RISE: Robust Individualized Decision Learning with Sensitive Variables
Anonymous Author(s)
AffiliationAddress email
Abstract
This paper introduces RISE, a robust individualized decision learning frameworkwith sensitive variables, where sensitive variables are collectible data and importantto the intervention decision, but their inclusion in decision making is prohibiteddue to reasons such as delayed availability or fairness concerns.
A naive baseline isto ignore these sensitive variables in learning decision rules, leading to significantuncertainty and bias.
To address this, we propose a decision learning framework toincorporate sensitive variables during offline training but do not include them inthe input of the learned decision rule during model deployment.
Specifically, froma causal perspective, the proposed framework intends to improve the worst-caseoutcomes of individuals caused by sensitive variables that are unavailable at thetime of decision.
Unlike most existing literature that uses mean-optimal objectives,we propose a robust learning framework via finding a newly defined quantile- orinfimum-optimal decision rule.
The reliable performance of the proposed methodis demonstrated through synthetic experiments and three real-data applications.
Introduction
Recently, there has been a widespread interest in developing methodology for individualized decisionrules (IDRs) based on observational data.
When deriving IDRs, some collectible data are importantto the intervention decision, while their inclusion in decision making is prohibited due to reasonssuch as delayed availability or fairness concerns.
For example, sensitive characteristics of subjectsregarding their income, sex, race and ethnicity may not be appropriate to be used directly fordecision making due to fairness concerns.
In the medical field especially for patients in severelife-threatening conditions such as sepsis, timely bedside intervention decisions have to be madebefore lab measurements are ordered, assayed and returned to the attending physicians.
However, dueto the delayed availability of lab results, most of the decisions are made with great uncertainty andbias due to partial information at hand.
We define sensitive variables as variables whose inclusion into decision rules is prohibited.
The formal definition of sensitive variables will be given in Section 3.
In this work we propose RISE ( R obust I ndividualized decision learning with SE nsitive variables), arobust IDR framework to improve the outcome of individuals when there are informative yet sensitivevariables that are either not available or prohibited from using during IDR deployment.
To achievethis, we propose to estimate the optimal IDR by optimizing a quantile- or infimum-based objective,respectively, for continuous or discrete sensitive variables.
Our idea falls along the lines of workthat considers algorithmic fairness [1] while extending it to the setting of causal inference [2] inthe sense that decisions are driven by causality rather than a general utility function.
We show inour empirical analyses that this leads to fairer and safer real-life decisions with little sacrifice ofthe overall performance.
This optimization problem is then shown to be equivalent to a weightedclassification problem where most existing machine learning classifiers can be readily applied .
Assuming that a large outcome value is preferable, optimal IDRs are traditionally derived throughmaximizing the mean outcome of the sample population.
In this paper, we are interested in a specificyet broadly applicable setting of learning that involves sensitive variables.
We consider offline learning where sensitive variables are collected and can be used in training the IDRs , but they cannotbe used as input in the resulting IDRs .
This is a setting commonly considered in the fairness andprivacy-related literature for classification (e.g., [3]), but not from a causal standpoint.
When thereexist important variables that are simply left out from training, the estimated IDR will be biased.
This bias can be removed if all important variables are used during training, which we will show in
Section 3.1 a mean-optimal approach.
The optimal action maximizes the mean outcome where themean is taken over the sensitive variables, conditioning on other variables.
This method, however,has no control of the disparity in sensitive variables.
Subjects with different sensitive values mayreport large outcome differences, hence unfairly or unsafely treated.
Therefore, objective functionswith robustness guarantees for sensitive variables are preferred, since they offer protection to subjectsin the lower tail of the outcome distribution with regards to the sensitive values.
For illustration, we consider a toy example with binary actions, A ∈ {− 1 , 1 } .
We note that thedecision can only be made based on the variable X whereas S is a sensitive variable.
The setupis shown in Table 1 and the oracle values under the mean-optimal rule and RISE 1 are given in Table 2.
The detailed setup can be found in Section 4.1 under Example 1.
We consider vulnerablesubjects as those with low outcome values, as highlighted in red in Table 1 (A full definition is givenin Section 3.2).
For X ≤ 0 .
5 , the mean-optimal rule would assign action A = 1 as it tries to achievethe largest average reward across S = 0 and S = 1 .
Recall that S is not available at the time ofdecision.
However, this action results in great harm for subjects with S = 1 as they could get theworst expected outcome of 0.
On the contrary, RISE improves the worst-case outcome by assigning A = − 1 , protecting the vulnerable subjects.
