Finding Regions of Heterogeneity in Decision-Making via Expected Conditional Covariance

1 Introduction . Understanding heterogeneity in decision-making is an established problem in medicine ( Birkmeyer et al. , 2013 ; Corallo et al. , 2014 ; De Jong et al. , 2006 ) , consumer choice ( Ortega et al. , 2011 ; Scarpa et al. , 2005 ) , and law ( Kang et al. , 2012 ; Kleinberg et al. , 2018 ; Arnold et al. , 2018 ) . In the context of medicine , this is referred to as the study of practice variation ( Atsma et al. , 2020 ; Cabana et al. , 1999 ) , where it is often observed that doctors , facing the same clinical context , will make different decisions . Likewise , in a legal context , judges often differ in their leniency in their decisions regarding bail ( Kleinberg et al. , 2018 ) , juvenile incarceration ( Aizer and Doyle Jr , 2013 ) , the use of alternatives to incarceration ( Di Tella and Schargrodsky , 2013 ) , and incarceration length ( Kling , 2006 ) . In some scenarios this variation may be justified : The best medical treatment may not be obvious . In others , it may be grossly unfair , as in the case of racial bias in bail decisions ( Arnold et al. , 2018 ) . ∗Equal contribution 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . In this work , we tackle the question of how to find and characterize this variation in the first place . In particular , we present a learning algorithm for identifying a “ region of heterogeneity ” , defined as a subset of all contexts ( e.g. , patients , cases ) for which the identity of the decision-maker substantially affects the decision . In medicine , a better understanding of treatment variation can inform the development and dissemination of clinical guidelines . In the legal domain , characterizing the cases where judges vary most in their leniency may help with investigating potential issues of fairness . We formalize characterizing the region of heterogeneity as a causal inference problem : We want to characterize examples where changing the decision-maker would have resulted in a different decision . The challenge is two-fold : First , we only observe a single decision-maker per example , so we can not directly observe how ( for instance ) multiple judges would have decided the same case . Second , our data on individual decision-makers is often scarce . For instance , in Section 5 , we consider a medical dataset with more than 400 doctors , each of whom has fewer than 9 samples on average . We will refer to decision-makers as “ agents ” , and our contributions are as follows : In Section 2 , we propose an objective defined in terms of counterfactual decisions across different agents , and show that this objective can be identified from observational data . Moreover , this objective does not require the use of agent-specific statistical models , making it amenable to our sparse setting . In Section 3 , we give an iterative algorithm to identify regions of disagreement by maximizing this objective and provide intuition ( in the form of a generalization bound ) for the factors that drive its performance . In Section 4 , we use a semi-synthetic dataset , derived from crowd-sourced recidivism predictions , to demonstrate that our algorithm recovers the correct region of heterogeneity accurately , even when there are many agents . Finally , in Section 5 , we apply our algorithm to a real-world healthcare dataset and confirm that it recovers intuitive regions of variation in first-line diabetes treatment . We conclude with a discussion of related work and implications . Our code is available at https : //github.com/clinicalml/finding-decision-heterogeneity-regions . Our algorithm does not determine whether variation is inherently good or bad or how it should be addressed . Rather , more careful study with domain experts would be required to determine if variation can ( or should be ) reduced and how . In addition , false discovery of variation is possible and could have a negative impact . We expect that validation on independent datasets would be required in real-world applications , using the regions identified by our method as plausible hypotheses to test . 2 Characterizing Heterogeneity from a Causal Perspective . 