Name-Value Arguments

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    Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

    Example: fairnessThresholder(Mdl,Tbl,"Gender","Smoker",BiasMetric="spd",BiasMetricRange=[–0.1 0.1]) specifies to find a score threshold so that the statistical parity difference for the nonreference group in the Gender sensitive attribute is in the range [–0.1, 0.1].

    Bias metric to use as a fairness constraint during the threshold optimization, specified as one of the metric names in this table.

    Metric NameDescription
    "DisparateImpact" or "di" (default)Disparate impact (DI)
    "AverageAbsoluteOddsDifference" or "aaod"Average absolute odds difference (AAOD)
    "EqualOpportunityDifference" or "eod"Equal opportunity difference (EOD)
    "StatisticalParityDifference" or "spd"Statistical parity difference (SPD)

    For more information on the bias metric definitions, see Bias Metrics.

    The fairnessThresholder function computes the bias metric for the nonreference group (that is, the complement of ReferenceGroups) and checks whether the value is within the bias metric bounds (BiasMetricRange).

    Example: BiasMetric="spd"

    Example: BiasMetric="EqualOpportunityDifference"

    Data Types: char | string

    Bounds on the bias metric to use as constraints during the threshold optimization, specified as a two-element numeric vector. This table describes the supported bias metric values and the default bias metric bounds for each bias metric.

    Metric NameSupported Bias Metric ValuesDefault BiasMetricRange Value
    "DisparateImpact" or "di"[0, ∞)[0.8, 1.25]
    "AverageAbsoluteOddsDifference" or "aaod"[0, 1][0, 0.05]
    "EqualOpportunityDifference" or "eod"[–1, 1][–0.05, 0.05]
    "StatisticalParityDifference" or "spd"[–1, 1][–0.05, 0.05]

    The fairnessThresholder function computes the bias metric (BiasMetric) for the nonreference group (that is, the complement of ReferenceGroups) and checks whether the value is within the bias metric bounds.

    Example: BiasMetricRange=[-0.1 0.1]

    Data Types: single | double

    Groups in the sensitive attribute to use as the reference group when computing bias metrics, specified as a scalar or a vector. By default, fairnessThresholder chooses the most frequently occurring group in the validation data as the reference group. Each element in the ReferenceGroups value must have the same data type as the sensitive attribute.

    The function uses a technique designed for binary sensitive attributes that contain a reference group and a nonreference group. Sensitive attribute groups not in the ReferenceGroups value form the nonreference group.

    Example: ReferenceGroups=categorical(["Husband","Unmarried"])

    Data Types: single | double | logical | char | string | cell | categorical

    Label of the positive class, specified as a numeric scalar, logical scalar, character vector, string scalar, cell array containing one character vector, or categorical scalar. PositiveClass must have the same data type as the true class label variable.

    The default PositiveClass value is the second class of the binary labels, according to the order returned by the unique function with the "sorted" option specified for the true class label variable.

    Example: PositiveClass=categorical(">50K")

    Data Types: single | double | logical | char | string | cell | categorical

    Loss to minimize during the threshold optimization, specified as "classiferror", "classifcost", or a function handle.

    This table lists the available loss functions. Specify one using its corresponding character vector or string scalar.

    ValueDescriptionEquation
    "classifcost"Observed misclassification cost

    L=∑j=1nC(kj,k^j)⋅I(yj≠y^j)n

    "classiferror"Misclassified rate in decimal

    L=∑j=1nI(yj≠y^j)n

    • C is the misclassification cost matrix, and I is the indicator function. If Mdl is a classification model object, the misclassification cost matrix corresponds to the Cost property of Mdl. If Mdl is a function handle, C is the default cost matrix, and the loss values for "classifcost" and "classiferror" are identical.

    • yj is the true class label for observation j, and yj belongs to class kj.

    • y^j is the class label with the maximal predicted score for observation j, and y^j belongs to class k^j.

    • n is the number of observations in the validation data set.

    To specify a custom loss function, you must specify Mdl as a classification model object. Use function handle notation (@lossfun), where the function has this form:

    lossvalue = lossfun(Class,Score,Cost)

    • The output argument lossvalue is a scalar.

    • You specify the function name (lossfun).

    • Class is an n-by-K logical matrix with rows indicating the class to which the corresponding observation belongs. n is the number of observations in Tbl or X, and K is the number of distinct classes in the response variable. The column order corresponds to the class order in Mdl.ClassNames. Create Class by setting Class(p,q) = 1, if observation p is in class q, for each row. Set all other elements of row p to 0.

