Assessing Fairness in the Presence of Missing Data

1 Introduction . Mounting evidence [ 20 , 8 , 26 , 42 , 43 ] has suggested that powerful machine learning algorithms can be unfair and lead to disproportionately unfavorable treatment decisions for marginalized groups . In recent years , there has been a growing body of research on addressing the unfairness and bias of machine learning algorithms [ 12 ] . Meanwhile , missing data are ubiquitous and present daunting challenges in real-world data analysis . Particularly , missing data , if not adequately handled , would lead to biased estimations and improper statistical inferences [ 29 ] . As such , analysis of incomplete data has been an active research area [ 4 , 33 ] . More recently , there is also a growing recognition that missing data may have deleterious impact on algorithmic fairness . For example , in medicine , bias caused by missing values in electronic health records is identified as a significant factor contributing to unfairness of machine learning ( ML ) algorithms used in medicine that may exacerbate health care disparities [ 36 , 20 ] . However , there has been little reported research on assessing fairness of statistical and machine learning models using datasets that contain missing values . In the presence of missing data , one popular approach for analysis , particularly in biomedical studies , is to use only the set of complete cases , i.e. , observations with all features observed , discarding incomplete cases . As a result , we can define two related yet distinct data domains , namely , the complete case domain and the complete data domain ( see Section 2.1 ) . In many biomedical applications , ∗This work is partly supported by NIH grant R01GM124111 and RF1AG063481 . The content is the responsibility of the authors and does not necessarily represent the views of NIH . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . samples in the complete data domain are considered to be randomly drawn from the target population of interest ; in other words , the complete data domain is defined by the target population , and the ultimate goal is to apply the trained models to the complete data domain . Under some missing data mechanisms ( see Section 2.1 ) , the complete case domain is biased for estimation of fairness in the complete data domain , i.e. , for the target population . As such , a fair algorithm in the complete case domain may show disproportionate bias towards marginalized groups in the complete data domain . The growing body of literature on algorithmic fairness has been primarily focused on two types of fairness definitions , group fairness and individual fairness [ 12 ] . Group fairness emphasizes that members from different groups ( e.g . gender , race etc . ) should be treated similarly , while individual fairness pays more attention to treatment similarity between any two similar individuals . In this work , we investigate group fairness in analysis of incomplete data . It has been noted that fairness definitions may not be compatible with one another in a sense that it is not possible to achieve fairness simultaneously under multiple definitions [ 19 ] . In binary classification problems , demographic ( or statistical ) parity [ 10 , 17 ] is a fairness notion that has been mostly studied . It states that the predicted outcome should be independent of sensitive attributes . However demographic parity can cause severe harm to prediction performance when the response is dependent of sensitive attributes . As an alternative , disparate mistreatment [ 47 ] states that misclassification level ( e.g. , in terms of overall accuracy , false negative rate , false discovery rate ) should be similar between two sensitive groups . Similarly , [ 24 ] proposed equalized odds , which requires both false positive rate ( FPR ) and false negative rate ( FNR ) to be the same between two groups . In the regression setting , fairness is usually associated with the parity of loss between two groups [ 1 , 35 , 16 ] . To fix ideas , we propose to use in this paper accuracy parity gap as the fairness notion in learning tasks including classification and regression . We consider the technique of re-weighting on assessing fairness of a given algorithm using complete cases , in which different complete cases are assigned different weights . We show that if the weights are properly chosen , such approach can mitigate the estimation bias induced by the difference of domains . It is worth noting that our results can be generalized to other fairness notions such as equal opportunity and prediction error parity with respect to mean square error . Several existing works [ 18 , 44 , 48 , 39 , 30 , 21 ] are related to our work , but there are a number of fundamental differences between these works and ours . [ 18 , 44 , 48 ] investigated the impact of missing data on the fairness of downstream prediction models . [ 39 ] investigated fairness across different domains and provided an upper bound of fairness in the target domain given fairness estimate in the source domain . But their work does not deal with missing data and associated challenges and does not consider the technique of re-weighting . In addition , they provided only upper bounds on transferring fairness . [ 30 ] empirically evaluated fairness in the presence of missing data . They did not observe consistent fairness results when comparing different methods for handling missing values such as imputation . We provide theoretical guarantees on assessing fairness of algorithms via analysis of incomplete data . [ 21 ] proposed a causal graph-based framework for modeling the data missingness to guide the design of fair algorithms . But they considered the case when the entire sample is missing , while we consider all three missing mechanisms . Our contributions : This work offers four novel contributions . First , it provides new insights for algorithmic fairness in analysis of incomplete data , which has not yet been well-investigated from a theoretical perspective . Second , we characterize the role and impact of the missing data mechanism on correction for the data domain shift in