# Unreading Race: Purging Protected Features
from Chest X-ray Embeddings

Tobias Weber  Department of Statistics, LMU Munich  Department of Radiology,
University Hospital, LMU Munich  Munich Center for Machine Learning (MCML)
Michael Ingrisch  Department of Radiology, University Hospital, LMU Munich
Munich Center for Machine Learning (MCML)  Bernd Bischl  Department of
Statistics, LMU Munich  Munich Center for Machine Learning (MCML)  David
Rügamer  Department of Statistics, LMU Munich  Munich Center for Machine
Learning (MCML)

## Abstract

#### Purpose

To analyze and remove protected feature effects in chest radiograph embeddings
of deep learning models.

#### Materials and Methods

An orthogonalization is utilized to remove the influence of protected features
(e.g., age, sex, race) in chest radiograph embeddings, ensuring feature-
independent results. To validate the efficacy of the approach, we
retrospectively study the MIMIC and CheXpert datasets using three pre-trained
models, namely a supervised contrastive, a self-supervised contrastive, and a
baseline classifier model. Our statistical analysis involves comparing the
original versus the orthogonalized embeddings by estimating protected feature
influences and evaluating the ability to predict race, age, or sex using the
two types of embeddings.

#### Results

Our experiments reveal a significant influence of protected features on
predictions of pathologies. Applying orthogonalization removes these feature
effects. Apart from removing any influence on pathology classification, while
maintaining competitive predictive performance, orthogonalized embeddings
further make it infeasible to directly predict protected attributes and
mitigate subgroup disparities.

#### Conclusion

The presented work demonstrates the successful application and evaluation of
the orthogonalization technique in the domain of chest X-ray classification.

##  1 Introduction

At the interdisciplinary intersection between computer science and
radiological imaging, the applications of deep learning for the analysis of
chest X-ray (CXR) images have garnered considerable attention. Besides the
fact that CXR imaging remains a cornerstone of diagnostic imaging, this can
largely be attributed to the availability of extensive, annotated datasets,
e.g., [1, 2, 3], which allows the utilization of deep learning algorithms for
a variety of research questions. Among others, these tasks include
classification [2], image generation [3, 4], image manipulation [5], report
generation [6], and self-supervised pre-training [7, 8]. With growing public
interest and application in practice, there is an escalating demand for
fairness, transparency, and explicability of such machine learning models.
This need is paramount in the context of clinical decision-making, where the
primary concern is the patient’s outcome.

Of particular concern is the potential for bias in predictive models, which
might lead to serious consequences in downstream applications. The work of [9]
[9] shows that deep learning models can predict a patient’s race in CXRs with
high certainty, even when clinical experts cannot. Building on this
observation, the recent study by [10] demonstrates that features like age,
sex, and race are already encoded in deep feature representations of neural
CXR classifiers and thus can have a strong influence on downstream prediction
tasks. Even more concerning, this property is not restricted to classifiers
but also extends to embeddings obtained by contrastive methods, e.g., Google’s
Chest X-ray Foundation Model [7, 11].

In the following, we define the term protected features as intrinsic
characteristics typically protected by law against discrimination and bias.
Not correcting for influences of protected features in predictions and
representations used for decision-making can have detrimental effects. This
could lead to CXR classifiers not fully generalizing to certain subgroups as
found in [12] or to exhibit fairness concerns [13]. As noted by [10], ensuring
fairness and equal treatment across all subgroups is a major challenge as
training datasets tend to be imbalanced for specific populations.

The purpose of this work is twofold: First, we evaluate whether protected
features significantly contribute to model predictions in classifiers based on
CXR embeddings using three publicly available models and two open datasets.
Second, we investigate whether the influence of the protected features can be
removed by means of a recently proposed orthogonalization procedure [14, 15],
resulting in unbiased deep representations of CXR images.