    • Score is an n-by-K numeric matrix of classification scores. The column order corresponds to the class order in Mdl.ClassNames. Score is a matrix of classification scores, similar to the output of predict.

    • Cost is a K-by-K numeric matrix of misclassification costs. For example, Cost = ones(K) – eye(K) specifies a cost of 0 for correct classification and 1 for misclassification.

    Example: LossFun="classifcost"

    Data Types: char | string | function_handle

    Maximum number of threshold values to evaluate during the threshold optimization, specified as a positive integer. fairnessThresholder uses a vector of min(n,MaxNumThresholds) threshold values as part of the optimization process, where n is the number of observations in the validation data.

    Example: MaxNumThresholds=250

    Data Types: single | double

    Properties

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    This property is read-only.

    Binary classifier, returned as a full or compact classification model object or a function handle.

    This property is read-only.

    Sensitive attribute, returned as a variable name, numeric column vector, logical column vector, character array, cell array of character vectors, or categorical column vector.

    • If you use a table to create the fairnessThresholder object, then SensitiveAttribute is the name of the sensitive attribute. The name is stored as a character vector.

    • If you use a matrix to create the fairnessThresholder object, then SensitiveAttribute has the same size and data type as the sensitive attribute used to create the object. (The software treats string arrays as cell arrays of character vectors.)

    Data Types: single | double | logical | char | cell | categorical

    This property is read-only.

    Groups in the sensitive attribute to use as the reference group, returned as a scalar or vector. (The software treats string arrays as cell arrays of character vectors.)

    The ReferenceGroups name-value argument sets this property.

    Data Types: single | double | logical | char | cell | categorical

    This property is read-only.

    Name of the true class label variable, returned as a character vector containing the name of the response variable. (The software treats a string scalar as a character vector.)

    • If you specify the input argument ResponseVarName, then its value determines this property.

    • If you specify the input argument Y, then the property value is 'Y'.

    Data Types: char

    This property is read-only.

    Label of the positive class, returned as a numeric scalar, logical scalar, character vector, cell array containing one character vector, or categorical scalar. (The software treats a string scalar as a character vector.)

    The PositiveClass name-value argument sets this property.

    Data Types: single | double | logical | char | cell | categorical

    This property is read-only.

    Score threshold, returned as a numeric scalar. The score threshold is the optimal score threshold derived by fairnessThresolder or the threshold input argument value.

    The ScoreThreshold property is empty when the original model predictions already satisfy the fairness constraints or when all potential score thresholds fail to satisfy the fairness constraints.

    Data Types: single | double

    This property is read-only.

    Bias metric, returned as a character vector.

    The BiasMetric name-value argument sets this property.

    Data Types: char

    This property is read-only.

    Bias metric value for the nonreference group, returned as a numeric scalar. Sensitive attribute groups not in the ReferenceGroups value form the nonreference group.

    fairnessThresholder computes the bias metric value by using the validation data set predictions, adjusted using the ScoreThreshold value.

    The BiasMetricValue property is empty when the original model predictions already satisfy the fairness constraints or when all potential score thresholds fail to satisfy the fairness constraints.

    Data Types: double

    This property is read-only.

    Bounds on the bias metric, returned as a two-element numeric vector.

    The BiasMetricRange name-value argument sets this property.

    Data Types: single | double

    This property is read-only.

    Validation classification loss, returned as a numeric scalar. fairnessThresholder computes the classification loss specified by the LossFun name-value argument. The function uses the validation data set predictions, adjusted using the ScoreThreshold value.

    The ValidationLoss property is empty when the original model predictions already satisfy the fairness constraints or when all potential score thresholds fail to satisfy the fairness constraints.

    Data Types: double

    Object Functions

    lossClassification loss adjusted by fairness threshold
    predictPredicted labels adjusted by fairness threshold

    Examples

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    Open Live Script

    Train a tree ensemble for binary classification, and compute the disparate impact for each group in the sensitive attribute. To reduce the disparate impact value of the nonreference group, adjust the score threshold for classifying observations.