Likewise, for X > 0 . 5 , the mean-optimal rule assigns A = − 1 while the RISE assigns A = 1 protecting those with S = [-
 thatcould-]0 that could have experienced anoutcome of 5 .
Compared to the mean-optimal rule, the proposed rule achieves a larger reward amongvulnerable subjects while maintaining a comparable overall reward.
[Table]
[Table]
Main contributions
Methodology-wise, 1) we propose a novel framework, RISE, to handlesensitive variables in causality-driven decision making.
Robustness is introduced to improve theworst-case outcome caused by sensitive variables, and as a result, it reduces the outcome variationacross subjects.
The latter is directly associated with fairness and safety in decision making.
Tothe best of our knowledge, we are among the first to propose a robust-type fairness criterion undercausal inference.
We introduce a classification-based optimization framework that can easilyleverage most existing classification tools catered to different functional classes, including state-ofthe-art random forest, boosting, or neural network models.
Application-wise, 3) the consideration ofsensitive variables in decision learning is important to applications in policy, education, healthcare,etc.
Specifically, we illustrate the application of RISE using three real-world examples from fairnessand safety perspectives where robust decision rules are needed, across which we have observedrobust performance of the proposed approach.
From a fairness perspective, we consider a job trainingprogram where age is considered as a sensitive variable.
From a safety perspective, we consider twoapplications to healthcare where lab measurements are considered as sensitive variables.
Related Work
Our work focuses on individualized decision rules, which aim at assigning treatment decision basedon subject characteristics.
Existing methods for deriving IDRs include model-based methods suchas Q-learning [4, 5, 6, 7, 8, 9] and A-learning [10, 11], model-free policy search methods such as[12, 13, 14, 15, 16, 17], and contextual bandit methods [18, 19].
In Appendix A.1, we provideadditional literature review on general IDRs under causal settings.
Fairness, safety and robustness are topics of interest that extend well beyond causal inference.
In the following, we provide a review ofthese areas, with focus given to work related to causal inference and IDRs.
Fairness and safety in IDRs.
The consideration of fairness and safety in machine learning has seenan explosion of interest in the past few years [1, 20, 21, 22, 23, 24, 25, 26], especially for solvingclassification and regression problems to help derive decisions that are not only accurate but also fair.
In these work, sensitive variables are also referred to as sensitive, protected, or auxiliary attributes.
We extend the definition of sensitive variables to include delayed information that is not available atdeployment as it is also suitable for this framework.
Among earlier work, preprocessing [3, 27, 28, 29] and inprocess training approaches [30, 23, 31]consider disentangling the input X from a known or unknown sensitive variable S so that thetransformed X does not contain any information related to S .
Due to the causal nature of IDRs,effect of IDRs cannot be estimated consistently when an informative S is left out and the resultingrule is suboptimal.
This follows from the classic argument that any unmeasured confounding (i.e., S ), if not accounted for, would lead to bias.
Similar issues persist in contextual bandits [32, 33].[34] considers reducing the impact of auxiliary variables on prediction under distributional shift. Although it is motivated from a causal idea, its main focus is still on prediction.
Inside the causalframework, [35, 36] extend fairness from prediction to policy learning using causal graphical modelsby incorporating fairness constraints.
[37] considers counterfactual fairness that seeks to achieveconditional independence of the decisions via data preprocessing.
Despite earlier efforts in bringingfairness into the causal framework, most of these approaches only ensure mean zero disparity in S but do not have robustness guarantees in the sense that the variance of the disparity in S is notcontrolled.
Besides, most examples consider a single categorical sensitive variable, but not multipleor continuous ones.
Robustness in IDRs.
Recently the statistical literature has witnessed a growing interest in developingrobust methods for estimating IDRs.
They introduce robustness into the objective function by usingquantile-optimal treatment regimes or mean-optimal treatment regimes under certain constraintsto improve the gain of individuals at the lower tail of the reward spectrum [38, 39, 40, 41, 42].
Inparticular, [38, 40] propose to estimate quantile- or tail-optimal treatment regimes.
[39] studies themean-optimal treatment regime under a constraint to control for the average potential risk.
[41]proposes a decision-rule-based optimized covariates dependent equivalent for tasks of individualizeddecision making.
[42] considers mean and quantile objectives simultaneously by maximizing theaverage value with the guarantee that its tail performance exceeds a prespecified threshold.
Robustness,in their sense, pertains to the outcome distribution subject to the sampling error.