##  2 Materials and Methods

###  2.1 Orthogonalization

Whereas the following introduces the orthogonalization more formally, a
geometric depiction of the approach is visualized in Figure 1. Let
$\bm{E}\in\mathbb{R}^{n\times d}$ be the embedding matrix of dimension $d$ for
$n$ samples, and $\bm{X}\in\mathbb{R}^{n\times p}$ a design matrix containing
the information of $p$ protected features for each sample. The ultimate goal
is to eliminate feature effects of $\bm{X}$ in $\bm{E}$. This can be achieved
by projecting $\bm{E}$ onto the space orthogonal to the space spanned by the
columns of $\bm{X}$. For this, we construct $\bm{Q}\in\mathbb{R}^{n\times q}$
using the QR-decomposition $\bm{X}=\bm{QR}\in\mathbb{R}^{n\times p}$. The
projection matrix $\mathcal{P}_{X}$, which projects an object, e.g. $\bm{E}$,
onto the space spanned by the columns of $\bm{X}$, is then given by
$\bm{Q}\bm{Q}^{\top}$, and its orthogonal complement
$\mathcal{P}_{X}^{\bot}=\bm{I}-\mathcal{P}_{X}$ with $n$-dimensional identity
matrix $\bm{I}$ [16]. Using this orthogonal projection matrix, we can obtain
the corrected embedding

$\widetilde{\bm{E}}=\mathcal{P}_{X}^{\bot}\bm{E}=(\bm{I}-\bm{Q}\bm{Q}^{\top})\bm{E}\,\text{.}$
(1)

The Frisch-Waugh-Lovell theorem [17] then guarantees that all
$\bm{X}$-influence is subtracted from $\widetilde{\bm{E}}$. In particular,
this will result in a zero influence of any of the features in $\bm{X}$ on
$\bm{E}$ when checking with a significance-test or linear model.

In order to apply this procedure for new unseen data $\bm{E}^{*}$, we derive
$\mathcal{P}_{X^{*}}=\bm{Q}^{*}\bm{Q}^{*\top}$ from the associated protected
features $\bm{X}^{*}$ and subsequently compute the orthogonalized embedding
$\widetilde{\bm{E}}^{*}=(\bm{I}-\mathcal{P}_{X^{*}})\bm{E}^{*}$.

###  2.2 Datasets

The methodology is retrospectively evaluated on the MIMIC [1] and CheXpert [2]
datasets (cf. Table 1), which both contain patients from US hospitals. The
racial groups are derived in the same fashion as in [9, 11]. We apply the
U-zeros strategy from [18] to define binary pathology labels.

#### MIMIC

We utilize the recommended official training (42,148 patients with 181,342
scans) and test splits (257 patients with 3,041 scans) [1], which are
contained in the published embeddings of the CXR foundation model [7]. The
dataset shows a substantial racial imbalance, with 75.77% / 79.77% of patients
being White, while Black and Asian patients constitute only 19.93% / 17.51 %
and 4.30 % / 2.71 % in the train and test, respectively. A similar racial
disparity is evident in the scan distribution. The age statistics reveal
variations among racial and gender subgroups. In the training set, Black
patients have a notably lower average age compared to White patients (62.6
versus 57.7 mean age).

#### CheXpert

Our CheXpert subset consists of the same train (25,730 patients with 76,205
scans) and test (12,866 patients with 38,240 scans) split previously employed
by [9] and [11] in their analyses. There is a notable disparity in patient
demographics across racial and gender categories. White patients constitute
the majority with 77.86% / 77.38 %, followed by Asians (15.33% / 15.79%),
while Black patients are underrepresented (6.81% 6.83%). Generally, Black
patients in both the training and test sets tend to be younger on average
(63.1 / 63.3 versus 55.7 / 57.4 mean age).

###  2.3 Analyzed Models

Our analysis includes the embeddings and feature representations of three
models, each compromising different architectures and training procedures. We
focus on publicly available and pretrained architectures and embeddings.

#### CXR Foundation Model (CFM; [7])

This EfficientNet-L2 model [19] is trained on over 800,000 CXR images from
India and the US using a supervised contrastive procedure [20] by
differentiating between normal and abnormal samples. The model is only
available using a restricted application-based API interface that offers
$1376$-dimensional feature embeddings. Thus, we utilize the publicly available
and packaged embeddings of the MIMIC
dataset.111https://doi.org/10.13026/pxc2-vx69

#### CheSS [8]

With a ResNet-50 [21] as a basis, CheSS is a contrastive self-supervised model
trained with the MoCo v2 strategy [22] on 4.8 million CXRs from South Korea.
The model yields $2048$-dimensional feature vectors.