2.1 Notation . Let the data be drawn from a distribution P ( X , A , Y ) , where X is a random variable representing context ( or features ) , A is a discrete agent , and Y ∈ { 0 , 1 } is the binary decision . The spaces of all X and A are denoted as X andA with realized values as lower case x and a , respectively . Indicator variables 1 [ · ] are one if the statement inside the brackets is true and zero otherwise . For a subset S ⊆ X , 1 [ x ∈ S ] is sometimes written as a function S ( x ) , where S : X → { 0 , 1 } . A subset S may have several disjoint regions . E [ Y |X ∈ S ] denotes the average Y across samples in S . For instance , if S = { X : X0 < 10 } , then E [ Y |X0 < 10 ] is a scalar average of Y among samples with X0 < 10 . 2.2 Heterogeneity as a Causal Contrast . Our conceptual goal is to identify a region S ⊆ X where different agents tend to make different decisions even when faced with the same context . We can formalize this in the language of potential outcomes from the causal inference literature ( Pearl , 2009 ; Hernán and Robins , 2020 ) , which for clarity we will refer to as potential decisions : In particular , we denote Y ( a ) to be the potential decision of agent a . The fundamental challenge of causal inference is that we do not observe all potential decisions { Y ( a ) : a ∈ A } for each sample , but only a single decision Y . With this in mind , we will make the following assumptions , standard in the literature on causal effect estimation . Assumption 1 ( Causal Identification Assumptions ) . ( i ) Consistency : Y = y , A = a =⇒ Y ( a ) = y , and ( ii ) No Unmeasured Confounding ( NUC ) : For all a ∈ A , Y ( a ) ⊥ A | X . Consistency links the potential Y ( a ) to the observed Y , and NUC says that there are no unobserved factors that influence both the assignment of agents and the decision itself . For instance , the quasirandom assignment of cases to judges conditioned on features X satisfies NUC ( Kleinberg et al. , 2018 ) . NUC may be violated if key aspects of the case ( e.g. , misdemeanor vs. felony ) are omitted as features . For instance , misdemeanor and felony cases may be seen by different judges and have different decision processes , but this variation is not due to agent preferences . Given these assumptions , we propose a causal measure of agent-specific bias , defined as a contrast between potential decisions . Definition 1 ( Conditional Relative Agent Bias ) . For an agent a ∈ A and a subset S ⊆ X , the conditional relative agent bias is defined as E [ Y ( a ) − Y ( π ( x ) ) | A = a , X ∈ S ] ( 1 ) where Y ( a ) is the potential decision of agent a , and Y ( π ( x ) ) B ∑ a′ E [ Y ( a′ ) | x ] π ( a′ | x ) denotes the expected potential decision under the agent assignment distribution π ( a′ | x ) B P ( A = a′ | X = x ) . Note that under Assumption 1 , Y ( π ( x ) ) = E [ Y | x ] ,2 but here we emphasize its causal interpretation as the expected decision of a random agent . Equation ( 1 ) represents the relative difference between the decision of an agent ( on their particular distribution of cases in the region ) and the potential decision of a random agent . In particular , Equation ( 1 ) can be written as follows under Assumption 1 E [ Y ( a ) − Y ( π ) | A = a , X ∈ S ] = ∫ x∈S E [ Y ( a ) − Y ( π ) | X = x ] p ( x | A = a , X ∈ S ) dx , ( 2 ) where we shorten Y ( π ( x ) ) to Y ( π ) . This is the average difference ( over p ( x | a ) , restricted to those x in the set S ) of the conditional expected difference between Y ( a ) and Y ( π ) . For example , suppose that the agent a is a judge who is particularly lenient on bail decisions for felony arrests ( the region S ) , and Y = 1 denotes granting bail . Then imagine the following counterfactual : Take the felony cases that are assigned to this judge and reassign each individual case , described by x , to a random judge a′ , proportionally to p ( a′ | x ) . We may then observe , on average , that the bail rate would decrease , because most judges are less lenient than judge a , corresponding to a positive value of Equation ( 1 ) . Equation ( 1 ) has the additional advantage of being easy to estimate : Under Assumption 1 , it can be rewritten3 as E [ Y−E [ Y | X ] | A = a , X ∈ S ] , the expected residual in predicting ( using the conditional expectation E [ Y | X ] ) the decision of an agent a across the context x typically seen by that agent . 