    Load the data census1994, which contains the data set adultdata and the test data set adulttest. The data sets consist of demographic information from the US Census Bureau that can be used to predict whether an individual makes over $50,000 per year. Preview the first few rows of adultdata.

    load census1994
head(adultdata)
        age       workClass          fnlwgt      education    education_num       marital_status           occupation        relationship     race      sex      capital_gain    capital_loss    hours_per_week    native_country    salary
    ___    ________________    __________    _________    _____________    _____________________    _________________    _____________    _____    ______    ____________    ____________    ______________    ______________    ______

    39     State-gov                77516    Bachelors         13          Never-married            Adm-clerical         Not-in-family    White    Male          2174             0                40          United-States     <=50K 
    50     Self-emp-not-inc         83311    Bachelors         13          Married-civ-spouse       Exec-managerial      Husband          White    Male             0             0                13          United-States     <=50K 
    38     Private             2.1565e+05    HS-grad            9          Divorced                 Handlers-cleaners    Not-in-family    White    Male             0             0                40          United-States     <=50K 
    53     Private             2.3472e+05    11th               7          Married-civ-spouse       Handlers-cleaners    Husband          Black    Male             0             0                40          United-States     <=50K 
    28     Private             3.3841e+05    Bachelors         13          Married-civ-spouse       Prof-specialty       Wife             Black    Female           0             0                40          Cuba              <=50K 
    37     Private             2.8458e+05    Masters           14          Married-civ-spouse       Exec-managerial      Wife             White    Female           0             0                40          United-States     <=50K 
    49     Private             1.6019e+05    9th                5          Married-spouse-absent    Other-service        Not-in-family    Black    Female           0             0                16          Jamaica           <=50K 
    52     Self-emp-not-inc    2.0964e+05    HS-grad            9          Married-civ-spouse       Exec-managerial      Husband          White    Male             0             0                45          United-States     >50K

    Each row contains the demographic information for one adult. The information includes sensitive attributes, such as age, marital_status, relationship, race, and sex. The third column flnwgt contains observation weights, and the last column salary shows whether a person has a salary less than or equal to $50,000 per year (<=50K) or greater than $50,000 per year (>50K).

    Remove observations with missing values.

    adultdata = rmmissing(adultdata);
adulttest = rmmissing(adulttest);

    Partition adultdata into training and validation sets. Use 60% of the observations for the training set trainingData and 40% of the observations for the validation set validationData.

    rng("default") % For reproducibility
c = cvpartition(adultdata.salary,"Holdout",0.4);
trainingIdx = training(c);
validationIdx = test(c);
trainingData = adultdata(trainingIdx,:);
validationData = adultdata(validationIdx,:);

    Train a boosted ensemble of trees using the training data set trainingData. Specify the response variable, predictor variables, and observation weights by using the variable names in the adultdata table. Use random undersampling boosting as the ensemble aggregation method.

    predictors = ["capital_gain","capital_loss","education", ...
    "education_num","hours_per_week","occupation","workClass"];
Mdl = fitcensemble(trainingData,"salary", ...
    PredictorNames=predictors, ...
    Weights="fnlwgt",Method="RUSBoost");

    Predict salary values for the observations in the test data set adulttest, and calculate the classification error.

    labels = predict(Mdl,adulttest);
L = loss(Mdl,adulttest)
    L = 
0.2080

    The model accurately predicts the salary categorization for approximately 80% of the test set observations.

    Compute fairness metrics with respect to the sensitive attribute sex by using the test set model predictions. In particular, find the disparate impact for each group in sex. Use the report and plot object functions of fairnessMetrics to display the results.

    metricsResults = fairnessMetrics(adulttest,"salary", ...
    SensitiveAttributeNames="sex",Predictions=labels, ...
    ModelNames="Ensemble",Weights="fnlwgt");
metricsResults.PositiveClass
    ans = categorical
     >50K 


    metricsResults.ReferenceGroup
    ans = 
'Male'

    report(metricsResults,BiasMetrics="DisparateImpact")
    ans=2×4 table
    ModelNames    SensitiveAttributeNames    Groups    DisparateImpact
    __________    _______________________    ______    _______________

     Ensemble               sex              Female        0.73792    
     Ensemble               sex              Male                1    


    plot(metricsResults,"DisparateImpact")

    Figure contains an axes object. The axes object with title Disparate Impact, xlabel Fairness Metric Value, ylabel sex contains 2 objects of type bar, constantline.

    For the nonreference group (Female), the disparate impact value is the proportion of predictions in the group with a positive class value (>50K) divided by the proportion of predictions in the reference group (Male) with a positive class value. Ideally, disparate impact values are close to 1.

