"Who experiences large model decay and why?" A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Authors: Harvineet Singh
Fan Xia
Alexej Gossmann
Andrew Chuang
Julian C. Hong
Jean Feng
Venue: ICML 2025 poster (2025)
Keywords: Distribution shift
algorithmic fairness
interpretability
explanation
decomposition
mediation analysis
Primary area: social_aspects->robustness
TL;DR: The paper introduces a hierarchical hypothesis testing method that identifies and explains heterogeneous model performance decay across subgroups, enabling targeted corrective actions to mitigate performance degradation.

Abstract: Machine learning (ML) models frequently experience performance degradation when deployed in new contexts. Such degradation is rarely uniform: some subgroups may suffer large performance decay while others may not. Understanding where and how large differences in performance arise is critical for designing *targeted* corrective actions that mitigate decay for the most affected subgroups while minimizing any unintended effects. Current approaches do not provide such detailed insight, as they either (i) explain how *average* performance shifts arise or (ii) identify adversely affected subgroups without insight into how this occurred. To this end, we introduce a **S**ubgroup-scanning **H**ierarchical **I**nference **F**ramework for performance drif**T** (SHIFT). SHIFT first asks "Is there any subgroup with unacceptably large performance decay due to covariate/outcome shifts?" (*Where?*) and, if so, dives deeper to ask "Can we explain this using more detailed variable(subset)-specific shifts?" (*How?*). In real-world experiments, we find that SHIFT identifies interpretable subgroups affected by performance decay, and suggests targeted actions that effectively mitigate the decay.

"Who experiences large model decay and why?"
A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Harvineet Singh
Fan Xia
Alexej Gossmann 2 Andrew Chuang'
Julian C. Hong Jean Feng
Abstract
performance degradation while others may experience very negligible differences, if at all (Yang et al., 2023). Under- standing the subgroups where shifts are most pronounced and providing subgroup-speciffr explanations is critical from the perspectives of algorithmic fairness (Mitchell et al., 2021) and backwards compatibility (Srivastava et al., 2020). Subgroup-leve explanations also help model developers design targeted corrective actions that only modify the algo- rithm's behavior in the most affected subgroups and limit any other unintended effects ("Don't 't fix what ain't 't broke") (Globus- Harris et al., 2022; Suriyakumar et al., 2023).
Machine learning (ML) models frequently experi- ence performance degradation when deployed in new contexts. Such degradation is rarely uniform: some subgroups may suffer large performance de- cay while others may not. Understanding where and how large differences in performance arise is critical for designing targeted corrective ac- tions that mitigate decay for the most affected sub- groups while minimizing any unintended effects. Current approaches do not provide such detailed insight, as they either (i) explain how average performance shifts arise or (ii) identify adversely affected subgroups without insight into how this occurred. To this end, we introduce a Subgroup- scanning Hierarchical Inference Framework for performance drifT (SHIFT). SHIFT first asks "Is there any subgroup with unacceptably large per- formance decay due to covariate/outcome shifts?" (Where?) and, if so, dives deeper to ask "Can we explain this using more detailed variable(subset)- specific shifts?" (How?). In real-world experi- ments, we find that SHIFT identifies interpretable subgroups affected by performance decay, and suggests targeted actions that effectively mitigate the decay.
For instance, suppose an ML algorithm for predicting un- planned readmission achieves overall accuracy of 85% in hospital A and 83% in hospital B. While the change in overall accuracy may not be clinically significant, the change within some subgroup may be sufficiently large to be deemed harmful. If so, it is natural to ask how this hetero- geneity in performance decay arose: was it due to a change in how certain diseases are recorded, which medications are prescribed for certain patients, or something else alto- gether? If we know the affected subgroup and why, we can specifically address the root cause, such by updating data pre- phe-processing and/or the algorithm within the subgroup.
As such, our goal is to simultaneously understand where an ML algorithm performs substantially worse and how it arose. Numerous methods have been developed to find subgroups where an ML algorithm performs poorly (d' Bon et al., 2022; Eyuboglu et al., 2022; Liu et al., 2023; Sub- baswamy et al., 2024), which can in principle be extended to identify subgroups with large model decay. Answering "how" is more tricky. We can obtain an approximate high- level answer by decomposing the average performance drop within an identified subgroup into the contribution from a shift in the marginal distribution of the input features X (covariate shift) versus shift in the conditional distribution of the target Y|X (outcome shift) QQuinonero-Candllaa et al., 2009; Cai et al., 2023).
1. Introduction
ML algorithms known to degrade in performance when applied in different contexts, which has led to extensive work on explaining how differences in an ML algorithm's ''s average performance arise (Cai et al., 2023; Zhang et al., 2023). However, performance differences are rarely uni- form in practice: some subgroups may experience severe
University of California, San Francisco, USA 'Independent re- searcher. Correspondence to: Jean Feng jeannfeng @ uesf.cdu.>
However, this is only a partial solution. For one, it misses situations where the subgroup of individuals experiencing severe covariate shifts is not the same as the one experienc- ing severe outcome shifts, and each subgroup may require different corrective actions. More importantly, we often
Proceedings of the 4gnd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Cooee is available http://gtt htttp://gtthbb.../ thub. jjfeng/ shi ft.

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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
want know precisely which subset of input variables were involved, many real-world shifts involve only few vari- ables (i.e. sparse) and can be fixed in a targeted manner (Castro et al., 2020; Finlayson et al., 2021). Existing meth- ods are currently insufficient, as they rely on assumptions that often do not hold in practice, e.g. the true causal graph known (Zhang et al., 2023; Quintas-Martin Martinez et al., 2024), the data follows simple parametric models (Baron & Kenny, 1 1986), or unrealistically large datasets (Singh al., 2024).
Aggregate Test: Where?
Detailed Tests: How?
Covarlate Shift in p())
Subgroup detected:
X
X
X3
X, <
Outcome Shift p(Y X))
No subgroup
detected
Subgroup-scanning Hierarchical Inference Framework for performance drifT (SHIFT) is a two-stage hypothesis testing procedure that first checks if there is subgroup with unacceptably large performance decay due to aggregate covariate and outcome shifts with respect to all X variables. If so, it checks if this can be explained by detailed variable((ubbstt)-seccii shifts. Red indicates the shift was flagged for further investigation. In this example, covariate shift is flagged because affected a subgroup and variables X1 and X3 were flagged as potential explanations.
Figure 1:
To overcome these limitations, we present a nonparametric Subgroup-scanning Hierarchical Inference Framework for performance drifT (SHIFT) (Fig 1). Whereas prior works have approached drift diagnosis primarily through the lens of estimation, SHIFT approaches this through hypothesis testing. The advantage that hypothesis tests answer simple yes/no questions, which is often more feasible in settings with limited data; in fact, we conduct omnibus tests, which require even less data as they do not need to identify the entire subgroup that is adversely affected. Furthermore, hypothesis tests allow us to check the very assumptions that other works have take on face value. The first stage of SHIFT performs a high-level analysis: decomposing distribution shift into an "aggregate" covariate shift with respect to all of X and an "aggregate" outcome shift with respect to all of X, SHIFT tests if either have led to un- acceptably worse performance in any meaningfully large subgroup (Where?). If so, the second stage drills down to test if this can be adequately explained by shift solely with respect to a sparse subset of variables in X (How?). The major contributions of this work are:
Maximum Mean Discrepancy (MMD) (Gretton et al., 201 2a; Luedtke et al., 2018). More recent works focus detecting only those that are harmful to overall performance (Pod- kopaev & Ramdas, 2022; Panda et al., 2024). No prior meth- ods have been developed to specifically detect distribution shifts that lead to disproportionate harm in a (sufficiently large) subgroup, which SHIFT aims to address.
Decomposing model performance. Various methods have been developed to quantify the contribution of each feature subset to the average performance (Budhathoki et al., 2021; Cai et al., 2023; Wu et al., 2021; Zhang et al., 2023; Quintas- Martinez et al., 2024) and, more recently, the variability of performance changes (Singh et al., 2024). Mathemati- cally, these methods rely on techniques similar to those used in mediation analysis for decomposing the average treat- ment effect into indirect and direct effects (Baron & Kenny, 1986) and variable importance (VI) methods for explaining the variability of the conditional average treatment effect (CATE) (Hines et al., 2023), respectively. Most of these methods either assume a parametric model or knowledge of the causal graph between individual variables. While VI methods that focus on decomposing variability have much weaker assumptions (Hines et al., 2023; Singh et al., 2024), they generally require large datasets and their confidence intervals (CIs) cannot be casily inverted to produce valid hy- pothesis tests (the influence function is degenerate because the estimand is at the boundary of the parameter space un- der the null, leading to inflated Type error rates) (Hudson, 2023). Through carefully framed hypothesis tests, SHIFT provides valid statistical inference without parametric as- sumptions or knowledge of detailed causal graph.
Introduction of novel hierarchical hypothesis test- ing framework that detects subgroups experiencing large performance decay due to aggregate-level covari- ate/outcome shifts, which are then explained using de- tailed variable variable (subset)-specfiic shifts.
SHIFT does not rely on strong assumptions and suitable for smaller datasets, making it broadly applicable to real- world scenarios.
Our simulations demonstrate that SHIFT correctly iden- tifies relevant shifts. Real-world experiments show that SHIFT can guide the design of model/data corrections that strictly improve performance.
2. Related Work
We briefly discuss the three most related areas below (sum- marized in Table 1). See Appendix E for more detailed discussion as well as other related areas.
Detecting distribution shifts. Many methods have been developed to detect any shift in marginal/conditional dis- tributions, such as Kolmogorov-Sminoo (KS) (Rabanser et al., 2019), kerne--bsed tests (Zhang et al., 2011), and
Discovering subgroups. Methods have been developed to identify subgroups with low performance within a single distribution (Eyuboglu et al., 2022; d'Eon et al., 2022; Ali et al., 2022; Feng et al., 2024a; Dong et al., 2024; Rauba
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Table I: Subgroup-scanning Hierarchical Inference Framework for performance drifT (SHIFT) compared to prior work
Category (Example methods)
Detects subgroup
Valid hypothesis test?
Avoids detailed
Detailed explanations for outcome/covariate shifts?
with large decay?
causal graph?
Detect any shift (Rabanser et al., 2019; Zhang et al., 2011) Detect loss shift (Podkopaev & Ramdas, No 2022)
No
Yes
Yes
Outcome only
Yes
Yes
No
Decompose average perf decay (Zhang No
Some methods
No
Covariate only
al., 2023; Cai et al., 2023; Quintas-
Martinez et al., 2024)
Decompose shift variability (Singh et al., No 2024)
No
Yes
Outcome & Covariate
Decompose ATE (Baron & Kenny, 1986) Decompose CATE variability (Hines al., 2023)
No
Parametric only
No
Covariate only Outcome only
No
No
Yes
Subgroup discovery (Byuboglu et al., No 2022; E'Eon al., 2022)
Some
Yes
No
SHIFT (Proposed)
Yes
Yes
Yes
Outcome Covariate
et al., 2024; Subbaswamy et al., 2024) and subgroups with large CATE (Athey et al., 2019). However, most meth- ods only provide point estimates and not statistical infer- ence (CIs/hypothesis tests). More importantly, existing methods can be directly adapted to explain how large per- formance decay arises across subgroups with respect to variable-specific shifts.
variable subsets X, and use p=(x) and ps to denote X, --seeciiic covariate and outcome shifts, respectively. We will present their exact definitions later.
SHIFT is hierarchical diagnostic framework that does more detailed analysis of performance drift compared to the standard two-way decomposition in (2) by accounting for heterogeneity of performance shifts. At the first level, SHIFT checks if the aggregate covariate and outcome shifts lead to subgroups with large performance decay. If so, SHIFT searches for more detailed explanation among candidate variable(suuseet-sseeii shifts.
3. Hierarchical Testing Framework
Given a set of features X and outcome Y, want to understand the difference in performance of an algorithm across source and target domains, denoted by d = 0 and d = 1 respectively. We refer to the joint distribution of (X.Y) Y) in each domain by Pa and its corresponding expecta- tion with Eg. Performance is quantified by a loss function L = {y. fx)) E IR. The average loss conditional on æ in domain d is denoted (x) Fulee(, [(X)) X = x for d E (0, 1}. Hat notation denotes estimate.
Throughout, SHIFT focuses only subgroups of individ- uals and performance shifts that are deemed large enough to be of practical interest, by requiring the domain expert to select a priori the minimum subgroup size e > 0 and mini- mum shift magnitude T 2 0. This is critical to ensure the practical usability of these methods, as alarms for negligible shifts lead to alarm fatigue (Cvach, 2012; Feng et al., 2025). The set A, denotes all subgroups whose prevalence in the source and target domains exceed e 0.
A shift in the joint distribution of (X, Y) can be decomposed into aggregate covariate and outcome shifts, which are de-
fined by the shifts Po (x) = P1 (x:) and po(ylæ) = P1 (ylæ), respectively. In this way, the shift from source to target can
The following two sections (Sec 3.1 and 3.2) introduce the aggregate and detailed hypothesis tests SHIFT and Section 4 describes the actual testing procedures.
be broken down into sequence of aggregate-level shifts:
po(x)po = pi (x (z) + pı (x)p
(1)
3.1. Aggregate tests:
Where?
and, correspondingly, the average performance change can be decomposed into:
SHIFT first tests if there exists a subgroup with large per- formance decay due an aggregate covariate shift and, likewise, subgroup impacted by an aggregate outcome shift. The impacts of these shifts within subgroup A E A,
E Eo
E [2, E, Znn - -o
Zo] outcome shift
covarinte shift
To generate even more detailed explanations of performance shifts, we will consider sparse shifts solely with respect to
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
is quantified using similar decomposition as (2), i.e.
Connection to mediation analysis. Prior works have high- lighted that the decomposition of average performance change into covariate and outcome shifts parallels the de- composition of the average treatment effect into indirect and direct effects, which is commonly analyzed in causal mediation analysis (Castro et al., 2020; Singh et al., 2024). As this work decomposes subgroup-speciftr performance changes, it parallels recent efforts in the nascent but growing field on analyzing the heterogeneity of causal effect decom- positions (Loh et al., 2020; Rubinstein et al., 2023). The omnibus tests developed in this work may thus be useful for testing heterogeneous indirect/direct effects, an area that has not been addressed thus far. We discuss these connections further in Appendix A.
E [ E 4| - Eu e X E A]
- - E A] + E EI - A].
