Penalized Fair Regression for Multiple Groups in Chronic Kidney Disease

Carter H. Nakamoto; Lucia Lushi Chen; Agata Foryciarz; Sherri Rose

stat.MEDec 19, 2025

Penalized Fair Regression for Multiple Groups in Chronic Kidney Disease

Authors: Carter H. Nakamoto, Lucia Lushi Chen, Agata Foryciarz, Sherri Rose

Organizations: Department of Health Policy, Stanford University, 615 Crothers Way Encina Commons MC6019 Stanford, CA 94305, USA · Psyfy Inc. · Department of Epidemiology and Population Health, Stanford University · Department of Health Policy, Stanford University

Abstract

Fair regression methods have the potential to mitigate societal bias concerns in health care, but there has been little work on penalized fair regression when multiple groups experience such bias. We propose a general regression framework that addresses this gap with unfairness penalties for multiple groups. Our approach is demonstrated for binary outcomes with true positive rate disparity penalties. It can be efficiently implemented through reduction to a cost-sensitive classification problem. We additionally introduce novel score functions for automatically selecting penalty weights. Our penalized fair regression methods are empirically studied in simulations, where they achieve a fairness-accuracy frontier beyond that of existing comparison methods. Finally, we apply these methods to a national multi-site primary care study of chronic kidney disease to develop a fair classifier for end-stage renal disease. There we find substantial improvements in fairness for multiple race and ethnicity groups who experience societal bias in the health care system without any appreciable loss in overall fit.

Figures & tables

Figure 1: Penalty Weights for Each Group by Fairness Score Weight. Bars represent the empirical 25th and 75th percentiles of replications. Simulations with 40 draws of candidate penalty weight sets are represented.

Figure 2: Fair Regression Estimator Accuracy and Unfairness for Different Fairness Score Weights by Simulation Setting. Bars represent the empirical 25th and 75th percentiles of replications. Simulations with 40 draws of candidate penalty weight sets are represented.

Figure 3: Fair Regression Estimator Accuracy and Unfairness for Different Fairness Score Weights by Multi-Group Unfairness Synthesis Function, Simulation Setting 2. The subfigures correspond to performance measures for fitted estimators, while colors correspond to how the fairness components of the score functions were calculated. Bars represent the empirical 25th and 75th percentiles of replications. Simulations with 40 draws of candidate penalty weight sets are represented.

Figure 4: Accuracy and Mean Fairness for Proposed and Comparison Fair Reductions Methods by Simulation Setting. Each plotted point corresponds to the result of a single simulation under one of the score function variations or enumerated fair reductions hyperparameters. Simulations with 40 draws of candidate penalty weight sets are represented.

	n	Prevalence	Sensitivity	Specificity	Accuracy
Black	$8178$	$0.07$	$0.38$	$0.95$	$0.92$
Hispanic non-white	$2695$	$0.07$	$0.40$	$0.95$	$0.91$
Asian	$1434$	$0.07$	$0.19$	$0.98$	$0.92$
White	$53\,793$	$0.04$	$0.38$	$0.97$	$0.95$
Total	$67\,901$	$0.04$	$0.37$	$0.97$	$0.95$

Table 1: Unpenalized Regression Performance by Race and Ethnicity Group.

			Unfairness
Method	Accuracy	Sensitivity	Mean	Black	Asian	HNW
Baseline	$0.95$	$0.38$	$0.07$	$0.06$	$0.15$	$0.08$
$\begin{aligned} \text{Pop }\alpha=\,&0.1,0.2,0.3,0.4\\ &0.5,0.6,0.7,0.8,0.9,0.99\end{aligned}$	$0.95$	$0.39$	$0.01$	$0.00$	$0.10$	$0.00$
Max $\alpha=0.2,0.3$ $\begin{aligned} \text{Group }\alpha=\,&0.1,0.2,0.3,0.4\\ &0.5,0.6\end{aligned}$	$0.95$	$0.32$	$0.04$	$0.04$	$0.04$	$0.02$
Max $\alpha=0.1$	$0.95$	$0.27$	$0.03$	$0.04$	$0.02$	$0.01$
$\begin{aligned} \text{Max }\alpha=\,&0.4,0.5,0.6,0.7\\ &0.8,0.9,0.99\end{aligned}$ Group $\alpha=0.7,0.8,0.9,0.99$	$0.90$	$0.31$	$0.05$	$0.06$	$0.00$	$0.03$

Table 2: Performance for Baseline Unpenalized Regression, Novel Penalized Regressions, and Fair Reductions Methods.

	Group A	Group B	Group C
Setting 1	32 (0.15)	20 (0.14)	37 (0.16)
Setting 2	1 (0.03)	20 (0.11)	37 (0.13)
Setting 3	2 (0.04)	2 (0.04)	3 (0.05)

Table 3: Group Size in Percents by Simulation Setting. Means and standard deviations (SD) are given across simulations.