    To try to improve the nonreference group disparate impact value, you can adjust model predictions by using the fairnessThresholder function. The function uses validation data to search for an optimal score threshold that maximizes accuracy while satisfying fairness bounds. For observations in the critical region below the optimal threshold, the function changes the labels so that the fairness constraints hold for the reference and nonreference groups. By default, the function tries to find a score threshold so that the disparate impact value for the nonreference group is in the range [0.8,1.25].

    fairnessMdl = fairnessThresholder(Mdl,validationData,"sex","salary")
    fairnessMdl = 
  fairnessThresholder with properties:

               Learner: [1×1 classreg.learning.classif.CompactClassificationEnsemble]
    SensitiveAttribute: 'sex'
       ReferenceGroups: Male
          ResponseName: 'salary'
         PositiveClass: >50K
        ScoreThreshold: 1.6749
            BiasMetric: 'DisparateImpact'
       BiasMetricValue: 0.9702
       BiasMetricRange: [0.8000 1.2500]
        ValidationLoss: 0.2017

    fairnessMdl is a fairnessThresholder model object. Note that the predict function of the ensemble model Mdl returns scores that are not posterior probabilities. Scores are in the range (-∞,∞) instead, and the maximum score for each observation is greater than 0. For observations whose maximum scores are less than the new score threshold (fairnessMdl.ScoreThreshold), the predict function of the fairnessMdl object adjusts the prediction. If the observation is in the nonreference group, the function predicts the observation into the positive class. If the observation is in the reference group, the function predicts the observation into the negative class. These adjustments do not always result in a change in the predicted label.

    Adjust the test set predictions by using the new score threshold, and calculate the classification error.

    fairnessLabels = predict(fairnessMdl,adulttest);
fairnessLoss = loss(fairnessMdl,adulttest)
    fairnessLoss = 
0.2064

    The new classification error is similar to the original classification error.

    Compare the disparate impact values across the two sets of test predictions: the original predictions computed using Mdl and the adjusted predictions computed using fairnessMdl.

    newMetricsResults = fairnessMetrics(adulttest,"salary", ...
    SensitiveAttributeNames="sex",Predictions=[labels,fairnessLabels], ...
    ModelNames=["Original","Adjusted"],Weights="fnlwgt");
newMetricsResults.PositiveClass
    ans = categorical
     >50K 


    newMetricsResults.ReferenceGroup
    ans = 
'Male'

    report(newMetricsResults,BiasMetrics="DisparateImpact")
    ans=2×5 table
        Metrics        SensitiveAttributeNames    Groups    Original    Adjusted
    _______________    _______________________    ______    ________    ________

    DisparateImpact              sex              Female    0.73792      1.0048 
    DisparateImpact              sex              Male            1           1 


    plot(newMetricsResults,"di")

    Figure contains an axes object. The axes object with title Disparate Impact, xlabel Fairness Metric Value, ylabel sex contains 2 objects of type bar. These objects represent Original, Adjusted.

    The disparate impact value for the nonreference group (Female) is closer to 1 when you use the adjusted predictions.

    Open Live Script

    Train a support vector machine (SVM) model, and compute the statistical parity difference (SPD) for each group in the sensitive attribute. To reduce the SPD value of the nonreference group, adjust the score threshold for classifying observations.

    Load the patients data set, which contains medical information for 100 patients. Convert the Gender and Smoker variables to categorical variables. Specify the descriptive category names Smoker and Nonsmoker rather than 1 and 0.

    load patients
Gender = categorical(Gender);
Smoker = categorical(Smoker,logical([1 0]), ...
    ["Smoker","Nonsmoker"]);

    Create a matrix containing the continuous predictors Diastolic and Systolic. Specify Gender as the sensitive attribute and Smoker as the response variable.

    X = [Diastolic,Systolic];
attribute = Gender;
Y = Smoker;

    Partition the data into training and validation sets. Use half of the observations for training and half of the observations for validation.

    rng("default") % For reproducibility
cv = cvpartition(Y,"Holdout",0.5);

trainX = X(training(cv),:);
trainAttribute = attribute(training(cv));
trainY = Y(training(cv));

validationX = X(test(cv),:);
validationAttribute = attribute(test(cv));
validationY = Y(test(cv));

    Train a support vector machine (SVM) binary classifier on the training data. Standardize the predictors before fitting the model. Use the trained model to predict labels and compute scores for the validation data set.

    mdl = fitcsvm(trainX,trainY,Standardize=true);
[labels,scores] = predict(mdl,validationX);

    For the validation data set, combine the sensitive attribute and response variable information into one grouping variable groupTest.

    groupTest = validationAttribute.*validationY;
names = string(categories(groupTest))
    names = 4×1 string
    "Female Smoker"
    "Female Nonsmoker"
    "Male Smoker"
    "Male Nonsmoker"

    Find the validation observations that are misclassified by the SVM model.

    wrongIdx = (validationY ~= labels);
wrongX = validationX(wrongIdx,:);
names(5) = "Misclassified";