=
oulcome shift
covariate shift
This leads to tests for the following null hypotheses:
Hypothesis 3.1 (Agg covariate shift), HØ¹ : For all sub- groups A E A.,, the performance drift in A due to the ag- gregate covariate shift is no larger than tolerance 2 0, i.e. (x) A] Eol (ol E A] ST.
Hypothesis 3.2 (Agg outcome shift). Hy YIX; For all subgroups A E A.. the performance drift in A due to the aggregate outcome shift is no larger than tolerance T 2 0. i.e. E (X) - Xx E T..
3.2. Detailed tests: How?
For each shift mechanism, rejection of the first-stage test im- plies that there is a subgroup for which performance change was large. The next step is to find a detailed explanation, by identifying the variables most likely to be responsible.
For each shift mechanism, rejection of the null means that there is a subgroup of concern and further investigation is warranted, thereby triggering a second stage of testing. Before diving into the second stage, we discuss connections between these aggregate tests and the existing literature.
SHIFT finds explanations by searching over suite of can- didate shifts with respect to individual variables or vari- able subsets. Because the true causal graph is not typically known in practice, the set of all possible variable(subset)- specific shifts is exponentially large and a comprehensive search over all such shifts is computationally intractable. As such, SHIFT considers a restricted set of detailed candi- date shifts potential explanations. In this work, given variable subset X,, we consider the following:
Connection to MMD. The proposed tests assess for distri- butional differences by comparing the maximum difference in the mean loss along the shift sequence in (I). This shares similarities to MMD, which also measures the distance be- tween two distributions in terms of the maximum difference in expected value over some function class (often referred to as the "critic") (Gretton et al., 2012a). To see the connec- tion more formally, we rewrite the above tests in terms of binary detectors where (X) = 1 { E A} for subgroup A. Define the critic function class to be the set of "filtered" loss functions {<x, 8) + h (X)(((()) y) : A E A. J. For the first two distributions in (1), MMD defines their distance as the maximum average difference of the filtered loss, i.e.
Outcome shift: We consider the candidate ps (ylrz) ==
Pi (o))), where po(x) = po(y = 1|z) is the out- come probability at the source. This is similar to shifts considered in model recalibration (Steyerberg, 2009),
where the shift is defined relative to the outcome's 's original conditional probability in the source domain.
sup Fio (o) AEA.
Foo [e(x, (X)),
Covariate shift: We consider the candidate p. == Pi (x=))o(r_..l Such a shift may occur, for instance, if X, precedes X_, causally and is commonly considered in prior works (Wu et al., 2021; Zhang et al., 2023; Singh et al., 2024).
where Ed, de indicates the expectation with respect to dis- tribution Pa, (Y)X). In contrast, the aggregate co- variate shift test can be viewed as measuring the maximum average difference of the conditional loss, i.e.
Other candidate shifts are certainly possible (see Sec F) and we leave them to future work. Critically, unlike prior works that offer variable-level explanations of performance decay assuming these candidate shifts are actually true (Wu et al., 2021; Zhang et al., 2023), SHIFT does not assume that these candidate shifts are correctly specified because everything is conducted through the lens of hypothesis testing. Instead, SHIFT tests whether a candidate offers good explanation.
Exole(X, (X) Eoo[E(X, (X)] (X))
sup AEA.
Foo (X))
A similar analogy holds for the aggregate outcome shift, which compares the last two distributions in (I). Thus, SHIFT can be viewed testing the Maximum conditional- Mean Discrepancy (McMD) rather than the MMD. Like MMD, McMD is zero when the compared distributions are equal. Unlike MMD, McMD can be large even when the mean difference is large in only a small subgroup, reflecting its priority placed on algorithmic fairness.
Given candidate shifts, we now quantify how well they explain the heterogeneous performance changes in the data. We say that an aggregate covariate shift is well-explained
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
by an X₂ --speciif covariate shift if the performance change induced by the former is welo-aaproximmate by the latter across all subgroups A, i.e. for all A E A..
3.3. Visualization of SHIFT
Results from SHIFT are visualized in a hierarchical plot (Fig 1), where "red" means "flagged" and "gray" means "not flagged." At the aggregate level, the covariatc/outcome shift is "flagged" if a subgroup was found to have large performance decay due to that shift mechanism (null was rejected). To interpret aggregate- test results, we sum- marize the detected subgroup using rule-based decision sets (Lakkaraju et al., 2016), although other ML explainability methods can be used instead. At the detailed level, we flag variable(subsstt--peecii covariate/outcome shifts that may offer a potential explanation of the heterogeneous perfor- mance shifts (null was not rejected). "Flag strength" is one minus the p-value for aggregate- tests and the p-value for detailed tests. Note that if none of the candidate sparse shifts are adequate explanations, one may need to explore alternative shift explanations (e.g. less sparse).
E X E A] - Zo X E A] ~ Zo E A] - Eol Zo X E A),
where (X) is with respect to an X--specific covari- ate shift. Similarly, an aggregate outcome shift is well- explained by an X--spiiific outcome shift if for all A E Ac,
E1 Zo|X E A = Ez - Zul X E A), Zu|Xe
(3)
where (X) is the expected loss under the candidate shift.
This is formalized in detailed tests of X Xs-peciff shifts with the following null hypotheses:
Hypothesis 3.3 (X,- ,-ppeciciic covariate shift). H.. For all subgroups A E Ac and tolerance т, the candidate X,-
specific covariate shift explains the performance change, i... Ex E A] E, (X) E EA] S.
4. Inference Procedure
Hypothesis 3.4 (X, (X,-specific outcome shift). H For all subgroups A E Ac and tolerance т, the candidate X.- specific outcome shift explains the performance change, Le. (X) - ( E S..
We now describe the inference procedure for tests intro- duced in the previous section. We begin with rewriting each hypothesis test terms of simple target of inference. This will illuminate the general approach we would like take, well as the technical challenges we will encounter.
If we fail to reject the null for an X--specific covariate or outcome shift, SHIFT flags as potentially important. Then for some prespecified > 0, the potentially important variable subsets for covariate and outcome shifts are
To illustrate, note that the aggregate outcome test can be equivalently expressed as testing the null hypothesis
H sup E1 (X) - Zo(X) - xX))(X)) SO. (5) AEA,
Šshift = p-vuue for Hgghit > a} Sn
(4)
target of inference
for shift = Y|X and shift = =X A human expert can then verify which variables in SchiEt are the true root cause(s) and design targeted corrective actions.
The target of inference can be interpretted as follows: ha is scaled by how much the difference in expected loss exceeds tolerance T, so the target of inference can be interpretted as the Maxinum Expected Exceedence (MEE) between the last two distributions in the shift sequence in (1). Similarly, the detailed outcome test can be rewritten as
Comparing the detailed and aggregate- tests, one may notice that they have nearly the same mathematical structure and yet are interpreted differently to answer differing ques- tions (where? versus how?). To see how this is possible, note that the tests could have been interpretted in the same way: aggregate-level tests check whether aggregate shifts are well-approximated by the zero function, i.e. whether E [2] - Zol X E A] x 0 and Zoo E A]- Zol E A] = O, while the detailed tests check if aggregate shifts are well-aappoximmated by candidate X, X,-specific shifts.
H YIX sup E, (X) 2, - h S 0. (6) AEA.
The aggregate and detailed covariate tests can be interpreted similarly, though the scaling term is not as clean:
H1N sup Eo ([Zo(X) 1) 7) hx(X)) 00 AEA.
Remark 3.1 (Modified covariate shift tests). When we have features that are independent of the loss function, covariate shifts in such features may still be flagged which is unde- sirable. This occurs due to collider bias since conditioning on the subgroup 1 (x E A} induces correlation between the independent features and the loss function. Section B gives more details. As a remedy, we filter features that are uncorrelated with the loss function as a data preprocessing step and then run the covariate shift tests as usual.
(7)
H sup Fo (Z)) - *x)a - -7) < heA.
(8)
where (x) =
Pa ( and (x)
Pi (o pa(a,) F. [X)] are scaled density ratios. Given this rewrit- ing of the MEE, we can now discuss two technical chal-
lenges that we can resolve, in part, through sample spliting.
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A Hierarchical Framework for Diagnosing Heterogencous Performmnce Drift
Data:
Train
Test
is maximized, the
optimal detector is h (X) = (X) - Zo(X) - >> 0}. Consequently, we can take a plug-in approach to construct candidate detector, i.e. (x)) = (1)))xx222(X)
Nuisance
Candidate Detectors h huggr hr
Parameters
Test statistic
eg El((' Z,(X) - x
2q 2,, 2, R,
0). A similar approach can be taken for the detailed version.
Step Fit candidate detectors and nuisance parameters
Step Run test on fitted detectors.
Estimating candidate detectors for the covariate shift tests is, however, not immediately obvious. For instance, the MEE in (7) cannot be maximized by individually maximiz- ing its conditional mean, because of the shared ratio term Ex])a(X)]] E, (x))) Instead, we find the optimal de- tector by solving the dual for a sequence of optimization problems. That is, we can reframe the task as solving
Figure 2: Overview of testing procedure
First, estimating a supremum over the infinite number of binary detectors A is computationally intractable. Never- theless, our goal is simply hypothesis testing, not estimation. We can accomplish this by sample splitting, where one por- tion of the data is for learning one (or a few) good candidate detector a and the remaining data is for testing the ex- pected exceedence for AA This can be viewed as running a restricted version of the original test, where we only test the MEE with respect to the singleton set { rather than all of A.. While this approach may be conservative, it provides statistical guarantees with fewer assumptions and better fi- nite sample behavior. The remaining question is how we can find good candidate detectors.
sup Eo [(2x(x)( - 1) 1)) ha(x) S Ei
A
(9)
s.t. w- =
for some w > O. Using the method of Lagrange mul- tiplicrs, the solution must have the form (wv.)) (X) 1 (2)(X) - >) (A(X)= - 1) 2 0} for some X 2 0. Thus we can estimate the optimal candidate detector by sweeping over grid of w and À values. We can estimate detectors for detailed covariate shifts in a similar manner.
Step 2. Conduct double-debiased tests held-out data. On the remaining data, we construct asymptotically linear estimators for the MEE with respect to the fitted candi- date detector(s), using the approach of one-step correction. This is relatively straightforward for (5), (7), and (8) by noting the mathematical similarities between MEE and di- rect/indirect effects in causal mediation analysis. However, one-step correction for the detailed outcome shift does not follow from standard recipes, which require the target of inference to be pathwise differentiable. The problem is that (6) involves (X), which is not pathwise differentiable because its definition involves indicator functions. Still, we can sidestep this issue by leveraging the binning trick in Singh et al. (2024). Rather than defining an outcome shift as function of po(z)), we define binned outcome shift that replaces all occurences of no with binned version. Assuming that the set of observations that fall exactly on the bin edges have measure zero, we can show that the MEE with respect to the binned outcome shift is pathwise differ- entiable, so to allow construction of an asymptotically linear estimator.
Second, the MEE involves unknown outcome models (Za) and scaled density ratio models (# and A),. which we collectively refer to as nuisance parameters. Prior works have shown that plug-in estimators, which use the same data to both train nuisance parameters and estimate targets of inference, are biased. Following results in double-debiased ML and semiparametric theory (Chernozhukov et al., 2018), we use sample splitting to remove some of this bias. The question is then how to remove the remaining bias for achiev- ing the desired Type I error control.
Given the benefits of sample-splittig, the overall testing procedure uses this as the basis: Step I estimates candidate detectors and nuisance parameters on a training partition and Step 2 uses the remaining data to conduct a restricted test with respect to the fitted models (Figure 2). For ease of ex- position, we describe the procedure for single sample-split, but it can be easily extended with cross-fitting to improve statistical efficiency (Kennedy, 2024). Here we describe each step broadly and highlight key innovations needed to address the technical challenges. The detailed testing procedure (including hyperparameter selection) is given in Section C of the Appendix.
Theoretical properties. Under the assumptions described in Appendix D, we can prove that the estimators for the MEE with respect to fitted detectors are asymptotically linear and their respective tests control the Type error and have power one, asymptotically. Consequently, for outcome and
Step 1. Estimate candidate detectors and nuisance pa- rameters using the training partition. The nuisance parameters can be estimated using MI. following stan- dard recipes (Kennedy, 2024). Estimating candidate de- tectors for the aggregate and detailed outcome shift tests is also straightforward. For the aggregate version, the es- timand in (5) is maximized when the conditional mean
Appendix C.3 discusses a more statistically efficient but more complex procedure involving the Maximum conditional Expecta- tion of the Exceedence (MeEE) rather than the MEE. We discuss testing the MEE the main manuscript for ease of exposition.
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
covariate shifts (shift Y|X and shi ft X), if there is candidate detailed shift with respect variable subset s ttshift that corresponds to the true shift, it will be flagged by SHIFT, i.e. P(s*sahirt ft & (thrrt) S a.
fitted to data from the source domain. We take 8000 points from both source and target domains, and split them into halves for training and evaluation.
Setup 1a/b (Compare agg-level outcomeccovariate tests): For X E R the shift only occurs in subgroup A = {a|2ı $ 3.533 Setup la only shifts the outcome logits per @1 (z) @0(2) 0.6zz 1 [x A}: Setup Ib only shifts the mean of the first covariate. To make the tests comparable, SHIFT tests for 0, c 0.05.
5. Results
We now validate SHIFT in simulation studies where the ground truth is known and two real-world case studies. For comprehensive validation, we vary the type and degree of shifts, the MI. algorithms under study, and the data sizes. We present a summary of the results here due to space constraints and provide full experiment details in Section G.
Setup 2 (Compare detailed outcome test): Φn(x)) = 0.8 + 0.5.z + z3 +0.6r, and Φ1 (I) = + + +0.6I4. The outcome shifts with respect to both X1 and X2, but the shift in X₂ is minimal and below tolerance т. Accuracy drops by 5.9%. SHIFT tests for 7 = 0.05, e = 0.05.
SHIFT. For all experiments, performance defined in terms of the 0-1 misclassification loss. We fit ML models (e.g. gradient boosting trees (GBT)) for the nuisance parameters and detectors, with hyperparameters chosen through cross validation. The significance level is set to = 0.05.
Setup 3 (Compare detailed covariate test): 22 2, 2) and
mo = (1,0,0,1)
mi = (0,0,0,0), = 2, 2). Both X1
and X2 shift but X4's shift is very small and below tolerance T. Accuracy drops by 5.4%. SHIFT tests for = 0.02, c 0.05.