2.3 Formalizing a Causal Objective . Our primary goal is to discover a region S where substantial heterogeneity exists across agents . To do so , we define an aggregate objective across a group G of agents , where G ( a ) ∈ { 0 , 1 } is an indicator function for membership . Q ( S , G ) B ∑ a : G ( a ) =1 P ( A = a | X ∈ S ) E [ Y ( a ) − Y ( π ) | A = a , X ∈ S ] , ( 3 ) We now show that this quantity can be identified and estimated from observational data without requiring agent-specific statistical models , before discussing the interpretation of this objective . Theorem 1 ( Causal Identification ) . Under Assumption 1 , Q ( S , G ) can be identified as Q ( S , G ) = ES [ Cov ( Y , G | X ) ] = ES [ ( Y − E [ Y | X ] ) G ] , ( 4 ) where ES [ · ] B E [ · | X ∈ S ] and Cov ( Y , G | X ) is the conditional covariance . Theorem 1 and other theoretical results are proven in Appendix A . The result follows from proving that the agent-specific bias ( Definition 1 ) is identifiable using the expected conditional covariance between Y and the binary indicator 1 [ A = a ] . With this in mind , we optimize the following objective , where the set S is constrained to be at least a certain size β and S is a hypothesis class of functions S . max S∈S , G Q ( S , G ) s.t. , P ( S ) ≥ β , ( 5 ) Interpretation : Intuitively , this objective measures the disagreement between the agents in the group G ( a ) = 1 and the overall average E [ Y | X ] on the region S . Hence , the choice of group is important for interpreting the objective : If G ( a ) = 1 for all agents , the objective will be zero for any set S , as can be seen from Equation ( 4 ) , applying the definition of the conditional expectation . 2See Proposition A1 for a short proof , and Proposition A2 for the derivation of Equation ( 2 ) . 3See Proposition A3 in Appendix A.1 . Accordingly , we seek a region S for which the partially maximized objective L ( S ) B maxG Q ( S , G ) is large : This partial maximization is obtained by taking G ( a ) = 1 whenever the conditional relative agent bias of agent a ( on the set S ) is non-negative . Thus , Equation ( 5 ) can be re-written as max G Q ( S , G ) = ∑ a∈A P ( A = a | X ∈ S ) |E [ Y ( a ) − Y ( π ) | A = a , X ∈ S ] |+ , ( 6 ) where |x|+ B max ( x , 0 ) , and this objective becomes an average over agents who have a positive bias . This population objective is also equivalent ( up to a constant factor ) to the ( weighted ) sum of the magnitude of each agent ’ s conditional relative agent bias . See Proposition A4 in Appendix A.1 . Lack of Overlap : We have not made the overlap or positivity assumption that P ( A = a | x ) > 0 for all x , a . While this assumption is required to identify conditional causal effects E [ Y ( a ) − Y ( a′ ) | X ] ( Nie and Wager , 2017 ; Wager and Athey , 2018 ; Shalit et al. , 2017 ) , it is not required for identifying our causal contrast . Our problem only requires each context has a positive probability of being seen by more than one decision maker . For instance , suppose that S contains both misdemeanors and felonies and there are four judges a0 , . . . , a3 . If judges a0 and a1 make bail decisions exclusively for felonies while judges a2 and a3 make bail decisions exclusively for misdemeanors , our measure captures disagreement between a0 and a1 and between a2 and a3 even though comparisons between a0 and a2 or other pairs are impossible to make . Thus , we have chosen to compare Y ( a ) to the decisions of viable alternative agents , weighted by their probability p ( a′ | x ) of being selected for such a case .

Summary: Summary:  This paper considers settings where decisions for individuals are each made by different decision makers (agents). The paper defines the problem of finding a region in feature space in which the agents administer highly varying decisions. For example, perhaps judges mete out different sentences for misdemeanors by young men. The paper defines new causal criteria to characterize such regions in the feature space, where the choice of agent has an effect on the decision. The paper introduces an iterative algorithm to maximize this criteria to identify regions of heterogeneity, and provides theoretical results about the optimality of the proposed algorithm. The paper empirically demonstrates the algorithm on a semi-synthetic dataset and real clinical data about doctor recommendations for diabetic patients.  Contributions:  + A new criteria for defining regions where there is heterogeneity in decisions administered across agents. + Results for estimating the criteria from observational data and an iterative algorithm for optimizing the criteria. + Theoretical results about the algorithm and empirical validation of the algorithm.