    Plot the validation data. The color of each point indicates the sensitive attribute group and class label for that observation. Circled points indicate misclassified observations.

    figure
hold on
gscatter(validationX(:,1),validationX(:,2), ...
    validationAttribute.*validationY)
plot(wrongX(:,1),wrongX(:,2), ...
    "ko",MarkerSize=8)
legend(names)
xlabel("Diastolic")
ylabel("Systolic")
title("Validation Data")
hold off

    Figure contains an axes object. The axes object with title Validation Data, xlabel Diastolic, ylabel Systolic contains 5 objects of type line. One or more of the lines displays its values using only markers These objects represent Female Smoker, Female Nonsmoker, Male Smoker, Male Nonsmoker, Misclassified.

    Compute fairness metrics with respect to the sensitive attribute by using the model predictions. In particular, find the statistical parity difference (SPD) for each group in validationAttribute.

    metricsResults = fairnessMetrics(validationAttribute,validationY, ...
    Predictions=labels);
metricsResults.ReferenceGroup
    ans = 
'Female'

    metricsResults.PositiveClass
    ans = categorical
     Nonsmoker 


    report(metricsResults,BiasMetrics="StatisticalParityDifference")
    ans=2×4 table
    ModelNames    SensitiveAttributeNames    Groups    StatisticalParityDifference
    __________    _______________________    ______    ___________________________

      Model1                x1               Female                     0         
      Model1                x1               Male               -0.064412         


    figure
plot(metricsResults,"StatisticalParityDifference")

    Figure contains an axes object. The axes object with title Statistical Parity Difference, xlabel Fairness Metric Value, ylabel x1 contains 2 objects of type bar, constantline.

    For the nonreference group (Male), the SPD value is the difference between the probability of a patient being in the positive class (Nonsmoker) when the sensitive attribute value is Male and the probability of a patient being in the positive class when the sensitive attribute value is Female (in the reference group). Ideally, SPD values are close to 0.

    To try to improve the nonreference group SPD value, you can adjust the model predictions by using the fairnessThresholder function. The function searches for an optimal score threshold to maximize accuracy while satisfying fairness bounds. For observations in the critical region below the optimal threshold, the function changes the labels so that the fairness constraints hold for the reference and nonreference groups. By default, when you use the SPD bias metric, the function tries to find a score threshold such that the SPD value for the nonreference group is in the range [–0.05,0.05].

    fairnessMdl = fairnessThresholder(mdl,validationX, ...
    validationAttribute,validationY, ...
    BiasMetric="StatisticalParityDifference")
    fairnessMdl = 
  fairnessThresholder with properties:

               Learner: [1×1 classreg.learning.classif.CompactClassificationSVM]
    SensitiveAttribute: [50×1 categorical]
       ReferenceGroups: Female
          ResponseName: 'Y'
         PositiveClass: Nonsmoker
        ScoreThreshold: 0.5116
            BiasMetric: 'StatisticalParityDifference'
       BiasMetricValue: -0.0209
       BiasMetricRange: [-0.0500 0.0500]
        ValidationLoss: 0.1200

    fairnessMdl is a fairnessThresholder model object.

    Note that the updated nonreference group SPD value is closer to 0.

    newNonReferenceSPD = fairnessMdl.BiasMetricValue
    newNonReferenceSPD = 
-0.0209

    Use the new score threshold to adjust the validation data predictions. The predict function of the fairnessMdl object adjusts the prediction of each observation whose maximum score is less than the score threshold. If the observation is in the nonreference group, the function predicts the observation into the positive class. If the observation is in the reference group, the function predicts the observation into the negative class. These adjustments do not always result in a change in the predicted label.

    fairnessLabels = predict(fairnessMdl,validationX, ...
    validationAttribute);

    Find the observations whose predictions are switched by fairnessMdl.

    differentIdx = (labels ~= fairnessLabels);
differentX = validationX(differentIdx,:);
names(5) = "Switched Prediction";

    Plot the validation data. The color of each point indicates the sensitive attribute group and class label for that observation. Points in squares indicate observations whose labels are switched by the fairnessThresholder model.

    figure
hold on
gscatter(validationX(:,1),validationX(:,2), ...
    validationAttribute.*validationY)
plot(differentX(:,1),differentX(:,2), ...
    "ks",MarkerSize=8)
legend(names)
xlabel("Diastolic")
ylabel("Systolic")
title("Validation Data")
hold off

    Figure contains an axes object. The axes object with title Validation Data, xlabel Diastolic, ylabel Systolic contains 5 objects of type line. One or more of the lines displays its values using only markers These objects represent Female Smoker, Female Nonsmoker, Male Smoker, Male Nonsmoker, Switched Prediction.