Baseline methods. There is no existing comparator that provides universal testing for all four types of shifts (ag- gregate/detailed and covariate/outcome) for the exact for- mulations used in SHIFT. Given these constraints, different comparators are used for different shift types and, when necessary, adapted to be as close possible.
SHIFT correctly identifies relevant shifts, achieves nom- inal type-I error rate, and is consistent. In Setups 1a/b, the aggregate-leve tests in SHIFT are considerably more powerful than KCI and MMD, which are both kernel- based methods that tend to do poorly in high dimensions (Table 2). In contrast, SHIFT takes advantage of flexible ML estima- tors, which allows it to recover the true subgroup with reasonable accuracy (73.7% and 41.9% in setups la and Ib, respectively). In Setups and 3, the aggregate-level tests in SHIFT also correctly flag outcome shifts (Fig 3a) and covari- ate shifts (Fig 3b). respectively. At the detailed level, SHIFT correctly flags variable X₃ as being a good explanation for the large performance shifts; the others are ignored because they either do not contribute or have negligible impacts. In Appendix we also show that SHIFT controls the Type-I error rate and is consistent (asymptotically power-one).
For aggregate shifts, we compare against Kernel indepen- dence tests KCT (Zhang et al., 2011) and MMD (Gretton et al., 2012b). For detailed outcome shifts, we compare against (a) TE-VIM (Hines al., 2023) which quantifies VI for ex- plaining conditional average treatment effect, (b) Pa rrmm which fits parametric regression model of the outcome Y given domain D, features X, and interaction terms DX and determines VI based on coefficients of the interaction terms, (c) Pa raamooss which is the same as Paramy except it regresses loss f, and (d) XCI (Zhang et al., 201 which is a kernel conditional independence test for D 1. X For detailed covariate shifts, we compare against (a) KS which is the classic gellmogrrov-mmmoon test for comparing two univariate distributions, (b) Score (Kulinski et al., 2020) which detects shifts in X,1 X_, via the Fisher score, and (c) KCI (Zhang et al., 2011) which is a kernel conditional independence test for D L XX,
Table 2: Aggregate tests. Power for detecting outcome or covariate shifts in a subgroup. Power is computed as the rejection rate among 25 random draws of the dataset. We observe that SHIFT has the highest power.
5.1. Simulations
Setup
SHIFT
KCI
MMD
Here we illustrate how SHIFT is more powerful and iden- tifies only relevant shifts, i.e. those that contribute to per- formance drifts of magnitude 2 r in some subgroup with prevalence 2
la Outcome
0.56 (0.42.0.7)
0.26 (0. (6.06.4.)
0.06 (0.02.0.16)
1b Covariate
0.94 (0.84.0.98)
0.0 (0.00.0))
0.46 (0.32.0.6)
Comparators do not flag the correct shifts. For com- parators in the detailed outcome test (Setup 2), we find the following: TE-VIM does not find any variable able to ex- plain heterogeneity of performance drift because it has weird behavior at the null. KCI can only check if the "marginal"
Data generating process. We generate variables X from multivariate normal distribution centered at m. and co- variance Σ for domain d and binary outcome Y per logit 6a(x). The ML algorithm is a logistic regression model
7

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A Hierarchical Framework for Diagnosing Heterogencous Performance Drift
Aggregate Coveriate Outcome
Datected large decay in 47.9%
distribution of the loss can be explained by individual vari- ables, i.e. if the loss distribution is independent of D given Xj. This is very specific type of explanation and does not hold, and so KCI fails to find any good explanation. ParamLoss is an incorrectly specified model and thus in- correctly flags none of the variables. ParamY is correctly specified model so it flags X1 and X₂ as shifting, which is correct though does not respect the specified tolerance. Similar issues are found in the detailed covariate test (Setup 3). KCI, KS, and Score all incorrectly flag both X and X4 even though X4 is irrelevant. This is because they check if X4 has shifted, but do not account for the fact that X₃ is not actually used by the model nor does it affect Y in any capacity.
target data subgroup:
0.00
1.00
IfX1 2.43 X1 (-2.31, OR 1.27
Detailed
Baselines Outcome
X1
0.16
0.00
0.49
1.00
0.00
X2-
0.00
0.00
0.56
1.00
0.00
x4
0.00
0.00
0.42
0.45
0.00
k
ParamLoss $ ParamY
TE-VIM 1
SHIFT
(a) Setup 2. only X1-specific outcome shift should be flagged
Detected decay 10.1% target data, e.g. subgroup:
Aggregele Covariste 1 Outcome
XX1
1.00
0:00
Baselines Covariate
Detailed
X1
0.00
1.00
1.00
0.38
5.2. Real-world case studies
X2-
0.00
0.52
0.51
0.00
Health insurance prediction across states. We study per- formance drift of an MLP trained to predict public health insurance coverage using census data from Nebraska, which is subsequently applied to Louisiana. Datasets have 3166 and 12000 points respectively and 34 features. Accuracy drops by 13.7% on average. At the aggregate-level, SHIFT finds that both outcome and covariate shifts affect subgroup- level accuracy (Fig 3c). For example, accuracy for the subgroup detected by the aggregate outcome test, compris- ing 50.6% of the target data, decays by 19.4%. Grouping the variables (34 in total) into 5 broad categories, we find from the detailed tests from SHIFT that shifts with respect to demographics can explain subgroup-level decay due to both shift types. Similar to that in simulations, the KCI baseline method struggles to find a good explanation while TE-VIM only flags employment-relbe relttrdl variables. Based on these findings, we compare three ways to fix the model: a standard non-targeted (Non-T) T) fix that retrains the model for everyone with respect all variables; fix that retrains the model for everyone with respect only the employment- related variables identified by TE-VIM; and very targeted fix that only updates the model for the subgroup and the de- mographic variables detected by SHIFT (Table 3). We find that the targeted fix does better than the non-targeted fixes, and the non- con-targeted fixes inadvertently decay performance in other subgroups.
X3-
0.00
0.50
0.48
0.00
X4-
0.00
1.00
1.00
0.00
KCi
KS
Score t
SHIFT
(b) Sctup 3, only X Xi-spccific covariate shift should he flagged
Detected decay in 14.5% source data, e.g. in subgroup: If Income < -6100
Aggreçate Covariste Outcome
Desected decay target data, a.g. subgroup:
1.00
1.00
Parents Unomplayed
Unemployedd OR
Baselines Covariate
Detailed
Detailed
Baselines
Outcome
Demo- 0.00
0.05
0.64
0.00
0.01
Education-
0.00
0.00
0.00
0.00
0.00
Employment-
0.00
0.00
0.08
Health- 0.00
0.29
0.00
0.00
0.00
Misc
0.00
0.02
0.00
0.00
0.01
KCI
kC
TE-VIM
SHIFT
(c) Insurance coverage prediction with 5 variable subsets
Aggregate Covariate Outcome +
Detected decay in 41.4 1.88% target data, e.g. subgroup: ED encounters be
Detected decay target data, e.g. subgroup:
0.98
Num encounters in (2.5]
Baselines Covariate
Detailed
Detailed
Baselines Outcome
Demo:
0.00
0.17
0.23
0.00
0.00
Encounters
0.44
0.86
0.52
0.04
0.16
Labs-
0.01
0.50
0.01
0.23
0.00
Vitals
0.06
0.19
0.24
0.00
0.00
KCI 1
kdit
TE-VIM 1
SHIFT
(d) Readmission prediction with 4 variable subsets
Readmission prediction across hospitals. The clinical Al field has developed numerous models to predict whether patients will be have an unplanned readmission after dis- charge from a hospital, which can be used to allocate extra resources to high-risk patients. We study GBT readmis- sion model trained on data from an academic hospital and transferred sefety-net hospital. Since the hospitals serve different populations, the goal is to understand which exact shifts contributed the most to accuracy changes, such as changes in how patient variables are measured or changes in how care is delivered. Datasets from the academic and
Figure 3: Hypothesis testing results for variable(subset)- specific shifts. SHIFT shown in outlined boxes: basclines for covariate and outcome shifts shown the bottom left and right, respectively. Null hypotheses either state that shift should be flagged (f), in which case we flag it in red if the p-value > 0.05 and show the p-value in the colored box, or that a shift should not be flagged (t), in which case we flag it the p-value S 0.05 and show 1- p-value. For synthetic Setups 2 and 3, we report median p-values over 50 randoely-sampled datasets.
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Table 3: Comparing model updates on insurance study. We report AUC and 95% CI for performance of the original model and targeted versus non-targeted model updates, as measured with respect to the overall population (left col- umn) and the subgroups where the original model (Org) and non-targeted model updates (Non-T T and TE-VIM) ex- perience large performance decays (right three columns). The targeted update based on SHIFT results performs better along all dimensions.
6. Conclusion
We propose hypothesis tests to identify subgroups where an ML model decays in performance due to distribution shift across two contexts. The tests can also explain how the de- cay arises by checking for variable subset-specific shifts that can explain the decay. The tests can be configured to detect only meaningfully large performance decay and can be im- plemented readily using off-the-shelf ML models. Despite using ML estimators, we show that the tests have controlled false detection rate and good power asymptotically. Al- though the experiments here primarily focus on tabular data, SHIFT can be extended to unstructured data such images and text by featurizing such data into concepts (Koh et al., 2020; Feng et al., 2024b). Our explorations with text data show that SHIFT provides a solid theoretical foundation on which future work can build.
Subgroup for models Non-T
Model
Overall
Org
TE-VIM
Original model,
69.2
59.0
Org
(67.0,71.4)
(55.3,62.3)
Non-targeted update, Non-T
73.0
67.3
41.8
(71.0,75.0)
(64.0,70.4)
(24.6,59.2)
Update as per
73.0
64.9
63.7
TE-VIM feats.
(71.0,75.1)
(61.6,67.9)
(56.6,70.5)
Targeted update as per SH H"1"
74.8
67.7
66.4
65.0
Acknowledgements
(72.8,76.8
(64.4,70.4)
(46.8.83.6)
(57,5,71.5)
We would like to thank Lucas Zier, Patrick Vossler, Avni Kothari, and Romain Pirracchio for their helpful input and comments on this work. We are especially grateful to Adarsh Subbaswamy, Nicholas Petrick, and Gene Pennello, who provided invaluable feedback on the project from its inception to completion. We thank them for their tireless commitment. This work was funded through a Patient- Centered Outcomes Research Institute@ (PCORI@) Award (ME-2022C1256119 The views presented in this work are solely the responsibility of the author(s) and do not nec- essarily represent the views of the PCORIO, its Board of Governors or Methodology Committee, and the Food and Drug Administration. JCH acknowledges support from the National Cancer Institute of the National Institutes of Health (R01CA277782), which had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
netety-net hospitals have 7468 and 6515 points, respectively, and 27 features. Accuracy on average decays by 6.1% when the model is transferred. SHIFT detects significant changes in subgroup-level accuracy due to both aggregate outcome and covariate shifts (Figure 3d). For instance, the subgroup detected by the aggregate covariate test (comprising 41.8% of target data) has 15.4% drop in accuracy. We find that the top feature highlighted by SHIFT for both covariate and outcome shifts is num ED encounters. When the same variable is highlighted for both covariate and outcome shifts, it can indicate that the definition of the variable has shifted. Investigating the data extraction procedure further, we in- deed find this to be the case: the encounters feature was extracted differently across the hospitals. After correcting the extraction of this feature, covariate shifts longer lead to significant subgroup-level accuracy drop (p-value for the aggregate covariate test is no longer significant). This illustrates how SHIFT can help bridge accuracy gaps.
Impact Statement
The methods in the work can identify subgroups that experi- ence overly large performance decay when ML algorithm is transferred across domains or used over time. Results from SHIFT can be used to suggest interventions that can improve the impacted subgroup's 's performance, without sub- stantively impacting other subgroups. We recommend work- ing with domain experts to define what constitutes a mean- ingfully large subgroup and performance decay, as these determine what the test will aim to detect. These thresholds impact the interpretation of the tests and subsequent actions taken for closing performance drops.
Application to unstructured and high-dimensional data. Although SHIFT is primarily designed for tabular data, its aggregate-Ievel tests are suitable for analyzing unstructured data; its detailed-level tests can also be used, if one has prespecified concepts (Koh et al., 2020). As an example, we apply SHIFT to the CivilComments dataset (Koh et al., 2021), which contains comments on online articles and are judged to be toxic or not. Given 168-dimensional embed- dings of the comments, SHIFT detects accuracy drops, as described in Section J.
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A Hierarchical Framework for Diagnosing Heterogencous Performance Drift
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Appendix
Table of contents
We summarize the notation used in Table 4.
Section A relates proposed tests to causal interaction and mediation analysis.
Section B describes the modified covariate shift test which removes features uncorrelated with loss as a preprocessing step.
We provide step-by-ster details of the testing procedure in Section C.
Section D presents theoretical results and their proofs.
Section E discusses related work in more detail and demonstrates challenges in testing with prior approaches.
We discuss some alternative definitions of detailed tests and demonstrate their drawbacks in Section F.
Sections G and H give experiment and implementation details.
We show more simulations for type-l error control and power in Section
We apply SHIFT to a high-dimensional text dataset from the WILDS benchmark in Section
Table 4: Summary of notation.
Notation
Mcaning
X,Y, f(X)
Covariates, Outcome, ML algorithm's 's prediction
X, DE (0,
Subset of covariates (or variables)
Domain or dataset, means source and means target domain Probability density (also used for distribution) in domain d Subgroup A C X' and binary indicator function for subgroup A Loss per data point (z, y)
Pa(X,Y)
A,hA
Ely. f(x))
Za(x), (0,1)
Average loss per data point [a( ((X)) = x] Expectation with respect to Pa for subgroup Tolerance specified by analyst to detect performance decay of magintude r or higher
EaI-X E A] T
e
Prevalence specified by analyst for the minimum group size detect Significance level the desired false rejection rate of the null hypothesis conditional outcome function in source domain, pa(x) Pa(Y = 1|X x)
Ma(ax)
Ps (3) MMa); (a(x)
Conditional outcome distribution shifted with respect to X, and the resulting loss
(z) T(x), (x)
Conditional loss function (z) = Eo[e( [z))
Density ratios (=) and I.. (x) Pols,
(=(x,)
(2( Eo (XX) and Eo (X) respectively (X)
T (s.