When sensitivevariables are present, we consider instead the robustness of the outcome distribution subject to theuncertainty due to sensitive variables, providing a more targeted way of ensuring robustness, which isdirectly related to fairness and safety.
Compared to algorithms based on explicit fairness constraints(for example [43, 44] in classification and [35, 37] in causal inference) that seek to remove thedisparity across different values of S , our method reduces the variance of disparity across S .
Inaddition, constraint-based approaches typically require specialized optimization procedures whereasour approach presents an elegant and systematic way for optimization.
To our knowledge, we are thefirst few to consider decision fairness via a robust objective under the causal framework.
Robust Decision Learning Framework with Sensitive Variables
3.1 Preliminaries
Notation.
We let random variables be represented by upper-case letters, and their realizations berepresented by lower-case letters.
Suppose there are n independent subjects sampled from a givenpopulation.
For subject i , let A i ∈ {− 1 , 1 } denote a binary treatment assignment and Y i denote thecorresponding outcome.
Without loss of generality, we assume a larger value of outcome is desirable.
Under the potential outcomes framework [45, 46], let Y i ( a ) be the potential outcome had the subjectbeen assigned to A = a for a = 1 or − 1 .
Let X i ∈
X be the feature vector and, for now, S i be asingle sensitive variable.
Extension to multiple sensitive variables is presented in Section 3.4.
Weconsider S ∈ S where S
= { 1 , . . .
, K } if S is discrete and S
= R if S is continuous.
Definition of sensitive variables.
We define sensitive variables that are important to the interventiondecision, but their inclusion in decision making is prohibited.
Formally, consider variables X and
S that are both available during model training and are both determinants of conditional average treatment effect [47].
While X and S may be both involved in training, the derived decision rule d ( · )precludes the input of S due to sensitive concerns.
Hence, the derived IDR is only a function withthe form d ( X ) :
X → A .
Following the above definition, we consider an offline learning frameworkwhere sensitive variables are collected and can be used in obtaining the IDRs, but they cannot be usedin the resulting IDRs.
A causal diagram and a decision diagram are provided in Appendix A.6.
Assumption 1.
Assume the following conditions hold:
1a Consistency: Y
=
Y ( − 1) 1 ( A = − 1)
+ Y (1) 1 ( A = 1) .
1b Positivity: 0 <
Pr ( A = 1 | X, S ) < 1 .
1c Unconfoundedness: { Y ( − 1) , Y (1) }
⊥ A |{ X, S } and { Y ( − 1) , Y (1) } ̸⊥ A | X .
Assumption 1a is the standard consistency assumption in causal inference and Assumption 1b statesthat every subject has a nonzero probability of getting the treatment.
Assumption 1c states thatgiven X and S , the potential outcomes are independent of the treatment assignments.
Besides,the unconfoundedness does not hold when only X is conditional, signifying the importance of S .
Under causal settings, Assumption 1c ensures that treatment effects cannot be non-parametricallyidentifiable without S [48, 47, 49, 50, 51].
Approaches such as disentanglement of X from S undersupervised learning settings mentioned in Section 2 will introduce bias towards estimating the IDR.
Naive approaches that omit sensitive variables.
When S is not available for future deployment, anaive approach is to maximize E X { E ( Y | X, A = d ( X )) } over d using ( X, A, Y ) during the trainingprocedure.
This approach introduces bias in the estimation of conditional treatment effect and leadsto a suboptimal IDR due to the unmeasured confounder S .
Mean-optimal approaches that use the sensitive variables.
It is thus important that one includes Sinto the training procedure.
For example, if we consider the value function framework (i.e., expectedoutcome) used by most existing works such as [52, 53], we can show that
[Equation]
where the third equality in (1) holds by Assumption 1 and the last inequality also indicates thenaive approaches without using S will in general fail.
Then one valid approach is to maximize
E X (cid:2) E S
| X { E ( Y | X, S, A = d ( X )) } (cid:3) over d using ( X, S, A, Y ) .
The optimal IDR under this criterion is, for every X ∈ X ,
[Equation]
d [Equation] sgn [Equation]
E S [Equation] E [Equation]
Y [Equation] S [Equation]
A
[Equation] [Equation] 1) } ) , which finds the treatment that maximizes the conditional expected outcome given X by averagingout the effect of the sensitive variable S . Mean-optimal approaches, however, fail to control thedisparities across realizations of the sensitive variables due to the integration over S , which may leadto unsatisfactory decisions to certain subgroups, as illustrated in the toy example in Section 1.