#### CheXNet Classifier (CLF, [18])

The CLF label represents a Densenet-121 classifier trained on MIMIC and
CheXpert, respectively. The models are publicly available and obtained from
the torchxrayvision library [23] yielding $1024$-dimensional last-layer
feature vectors.

###  2.4 Statistical Analysis

Our analysis consists of three parts, each evaluated on the original and
orthogonalized embedding: Influence of protected features on pathology
classification, direct prediction of protected information, and downstream
task performance.

#### Evaluating the Influence of Protected Features

We employ a linear evaluation protocol, using the embedding as the only
feature to predict the different pathologies. Our prediction model of choice
is a logistic regression, optimized in a gradient-based iterative fashion due
to its capability of handling high-dimensionality in the feature embeddings.
The implementation is done in PyTorch v2.0 [24]. The network is trained for 10
epochs with a learning rate of $0.0001$ and batch size of $256$. All metrics
are evaluated over 10 random initializations. We do not conduct a
hyperparameter search as the goal of this paper is not to beat the current
state-of-the-art but to analyze and examine the effects of protected features
on predictive performance.

In order to check the influence of protected features in the prediction model,
we use a second linear model (the _evaluation model_) that estimates the
influence of protected features on the predicted logits from the _prediction
model_. From the evaluation model, we obtain regression coefficients for age,
race, and sex to quantify their influence and also to assess their
significance via the provided t-statistics and -tests.

#### Extracting Protected Features

While the previous evaluation checks the influence of protected features on
the model’s prediction when the goal is to classify pathologies, we
additionally employ three (generalized) linear models to directly predict the
protected information using the embedding. A linear model is used for the
prediction of age, a logistic regression for the prediction of sex, and a
multinomial regression to predict the three different race outcomes. All
models are again trained using a gradient-based iterative procedure in PyTorch
v2.0.

#### Downstream Prediction Performance

Applying the orthogonalization method results in the transformation of the
embedding vector. We investigate how the subsequent downstream pathology
prediction performance is impacted, when removing protected feature
information, using the previously trained CXR classifier models.

##  3 Results

###  3.1 Influence of Protected Features on Predictions

The estimated influence of protected features for the labels Pleural Effusion,
Cardiomegaly, and No Finding is given in Table 2. Extended results including
the distribution of coefficients and p-values over multiple runs are found in
Supplement A. The three labels are chosen due to their prevalence in the
datasets.

#### Without Orthogonalization

The majority of coefficients exhibit extremely small p-values, signaling a
significant influence on the predictions. This highlights the substantial
influence that protected features have on the model outputs. For example, in
the case of the MIMIC CFM embeddings and Pleural Effusion, an increase of
patient age of ten years results in a significant increase of $0.1\cdot
4.246=0.42$ in the predicted logits, or equivalent, a multiplicative increase
in log-odds of almost 52% for the prediction model.222Note that age is divided
by 100 in our encoding, hence an increase in one year is equivalent to 0.01.

Moreover, race also shows a significant influence, in particular being a Black
instead of White patient, resulting in a reduction of the predicted
probability for Pleural Effusion. The respective estimated additive effect
$-0.694$ results in logit values that imply a reduction of 50% in the log-odds
for the prediction model. While being Black in the MIMIC CFM embedding is
associated with a risk-reducing factor for Pleural Effusion, it increases for
Cardiomegaly with a coefficient of $0.145$.

Being female is generally associated with a negative coefficient with the
exception of Pleural Effusion in the CheXpert CheSS embeddings ($0.135$). The
coefficients for Asians exhibit relatively small values and, in many cases,
p-values greater than 0.05, indicating a lack of statistical significance.

An interesting case is the label No Finding, which indicates the presence of a
patient with no abnormalities. The obtained coefficients confirm our
methodology, as their magnitudes are similar to Pleural Effusion and
Cardiomegaly but are equipped with a flipped sign.