Scaled versions of (Ix) and (2) by the factors
En (X))
A. Connection to causal mediation analysis
Figure 4 summarizes the correspondence between the proposed tests and the tests for interaction and mediation in the causal inference literature. Proposed hypothesis tests could benefit heterogeneous mediation analysis that identifies meaningful subgroups, as opposed to prespecified or model-driven latent subgroups. Specifically, the proposed tests can be extended to determine whether a subgroup with large direct and indirect effects exists. The extension is straightforward when subgroups are defined using pre-exposure covariates, as in conventional mediation analysis. When subgroups are defined using potential mediators, further identification assumptions are required before the proposed tests can be applied.
B. Modified covariate shift test
Counter-intuitiven the tests as defined in Hypotheses 3.1 and 3.3 may flag shifts in covariates that are unrelated to the loss function. That is, for the aggregate covariate shift test, we may find group for which
E (X)) A] E
A] >
even when (x) = (X). This can happen because we are conditioning on X e which introduces a shift in { e A by conditioning on the collider for effect of f. A simple fix to restrict the detectors depend only on features that
14

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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
D
D
Interaction
Mediation
D
X
X
Zd
Zo
Y
X
Y
X
X
X
Figure 4: Graphical description of the hypothesis tests. Differences in distributions of and Y (and hence average performance model) across the two domains = 0, 1 represented by effect variable D on X, (left) Performance can vary due to changes in conditional outcome distribution (orange edge, outcome shift) and/or changes in the covariate distribution (green path, covariate shift). Variables Za, d E (0, 1) represents conditional loss in the domains described in text. Firstly, test whether each of the effects is zero. (middle) When test for outcome shift is rejected, identify feature subsets X, such that the complement X-, do not interact with the effect of D on Y (orange dashed line). (right) When test for covariate shift is rejected, we identify feature subsets X, such that X-, which do not mediate the effect of D on Y (green dashed path).
D
X
Y
A
X₂
Figure 5: Modified covariate shift test. Graph provides an example data generating mechanism where X1 is marginally independent of Y. The feature X₂ influences but does not shift. Conditioning on A introduces collider on the path from D to (and hence 2). Therefore, the loss affected by the covariate shift even though only depends on a feature that does shift. In the modified test, we remove the feature X2 before testing for covariate shifts,
15

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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
correlated with the loss.
For the covariate shift tests, subset the covariates to keep those that are correlated with the loss, Xeorr [X, + f E f1,2...., (x The null hypotheses 3.1 and 3.3 are defined only with respect to the correlated features, X == X Xoorr. To avoid 'peeking into the data', we filter uncorrelated features based on the training data split. Remaining part of the tests are the same as earlier.
C. Step-by-step testing procedures
Source and target domains comprise of i.idd observations of (ar, y). We denote empirical average over data points from domain dl by the operator Pa,,. For example, Po.. refers to taking the empirical average for the evaluation dataset of the source domain. We use Pr, to refer to an average over the pooled source and target data. For case in notation, we write the same number of samples n for each dataset. However, they could be different. In fact, our hypothesis tests are particularly beneficial for cases when the target datasets are too small for point-identifyin performance shifts. We split the datasets into two parts, one is used for fitting all nuisance parameters and the detectors, and other is used to evaluate the MEE and for inference.
C.1. Fitting nuisance parameters
We require seven nuisance parameters denoted by v) = (Zo; Z1 Zs: Zo.s; गI, Txz TV). We denote their estimates by ij. The nuisance parameters can be broadly categorized as outcome and density ratio models.
Outcome models. To fit the conditional loss funetions Zu and 21, we first fit the conditional outcome probabilities (o(2)z) and in (y/æ:) by regressing y on Then, conditional loss is simply an expectation over the two possible outcome Za(z) = (y) (y, (()))) For 2,, we fit the X,, conditional outcome probability p. by regressing y on Te. (o(2)) and taking an expectation over y. That is, (xx = Eye(o.1) L(y (y) Po())) Similarly, the conditional loss function Zo. (x) Eo hx()) can be estimated through regressing on F., hx(x).
Ez) (X)) and (z) polæ, (b, (X),
(r.) (X)] by
Density ratio models. We estimate the scaled density ratios A = po(2) [X)]
first fitting models for the density ratios and plugging them into the definitions. We also need a density ratio model for the detailed outcome test, TV (X, = I.. X_ = x=.) pi -=(A_,=E_,Ipo (=x)))= probabilistic classifier estimate the probability a data point belongs to target domain from the pooled source and target data. Density ratios are computed odds ratio for the classifiers. Please refer to Sugiyama et al. (2007) for more details,
(x
Density ratios are fit using
Remark. We note that the scaled density ratio models depend on the detectors h which we haven't 't fit yet. To break this cyclical dependence, we first fit detectors (as shown in the next section) for the unscaled density ratios and then compute the scaled density ratio models.
C.2. Fitting detectors
Recall that detectors h (z) are binary functions meant to detect the data points with high mean exceedence. Fitting procedures for detectors vary for the tests since the exceedence is defined differently. All detectors can be implemented using off-the-shelf ML regression libraries.
Aggregate test. For the outcome test, the plug-in approach hx(x)) 1(21(x) -0(x)) - 0). Alternatively, can fit a detector by regressing Zx(x) from J,
A E arg min Pi.. (I. -0(()) 7) -((x))²
(10)
For the covariate test, we solve the optimization problem (9) after plugging in 2o and म. We find values of w, A by grid search that maximize the objective for detectors of the form,
A/ (w.)) = {xr (Zo(z) - (f(r)
(11)
Detailed test. For outcome test, simply plug-in the estimated nuisance parameters into the MEE and threshold it,
ha(x) 1() 2x(x) 0}.
(12)
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
For the covariate test, grid search for wi, À values to optimize the detailed test counterpart of (9) with detectors of the form,
h ( = =
-x, (1))
(13)
C.3. Estimating the Maximum conditional Expectation of the Exceedence (McEE)
The main manuscript discusses the testing procedure terms of the MEE for ease of exposition. practice, to maximize statistical power for the hypothesis tests in SHIFT, it is better to construct estimators of the Maximum conditional Expectation of the Exceedence (McEE), as its efficient estimator has lower variance. Below, present the inference procedure for McEE on the held-out data, which is slightly more complex than that for MEE because it involves ratios instead.
We first present plug-in in estimators for McEE, which are key quantities in the tests (Table 5). However, as noted earlier, plug-in estimates have been shown to give biased estimates for estimands involving infinite-dimensional quantities like outcome models or density ratios, and using flexible ML estimators for the nuisance parameters do not help remove the bias (Chernozhukov al., 2018). For this reason, the plug-in estimators cannot be readily used to perform valid tests. We propose debias the estimates by one-step correction, also known as double/debiased estimation. Debiased estimators can be shown to be asymptotically normal and give valid tests. Note that the estimators for aggregate special cases of the ones for detailed tests with X, set to empty set. Therefore, we only present the estimators for the detailed tests and then analyze their theoretical properties.
Table 5: Plug-in in and debiased estimates restricted MEE for detector given the fitted nuisance parameters r). We recommend using the debiased estimates because they can give valid tests.
Type of test Aggregate outcome
Plug-in estimator
Debiased estimator
same McEEy(0)
P ((,,((((,,((X)) (1Xx))))M)((
Aggregate covariate
same McEEx(0)
Pl, -ttp:a/aaaaaPttaaaaaeaaoaaaaaaa-
Detailed McEEy(s)
outcome,
given in (14)
PI,, -
Detailed McEEx(s)
covariate,
given in (17)
P PI, (e)
Detailed test for outcome shift. As discussed Sec 4, the MEE in the case of detailed outcome shift is not pathwise differentiable because it depends on an indicator function (in Z, (z))). Therefore, we follow the binning trick Singh et al. (2024) to change MEE into a pathwise differentiable quantity. Recall that -s-seciciic outcome shifts are defined as function po(r). We change
depend on binned version, Pain (x) 1/4 Luo (x) B 1J for some fixed B E Z4. It discretized (x)) into B equally spaced bins in j0, 1]. Thus, the binned MEE is defined
function of Z ((v, J [(x) (p,)))x((())))) (2,, (2:).. As long as the binned jhin does not fall on the bin edges (almost surely), the indicator involved in the binned MEE is zero (almost surely). Thus, ensuring that MEE is pathwise differentiable. We expect the binned and original version of MEE to be similar for a large enough number of bins. We can now define the estimator.
E,
The debiased estimator for MEE has the form of a V-statistic. We denote V-statistic by the operator 1i. which takes an average over all pairs of observations with replacement. That is, V-statistic over observations O₁, Oz.... On
17

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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
computed +.4 Σ شز (O,, for some function v. The tilde notation Pi.. is used to distinguish between the
arguments to v. Debiased estimator for MEE given the fitted nuisance parameters r is expressed as ratio McEEy(s)
nim McEEy (s)/Pn[haa where McEEy
mum
(s) is
McBEy (8) = PI,MPi (Y,J( (X))((X X- K & (iiin))))
(14)
+ 1, Σ fly (XX)) Poin (X))
(15)
P Σ XX, & -.)vv((aa
8 (X)) Arsin (X))
(16)
Detailed test for covariate shift. Define the conditional loss function (x.)(x) - Eo h (arr and its estimate as Zo.e- Debiased estimator for MEE the case of covariate shift given the fitted nuisance parameters i) expressed as a ratio num
num (cEEx(s) McEExx (s)/PPn(aate where McEEx
(s) is
McEEx (8) = Pi,
(0(2)) 1
- Po, (Ctt,
(17)
Po,nI(e 70 (
(18)
+ Ponn (2. (z) - Pi,n Pi,nI [x)]
(19)
C.4. Inference
For inference, we use the Gaussian multiplier bootstrap method to construct bootstrap distributions for the test statistic (McEF) under the null (Hsu, 2017). Each bootstrap sample, involves sampling n variables distributed as standard normal, Ei...., En N (0, 1) and recomputing the centered McEE with per-sample weights. We construct the p-value the proportion of times the bootstrap test statistics exceeds the observed test statistic.
D. Theoretical results
Here, we provide details of type-l error and power guarantees of our testing procedures for the four hypotheses, two aggregate (covariatc/outcome)) and two detailed level (covariatc/outcomee for a subset). Specifically, we analyze the McEE for the restricted version of the tests with a singleton detector {h Extension to maxima over multiple detectors follows from Gaussian approximations of maxima over averages (e.g. Chernozhukov al. (2013)).
Outline. At high-level, we show that our debiased estimators of McEE are asymptotically linear. It implies that they converge to normal distribution centered at the true McEE. Because of the normality, we can conduct inference using standard tests such as 7-test or Wald test for the null hypothesis McEE < O. Properties of type-I error control and power will follow. So, the key to show asymptotic linearity of the estimators.
We first give an outline of the linearity analysis, which applies the techniques developed for analyzing V-statistics (van der Vaart, 1998). Our estimators of McEE, (14) and (17), are one-step corrected estimators which start from a plug-in estimate, McEE(P), and debias it by adding its canonical gradient IP,, (o; P) (also called an influence function).
McFE(P) P,, (P;
Here, the canonical gradient e a function of the probability distribution P over the observations (X, Y, D). Following the von-Mises expansion of McEE (P). the bias of the one-step corrected estimator can be decomposed into three terms,
(M) + P., (o: (P) McEE(P)
= =(P, - P) PP) + R(P,
= =PP, P);oo (P, ())()) ((a: -) + R(P.PP
for second-order remainder term A((², IP). The goal of analysis is to show that each of the terms is negligible at (1/2) rate such that we get asymptotically linear representation of the one-step corrected estimator.
(cEE((P) +
PP (o:
McEE(P) Parb(o; P) Op(n 1/2).
-1/2)
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A Hierarchical Framework for Diagnosing Heterogencous Performance Drift
The form implies that the one-step corrected estimator is asymptotically normal with mean Mc and variance var(bbo; P))/n, which allows constructing valid tests such as z-test Wald test.
We present the theoretical results for detailed tests in the next section.
D.I. Detailed test of outcome shift
For pathwise differentiability of the McEE, we require mild condition to hold for the binned outcome shift Jbin- It states that the true shifted probabilities /bin lie inside the bins almost surely. Additionally, we require that the nuisance parameters are consistently estimated. It is possible to ensure consistency by using nonparametrie ML estimators, Most importantly, we require that the product of estimation emross in the V-statistic density ratio model fy and the -specifii outcome shift rate. In the average treatment effect literature, similar assumptions made for the outcome and treatment propensity models (Chernozhukov al., 2018). Note that this condition is significantly milder than requiring that both the models have fast convergence. The condition holds long for the models.
model converges at
guarantee Op (1/4) rate of convergence
Condition D.1. For variable subset assume the following holds:
(binning) For all bin edges b of (2) except {0, 13. the set {x : [pu()) - bl S c) is measure zero for some c. (consistency) Nuisance parameter estimates Moin: #V. Pg are consistent.
(product of errors) P1 (ft- - πy { Ps) = (n-12).
Theorem D.2. Suppose Condition D.1 holds. Then the estimator McEEy(s) follows a normal distribution asymptotically, centered at the estimand Ms)E
To prove the above theorem, recall that McEEy(s) is defined as the ratio, McEET (s) The following lemma will first show that the estimator for the numerator is asymptotically linear. Denominator be estimated simply an empirical average of A (z), hence it is linear. Then, we will use the Delta Method (van der Vaart, 1998) estimate the ratio and prove it has a normal distribution asymptotically.
Suppose the nuisance parameters in McHEy (s) are rly,s (TTV &hins Ps). We write the one-step corrected estimate of McEEy (s) as a V-statistic.
McEEy (s) = PI,, I(X,
x 8-
+ PIImPii, 1.m Σ f()))) ( (xinnx (X))
-PImPI, i. Σ ely. f(x. )h X (X X el (i)P)(()))) X Aran Abmn(X) (X))
Pi, u y: (mmm)
(20)
Lemma D.3. Assuming Condition D.1 holds, McEEy (s) is an asymptotically linear estimator for McEEy (s)
McEEy
Rum (8) McEEEy (s) Pi,n y. (
(1122
for the influence function
(xx (z, y ) P Pil(y, (
X-s (malbii)
(21)
Σ (1:)) Mbin
(22)
Σ E(ū, (z.,,
(X.r X- -lMonn (x) ()))p(()) x. Moin (x))
(23)
(24)
Proof. Define the symmetrized version of v in (20) Usyn (X, X.V,X,Y P) estimator
Then, rewrite the
2
num McEEy
(s) = Fi,, P Usym
(X,Y,8, caum
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Following Theorem 12.3 (van der Vaart, 1998), the Häjek projection of McEEy (s) is
num
mum yy
(s)
P1 P XX, :
McEEy (s) X Y
Pi Veym (X,, X()), (2)); , (mt.)))