3.2 Robust Optimality with Sensitive Variables
Driven by the limitation of existing approaches, our goal is to derive a robust decision rule thatmaximizes the worst-case scenarios of subjects when some sensitive information is not available atthe time of deploying the decision rule.
Specifically, our robust decision learning framework drawsdecisions based on individuals’ available characteristics summarized in the vector X without thesensitive variable S , while improving the worst-case outcome of subjects in terms of the sensitivevariable in the population.
Formally, given a collection D of all treatment decision rules dependingonly on X , the proposed RISE approach estimates the following IDR, which is defined as
[Equation]
where G S | X ( · ) could be chosen as some risk measure for evaluating E ( Y | X, S, A = d ( X )) for each
S ∈ S .
Examples include variance, conditional value at risk, quantiles, etc.
In this paper, we consider
G S | X as the conditional quantiles (for a continuous S ) or the infimum (for a discrete S ) over S .
Specifically, for a discrete S , G S | X is considered as an infimum operator of E ( Y | X, S, A = d ( X )) over S .
We thus aim to find
[Equation]
where inf is the infimum taken with respect to E ( Y | X, s, A = d ( X )) over s ∈ S .
This implies that for a given X , d ∗ ( X ) assigns the treatment that yields the best worst-case scenario among all possible values of S for every X ∈ X , or equivalently,
[Equation]
For a continuous S , we consider G S | X { E ( Y | X, S, A = d ( X )) } as Q τS | X { E ( Y | X, S, A = d ( X )) } , which is the τ -th quantile of { E ( Y | X, S, A = d ( X )) } and τ ∈ (0 , 1) is the quantile level of interest.
Specifically, Q τS | X { E ( Y | X, S, A
= d ( X )) }
= inf { t : F ( t ) ≥ τ } with F denoting the conditional distribution function of E ( Y | X, S, A = d ( X )) given X and d .
Note the randomness behind E ( Y | X, S, A = d ( X )) given X and d is fully determined by the sensitive variable S .
Then optimal IDR under this criterion is defined as
[Equation]
This implies that for a given X , d ∗ ( X ) assigns a treatment that yields the largest τ -th quantile of the outcome over the distribution related to S , or equivalently,
[Equation]
We let τ = 0 .
25 throughout the paper and suppress τ for simplicity.
Results on varying the value ofτ is provided in Appendix; see Section 4.1 for details.
Identifying vulnerable subjects.
Our RISE framework provides a natural way to define vulnerablegroups .
Specifically, for a discrete S , if inf S { E ( Y | X, S, A = 1) } > inf S { E ( Y | X, S, A = 0) } ,then arg inf S { E ( Y | X, S, A = 0) } is vulnerable given X , otherwise arg inf S { E ( Y | X, S, A = 1) }is vulnerable.
In other words, the vulnerable subjects are those in the worst-off group that needsprotection.
Similarly, for a continuous S , if Q S { E ( Y | X, S, A = 1) } > Q S { E ( Y | X, S, A = 0) } ,then the set { S : E ( Y | X, S, A = 0) ≤ Q S { E ( Y | X, S, A = 0) }} defines the vulnerable subjectsgiven X , otherwise this group is defined as { S : E
( Y | X, S, A = 1) ≤ Q S { E ( Y | X, S, A = 1) }} .
3.3 Estimation and Algorithm
Here we provide a transformation of the proposed RISE from an optimization problem to a weightedclassification problem.
There are several advantages to this conversion: 1) The optimization problemdefined in (2) involves a nonsmooth and nonconvex objective function that could lead to computationalchallenges.
2) With multiple powerful statistical and machine learning toolbox to choose from, aclassification problem can be more readily solved in practice.
Hyperparameter tuning and modelselection could be conducted to further boost performance.
Compared to a direct optimizationof (2), a classification-based optimizer allows the use of off-the-shelf software packages that can betailored to different functional classes or incorporate different properties such as model sparsity.
Proposition 1.
Maximizing the objective function in (2) is equivalent to maximizing
[Equation]
− 1) } ] (cid:9) .
With Proposition 1 and a proper estimator of the outcome model E ( Y | X, S, A ) using training data D n = { X i , S i , A
i , Y i } ni
=1 , we replace the expectation of Y i by its estimate ˆ Y i and solve the following,
[Equation]
where g 1 ( x i ) = G s | x { ˆ Y i ( x i , s, a i = 1) } and g 2 ( x i ) =
G s | x { ˆ Y i ( x i , s, a i = − 1) } .