Overall, the analysis of MIMIC results in generally higher coefficients than
CheXpert. When evaluating the CheSS embedding, e.g., the age coefficient for
Pleural Effusion is $3.529$ versus $1.679$. This is also reflected in other
pathologies and protected features, except for the coefficient of the
attribute Black, which increases from $0.088$ to $0.362$ between the two
datasets.

In summary, the obtained coefficients express and state the existence of
biases in downstream predictions.

#### Applying Orthogonalization

However, after applying orthogonalization, the resulting regression
coefficients are estimated to be zero and exhibit p-values of one, implying
that the protected feature information is successfully removed in the
corrected embedding.

We visually inspect the effects of the orthogonalization in Figure 2 using the
first two PCA components of the embedding before and after orthogonalization.
We do this by exemplary comparing the marginal distributions of these two
components for the two available sex labels (M/F) across all evaluated
modalities. This visual inspection reveals that the orthogonalization
procedure does not change the directions of the highest variance in the
datasets and the general structure of the embedding space is preserved.
However, when examining the differences in the two sex labels, an alignment in
the marginal distributions can be detected. This is particularly striking for
the marginals of the second principal component in the MIMIC CFM and CheSS
embeddings. After applying orthogonalization, the shift between those
distributions is removed, resulting in a loss of discriminative power and
elimination of bias regarding this feature. In contrast, there are no or only
minor visual changes in the marginals of the first two PCA components for
Pleural Effusion (cf. Figure 2).

###  3.2 Predicting Protected Information

Apart from their influence on pathology classification, we investigate how
well the protected information can be predicted directly from the embeddings
themselves before and after orthogonalization. As depicted in Figure 3 and
Table 4, identification of protected features in deep embeddings can be
achieved with high accuracy, e.g., a patient’s sex with an AUC of 0.979 in
MIMIC CFM. The orthogonalization removes the respective feature information, a
classification is then not feasible anymore and results in random guessing.
This does not only apply to sex and race prediction but also to the patient’s
age. In the case of CheXpert‘s CheSS embeddings, linear regression yields an
$R^{2}$ of $0.529$, whereas, after correction, the $R^{2}$ is $-0.009$.
Generally, the two CLF models show the lowest AUCs when predicting sex or
race.

###  3.3 Downstream Prediction Performance

Lastly, another important question is how orthogonalization influences
embeddings with respect to their predictive power. Table 3 contains an
overview of the estimated downstream prediction performances when training a
logistic regression on both, the original and corrected embeddings,
separately. The MIMIC CFM embedding has a notably higher macro-averaged AUC in
comparison to CheSS and CLF ($0.789$ versus $0.695$ and $0.679$.) On CheXpert,
the CLF embedding with an AUC of $0.720$ yields a slightly better result than
CheSS with $0.710$. The orthogonalized MIMIC embeddings perform generally
worse than their uncorrected counterparts containing the protected features.
For CheSS and CLF, the difference in performance is $-0.14\%$ and hence
negligible, while for the CFM embeddings, the decrease is $-1.39\%$. However,
in the case of CheXpert, the removal of the protected feature even results in
a performance boost, e.g., $+1.83\%$ for CheSS. Removing biases hence does not
automatically imply a loss in performance.

Another question concerns whether removing protected features also influences
subgroup disparities. Figure 4 depicts the deviation of the subgroup AUC from
the overall AUC for Pleural Effusion and Cardiomegaly. In general, the MIMIC
dataset exhibits stronger disparities than CheXpert, especially in the
underrepresented Asian class. Using the orthogonalization, in fact, mitigates
the disparity in most cases, e.g., the Pleural Effusion class in MIMIC Chess
from $-12.54\%$ to $-7.76\%$. Inspecting the CheXpert CheSS and CLF embeddings
for Cardiomegaly, the correction results in the full mitigation of racial bias
towards Asian and Black.

##  4 Discussion

Our analysis investigates the implicit encoding of protected features age,
sex, and race in three independent neural representations of MIMIC and
CheXpert, and shows a significant influence of these on model predictions.
Using an orthogonalization allows us to correct this effect. Our findings
indicate that, post-correction, recovering protected features from the
embeddings is not possible anymore and their influence on predictions is
eliminated. Orthogonalization further allows to reduce subgroup disparities.