McEEy (8) Xi, Y
Σ Usym,1 (Xi, (8,0)
where McHEy (8) = P1 Usyin
Xx (x,,, X" ; 178.50)
Decompose the bias of the estimator as,
McBBy (s) McEEy"" (s) =P,,
mmm
1, PP Uym
X, X,Y,X,Y, 1/Y ; smum
IP1 P1 Uxym Xx Y, X,Y, y; r/y num
1, Vegm
X, (X,, Y; mmu
IP i,n hsymm (X, Y; ;YT.s) + McEEy
mum
(s)
(25)
num
+ (Fi, IPi)
Vaym,1 (X, 18.s5
+ McEEy (8) - Usym,1 (X,Y) McEEE
(26)
(Fi. PP)) (Vsym, (X,Y) + McEEqqu (s))
(27)
mum
+ Pi (osym,1 (X, Y: ( (
McEEy (s) Usgm.11 (( McEE (s)
(28)
Now, we consider the four terms of the decomposition one-by-one.
Term (25): Suppose Pi 2 (X, Y, X, Y; (Y..)) van der Vaart (1998), one can show that
oc. Via straightforward extension of the proof in Theorem 12.3 in
var
I,n 1i.
X,Y,X, Y; 1/Y
cnum
-p 1.
var PPinneeym.
X, Y NY (yy.
Then by Theorem 11.2 in van der Vaart (1998) and Slutsky's lemma, have
mum (s) = =0p (n-1/2)
NY 8; 18..)) - PP., Usym,,1 (X, Y; ( + McFEy PPi, PP. Usym (x,Y, Y,
Term (26): This term is asymptotically negligible since we perform sample splitting to estimate the nuisance parameters and evaluate the estimator for McEE, (s) on separate datasets. Then by Lemma in Kennedy (2024), we have that
(Pi.n Pi) (1sym,1 (X, Y; ( + McEEy (s)
Usym,1 1 (X, Y; (K m
McEE (s) On(nn 1/22 (n-122
as long as the estimators for the nuisance parameters are consistent.
Term (27): This term is the difference between empirical and population average of a population- quantity, hence follows normal distribution asymptotically by standard CLT.
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Term (28): We show that the term is asymptotically negligible well given the condition on nuisance estimation errors.
PIPi (Usyn
X, X,Y, ( + McEBy
num
(6) Vsym (X,,, (iKYy (Y
McBEY (s)
=P P1 (XX,, Y, X, Y; (y,55) - 2 (X, Y, Y; 7/.
(29)
-P, Pi EY, ((X (x,,) x-.) (+t (#v( - -lbbii))) - TV =vv (Xs, X- Mbii))
+ P1 Σ EVu, (()]( (01 (v/l Xx, A (xm(()) pr (y)l X
Moin (X))
P P Mx.xxxxx 8-s)
(30)
x ( X- Arbin (X) pi (y) Xx (X)) (xlmnnxx)))) (1l)))/(l/X..)) (2) Main (X)n(()))
(31)
By the law of iterated expectation, can expand the first summand as,
= (x,,
x) (Nx, X- /Abin) (yl Main TV Σ (())MM)) (u) (xnin(X) Pı (vl Mbin (X)))
TV (X 8 s//bmnn )p (8lX,,, X 14bin (x))
1
y
Pi PI, Im Σ Cly, f(X:: 8 he 8s))
(32)
(X) (2) (X)n TV X-ol/onn (oommix))))())) (1()))p((((( (v) Mbin (X))
Cancelling terms from the first and third estimand gives us,
= =- --P,PieC,,,XX (x))xXXX...) ( 8- ( (0r (31l X.s Poin (X) (21 Main (X)))
+ + Σ (v. (X))) (fr Xs, Abii (X)) pr (1( X Mhin ()))
9
By the definition of the density ratio TV, the second summand written as V-statistic.
-P f(x. (-) { (er Xy) Pronn (X)) - P1 (v1.x,, Abin
X Mbin (X)))
+ P. Σ Cv. (X)) (X,, x- Meiia (X)) (fr (y/ Abin (X)) p1 (3) X 1Abin (X)))
Collecting the common terms, the sum simplifies to
Pi Σ Cly ((X))a(())
(33)
x (-s #y (xx (pi(ulX,, - (xin (X)))
which is on by Condition (D.1) on product of errors in ît and p. models.
Hence, the bias the estimator is (n 1/2). -112).
Proof of Theorem D.2. Applying Delta Method to the numerator and denominator estimates, which we know to be asymp- num
totically linear from Lemma D.3, get that the estimator McEEy (s) McEEy (s)/Pi,n [xx))) is asymptotically linear,
McEEy (s) (s)/PI,,(ha(x))
McEE
=P PY, (X, ( op 1/2)).
with the influence function
McER (s)
Vy,, (X, Y.
V (X, Y: (
(P)(h ha(z)
ha(x))
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
where V. (X, Y; num) is defined in (21).
Yy,))
As consequence, the estimator is asymptotically normal,
V (McEEy(s) - (s)Ey(s)) -d (0, var(vy,. Y3; (s,)))
(34)
We estimate the variance from the data to construct valid tests, for example 7-test Wald test. Alternatvell,, can use multiplier bootstrap which bypasses estimating the variance.
D.2. Detailed test of covariate shift
The form of the debiased estimator McEEx is similar to those in indirect effects literature and follow straightforwardly from the approach outlined in the previous section. Therefore, omit derivations for asymptotic lincarity. See Lemma G.2 in Singh et al. (2024) for an illlustration.
We require the following conditions for asymptotic linearity. For the density ratios т T, Ts,4 to be well-defined, we need the support of covariates in the source domain to be larger than the support in the target domain. This can be ensured by restricting the target domain to the common support (Cai et al., 2023). Importantly, we require that only the product of estimation errors of the nuisance parameters is asympotically negligible op(n 11/)). Therefore, even though the estimator involves fitting many models, debiasing relaxes the conditions required their estimation errors.
Condition D.4. Assume the following holds,
(consistency) Nuisance parameter estimates Zo. s0 T. A..A are consistent.
(contiguity) Po (xr) > whenever pl (x)) > 0. (product of errors in Zo and At (o((Z(()) Zo(z)) ( Tx(x)) op(nn 1/2)
(product of errors in Zu,s and *,,A) Po( Zo.. - Zo. (ft A - T.,A) = (n(n=112)
Theorem D.5. Given Condition (D.4) holds, the estimator McEEx(() = McEEx (s)/PP,(ha()) defined in (17) follows normal distribution asymptotically, centered at the estimand (s)Ex
mum
E. More details on related work
Hypothesis tests for distribution shift. Rabanser et al. (2019) compares multiple distribution shift detection methods and find that two-sample testing on learned representation works well. Kulinski et al. (2020) finds features that shift via conditional independence tests using a divergence measure based score function. However, it requires specifying a density model, challenging problem in high-dimensions, and does not provide inferential guarantees. Hindy et al. (2024) proposes a conformal prediction method to test for distribution shift over time. They focus on finding shifts in intermediate steps of a multi-step process instead of finding feature subsets. Much of the shift detection literature does not focus on finding shifts that significantly vary performance. In contrast, Panda et al. (2024) proposes hypothesis test to find contributing features for a performance change. To determine whether a feature important, they test whether the effect of adding the feature to the algorithm on its performance is the same across the two distributions. However, the specifics of how they add/remove features, essentially by zero-ing in the features, do not respect feature correlations. They do not show whether the tests have controlled type-I error rate, which means the test could falsely identify many features important.
Hypothesis tests for unfairness. One application of SHIFT is to test for disparities in an algorithm's performance (unfairness) by considering the source and target datasets to be data for the two protected groups. Hence, literature on auditing unfairness is relevant. Zahn et al. (2023) searches through the covariate space to find subgroups with high unfairness. The tests are limited to certain fairness notions such as demographic parity and equalized odds, and make parametric assumptions for inference since they use a x2 test. We note that an extensive literature exists on multi-calibration or multi-accuracy which aims to train models that are accurate for all computationally iddetiiiiiieeeeeeellllee groups (Kearns et al., 2018; Hebert-Johnson et al., 2018; Kim et al., 2019). Apart from similarities in the focus on identifiable subgroups, the objectives of the work are different. Such methods are complementary to our testing approach and can be used if the tests suggest to retrain the model.
Challenges in testing when performance decay is zero. Prior works (Hines et al., 2023; Quintas-Martinez Martinez et al., 2024; Singh al., 2024) on VI for treatment effects propose tests that suffer from undercoverage when the null hypothesis
22

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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Null-importance test for TE-VIM of X1 at level 0.90
1.0
0.8
Coverage rate
0.6
0.4
0.2
0.0
2.5
10.0
15.0 17.5
Shift CATE wrt
Figure 6: Testing VIs boundary of parameter space. Existing methods for the related problem of explaining treatment effect heterogeneity (Hines et al., 2023; Quinzan et al., 2023; Williamson al., 2021) do not give valid tests for importance of variable. We observe that the coverage rate of confidence intervals for their test statistics (specifically, TE-VIM) severely decreases below the desired rate when the variable importance goes to zero.
holds (Figure 6). Therefore, they do not give valid tests for importance of a variable. We observe similar behavior for the comparator TE-VIM experiments.
F. Options for defining detailed shifts with respect to a subset
We discuss some alternatives to defining the detailed level tests.
Detailed outcome shifts. Another choice for the null hypothesis H YIX that
e sup ElZ((x) - Zo( E
where (z,z) Ez]] Zo [z).
This test, however, will fail flag subsets important explanations. To illustrate, consider simple example where the conditional loss function does not change with respect to a subset but still the hypothesis will be rejected.
Example 1. Assume the conditional loss functions for the domains for the sigmoid function or (z) 1/(1 exr((=z)) Even though there is change in loss functions with respect to 42, the null hypothesis does not hold for xi since the difference of sigmoids is still a function of æ1. However, the target loss can be defined as a function of 22 and (z) alone, Z1 (x) afa (Zo(z)) + 0.5z2).
7 (x1 0.5xz) and Zı (I) a (x1 x2))
The unintuitive behavior in the example occurs because we explain the discrepancy between Z and Zo by taking their difference as opposed to any other scale.
Detailed covariate shifts. natural choice for the null hypothesis is to posit that (z( = Zo (o(a)( for all æ. However, this formulation would attribute performance change to covariate shifts in variables which are irrelevant to the loss function. Consider a simple example to illustrate the issue.
Example 2. Assume conditional loss and one the two features are distributed the same in source and target, Y|X = (x1, F2) ~ ((1,, 1), In NO, 1) in Po and P,. For simplicity, consider Y. Second feature has a covariate shift from
23

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A Hierarchical Framework for Diagnosing Heterogeneous Performmnce Drift
a2 N(0, 1) in Po 22 (1,, 1) in P1. That is, (1) & Po(x). Since Zo(x) Zz( and 2i does not shift, there is
no performance change. Even though 22 is irrelevant to the loss, the shift in 22 will leads to reject the null and attribute performance change to covariate shift.
As the example suggests, one way to address this undesired behavior is to focus only on shifts in variables important to the conditional loss Zo(x)) instead of all variables x.
G. Experiment details
Code to reproduce the experiments is at the link http:://github..om//fffeegg details of the experiments and the implementation.
We describe important
G.I. Simulations
For simulation setups 2 and 3, we generate n 8000 data points in both the source and target domains. We split datasets into equal halves for training and evaluation. For Setup I, we reduce data points to 2000 for all methods since the kernel-based baselines take a long time. We split the sample in the ratio 20-80% for SHIFT to ensure more data for evaluation. The ML algorithm is a logistic regression model fitted on a separate sample of n = 10000 points from the source domain.
Setup la (Outcome shift in subgroup): Define a subgroup A = (a E R & [-3.5,355]]. ma = #1 (0, 0, O, O, O, 0, 0), En = En (2, 2, 2, 2, du = 011,0,,1,111,,00,111111111111111 logit for target domain is Φi (0.2, 0.5, 1, 01,10011,11,11,,1,,11,1,1111, when x E A and (0 otherwise. Accuracy overall is similar across source and target (drops by 0.5%). Since the baselines test for any difference in conditional outcome distribution, SHIFT similarly tests for r = 0. Given the subgroup A is 7.8% of the data, SHIFT tests for minimum prevalence of e = 0.05.
The
Setup lb (Covariate shift in a subgroup): Logits are same as Setup la Φ1 = do = 0.5, 0,11,,,,01,,11,1111,,, 0.1,0,,,,0100,,0...
We vary the mean of the first covariate from to in the subgroup defined as A = fa E 0 | E [-4,4]]. Covariance matrices are the same as Setup la. SHIFT tests for tolerance T 0 and prevalence of 0.05.
Setup 2 (Outcome shift only): m. μ11 (0,0,0,0), Mi Σn diag 2, (z) = 0.8z, + 0.5x. + 2z + 0.62z and Φ1 (z) 0.2.z + 0.422 + 23 + 0.6.r.. Therefore, the outcome shifts with respect to both Xı and X2, but the shift in X2 is minimal and below tolerance T. Accuracy of the ML algorithm by 5.9% from 83.8% in source domain to 77.9% in target domain. SHIFT tests for r = 0.05, c = 0.05.
Setup 3 (Covariate shift only): ma = (1,0,0,1) ), Eo = diag(2. 2,2,, and μn = (0,0,0,0). Σ, diag(1, 2, 2, 2). dn- = Φ1 2.5z + x2 + 0.522 + 0.121 Both X1 and X, shift but X1's shift very small and below tolerance т. Accuracy drops by 5.4% from 90.9% 85.5%. SHIFT tests for 0.02, 0.05.
G.2. weal-world case studies
Background of case studies. We chose the case studies to mirror the real-world application of the framework. They consist of settings where covariate outcome shifts impact performance and domain experts do not know which shifts are detrimental. Such settings are highly prevalent in healthcare where ML performance varies widely across hospitals and time.
The first case study is based on a systematic analysis in Liu et al. (2023) that analyzed performance drops of an algorithm for predicting insurance coverage across different US states in the American Community Survey dataset. Among many state pairs, Liu et al. (2023) primarily found a large decay when transfering the algorithm from Nebraska to Louisiana. We decided to dive deeper into this analysis by identifying which subgroups were affected and why. SHIFT detected that people who are unemployed or whose parents are not in the labor force experience a large decay (Fig 3c). Since health insurance coverage is tied to employment in the US and insurance rates and incomes differ between the states, such decay is expected.