We have thefollowing proposition to address noncontinuity in (3) and transform it into a classification problem.
Define F as a class of all measurable functions over X .
Proposition 2.
Maximizing the empirical objective in (3) is equivalent to a weighted classificationproblem of minimizing over f ∈ F ,
[Equation]
with features x i , a label sgn { g 1 ( x i ) − g 2 ( x i ) } and the sample weight | g 1 ( x i [Equation] i
[Equation] | for 1 ≤ [Equation] With Proposition 2, we have transformed the optimization problem (2) into a weighted classificationproblem (4) where for subject i with features x i , the true label is sgn { g 1 ( x i ) − g 2 ( x i ) } and thesample weight is | g 1 ( x i ) − g 2 ( x i ) | .
The estimated optimal decision rule by (4) is then given byˆ d ( x ) =
sgn { ˆ f ( x ) } 2 .
The proof of Proposition 1 and Proposition 2 is presented in Appendix {+A.2.
+}
If ˆ f ( x ) = 0 , assign a random treatment.
Input: Training data D n = { Y i , A i , X i , S i } ni =
Output: Estimated decision rule ˆ d for i ← 1 to n do ˆ
Y i ( x i , s i , a i ) ← Model E ( Y | X, S, A = a ) using D n with a = 1 and a = − 1 , respectively; if S is continuous then g 1 ( x i ) ← Model Q S | X,A { E ( Y | X, S, A = a ) } via quantile regressions of ˆ Y i ( x i , s i , a i ) on x i , for D n with a = 1 ; g 2 ( x i ) ← Model Q S | X,A { E ( Y | X, S, A = a ) } via quantile regressions of ˆ Y i ( x i , s i , a i ) on x i , for D n with a = − 1 ; else then g 1 ( x i ) ← Compute inf s ∈ S { ˆ Y
i
( x i , s, a i = 1) } ; g 2 ( x i ) ← Compute inf s ∈ S { ˆ Y i ( x i s a i = − 1) } ; end d a weighted classification model with features x i , label sgn { g 1 ( x i ) − g 2 ( x i ) } , and sample weight | g 1 ( x i ) − g 2 ( x i ) | i ≤ n ; return d . Algorithm 1 provides an algorithmicoverview of RISE.
The inner expectation E ( Y | X, S, A ) can be modeled asˆ Y ( X, S, A ) using a twin model separatedby the treatment and control groups.
Fora continuous S , we propose to estimate G ( X, A ) =
Q S | X,A { E ( Y | X, S, A ) }via a quantile regression of ˆ Y on Xbut without S .
For a discrete S ,we propose to obtain an estimate of G ( X, A ) =
inf S { E ( Y | X, S, A ) } by finding the minimum among { E ( Y | X, S =1 , A ) , . . .
, E ( Y | X, S = K, A ) } .
The estimated decision rule can then be obtainedfrom the weighted classification.
In ourimplementation, neural networks are usedto fit models in the training data sets.
Thedetails on modeling and hyperparametertuning via cross-validations are given in Appendix A.3.
A Python package risebased on neural networks is built.
Notethat the model choices are flexible.
3.4 Extension to Multiple Discrete Sensitive Variables
For multiple discrete sensitive variables, similar estimation procedure can be conducted as outlinedin Section 3.3.
Suppose there are L discrete sensitive variables,
i.e., S =
{ S 1 , S 2 , . . .
, S L } .
Theinner expectation E ( Y | X, S 1 , . . .
, S L , A ) can be obtained with a twin model of Y on X and all Sfor each treatment level.
The infimum over S is obtained by finding the minimum iterating spaceof possible parameter values for each sensitive variable.
See Section 4.2 for an example of usingmultiple discrete sensitive variables.
We will discuss in Section 5 the challenges and future workrelated to scenarios with multiple continuous sensitive variables or that with a mixture of continuousand discrete sensitive variables.
Numerical Studies
In this section, we perform extensive numerical experiments to investigate the merit of robustness ofthe proposed framework via simulations and three real-data applications.
The results demonstratethat the proposed rules achieve a robust objective with sensitive variables unavailable at the time ofdecision while maintaining comparable mean outcomes.
Compared approaches.
For comparison, we consider the naive and mean-optimal approachesdescribed in Section 3.1, which correspond to different choices of G ( · ) functions.
The naive decisionrule that simply disregard information of S , denoted as Base , can be formulated in our optimizationframework of (2) by letting G ( X, A ) =
E ( Y | X, A ) .