The phenomenon of bias in CXR classifiers is a highly active research area.
[25] identify various sources of bias during all stages of model development
and [26] account shortcut learning as a possible origin of such biases.
Further, [27] detect an underdiagnosis bias in underrepresented patient
populations. [10] warn about encoded protected features in latent
representations of classifiers. More in-depth analyses about subgroup
disparities are conducted in [13, 12]. We contribute to this discussion by not
only stating the existence of biases but also by proposing additional measures
to effectively remove them.

Previous work [11] already addressed potential biases in the CheXpert
embeddings of CFM by testing for differences in the mean of specific subgroups
in the first four PCA components. As this comes with the downside of
evaluating only one protected feature isolated at a time and does not directly
quantify their influence in a prediction model, we extend this idea and
evaluate to what extent all protected characteristics jointly influence
downstream predictions. Additionally, we discuss orthogonalization as a
potential solution for the embedding bias. The authors of [28] tackle
debiasing CXR classifiers by adapting the model used for prediction. Our
approach, in contrast, is model-agnostic and can correct any neural
representation.

The implicit usage of protected features in predictive models prompts a
discussion about their role as potential biomarkers or indicators in the
context of training datasets. However, if an accurate medical prediction
supported by protected information is of importance, we argue that this
relationship should be modeled by including the protected information as a
predictor. For this purpose, orthogonalization helps to disentangle the
influence of protected features and other factors and makes the influence of
race, sex, age, or other information explicit instead of concealing it within
the neural representations. By explicitly including metadata as separate
predictors, as demonstrated in semi-structured models [15, 14], the influence
of protected features can be accurately quantified and considered in the
decision-making process. This approach enables a more precise and nuanced
understanding of how protected features impact predictions, which can be
particularly crucial for ensuring fairness and avoiding unintended biases in
medical applications.

Our study has several limitations. The downstream classifier is reduced to a
specific model and parameter setting, which yields a fair comparison between
datasets and embeddings. We did not conduct a hyperparameter search that might
enable better performance metrics; instead, our study focused on the influence
of protected features concealed in a deep representation and the removal
thereof. Our analysis is further based on a linear evaluation protocol that
allows the application of the orthogonalization method. An extension of this
method to incorporate non-linearities is non-trivial and is subject to future
work. Lastly, even with the orthogonalized embedding, subgroup disparities
still exist, implying more complex relationships between protected features
and outcomes.

To conclude, our study confirms the results of previous research [11, 10],
which addresses the existence of biases in deep feature embeddings of CXR
images. We propose an orthogonalization as a way to isolate and remove the
effects of protected features, providing a post-hoc measure to cleanse
embeddings. Despite the removal of information in the embedding, downstream
predictive power does not necessarily deteriorate and can even improve
performance in some cases.

## Acknowledgments

The authors gratefully acknowledge LMU Klinikum for providing computing
resources on their Clinical Open Research Engine (CORE). This work has been
partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) as part of BERD@NFDI - grant number 460037581.