    Open Live Script

    The fairnessThresholder function uses a technique designed for binary sensitive attributes that contain a reference group and a nonreference group. This example shows how to use the function when the sensitive attribute contains more than two groups.

    Read the sample file CreditRating_Historical.dat into a table. The predictor data contains financial ratios for a list of corporate customers. The response variable contains credit ratings assigned by a rating agency. Consider the industry sector information as a sensitive attribute.

    creditrating = readtable("CreditRating_Historical.dat");

    Because each value in the ID variable is a unique customer ID—that is, length(unique(creditrating.ID)) is equal to the number of observations in creditrating—the ID variable is a poor predictor. Remove the ID variable from the table, and convert the Industry variable to a categorical variable.

    creditrating.ID = [];
creditrating.Industry = categorical(creditrating.Industry);

    In the Rating response variable, combine the AAA, AA, A, and BBB ratings into a category of "good" ratings, and the BB, B, and CCC ratings into a category of "poor" ratings.

    Rating = categorical(creditrating.Rating);
Rating = mergecats(Rating,["AAA","AA","A","BBB"],"good");
Rating = mergecats(Rating,["BB","B","CCC"],"poor");
creditrating.Rating = Rating;

    Partition the data into a training set, validation set, and test set. Use approximately one third of the observations to create each set.

    rng("default")
cv1 = cvpartition(creditrating.Rating,"Holdout",1/3);
tblNotForTest = creditrating(training(cv1),:);
tblTest = creditrating(test(cv1),:);

cv2 = cvpartition(tblNotForTest.Rating,"Holdout",1/2);
tblTrain = tblNotForTest(training(cv2),:);
tblValidation = tblNotForTest(test(cv2),:);

    In this example, consider industries with high ratios of good to poor ratings as reference groups in the Industry sensitive attribute. Compute the ratios using the training data set tblTrain and the grpstats function.

    info = grpstats(tblTrain,["Industry","Rating"]);
goodInfo = info(info.Rating == "good",1:3);
poorInfo = info(info.Rating == "poor",1:3);
goodToPoorRatio = goodInfo.GroupCount./poorInfo.GroupCount
    goodToPoorRatio = 12×1

    2.0000
    1.5122
    2.1212
    1.3061
    1.7778
    2.5152
    2.4118
    1.9394
    1.4186
    1.1875
    2.8846
    2.1667
      ⋮

    Define the well-rated industries as those with goodToPoorRatio values greater than 2.5. Consider the industry with the highest goodToPoorRatio value as the best-rated industry.

    wellRatedIndustries = goodInfo.Industry(goodToPoorRatio > 2.5,:)
    wellRatedIndustries = 2×1 categorical
     6 
     11 


    maximumRatio = max(goodToPoorRatio);
bestRatedIndustry = goodInfo.Industry(goodToPoorRatio == maximumRatio,:)
    bestRatedIndustry = categorical
     11

    Compute fairness metrics with respect to the sensitive attribute by using the training data. In particular, find the statistical parity difference (SPD) for each group in Industy. Specify a good rating as the positive class, and specify the best-rated industry (11) as the reference group. Use the report and plot object functions of fairnessMetrics to display the results.

    dataMetricsResults = fairnessMetrics(tblTrain,"Rating", ...
    SensitiveAttributeNames="Industry", ...
    PositiveClass="good",ReferenceGroup=bestRatedIndustry);
report(dataMetricsResults,BiasMetrics="StatisticalParityDifference")
    ans=12×3 table
    SensitiveAttributeNames    Groups    StatisticalParityDifference
    _______________________    ______    ___________________________

           Industry              1                -0.075908         
           Industry              2                 -0.14063         
           Industry              3                -0.062963         
           Industry              4                  -0.1762         
           Industry              5                 -0.10257         
           Industry              6                -0.027057         
           Industry              7                -0.035678         
           Industry              8                 -0.08278         
           Industry              9                 -0.15604         
           Industry              10                -0.19972         
           Industry              11                       0         
           Industry              12               -0.058364         


    plot(dataMetricsResults,"StatisticalParityDifference")

    Figure contains an axes object. The axes object with title Statistical Parity Difference, xlabel Fairness Metric Value, ylabel Industry contains 2 objects of type bar, constantline.

    For each group g in the sensitive attribute, the SPD value is the difference between the probability of being in the positive class (good) when the sensitive attribute value is g and the probability of being in the positive class when the sensitive attribute value is the reference group value (11). Ideally, SPD values are close to 0.