The IDR can be estimated directly by fittinga model of Y on X in each treatment arm.
The resulting IDR is not sensitive variables-aware andis biased due to confounding, as discussed.
Another IDR that resembles traditional mean-optimaldecision rules, denoted as Exp , can be formulated as G ( X, S, A ) =
E ( Y | X, S, A ) .
This can beobtained by training a classification model without S , i.e., only using X , after obtaining an outcomemodel for the inner expectation E ( Y | X, S, A ) .
Note that this approach is not robust to extremebehaviors in S .
The modeling approaches described in Appendix A.3 applies to here.
We also includethe double robust [54] versions of Base and Exp, respectively, by adapting Policytree (PT) [55, 56],the latest state-of-the-art policy learning method for maximizing the expected values. The two newmethods are termed PT-Base and PT-Exp .
Evaluation metrics.
Objective: the quantile objective is estimated and reported for a continuous
S and the infimum objective is for a discrete S .
The objective, when τ < 0 .
5 , (here τ = 0 .
25 )represents the value of the “low performers” among all possible value of S under a given d . 2)
Value:the value function, or expected reward used by the most existing methods, such as [52, 53], is definedas V ( d ) =
E { Y ( d ) } .
It represents the “average performers”.
For randomized trials, an unbiased estimator of V ( d ) is given by ˆ V ( d ) = { (cid:80)
Ti =1 Y
i 1 ( A i = ˆ d ( X i )) /π
( A i , X i ) } / { (cid:80)
Ti =1 1 ( A i = ˆ d ( X i )) /π
( A i , X i ) } [57], where T is the sample size of the test data.
For observational studies, thevalue is estimated with ˆ V ( d ) =
T − 1 (cid:80) Ti
=1 ˆ Y
i
( x i , s i , a i = ˆ d ) .
We report the metrics among allsubjects and among the potential vulnerable subgroup, respectively.
For simulation, we considertraining data and the testing data, respectively, with sample sizes of 5,000.
For real-data applications,we consider a 80-20 split of the dataset into a training data and a testing data.
Continuous covariatesare standardized before the estimation.
All results are based on 100 replications.
Experiments areperformed on a 6-core Intel Xeon CPU E5-2620 v3 2.40GHz equipped with 64GB RAM.
Pythoncode is provided as part of Supplementary Materials.
4.1 Simulation Studies
Example 1.
Here we provide details for the simulation of the motivating example introduced in
Section 1.
The outcome is generated according to: Y i =
( X i > 0 . 5) { 5 + 10 {+
+} ( A i = 1) + 22 S
i −24
( A i = 1) S i } + 1 ( X i ≤ 0 . 5) { 11 + 19 1 ( A i = 1) + 2 S
i − 32
( A i = 1) S i } + ϵ i , where thecovariate X i ∼ Unif [0 , 1] , treatment assignment A
i ∼ Bernoulli (0 . 5) , and the noise ϵ i ∼ N (0 , 1) .
For a discrete type S , S i ∼ Bernoulli (0 . 5) .
For a continuous type S , S i is generated from a mixtureof beta distributions, Beta (4 , 1) and Beta (1 , 4) , with equal mixing proportions.
Example 2.
We generate Y using the following model: Y i = { 0 .
5 + 1 ( A i = 1) + exp( S i ) −2 .
5 S
i 1 ( A i = 1) }{ 1+ X i 1 − X i 2 + X 2 i 3 +exp( X i 4 ) } + { 1+2 1 ( A i = 1)+0 .
2 exp( S i ) − 3 .
5 S i 1 ( A i =1) }{ 1 + 5 X i 1 − 2 X i 2 + 3 X i 3 + 2 exp( X i 4 ) } + ϵ , where X ij ∼ U (0 , 1) , j = 1 , . . .
, 6 , A satisfieslog { P ( A i = 1 | X i ) /P
( A = 0 | X i ) }
= 0 .
6( − S i + X
i 1 − X i
+ X
i 3 − X i
+ X
i 5 − X i 6 ) , and ϵ i ∼ N (0 , 1) .
For a continuous type S , S i is generated from a mixture of beta distributions, Beta (4 , 1)and
Beta (1 , 4) , with equal mixing proportions; for a discrete type S , we consider a binary S
i thatsatisfies log { P ( S i =
| X i ) /P
( S i =
| X i ) }
= − 2 .
5 + 0 .
8( X i
+ X
i
+ X
i
+ X
i
+ X i 5 + X i 6 ) .