Panel of Figure 5 — Figure 5: Fair Regression Estimator Accuracy and Unfairness by Number of Candidate Unfairness Penalty Weight Draws Across Simulation Settings. Bars represent the empirical 25th and 75th percentiles of replications. Simulations with 40 draws of candidate penalty weight sets are represented.

Panel of Figure 6 — Figure 6: Fair Regression Estimator Accuracy and Fairness for Correctly Specified and Misspecified Fair Regression Estimators by Simulation Setting. Each plotted point corresponds to the result of a single simulation under one of the score function variations. Simulations with 40 draws of candidate penalty weight sets are represented.

Panel of Figure 7 — Figure 7: Fair Regression Estimator Accuracy and Unfairness for Different Fairness Penalty Score Weights by Multi-Group Unfairness Synthesis Function for Simulation Setting 1. Bars represent the empirical 25th and 75th percentiles of replications. Simulations with 40 draws of candidate penalty weight sets are represented.

Panel of Figure 8 — Figure 8: Fair Regression Estimator Accuracy and Unfairness for Different Fairness Penalty Score Weights by Multi-Group Unfairness Synthesis Function for Simulation Setting 3. Bars represent the empirical 25th and 75th percentiles of replications. Simulations with 40 draws of candidate penalty weight sets are represented.

Panel of Figure 9 — Figure 9: Fair Regression Estimator Accuracy and Unfairness for Different Fairness Score Weights by Simulation Setting (Error Rate Disparity Fairness Measure with Random Forest Estimator). Bars represent the empirical 25th and 75th percentiles of replications. Simulations with 40 draws of candidate penalty weight sets are represented.

Panel of Figure 10 — Figure 10: Accuracy and Mean Fairness for Proposed and Comparison Fair Reductions Methods by Simulation Setting (Error Rate Disparity Fairness Measure with Random Forest Estimator). Each plotted point corresponds to the result of a single simulation under one of the score function variations or enumerated fair reductions hyperparameters. Simulations with 40 draws of candidate penalty weight sets are represented.

Panel of Figure 11 — Figure 11: Fair Regression Estimator Accuracy and Unfairness by Number of Candidate Unfairness Penalty Weight Draws Across Simulation Settings (Error Rate Disparity Fairness Measure with Random Forest Estimator). Bars represent the empirical 25th and 75th percentiles of replications.

Diagnosis	Search Method	Query
ESRD	Text Query	“End stage renal disease” or “End stage renal failure”
eGFR	Preexisting	See repository
HbA1c	OMOP Concepts: Measurement	4184637, 36304734
Bilirubin	OMOP Concepts: Measurement	4118986, 3024148, 4216632, 3024128
Diastolic BP	OMOP Concepts: Measurement	4298393, 3012888
Systolic BP	OMOP Concepts: Measurement	4152194, 3004249

Table 4: Identification Strategy for ABFM Variables. For most variables, we queried for the enumerated OMOP concepts in the enumerated fields. For other variables, we queried the OMOP concept lookup table for the OMOP ABFM data with the given text.

	Missing	Overall	No ESRD	ESRD
n	0	67901	65084	2817
Measurements
Minimum eGFR, mean (SD)	37179	48.3 (18.7)	49.5 (17.9)	22.3 (16.0)
Mean Diastolic BP, mean (SD)	0	75.3 (7.9)	75.4 (7.9)	74.2 (8.8)
Mean Systolic BP, mean (SD)	0	136.6 (11.9)	136.5 (11.7)	140.0 (14.1)
Mean HbA1c, mean (SD)	44687	6.8 (1.4)	6.8 (1.4)	6.9 (1.6)

Table 5: Patient Characteristics in 2017-2018 by ESRD Status in 2019.

Figure 12: Coefficients for Baseline Unpenalized and Novel Penalized Regressions. Predictors have been standardized to facilitate comparison of coefficient magnitudes.

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Demographic parity (DP) is a widely used group fairness criterion requiring predictive distributions to be invariant across sensitive groups. While natural in classification, full distributional DP is often overly restrictive in regression and can lead to substantial accuracy loss. We propose a relaxation of DP tailored to regression, enforcing parity only at a finite set of quantile levels and/or score thresholds. Concretely, we introduce a novel (

{\ell}

, Z)-fair predictor, which imposes groupwise CDF constraints of the form F f |S=s (z m ) =

{\ell}

m for prescribed pairs (

{\ell}

m , z m ). For this setting, we derive closed-form characterizations of the optimal fair discretized predictor via a Lagrangian dual formulation and quantify the discretization cost, showing that the risk gap to the continuous optimum vanishes as the grid is refined. We further develop a model-agnostic post-processing algorithm based on two samples (labeled for learning a base regressor and unlabeled for calibration), and establish finite-sample guarantees on constraint violation and excess penalized risk. In addition, we introduce two alternative frameworks where we match group and marginal CDF values at selected score thresholds. In both settings, we provide closed-form solutions for the optimal fair discretized predictor. Experiments on synthetic and real datasets illustrate an interpretable fairness-accuracy trade-off, enabling targeted corrections at decision-relevant quantiles or thresholds while preserving predictive performance.

Abstract

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