fairness estimation through the analysis of incomplete data . Third , there has been limited theoretical work on domain adaptation problems when estimated weights are used to correct for domain shift . [ 13 ] considers the setting where true inverse probability weights are known , [ 41 ] considers the case when weights are estimated , but their analysis of algorithm ’ s generalization performance is compared with a clipped empirical risk and does not provide any lower bound . In this work we investigate the setting where the weights are estimated from the data based on correctly or incorrectly specified propensity score models under a given missing data mechanism . Besides , both upper and lower bounds for fairness estimation bias are provided . Fourth , while the estimand that is of primary interest in domain adaptation literature is the prediction accuracy , the estimand of our interest takes the form of absolute difference between two prediction accuracy terms . To obtain a tighter upper bound , we conduct proper relaxations and incorporate Bennett ’ s inequality in the proof of Theorem 1 , which is more nuanced . In addition , novel proof techniques are also presented in the proof of Theorem 2 after applying Bernstein ’ s inequalities , which does not appear in existing works for establishing lower bounds in domain adaptation problems such as [ 13 ] . 2 Problem Formulation . 2.1 Preliminaries on missing data . The data structure and notation used in this paper mostly follow the convention in the missing data literature [ 29 ] and are summarized in Figure 1 . Suppose we have a random sample of n observations from a target population of interest . If there were no missing values , each observation/case zi : = { xi , yi } ∈ X × Y ( i = 1 , . . . , n ) consists of predictors xi ∈ X and label ( response variable ) yi ∈ Y . Denote the complete data matrix by Z in which the ith row is denoted by zi . Some entries of Z are missing and we define the indicator for observing zij or not as rij = 1zij is observed , where zij is the j-th feature in zi . Denote the corresponding indicator matrix by R. Let z ( 1 ) i denote the components of zi that are observed for observation i , and z ( 0 ) i denote the components of zi that are missing for observation i . For example , consider the case when there are two predictors and one response ; if only zi1 is observed , then z ( 1 ) i = zi1 , z ( 0 ) i = ( zi2 , zi3 ) . We then define the observed data Z ( 1 ) as the collection of the observed components from all n observations , { z ( 1 ) i , i = 1 , . . . , n } and the missing data Z ( 0 ) as the collection of all the missing components , { z ( 0 ) i , i = 1 , . . . , n } . There are three primary missing data mechanisms , namely , missing completely at random ( MCAR ) , missing at random ( MAR ) and missing not at random ( MNAR ) [ 29 ] . Data are said to be MCAR if the distribution of R is independent of Z . For MAR , the distribution of R depends on Z only through its observed components , i.e. , R ⊥ Z ( 0 ) |Z ( 1 ) . For MNAR , the distribution of R depends on the missing components of Z . We seek to investigate fairness guarantee under all three mechanisms . In the presence of missing data , observation i is said to be a complete case if it is fully observed ( i.e . z ( 1 ) i = zi ) . Let Ri : = 1zi is fully observed denote the indicator of complete cases . We can then define two different data domains ( distributions ) and we use the two terms , domain and distribution , interchangeably in the remainder of the paper . The complete data domain , denoted by DT , is the distribution of zi in the joint space X × Y . The complete case domain , denoted by DS , is the distribution of observations that have all variables fully observed : zi|Ri = 1 . There is an important connection between these two domains and the aforementioned Z ( 1 ) and Z ( 0 ) . Z ( 1 ) contains the data in all the complete cases and the observed data in the incomplete cases . Combining Z ( 1 ) and Z ( 0 ) yields the complete data , which follows the distribution DT ( Figure 1 ) . Under MCAR , the distribution of the complete cases in DS is the same as the distribution of the complete data in DT . However under MAR or MNAR , the data distributions in DS and DT can be different . For example , if missingness depends on gender and other features associated with gender , then females may have a substantially higher proportion of missing values and the feature distribution in females in DS may be very different from that in DT . As a result , an algorithm that is trained to be fair in DS may not be fair when evaluated in DT , noting again that in practice we typically are more interested in fairness guarantee in DT . We can view this as a domain shift from DS to DT . This shift in the setting of incomplete data can be characterized by the true propensity score ( a.k.a , the probability of being a complete case ) [ 23 ] . To mitigate the fairness estimation bias caused by the shift , different weights can be assigned to the data in the fairness estimator . Missing data mechanism plays a vital role in obtaining the ‘ right ’ weight . For MCAR , uniform weights are equivalent to the true weights . For MAR , one can train popular nonparametric models to obtain a consistent estimator for the true weights . For MNAR , it ’ s much more difficult to obtain consistent estimators for the true weights . Hence one of our contributions is to characterize the role and impact of the missing data mechanism on bias correction in fairness estimation through the analysis of incomplete data .

Summary: This paper discusses how to estimate fairness measures in the presence of missing values. Especially, it compares the fairness measure for the complete data domain (data distribution assuming no missing values) and the fairness measure estimated with reweighing on the complete case domain (data distribution after dropping all the data points that have missing values). They provide an upper bound and a lower bound on the difference of the two. They show experimental results of estimating the fairness measures on synthetic data and two real-world datasets to evaluate.