The readmission case study analyzes an algorithm to predict readmission that is trained on a well-resourced academic hospital and applied to a safety-net hospital Since safety-net hospitals serve patients regardless of their ability to pay, their populations are quite different. SHIFT detected that patients with many emergency encounters experience a large decay (Fig 3d), which expected because safety-net hospital patients seek care from emergency departments for very different
24

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A Hierarchical Framework for Diagnosing Heterogencous Performance Drift
reasons than academic hospitals. Thus, SHIFT helps detect subgroups in realistic benchmarks.
Health insurance prediction. We extract datasets for the two states from the 2018 yearly American Community Survey. It is available to download using the folktables package (Ding et al., 2021). Source (Nebraska) and target (Louisiana) domains have 3166 and 12000 points respectively, and 34 features. For the covariate shift tests, we filter out features that are uncorrelated with the loss function, resulting in a total of 18 features. Outcome shift tests keep all features. The features are categorized into broad categories (Table 6). Outcome is whether the person has public health insurance. We train a multi-layer perceptron on separate datasets of 3166 points from source domain. Public health insurance rate (class prevalence) increases drastically from 19.9% 40.4% in Louisiana. SHIFT tests for tolerance 7 0.05 and prevalence 0.05.
Table 6: Featues for health insurance prediction. Demo refers to demographics and Misc refers to Miscellaneous features.
Feature name
Category
SHAP importance for ML algorithm
Sex
Demo
0.072
Citizenship status not citizen Citizenship status naturalized Citizenship status born abroad Citizenship status born in Puerto Rico, Guam Citizenship status born in US
Demo
0.010
Demo
0.001
Demo
0,001
Demo
0.000
Demo
0,003
Race White
Demo
0.003
Never married or under 15 years
Demo
0,006
Divorced
Demo
0.043
Widowed
Demo
0.030
Married
Demo
0.062
Separated
Demo
0,002
Nativity
Demo
0,021
Ancestry
Demo
0.023
Age
Demo
0,018
Cognitive difficulty
Health
0.016
Vision difficulty
Health
0,007
Hearing difficulty
Health
0.001
Disability
Health
0,074
Gave birth to child in past 12 months
Misc
0,006
Military service
Health
0,054
Mobility status
Health
0.052
Employment status of parents Employment status armed Employment status unemployed Employment status partial employed Employment status employed Employment status not in labor force Employment status partial armed Total person's 's income
Health
0.022
Employment
0,000
Employment
0.009
Employment
0.001
Employment
0.008
Employment
0.012
Employment
0.000
Employment
0.244
School at least high school or GED
Education
0.000
School at least bachelor
Education 0.010
Educational attainment
Education
0.186
School postgrad
Education
0.001
Readmission prediction. We access electronic health records from an academic and a safety-net hospital and extract datasets for predicting readmission within 30-day of discharge for patients diagnosed with heart failure. Source (academic hospital) and target (safety-n net hospital) domains have 7468 and 6515 points respectively, and 27 features. Covariate shift tests are run after filtering out features uncorrelated with the loss function, resulting in total of 20 features. The features
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
belong 4 broad categories (Table 7). We train random forest on a separate dataset of 22403 points from source domain. Readmissions increase considerably from 12.4% to 18.5% in the safety-net hospital. SHIFT tests for tolerance 0.02 and prevalence 0.05.
Table 7: Features for readmission prediction. Demo refers to demographic variables.
Feature name
Category
SHAP importance for ML algorithm
Num ED encounters
Encounters
0.499
Firstrace Decline/Other/nnnnown
Demo
0.005
Sex Female
Demo
0.004
First race White
Demo
0.000
First race Native Hawaiian or Other Pacific Islander
Demo
0.000
Sex Male/Other
Demo
0.004
First race Black or African American
Demo
0.007
First race Asian
Demo
0.003
Age
Demo
0.026
First race Native American or Alaska Native
Demo
0.000
Vital Weight/Scale
Vitals
0.003
Vital Pulse
Vitals
0.007
Labs Calcium, total, Serum Plasma
Labs
0.001
Labs Hemoglobin
Labs
0.233
Labs Potassium, Serum Plasma
Labs
0.032
Labs MCV
Labs
0.004
Labs Chloride, Serum / Plasma
Labs
0.003
Labs WBC Count
Labs
0.015
Labs Lactate, whole blood
Labs
0.020
Labs Carbon Dioxide, Total
Labs
0.000
Labs eGFR low estimate Labs eGFR high estimate Labs Phosphorus, Serum / Plasma Labs Sodium, Serum / Plasma
Labs
0.021
Labs
0.023
Labs
0.016
Labs
0.022
Labs Magnesium, Serum / Plasma
Labs
0.022
Labs Creatinine
Labs
0.017
Labs Anion Gap
Labs
0.011
Model update experiment. We demonstrate the actionability of the SHIFT results on health insurance prediction. The detailed outcome shift test in Figure 3c detects a subgroup affected by outcome shifts and identifies demographic variables as a potential explanation. Based on this, SHIFT allows targeted update to mitigate the decay in the detected subgroup. To address outcome shift, we fine-tune the MI algorithm with respect to demographie variables by fitting that takes original algorithm's predictions and demographic variables predict the outcome in target data. The updated algorithm applies the new model only for the detected subgroup and defaults to the original model otherwise. Thus, the targeted update addresses the decay in affected subgroup while keeping the algorithm behavior the same for everyone else.
model
We compare the targeted update to a standard practice in response to distribution shifts that is to retrain the model on all features and use for everyone. We call non-targeted update. Additionally, we update the model on only the employment-related features identified by the TE-VIM method. Table reports the AUC of the targeted and the two non-targeted updates. For the update, we take 2000 points from target domain held out from earlier tests and fit an MLP model. We evaluate the updates (on subgroups) in another held-out 2440 points from target domain. We observe that the targeted update achieves around the same performance on the detected subgroup as the non-targeted one. However, the non-targeted have the unintended effect of reducing the performance on another subgroup where the original model performed better. We find the other subgroup affected by the non-targeted updates by again applying SHIFT. Hence, the experiment demonstrates that SHIFT can help to adapt ML algorithms to new settings while making minimal changes to them.
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H. Implementation details
Estimation of nuisance parameters. We use sample-splittin to estimate all nuisance parameters on 50% of the source and target domain data and evaluating the test statistics on the remaining data. We use implementations provided in seikit-leaad package to fit ML models (Pedregosa et al., 2011). For outcome models, we search through grid of estimators and their hyperparameters through cooss-validttion, which includes logistic regression, random forest, and gradient boosting classifiers. Density models are chosen from logistic regression and calibrated random forest. We clip the density ratios by restricting predicted probabilities from the density models between [10-3, 1 - 10-3) to avoid high variance in our McEE estimates. We bin the outcome shift Jain into B = 40 equally-spaced intervals in [o, 1] for detailed outcome shifts in all experiments.
Setting testing hyperparameters 7 and 6. The hypotheses require specifying two hyperparameters, namely, minimum subgroup size and minimum shift magnitude т. Both are intuitive and can be set by anyone, but they are domain-specifie. The 7 specifies the smallest performance change that is deemed to be meaningful to detect. For safety-critical settings like healthcare, even a 1-2 % change might be meaningful. Accuracy changes below are part of the null hypotheses and will not be flagged, hence allowing the domain expert to ignore very small changes.
The e allows us to only consider shifts that affect subgroups of at least size e. If the prevalence of the subgroup experiencing performance change goes below e, the null hypothesis would be true and this tiny subgroup would no longer be of interest. In general, the power of SHIFT decreases as the prevalence of the shifting subgroup decreases. Existing tests also decrease in power but they set the minimum threshold to zero, stating that all shifts are of practical interest and yet have limited power to detect them. We show in experiments (Figures 8 and 7) that SHIFT can detect changes in datasets with low 500 data points (c = 5% would mean subgroups of 25 data points), but we recommend setting to have at least 100 data points to have decent power.
Together, - and e ensure that the tests are practical and reduce alarm fatigue caused by detecting negligible shifts.
Computational complexity. SHIFT runs in under 10 minutes for the real-world datasets with around 10,000 points. The bulk of the computation is dedicated to fitting the nuisance models, the runtime is O(V) where V is the number of cross-validation folds. Moreover, fitting these nuisance models is easily parallelizable.
Visualizing the fitted detectors. We explain the detectors by fitting decision sets using the MLIC package (Ghosh et al., 2022). Decision sets output propositional logic statements on the features, such as (If X1 > 0 AND X₂ < 1), to classify points into detected and not detected. We fit decision sets to classify detected subgroups in either source or target data, and report the sets for the detected class in Figures 3a, 3b, 3c, and 3d.
Comparators. We give details for the kernel-based comparators for the aggregate tests.
Aggregate tests: For outcome shift tests, KCI tests for D I X MMD typically is used to test unconditional independence between two variables. We repurpose MMD for conditional outcome shifts by testing D .. (e, X). Since the joint distribution factorizes into p(l, X, D) = p(]]X, D)x(X D) and we know that covariate distribution does not vary in Sctup 1, the null hypothesis is equivalent to D 1 I|X.
For the comparators of the detailed tests, X, is a good explanation when the null rejected whereas is likely to be a good explanation when we fail to reject.
Detailed outcome: Tests are set up such that the subset explains the outcome shift if we fail to reject the null for (a) and (d), and if we reject the null in the case of (b) and (c).
Detailed covariate: Tests are set up such that the subset explains the covariate shift if we fail to reject the null for (c), and if we reject the null in the case of (a) and (b).
I. Additional simulation results
We validate type-I and power properties of the tests by repeating them N = 50 random draws of the dataset with varying sample size and reporting the rejection rates.
Recall that Setup 2 only has conditional outcome shifts with respect to variables X, and Xg. Aggregate tests reject the outcome shift test with power tending to 1 as sample size increases (Figure 8). Rejection rate for agg X-test, which corresponds to an estimate of type-I error, is within a = 0.05. Since X, is the variable with the most shift, the test for
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
subset X, {X1} } has low rejection rate. The test for subset (X1; X2) achieves type-I error control. Alone neither of X2- X3, and X explains the shift, thus, they are correctly rejected. The rejection rates tend even with 2000 samples. Figure 9a shows that none of the comparators flags the shifting subset.
The advantage of double/debiased inference is evident from the results for Setup 3 (Figure 7). Recall that covariate distribution shifts only for X1 and X in Setup 3. For the aggregate tests, plug-in rejects agg Y-test even when there is no outcome shift. On the other hand, one-step achieves type-I error control. Similarly, we observe that detailed test for subset {X1,X4} achieves type-I error control. However, the type-I error control comes the cost of lower power in case of the remaining subsets. Thus, the one-step tests are likely to flag more subsets candidates than needed to explain the performance shift. It could be desirable to have such conservative tests if we do not want miss potential ways to fix the performance shift. Figure 9b shows that KCI identifies the shifting subset correctly while other comparators testing subsets either in principle (KS is for univariate samples) or in their implementation (Score).
J. Additional case study
Although SHIFT is primarily designed for tabular data, its aggregate-Ievel tests are suitable for analyzing unstructured data; its detailed-level tests can also be used, if one has prespecified concepts. As an example, we apply SHIFT to the CivilComments dataset (Koh et al., 2021), which contains comments online articles and are judged to be toxic or not. We consider a DistilBERT-base --t--b--s-uuuceeed model (Sanh et al., 2019) fine-tuned to classify toxic comments. Given the 768-dimensional embeddings from this BERT model, we can apply SHIFT to understand differences in accuracy when classifying comments that mention the female gender (target domain) versus the remaining (source). Accuracy of the model drops by 1.3% in the target. Results from SHIFT's aggregate-leve test find evidence for covariate shift, i.c. there exists a subgroup of size 2 5% that experiences an accuracy drop greater than 5% due to covariate shift (Table 8).
Table 8: Results on a high-dimensional text dataset. We report p-values from the aggregate tests on CivilComments data which consists of internet comments and their toxicity labels. SHIFT finds evidence for accuracy drop due to covariate shift.
Aggregate test
p-value
Covariate shift
0.00
Outcome shift
0.83
To run detailed-level tests in SHIFT, we require variables to be interpretable. Given unstructured data, one solution is to combine SHIFT with concept bottleneck models (Koh et al., 2020). We note that another solution, if one does not need statistical inference at the detailed level, is to simply analyze differences between the comments from the detected subgroup from SHIFT in the source and target domains. Using a combination of GPT-4o (OpenAl et al., 2024) and manual review, we found that in the subgroup where the toxicity classifier experienced performance decay at the target domain, the comments tended discuss politics, society, race, and identity more. This shift in topics may explain the performance drop. For instance, the combination of female references with discussions of race and political ideology might compound biases that the classifier has inadvertently Iearned.
Finally, we note that although the proofs for SHIFT's validity require sufficiently fast estimation rates for the nuisance parameters which tend to slow down as dimensionality increases, estimation rates for nuisance parameters may still be sufficiently fast if these parameters are sparse (e.g. (Wager& & Walther, 2015; Belloni & Chernozhukov, 2011)).
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Aggregate
Rejection Rate Rejection Rate
Subset = X, tolerance = 0.02
est
1.0
plugin
0.5
onestep
0.0
Subset Y, tolerance 0.02
1.0
0.5
500
2000
4000
8000
Detailed
Rejection Rate
Subset = {x1), tolerance = 0.02
est
plugin onestep
0.5
0.0
Rejection Rate
1.Subset = (X2), tolerance = 0.02
0.5
0.0
Rejection Rate
1.
Subset = (X3), tolerance 0.02
0.5-
0.0
Rejection Rate
1.Subset (X4), tolerance 0.02
0.5
Rejection Rate
Sybset (X1,X4), tolerance 0.02
0.5
0.0
500
2000
4000
8000
Figure 7: Covariate shift setup. Covariate test, tolerance 0.0 and prevalence 0.05.
29

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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Aggregate
Rejection Rate Rejection Rate
Subset = X, tolerance = 0.05
est
1.0
plugin
0.5
onestep
0.0
Subset Y, tolerance 0.05
1.0
0.5
0.0
500
2000
4000
8000
Detailed
Rejection Rate
Subsett = (x1), tolerance 0.05
est
plugin onestep
0.5
0.0
Rejection Rate
1.Subset = (X2), tolerance = 0.05
0.5-
0.0
Rejection Rate
Subset = (X3), tolerance 0.05
0.5-
0.0
Rejection Rate Rejection Rate
1.Subset (X4), tolerance 0.05
0.5-
Sybset (X1,X2), tolerance 0.05
0.5
0.0
500
2000
4000
8000
Figure 8: Outcome shift setup. Outcome test, tolerance 0.05 and prevalence 0.05.