Table 3 summarizes the performance of the proposed IDRs compared to the mean criterion for
Example 1 and Example 2.
The proposed RISE achieves the largest objectives and improves thevalue among vulnerable subjects, while maintaining comparative overall values.
As for the objective,intuitively, the proposed rule is expected to have a larger objective.
We also point out that there is nodirect relationship between the objective among all subjects versus the objective among vulnerablesubjects.
For example, using the toy example with setup in Table 1, and limiting to subjects with X ≤ 0 .
5 only, S = 1 is vulnerable and is assigned A = − 1 by the proposed RISE.
The objectiveamong S = 1 is 13 but the objective among both S
= 0 and S = 1 is 12 =
(11 + 13) / 2 , which issmaller than that among the vulnerable group.
In other words, by protecting the vulnerable subjects,the proposed rule may lead to an increase in the outcome of the vulnerable group, and the gain mayresult in a higher outcome than the overall mean outcome.
PT-Exp method tends to show the bestimprovement in terms of the overall value, as the doubly robust-based estimators tend to reducevariance in value estimation.
However, PT-Exp is shown to have minimal benefits for vulnerablesubjects.
RISE still shows the largest gain in the objective and value among vulnerable subjectsamong all compared methods.
In the appendix, we consider a continuous S for different quantile criteria τ = 0 .
1 and 0 . 5 to test therobustness of the proposed RISE.
Results show that when τ is small, there is more strength in theproposed method, as the algorithm aims to improve the worst-outcome scenarios.
The proposed RISEhas the largest gain in objective and value among vulnerable subjects when τ is 0.1, and has similarperformance as the compared approaches when τ is 0.5.
Also, we consider a scenario where S is notinvolved in the data generation of Y , i.e., Assumption 1c is simplified as { Y ( − 1) , Y (1) }
⊥ A | X .
Theestimated objective and value function are similar across all compared approaches, which indicates therobustness of RISE.
Finally, we study the performances of our method when Assumption 1b is nearlyviolated or Assumption 1c is violated.
Similar patterns have been observed that the proposed RISEachieves the largest objectives and improves the value among vulnerable subjects, while maintainingcomparative overall values.
The Details can be found in Appendix A.4.
4.2 Real-data Applications
We present three real-data examples to showcase the robust performance of RISE.
These applicationsconsider either fairness or safety in the context of policy [58] and healthcare [59, 60] where sensitivevariables exist.
The information of data access is provided in Appendix A.5.
Fairness in a job training program.
To illustrate the implication of the proposed method from afairness perspective, we consider the National Supported Work (NSW) program [58] for improving [Table] personalized recommendations of a job training program on increasing incomes.
This programintended to provide a 6 to 18-month training for individuals in face of economic and social problemssuch as former drug addicts and juvenile delinquents.
The original experimental dataset consistsof 185 individuals who received the job training program ( A = 1 ) and 260 individuals who did not( A = − 1 ).
The baseline covariates are age, years of schooling, race (1 = African Americans or
Hispanics, 0 = others), married (1 = yes, 0 = no), high school diploma (1 = yes, 0 = no), earning in1974, and earning in 1975.
The outcome variable is the earning in 1978.
In the exploratory analysisusing causal forest [61], we observe that age may play an important role in the causal effect of thejob training program on the long-term post-market earning.
In the following data example we use ageas the sensitive variable S and other baseline covariates as X .
The earnings in years 1974, 1975, and1978 are transformed by taking the logarithm of the earning plus one.
Improvement of HIV treatment.
To illustrate the implication of the proposed method from asafety perspective when there is delayed information, we consider the ACTG175 dataset among HIVpositive patients [59].
The original study considers a total of 2,139 patients who were randomlyassigned into four treatment groups.
In this data application, we focus on finding the optimal IDRsbetween two treatments: zidovudine combined with didanosine ( A = − 1 ) and zidovudine combinedwith zalcitabine ( A = 1 ).
The total number of patients receiving these two treatments is 1,046.
Thebaseline covariates we consider are age, weight, CD4 T-cell amount at baseline, hemophilia (1 = yes,
= no), homosexual activity (1 = yes, 0 = no), Karnofsky score, history of intravenous drug use (1 =yes, 0 = no), gender (1 = male, 0 = female), CD8 T-cell amount at baseline, race (1 = non-Caucasian,0 = Caucasian), number of days of previously received antiretroviral therapy, use of zidovudine in the30 days prior to treatment initiation (1 = yes, 0 = no), and symptomatic indicator (1 = symptomatic, 0= asymptomatic).