    Visualize the distribution of SPD values by using a box plot.

    boxchart(dataMetricsResults.BiasMetrics.StatisticalParityDifference)
ylabel("Statistical Parity Difference")
legend("Training Data")

    Figure contains an axes object. The axes object with ylabel Statistical Parity Difference contains an object of type boxchart. This object represents Training Data.

    The median SPD value is around –0.08.

    Train a binary tree classifier using the training data set. Use the trained model to predict labels and compute the classification error on the test data set.

    predictorNames = ["WC_TA","RE_TA","EBIT_TA","MVE_BVTD","S_TA"];
treeMdl = fitctree(tblTrain,"Rating", ...
    PredictorNames=predictorNames);
treePredictions = predict(treeMdl,tblTest);
L = loss(treeMdl,tblTest)
    L = 
0.1107

    You can adjust model predictions by using the fairnessThresholder function. The function uses the validation data to search for an optimal score threshold that maximizes accuracy while satisfying fairness bounds. Use the ReferenceGroups name-value argument to specify the well-rated industries (6 and 11) as the reference group. All other industries form the nonreference group. Specify the bias metric as the statistical parity difference and the bias metric range as [–0.005,0.005]. Note that these bounds apply to the SPD value for the collective nonreference group, not individual industries in the sensitive attribute.

    fairnessMdl = fairnessThresholder(treeMdl,tblValidation, ...
    "Industry","Rating", ...
    PositiveClass="good",ReferenceGroups=wellRatedIndustries, ...
    BiasMetric="StatisticalParityDifference", ...
    BiasMetricRange=[-0.005 0.005])
    fairnessMdl = 
  fairnessThresholder with properties:

               Learner: [1×1 classreg.learning.classif.CompactClassificationTree]
    SensitiveAttribute: 'Industry'
       ReferenceGroups: [2×1 categorical]
          ResponseName: 'Rating'
         PositiveClass: 'good'
        ScoreThreshold: 0.5444
            BiasMetric: 'StatisticalParityDifference'
       BiasMetricValue: 0.0034
       BiasMetricRange: [-0.0050 0.0050]
        ValidationLoss: 0.1198

    fairnessMdl is a fairnessThresholder model object.

    Adjust the test set predictions by using the new score threshold, and calculate the classification error.

    newPredictions = predict(fairnessMdl,tblTest);
newL = loss(fairnessMdl,tblTest)
    newL = 
0.1183

    The new classification error is similar to the original classification error.

    Compare the SPD values across the two sets of test predictions: the original predictions computed using treeMdl and the adjusted predictions computed using fairnessMdl. Specify a good rating as the positive class, and specify the best-rated industry (11) as the reference group. Use the report and plot object functions of fairnessMetrics to display the results.

    predMetricsResults = fairnessMetrics(tblTest,"Rating", ...
    SensitiveAttributeNames="Industry", ...
    Predictions=[treePredictions,newPredictions], ...
    PositiveClass="good", ...
    ModelNames=["Original Model","Adjusted Model"], ...
    ReferenceGroup=bestRatedIndustry);
report(predMetricsResults,BiasMetric="DisparateImpact")
    ans=12×5 table
        Metrics        SensitiveAttributeNames    Groups    Original Model    Adjusted Model
    _______________    _______________________    ______    ______________    ______________

    DisparateImpact           Industry              1          0.96499           0.95014    
    DisparateImpact           Industry              2           1.0755            1.0634    
    DisparateImpact           Industry              3          0.94643           0.94643    
    DisparateImpact           Industry              4           1.0541            1.0392    
    DisparateImpact           Industry              5           1.0262            1.0132    
    DisparateImpact           Industry              6           1.0186            1.0186    
    DisparateImpact           Industry              7          0.99692           0.96067    
    DisparateImpact           Industry              8            1.077             1.077    
    DisparateImpact           Industry              9           1.0392            1.0103    
    DisparateImpact           Industry              10          1.0781            1.0635    
    DisparateImpact           Industry              11               1                 1    
    DisparateImpact           Industry              12          1.0392            1.0225    


    plot(predMetricsResults,"spd")

    Figure contains an axes object. The axes object with title Statistical Parity Difference, xlabel Fairness Metric Value, ylabel Industry contains 2 objects of type bar. These objects represent Original Model, Adjusted Model.

    Visualize the two distributions of SPD values by using box plots.

    boxchart(predMetricsResults.BiasMetrics.StatisticalParityDifference, ...
    GroupByColor=predMetricsResults.BiasMetrics.ModelNames)
ylabel("Statistical Parity Difference")
legend

    Figure contains an axes object. The axes object with ylabel Statistical Parity Difference contains 2 objects of type boxchart. These objects represent Original Model, Adjusted Model.