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A Hierarchical Framework for Diagnosing Heterogeneous Performance Drift
Aggregate Covaritte Outcome :
Detected large decay
e.g. subgroup:
Agorogate
0.00
X1 -2 43 X1 in -1.28) 1,27
Detected decay in 10.1% of target data. g in subgroup:
X1 2.01
Covariate Outcome
Detailed
Baselines
Outcome
0.75
0.00
0.43
0.00
Baselines Covariate
Detailed
X2
0.00
0.00
0.85
1.00
0.00
0.76
0.00
0.58
0.00
0.00
0.50
0.72
X4
0.00
0.00
0.43
0.93
0.00
X3
0.00
0.31
0.93
0.01
X1,X2
0.88
1.00
0.00
X1.X4
0.81
X3,x4
0.00
0.00
0.00
X2.X3
0.00
0.01
SHFT
KCI
ParamiLas $ Param'Y
TE-VIM
kClt
Score $
SHIFT
(a) Setup 2. Outcome-only shift
(b) Setup 3. Covariate-only shift
Figure 9: Detailed tests for variable subsets in simulation setups.
31


Reviews:
[{"review_id": "TSujh7Ic6K", "reviewer": "Reviewer_GEzi", "summary": "This paper proposes a method (SHIFT) for diagnosing performance drift in machine learning models that are transferred from a “source” to a “target” domain. Specifically, it aims to identify where (i.e., in which subgroups) a model’s performance decays the most and how such decay arises, distinguishing between subgroup-specific covariate shifts versus outcome shifts. By framing these questions as hierarchical hypothesis tests and using sample splitting plus flexible ML estimators, SHIFT produces valid inferences (i.e., with Type I error control) while preserving decent statistical power. The paper provides both theoretical guarantees and empirical evaluations on synthetic and real-world datasets (public-health insurance coverage and hospital readmissions).", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for carefully reading the work and for appreciating its practical relevance. Indeed, heterogeneity in ML performance is a major safety concern in high-risk applications and there is no unified test to identify the sources of heterogeneity. \n\n---\n\n**Why SHIFT outperforms other methods** There are two reasons why SHIFT is better. First, SHIFT is much more targeted: whereas methods like MMD and KCI try to detect *any* shift which dilutes their power to find the more relevant shifts, SHIFT is only interested in finding *subgroups* with a performance drop and the subgroups have to satisfy minimum size and magnitude requirements. Second, SHIFT uses ML techniques to perform the test, whereas the comparators are kernel-based and do not scale well to high dimensions.\n\nAnother advantage of a unified framework is that the test results across the aggregate and detailed levels will not conflict with each other. The current approach in the literature is to use a mix of different tests for the two levels, which may lead to conflicting results that are challenging to reconcile since the tests often have different null hypotheses.\n\n---\n\n**Background of case studies** We chose the case studies to mirror the real-world application of the framework. They consist of settings where covariate or outcome shifts impact performance and domain experts do not know which shifts are detrimental. Such settings are highly prevalent in healthcare where ML performance varies widely across hospitals and time. We do not use the same data as the TE-VIM baseline since it is for causal inference and only assesses outcome shifts. We will update the paper with the following background. \n\nThe first case study is based on a systematic analysis in Liu et al. 2023 that analyzed performance drops of an algorithm for predicting insurance coverage across different US states in the ACS dataset. Among many state pairs, Liu et al. primarily found a large decay when transfering the algorithm from Nebraska to Louisiana. We decided to dive deeper into this analysis by identifying which subgroups were affected and why. SHIFT detected that people who are unemployed or whose parents are not in the labor force experience a large decay (Fig 3c). Since health insurance coverage is tied to employment in the US, and insurance rates and incomes differ between the states, such a decay is expected. \n\nThe readmission case study analyzes an algorithm to predict readmission that is trained on a well-resourced academic hospital and applied to a safety-net hospital. Since safety-net hospitals serve patients regardless of their ability to pay, their populations are quite different. SHIFT detected that patients with many emergency encounters experience a large decay (Fig 3d), which is expected because safety-net hospital patients seek care from emergency departments for very different reasons than at academic hospitals. Thus, SHIFT helps detect subgroups in realistic benchmarks. \n\n---\n\n**Fix issues in encounters feature** We processed the readmission data again to correct the encounters feature. After correction, covariate shifts no longer lead to a significant subgroup-level accuracy drop (p-value goes from 0.00 to 0.69). Thus, SHIFT has helped bridge the accuracy gap.\n\n---\n\n**Effect of scaling subgroup size** As the prevalence of the subgroup experiencing performance decay decreases, the power of SHIFT decreases. If the prevalence of the subgroup goes so low to be below the specified minimum threshold in SHIFT, the null hypothesis would be true and this tiny subgroup would no longer be of interest. Thus the \"breaking point\" of SHIFT is the specified minimum threshold, but this is *by design*. Other tests also decrease in power as the subgroup experiencing the shift decreases in prevalence. But existing tests set the minimum threshold to zero, stating that all shifts are of practical interest and yet have limited power to detect them. For empirical results of SHIFT for small subgroups, please refer to Fig 6, 7 in Appendix.\n\n---\n\n**Effect of dimensionality** The theoretical results primarily rest on one's ability to estimate the nuisance functions at a sufficiently fast rate. When dimensionality increases, estimation rates for nuisance functions tend to slow down, though it may still be sufficiently fast if these functions are sparse [see e.g. 1,2]. To test SHIFT empirically, we applied our method to a text classification problem with 768-dimensional embeddings and was able to detect shifts. For the revised manuscript, we will include more validation of SHIFT in higher-dimensional datasets.\n\n---\n\n**Type I error** Please refer to Fig 6, 7 in Appendix, where we confirm that Type I error of SHIFT is controlled.\n\nWe hope that our responses clarify the concerns raised.\n\n---\n\n[1] Wager et al. Adaptive concentration of regression trees... arXiv 2015\n\n[2] Belloni et al. $\\ell_1$-penalized quantile regression... Ann Statist 2011", "claims_and_evidence": "- SHIFT detects subgroups with large performance decay due to distribution shifts, specifically distinguishing whether the shift is driven by covariates or by the outcome distributions.\n- SHIFT can then explain decay with sparse variable subsets, checking if these smaller shifts can plausibly account for the large performance drop in the discovered subgroups.\n- Tests have valid Type I error control and good power asymptotically, meaning SHIFT will not flag a nonexistent shift too often, and will detect real shifts with high probability given enough data.\n- SHIFT helps practitioners mitigate shifts more effectively than blanket retraining, by highlighting targeted fixes that resolve problems within critical subgroups without negatively impacting performance elsewhere.", "methods_and_evaluation_criteria": "The proposed methods make sense for the problem at hand given the decomposition of distribution shift into these constituent components at the covariate and outcome level. Meanwhile the evaluation criteria seems applicable for driving application but less clear how this fits into the benchmarks used in other related works. For example, when comparing TE-VIM to SHIFT, the real world datasets are not those used in the original paper. The synthetic experiment does a good job of making it clear why and when SHIFT will outperform methods tailored to each type of shift. It is much less clear why the real-world case studies are relevant in the context of the prior work.", "theoretical_claims": "I briefly skimmed the the theoretical results in Appendix C. Primarily focusing Theorem C.2.", "experimental_designs_or_analyses": "Yes, I went through Section 5 in detail. My main concern with the experimental design is the choice of case study datasets. It’s unclear why these datasets were chosen and what underlies these datasets that we should expect their to be distribution shifts to test for. Furthermore, the readmission case study is unclear as to who the subgroups are.", "relation_to_broader_scientific_literature": "The key contribution of this paper in relation to the existing literature on distribution shift detection and heterogeneous subgroup performance is the formalization of the sources of heterogeneity and a universal test to distinguish both issues. Much of the literature has been disparate in aiming to develop statistical tests for either issue. To my knowledge, this is the first test to present this hierarchical test.", "essential_references_not_discussed": "There are no additional related works that I believe need to be cited.", "other_strengths_and_weaknesses": "Strengths:\n\n- Clear formalization of the testing framework and its relation to existing literature\n- Test unifies two areas of research producing a more useful test for practitioners\n\nWeaknesses:\n\n- Applied case studies section lacks clarity and unconvincing of the methods utility in settings outside of the synthetic setups\n- Synthetic setups could be more tailored to demonstrate why SHIFT outperforms prior methods\n- Theory is derived to show bounds on the Type I error but this isn’t measured in the empirical results", "other_comments_or_suggestions": "None.", "questions_for_authors": "Below is a list of questions for which I would engage in a discussion that would potentially lead to increasing my score to a weak accept or accept:\n\n1. Can you describe why this method outperforms other methods on the respective two tasks presented? If the claim is not the this method outperforms those but its utility is in being able to use it for both tasks at once then how should I think about that efficiency gain? It’s unclear to me how much more efficient it would be than just using two separate tests – if I knew which ones to use of course.\n2. I would like a much more detailed description and understanding of the case studies. What background underlies the choice in subgroup? Do we expect their to shifts in their distributions? What are the subgroups for readmission? \n3. If you fix the issues in the encounters feature do you actually see the accuracy gaps close? \n4. How does this method work when we scale the size of subgroups that experience a drop in accuracy? Given the epsilon parameter defines the size of the subgroup there are an arbitrary number of subgroups one can construct of that size. I assume there is a breaking point at which the method fails to detect if that epsilon is too small. \n5. Finally, I’m curious about how the dimensionality of the problem would factor into your theoretical and empirical results. Currently, I don’t see any dimensionality issues in the theory which makes sense given the nature of the test. Empirically though, the experiments are all in quite low dimensional settings. I’d be interested in understanding at what X variable subsets the test struggles.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fv2AXaxI7k", "reviewer": "Reviewer_JHtq", "summary": "The paper titled \"Who experiences large model decay and why?\" introduces a hierarchical framework called SHIFT (Subgroup-scanning Hierarchical Inference Framework for performance drifT) to diagnose heterogeneous performance drift in machine learning models. The framework aims to identify subgroups that experience significant performance decay due to covariate or outcome shifts and provides detailed explanations for these shifts. The goal is to enable targeted corrective actions that mitigate decay for the most affected subgroups.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments and are heartened to hear they appreciate the novelty of SHIFT and its ability to provide detailed explanations for subgroup-specific shifts. Indeed, SHIFT addresses a **critical and very practical question**: when performance of an ML algorithm drops in a new application context, which subgroups are adversely affected and why? Such settings are prevalent in healthcare, where ML algorithms differ in performance widely across demographic groups, time, and geographies [Finlayson et al. 2021]. \n\nDiscovering the subgroups experiencing performance drop is **highly important to catch hidden failures of the algorithm and to develop targeted fixes** to the algorithm for the affected subgroups without sacrificing its performance elsewhere. Current methods to study performance drops either do not focus on subgroup-level performance or do not quantify uncertainty in the discovered subgroups. SHIFT addresses this methodological gap via **statistically principled and computationally scalable** methods. Furthermore, we believe SHIFT provides a **solid theoretical foundation** on which future work can build on, as discussed below.\n\n---\n\n**Extensions to unstructured data**: Although SHIFT is primarily designed for tabular data, its aggregate-level tests are suitable for analyzing unstructured data; its detailed-level tests can also be used, if one has prespecified concepts. In the revised manuscript, we will include applications of SHIFT to both text and image datasets. \n\nAs an example, we have applied SHIFT to the CivilComments dataset [Koh et al. 2021], which contains comments on online articles and are judged to be toxic or not. We consider a DistilBERT-base-uncased model fine-tuned to classify toxic comments. Given the **768-dimensional** embeddings from this BERT model, we can apply SHIFT to understand differences in accuracy when classifying comments that mention the female gender (target domain) versus the remaining (source). Accuracy of the model drops by 1.3\\% in the target. Results from SHIFT's aggregate-level test find evidence for covariate shift, i.e. there exists a subgroup of size $\\ge$ 5\\% that experiences an accuracy drop greater than 5\\% due to covariate shift. \n\n| Test            | p-value |\n|-----------------|---------|\n| Covariate shift | 0.00    |\n| Outcome shift   | 0.83    |\n\nTo run detailed-level tests in SHIFT, we require variables to be interpretable. Given unstructured data, one solution is to combine SHIFT with concept bottleneck models [Koh et al. 2020]. We will include such an example in the revised paper. We note that another solution, if one does not need statistical inference at the detailed level, is to simply analyze differences between the comments from the detected subgroup from SHIFT in the source and target domains. Using a combination of GPT-4o and manual review, we found that in the subgroup where the toxicity classifier experienced performance decay at the target domain, the comments tended to discuss politics, society, race, and identity more. This shift in topics may explain the performance drop. For instance, the combination of female references with discussions of race and political ideology might compound biases that the classifier has inadvertently learned.\n\n---\n\n**Computational complexity** SHIFT is very **fast**. SHIFT runs in under 10 minutes for the real-world datasets with around 10,000 points. The bulk of the computation is just fitting the nuisance models, so the runtime is just O(V) where V is just the number of cross-validation folds. Moreover, fitting these nuisance models is easily parallelizable.\n\n---\n\n**Complexity in practice and open-source tools** We plan to publish an **open-source Python package** for running SHIFT that will provide simple and intuitive APIs. That way, users of the package can run SHIFT with just one or two lines of code.\n\n---\n\n**Multiple subgroups experience overlapping shifts** SHIFT is designed to handle situations where the subgroup experiencing covariate shift does or does not overlap with the subgroup experiencing outcome shift. When there are multiple subgroups experiencing covariate shift, SHIFT groups them together into one large subgroup and performs an omnibus test. SHIFT handles multiple subgroups experiencing outcome shifts similarly.\n\n---\n\n**Sensitivity to test parameters** We find that SHIFT is not very sensitive to the two test parameters: minimum subgroup size and shift magnitude. Moreover, these two parameters are very **intuitive**. While one could have a domain expert set them, they are intuitive parameters that can be easily selected by anyone and should simply reflect one's own tolerance for performance drift. We will include guidance on setting the parameters in the two case studies.\n\nWe hope that our responses address the reviewer's concerns and encourage them to reconsider their score.