The outcome variable is the CD4 T-cell amount at 96 ± 5 weeks from the baseline.
We consider CD8 T-cell amount at baseline as the sensitive variable.
The response of CD8 T-cellamong HIV positive patients has not been fully understood [62].
Clinically, it is plausible that only
CD4 is measured in clinical visits where treatments are based on, hence CD8 might not be measuredand not used in decision making.
As our exploratory analysis using causal forest shows, CD8 T-cellamount may play an important part in the treatment effect of the outcome.
Safe resuscitation for patients with sepsis.
For this application, we apply the proposed method totreating sepsis, a life-threatening disease.
This application intends to provide an example to applyour method with multiple categorical sensitive variables in the scenario where there is missing yetimportant information at the time of decision making.
We apply the proposed method to a sepsis studyfrom the University of Pittsburgh Medical Center (UPMC).
The original study cohort includes 30,687patients with Sepsis-3 [60] within 6 hours of hospital arrival from 14 UPMC hospitals between 2013and 2017.
For our data analysis, we consider X to be baseline patient characteristics 4 hours beforesepsis onset, which includes patient demographics of age, gender (1 = male, 0 = female), race (1 =
Caucasian, 0 = others), and weight, and vital signs of usage of mechanical ventilation (1 = yes, 0 =no), respiratory rate, temperature, intravenous fluids (1 = yes, 0 = no), Glasgow Coma Scale score,platelets, blood urea nitrogen, white blood cell counts, glucose, creatinine.
We consider two sensitivevariables, lactate and Sequential Organ Failure Assessment (SOFA) score 4 hours before sepsis onset.
Lactate and SOFA score have been two important indicators of sepsis severity [63, 64, 65].
Differentfrom the baseline patient demographics or common vital signs that are typically obtained at theadmission of patients, SOFA score combines performance of several organ systems in the body [60],which requires additional calculation and cannot be obtained directly.
Lactate labs measures the levelof lactic acid in the blood [66] and are less common in routine examination, which could be delayed inordering.
Hence, their measurements are obtained retrospectively after treatment decisions have beenmade and are not available at times of decision.
We discretize lactate level at clinically meaningfulvalue of 2 mmol/L [65], and SOFA score at value of 6 for analysis [67, 68].
The treatment option iswhether the patient took any vasopressors during the first 24 hours after sepsis onset.
The outcome ispatient survival at day 90.
The analysis cohort contains 6,539 patients in total.
We are interested inmaking decision about whether to treat patients with vasopressors in the first 24 hours after sepsisonset given the measurements of lactate and SOFA are not available at the time of decision making.
Additional rationale and background on this example are provided in Appendix A.5.
Results.
Table 4 presents the performance of various IDRs on the three applications.
As expected,
RISE has the largest objective as well as value among vulnerable subjects.
The patterns are similar tothat in Section 4.1.
We apply the Shapley additive explanations (SHAP) approach [69] to help visualize and interpret the important covariates in the decision rules given by RISE and Exp, respectively,in Appendix A.5.
The SHAP approach provides unified values to describe the correlation betweeneach feature and the predicted decision rule, respectively [69].
Overall, the direction of correlationsis similar for RISE and Exp, but their ranking of feature importance may be different.
[Table]
Discussion
We have proposed RISE, a robust decision learning framework with a novel quantile-, or infimumoptimal treatment objective intended to improve the worst-case scenarios of individuals when decisions with uncertainty are needed to be made with sensitive yet important information missing.
Our approach can be applied to a broad range of applications, including but not limited to policy,education, healthcare, etc.
For multiple continuous sensitive variables, the estimated decision ruleby RISE can be easily obtained by fitting similar quantile regression as described in Section 3.3.
For a mixture of continuous and discrete sensitive variables, the estimated rule can be obtained bytaking infimum over the discrete ones, then obtaining quantile over the continuous ones.
However,challenges remain in finding vulnerable subjects described in Section 3.2 under these settings asit may be computationally difficult to find a vulnerable set of S when they are multi-dimensional.
Another future work includes extension of the current binary treatment option to a multi-treatmentoption.
It is also worth mentioning that our work can be naturally extended to the scenario wherethere exist unmeasured confounders.
As long as the conditional average outcome given observedcovariates can be identified (e.g., via instrumental variables [70]), our method can be applied.
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[Table] applications are publicly available data, and the third application uses deidentified datathat all personally identifiable information has been removed. [Table]
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