    The SPD values for the original test set predictions are close to 0, with a median value of approximately 0.02. The SPD values for the adjusted test set predictions have a median value that is slightly closer to 0.

    Open Live Script

    Train a logistic regression model using the fitglm function. To adjust the score threshold for classifying observations, pass the model as an input to fairnessThresholder using a function handle.

    Load the patients data set, which contains medical information for 100 patients. Convert the Gender and Smoker variables to categorical variables. Specify the descriptive category names Smoker and Nonsmoker rather than 1 and 0.

    load patients
Gender = categorical(Gender);
Smoker = categorical(Smoker,logical([1 0]), ...
    ["Smoker","Nonsmoker"]);

    Create a table containing the continuous predictors Diastolic and Systolic, the sensitive attribute Gender, and the response variable Smoker.

    Tbl = table(Diastolic,Systolic,Gender,Smoker);

    Partition the data into training and validation sets. Use half of the observations for training and half of the observations for validation.

    rng("default") % For reproducibility
cv = cvpartition(Tbl.Smoker,"Holdout",0.5);
trainTbl = Tbl(training(cv),:);
validationTbl = Tbl(test(cv),:);

    Train a logistic regression model using the training data trainTbl and the fitglm function.

    modelspec = "Smoker ~ Diastolic + Systolic";
glmMdl = fitglm(trainTbl,modelspec,Distribution="binomial")
    glmMdl = 
Generalized linear regression model:
    logit(P(Smoker='Nonsmoker')) ~ 1 + Diastolic + Systolic
    Distribution = Binomial

Estimated Coefficients:
                   Estimate      SE        tStat      pValue  
                   ________    _______    _______    _________

    (Intercept)      116.98     44.939     2.6032    0.0092356
    Diastolic      -0.54261    0.21577    -2.5147     0.011913
    Systolic       -0.57999    0.28697    -2.0211     0.043268


50 observations, 47 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 54, p-value = 1.89e-12

    As indicated in the linear regression model equation, Nonsmoker is the positive class. That is, an observation with a predicted score greater than 0.5 is predicted to be a nonsmoker.

    Create a function handle to the predict function of the GeneralizedLinearModel object glmMdl.

    f = @(T) predict(glmMdl,T);

    Create a fairnessThresholder object by using the function handle f and the validation data validationTbl. The function searches for an optimal score threshold to maximize accuracy while satisfying fairness bounds. Specify the bias metric range so that the disparate impact value for the nonreference group is in the range [0.9,1.1].

    When you pass a classification model as a function handle, you must specify the positive class.

    fairnessMdl = fairnessThresholder(f,validationTbl, ...
    "Gender","Smoker", ...
    BiasMetricRange=[0.9 1.1], ...
    PositiveClass=categorical("Nonsmoker"))
    fairnessMdl = 
  fairnessThresholder with properties:

               Learner: @(T)predict(glmMdl,T)
    SensitiveAttribute: 'Gender'
       ReferenceGroups: Female
          ResponseName: 'Smoker'
         PositiveClass: Nonsmoker
        ScoreThreshold: 0.8087
            BiasMetric: 'DisparateImpact'
       BiasMetricValue: 0.9538
       BiasMetricRange: [0.9000 1.1000]
        ValidationLoss: 0.1600


    omega = fairnessMdl.ScoreThreshold
    omega = 
0.8087

    fairnessMdl is a fairnessThresholder model object. For each observation with a score in the range (1–omega,omega), the predict function of the fairnessMdl object adjusts the prediction. If the observation is in the nonreference group (Male), the function predicts the observation into the positive class (Nonsmoker). If the observation is in the reference group (Female), the function predicts the observation into the negative class (Smoker).

    Adjust the predictions for the entire data set Tbl by using the new score threshold.

    fairnessLabels = predict(fairnessMdl,Tbl)
    fairnessLabels = 100×1 categorical
     Smoker 
     Nonsmoker 
     Smoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Smoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Smoker 
     Nonsmoker 
     Smoker 
     Smoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Smoker 
     Smoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Nonsmoker 
     Smoker 
      ⋮

    Algorithms

    expand all

    References

    [1] Kamiran, Faisal, Asim Karim, and Xiangliang Zhang. "Decision Theory for Discrimination-Aware Classification." 2012 IEEE 12th International Conference on Data Mining: 924-929.

    Version History

    Introduced in R2023a

    See Also

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