\n\n---\n\n[1.] Koh et al. WILDS: A Benchmark of in-the-Wild Distribution Shifts. ICML 2021", "claims_and_evidence": "The authors claim that existing methods do not provide detailed insights into subgroup-specific performance decay. SHIFT is proposed as a solution to identify and explain large performance decay in subgroups. The paper provides evidence through simulations and real-world experiments, demonstrating that SHIFT can identify relevant shifts and guide model corrections effectively.", "methods_and_evaluation_criteria": "SHIFT is a two-stage hypothesis testing framework. The first stage identifies subgroups with large performance decay due to aggregate covariate or outcome shifts. The second stage provides detailed explanations by testing variable(subset)-specific shifts. The evaluation criteria include the ability to detect meaningful performance decay and provide valid statistical inference without strong assumptions.", "theoretical_claims": "The paper claims that SHIFT provides valid statistical inference through hypothesis testing, even with limited data. It does not rely on strong assumptions like knowledge of the true causal graph or large datasets. The theoretical properties of the framework are supported by asymptotic normality and controlled Type I error rates.", "experimental_designs_or_analyses": "The experiments include simulations and real-world case studies. Simulations vary the type and degree of shifts, ML algorithms, and data sizes to validate SHIFT's performance. Real-world case studies involve health insurance prediction across states and readmission prediction across hospitals. The experiments demonstrate SHIFT's ability to identify relevant shifts and guide targeted model updates.", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "Strengths of the paper include the novel hierarchical framework for diagnosing performance drift, the ability to provide detailed explanations for subgroup-specific shifts, and the use of hypothesis testing for valid statistical inference. \n\nWeaknesses may include the complexity of the framework and the potential challenges in implementing it in practice. The reliance on domain experts to set parameters like minimum subgroup size and shift magnitude may also be a limitation.", "other_comments_or_suggestions": "The paper could benefit from a more detailed discussion on the computational complexity of the framework and potential strategies for efficient implementation. Additionally, providing more examples of real-world applications and their outcomes could enhance the practical relevance of the framework.", "questions_for_authors": "How does SHIFT handle cases where multiple subgroups experience overlapping shifts?\nCan SHIFT be extended to handle unstructured data like images or text, and if so, how?\nWhat are the computational requirements for implementing SHIFT in large-scale datasets?\nHow sensitive is SHIFT to the choice of parameters like minimum subgroup size and shift magnitude?\nAre there any plans to make the SHIFT framework available as an open-source tool for broader use?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bouYjpqu1F", "reviewer": "Reviewer_g6ej", "summary": "This paper introduces SHIFT, a hierarchical hypothesis-testing framework designed to identify subgroups experiencing significant performance degradation in machine learning models due to distribution shifts. SHIFT first tests for the presence of large performance decay due to aggregate covariate and outcome shifts, and subsequently identifies specific subsets of input variables responsible for the observed decay. Unlike previous approaches, SHIFT does not rely on strong parametric assumptions or detailed causal knowledge, making it suitable for scenarios with limited data. For experiments, SHIFT is validated at the simulation level and also for shifts in real-world data.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive feedback appreciating the practical utility of the methods and for providing helpful suggestions on additional benchmarks. Indeed, SHIFT addresses a **critical and very practical question**: when performance of an ML algorithm drops in a new application context, which subgroups are adversely affected and why? Such settings are prevalent in healthcare, where ML algorithms differ in performance widely across demographic groups, time, and geographies [Finlayson et al. 2021]. \n\nDiscovering the subgroups experiencing performance drop is **highly important to catch hidden failures of the algorithm and to develop targeted fixes** to the algorithm for the affected subgroups without sacrificing its performance elsewhere. Current methods to study performance drops either do not focus on subgroup-level performance or do not quantify uncertainty in the discovered subgroups. SHIFT addresses this methodological gap via statistically principled and computationally scalable methods. Furthermore, we believe SHIFT provides a **solid theoretical foundation** on which future work can build on, as discussed below.\n\n---\n\n**Computational complexity** SHIFT is very **fast**. SHIFT runs in under 10 minutes for the real-world datasets with around 10,000 points. The bulk of the computation is just fitting the nuisance models, so the runtime is just O(V) where V is just the number of cross-validation folds. Moreover, fitting these nuisance models is easily parallelizable.\n\n---\n\n**Extensions to high-dimensional, non-tabular data e.g. WILDS** Although SHIFT is primarily designed for tabular data, its aggregate-level tests are suitable for analyzing unstructured data; its detailed-level tests can also be used, if one has prespecified concepts. In the revised manuscript, we will include applications of SHIFT to both text and image datasets. As an example, we have applied SHIFT to the CivilComments dataset [Koh et al. 2021], which contains comments on online articles and are judged to be toxic or not. We consider a DistilBERT-base-uncased model fine-tuned to classify toxic comments. Given the **768-dimensional** embeddings from this BERT model, we can apply SHIFT to understand differences in accuracy when classifying comments that mention the female gender (target domain) versus the remaining (source). Accuracy of the model drops by 1.3\\% in the target. Results from SHIFT's aggregate-level test find evidence for covariate shift, i.e. there exists a subgroup of size $\\ge$ 5\\% that experiences an accuracy drop greater than 5\\% due to covariate shift. \n\n| Test            | p-value |\n|-----------------|---------|\n| Covariate shift | 0.00    |\n| Outcome shift   | 0.83    |\n\nTo run detailed-level tests in SHIFT, we require variables to be interpretable. Given unstructured data, one solution is to combine SHIFT with concept bottleneck models [Koh et al. 2020]. We will include such an example in the revised paper. We note that another solution, if one does not need statistical inference at the detailed level, is to simply analyze differences between the comments from the detected subgroup from SHIFT in the source and target domains. Using a combination of GPT-4o and manual review, we found that in the subgroup where the toxicity classifier experienced performance decay at the target domain, the comments tended to discuss politics, society, race, and identity more. This shift in topics may explain the performance drop. For instance, the combination of female references with discussions of race and political ideology might compound biases that the classifier has inadvertently learned.\n\n---\n\n**Observations under very small sample scenarios** SHIFT performs well at small sample size. Please refer to Fig 6, 7 in Appendix, where we show that SHIFT maintains the specified type-I error rate and has good power as sample size decreases to 500.\n\n---\n\n**Iterative subgroup discovery is not explored** Our tests provide evidence that there is an affected subgroup with statistical significance. We can then use existing subgroup discovery methods to explore all subgroups [Eyuboglu et al. 2022, d'Eon et al. 2022]. Such methods, thus, are complementary to ours but do not provide statistical significance.\n\n---\n\n[1.] Koh et al. WILDS: A Benchmark of in-the-Wild Distribution Shifts. ICML 2021", "claims_and_evidence": "The claimed are backed by theoretical proofs. Empirically, it was shown via experiments (both simulated and real) that SHIFT can identify relevant shifts and improve subgroup accuracy.\n\nThe paper could be further strengthened by demonstrating SHIFT's robustness across a wider array of distribution shifts such as  high-dimensional, non-tabular data. Additionally, observations under extreme sparsity or very small sample scenarios would be informative.", "methods_and_evaluation_criteria": "The proposed method’s two-stage testing approach makes sense. Also, the experimental evaluation is sound, using both controlled simulation and real world case studies encompassing realistic shifts (insurance across states, hospital readmission across institutions). If possible, additional real world studies regarding a different domain would be very informative, but it could be hard to conduct such experiments.", "theoretical_claims": "As I am not currently well-versed in the theoretical aspects of the related literature, I must defer the analyses and verifications of the theoretical claims of the paper to other reviewers.", "experimental_designs_or_analyses": "Strengths:\n\n- Clear ground-truth validation of subgroup detection and shift attribution via simulations.\n- Also demonstrates practical utility via real-world studies, with SHIFT-driven fixes improving performance.\n\nWeaknesses:\n\n- Scalability - no discussion of runtime or feasibility in high-dimensional settings.\n- SHIFT focuses on one affected subgroup per shift type, but iterative subgroup discovery is not explored.", "relation_to_broader_scientific_literature": "The proposed method is closely related to distribution shifts, robust machine learning, bias and fairness in machine learning. Although the paper primarily focuses on tabular data, these concepts are also important for large scale models (if the proposed method could be effectively scaled so such magnitudes).", "essential_references_not_discussed": "It could benefit the paper if the WILDS dataset (Koh et al., 2021), real-world distribution shift benchmark, is included.", "other_strengths_and_weaknesses": "My thoughts on the strengths and weakness of the paper are discussed above.", "other_comments_or_suggestions": "Currently, I have no other comments or suggestions.", "questions_for_authors": "Currently, I have no other questions. If it is possible, I wish to confer with other reviewers regarding the theoretical aspects of the manuscript.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vv6G4qvhuJ", "reviewer": "Reviewer_buFa", "summary": "This paper proposes a nonparametric Subgroup-scanning Hierarchical Inference Framework for performance drifT (SHIFT) to use hypothesis testing for drift diagnosis. The SHIFT first decides if any subgroup experiences significant performance decay from drift, then checks the specific shift that explains the decay. In this way, SHIFT enables the explainable detection of subgroup shifts.\n\n## update after rebuttal\nThe rebuttal from the authors solved my concerns and provided insightful experiment and discussion. I decide to keep my original positive rating.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive feedback and are glad to know that they found the work to be well-written and to have **great potential in practical applications**. Indeed, SHIFT addresses a **critical and very practical question**: when performance of an ML algorithm drops in a new application context, which subgroups are adversely affected and why? Such settings are prevalent in healthcare, where ML algorithms differ in performance widely across demographic groups, time, and geographies [Finlayson et al. 2021]. \n\nDiscovering the subgroups experiencing performance drop is **highly important to catch hidden failures of the algorithm and to develop targeted fixes** to the algorithm for the affected subgroups without sacrificing its performance elsewhere. Current methods to study performance drops either do not focus on subgroup-level performance or do not quantify uncertainty in the discovered subgroups. SHIFT addresses this methodological gap via statistically principled and computationally scalable methods. Furthermore, we believe SHIFT provides a **solid theoretical foundation** on which future work can build on, as discussed below.\n\n---\n\n**Extending SHIFT to image/text data** Although SHIFT is primarily designed for tabular data, its aggregate-level tests are suitable for analyzing unstructured data; its detailed-level tests can also be used, if one has prespecified concepts. In the revised manuscript, we will include applications of SHIFT to both text and image datasets. As an example, we have applied SHIFT to the CivilComments dataset [Koh et al. 2021], which contains comments on online articles and are judged to be toxic or not. We consider a DistilBERT-base-uncased model fine-tuned to classify toxic comments. Given the **768-dimensional** embeddings from this BERT model, we can apply SHIFT to understand differences in accuracy when classifying comments that mention the female gender (target domain) versus the remaining (source). Accuracy of the model drops by 1.3\\% in the target. Results from SHIFT's aggregate-level test find evidence for covariate shift, i.e. there exists a subgroup of size $\\ge$ 5\\% that experiences an accuracy drop greater than 5\\% due to covariate shift. \n\n| Test            | p-value |\n|-----------------|---------|\n| Covariate shift | 0.00    |\n| Outcome shift   | 0.83    |\n\nTo run detailed-level tests in SHIFT, we require variables to be interpretable. Given unstructured data, one solution is to combine SHIFT with concept bottleneck models [Koh et al. 2020]. We will include such an example in the revised paper. We note that another solution, if one does not need statistical inference at the detailed level, is to simply analyze differences between the comments from the detected subgroup from SHIFT in the source and target domains. Using a combination of GPT-4o and manual review, we found that in the subgroup where the toxicity classifier experienced performance decay at the target domain, the comments tended to discuss politics, society, race, and identity more. This shift in topics may explain the performance drop. For instance, the combination of female references with discussions of race and political ideology might compound biases that the classifier has inadvertently learned.\n\n---\n\n**Computational cost** SHIFT is very **fast**. SHIFT runs in under 10 minutes for the real-world datasets with around 10,000 points. The bulk of the computation is just fitting the nuisance models, so the runtime is just O(V) where V is just the number of cross-validation folds. Moreover, fitting these nuisance models is easily parallelizable.\n\n---\n\n**Sensitivity to test parameters** We find that SHIFT is not very sensitive to the two test parameters: minimum subgroup size and shift magnitude. Moreover, these two parameters are very **intuitive**. While one could have a domain expert set them, they are intuitive parameters that can be easily selected by anyone and should simply reflect one's own tolerance for performance drift. We will include guidance on setting the parameters in the two case studies.\n\nWe hope that the responses address your concerns.\n\n---\n\n[1.] Koh et al. WILDS: A Benchmark of in-the-Wild Distribution Shifts. ICML 2021", "claims_and_evidence": "The claims are successfully supported by experiments on tabular data.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria make sense under the problem setting and tabular data.", "theoretical_claims": "The theoretical claims focus on the formulation of the hypothesis testing.", "experimental_designs_or_analyses": "The experiments are extensive and support the proposed claims.", "relation_to_broader_scientific_literature": "Previous work proposes to decompose the average performance drop within an identified subgroup into covariate shift and outcome shift. This work aims to further identify the related input variables. In addition, the prior work on drift diagnosis is primarily based on the estimation, while this work uses hypothesis testing.", "essential_references_not_discussed": "I didn't identify any such references.", "other_strengths_and_weaknesses": "Strengths:\n\nThis paper is well-written and easy to follow. The work proposes a framework to detect the subgroup with performance decay and identify the key variables for the performance decay, which is explainable and has great potential in practical applications. The application of SHIFT in practice is also well-discussed in this paper.\n\nWeaknesses:\n\nThis paper mainly focuses on the experiments with tabular data. Despite the conclusion mentions it can be applied on image/text data, it might bring new challenges, such as computational cost and the categorization of subgroups.", "other_comments_or_suggestions": "NaN", "questions_for_authors": "1. How is the efficiency or computational cost of the hypothesis testing in SHIFT?\n\n2. Will there be new challenges when extending SHIFT to image/text data? If so, is it possible to address these challenges?\n\n3. SHIFT requires domain experts to select the minimum subgroup size and minimum shift magnitude and how sensitive is it to the choices of the hyperparameters?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This work studies subgroup performance degradation through their new method SHIFT. The reviews were largely positive (2,3,3,3) with reviewers noting the strong theoretical and empirical claims. I want to commend the authors for their detailed and clear responses to reviewer concerns, including the relevance of the real-world case studies (GEzi), effects on high-dimensionality (GEzi), and runtime and compute costs (GEzi, buFa, g6ej). The most negative review came from JHtq, who did not engage with the authors following their detailed responses and whose review was quite short and vague. 

The contribution appears sound and well-motivated. As a result, I recommend accept.
Decision: Accept (poster)