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MIMIC
---
Training set
| All | White | Black | Asian | Male | Female
Patients | 42,148 | 31,936 (75.77%) | 8,398 (19.93%) | 1,814 (4.30%) | 20,123 (47.74%) | 22,025 (52.26%)
Scans | 181,342 | 140,445 (77.45%) | 33,906 (18,70%) | 6,991 (3.86%) | 97,361 (53.69%) | 83961 (46.21%)
Age | $62.6\pm 16.6$ | $63.9\pm 16.3$ | $57.7\pm 16.7$ | $62.1\pm 17.8$ | $62.32\pm 15.8$ | $63.0\pm 17.5$
Test set
| All | White | Black | Asian | Male | Female
Patients | 257 | 205 (79.77%) | 45 (17.51%) | 7 (2.72%) | 141 (54.86%) | 116 (45.14%)
Scans | 3,041 | 2,235 (73.50%) | 676 (22.22%) | 130 (4.27%) | 1,658 (54.52%) | 1,383 (45.48%)
Age | $65.8\pm 12.1$ | $66.2\pm 12.3$ | $64.1\pm 11.9$ | $67.4\pm 9.5$ | $66.0\pm 11.6$ | $65.4\pm 12.8$
CheXpert
Training set
| All | White | Black | Asian | Male | Female
Patients | 25,730 | 20,034 (77.86%) | 1,751 (6.81%)) | 3,945 (15.33%) | 14,165 (55.05%) | 11,565 (44.95%)
Scans | 76,205 | 59,238 (77.73%) | 5,596 (7.34%) | 11,371 (14.92%) | 44,774 (58.75%) | 31,431 (41.25%)
Age | $63.1\pm 17.4$ | $64.3\pm 17.2$ | $55.7\pm 17.4$ | $61.6\pm 17.4$ | $62.5\pm 17.0$ | $63.8\pm 17.9$
Test set
| All | White | Black | Asian | Male | Female
Patients | 12,866 | 9,956 (77.38%) | 879 (6.83%) | 2,031 (15.79%) | 7,091 (55.11%) | 5,775 (44.89%)
Scans | 38,240 | 29,844 (78.04%) | 2746 (7.18%) | 5,650 (14.278%) | 22,265 (58.22%) | 15,975 (41.78%)
Age | $63.3\pm 17.2$ | $64.2\pm 17.1$ | $57.4\pm 16.3$ | $61.1\pm 17.6$ | $62.8\pm 16.4$ | $63.9\pm 18.3$

Table 1: Statistics of the utilized MIMIC and CheXpert subsets per split and
subgroups.

| Pathology | Orthogonalized? | Age | Sex[Female] | Race[Black] | Race[Asian]
---|---|---|---|---|---|---
MIMIC CFM | Pleural Effusion | ✗ |  $\pagecolor{cb-burgundy!15}4.246$ ($<$210-16$$) |  $\pagecolor{cb-burgundy!15}-0.143$ ($<$210-16$$) | $-0.694$ ($<$210-16$$) |  $0.029$ ($0.386$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
Cardiomegaly | ✗ | $3.490$ ($<$210-16$$) | $-0.078$ ($<$210-16$$) | $0.145$ ($<$210-16$$) |  $-0.017$ ($0.428$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
No Finding | ✗ | $-3.438$ ($<$210-16$$) | $0.216$ ($<$210-16$$) | $0.307$ ($<$210-16$$) |  $0.157$ ($0.350$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
MIMIC CheSS | Pleural Effusion | ✗ | $3.529$ ($<$210-16$$) |  $-0.017$ ($0.143$) | $-0.277$ ($<$210-16$$) | $0.070$ ($510-4$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
Cardiomegaly | ✗ | $2.658$ ($<$210-16$$) |  $-0.007$ ($0.321$) | $0.088$ ($<$210-16$$) |  $0.013$ ($0.386$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
No Finding | ✗ | $-2.974$ ($<$210-16$$) | $0.250$ ($<$210-16$$) | $0.221$ ($<$210-16$$) |  $-0.024$ ($0.150$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
MIMIC CLF | Pleural Effusion | ✗ | $3.582$ ($<$210-16$$) | $-0.193$ ($<$210-16$$) | $-0.275$ ($<$210-16$$) | $0.069$ ($0.001$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
Cardiomegaly | ✗ | $2.992$ ($<$210-16$$) | $-0.099$ ($<$210-16$$) | $0.136$ ($<$210-16$$) |  $-0.018$ ($0.255$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
No Finding | ✗ |  $\pagecolor{cb-burgundy!15}-2.907$ ($<$210-16$$) | $0.217$ ($<$210-16$$) | $0.207$ ($<$210-16$$) |  $-0.018$ ($0.279$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
CheXpert CheSS | Pleural Effusion | ✗ | $1.679$ ($<$210-16$$) | $0.135$ ($<$210-16$$) | $-0.072$ ($210-5$) | $0.051$ ($110-4$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
Cardiomegaly | ✗ | $1.278$ ($<$210-16$$) |  $0.004$ ($0.511$) | $0.362$ ($<$210-16$$) | $0.112$ ($<$210-16$$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
No Finding | ✗ | $-2.347$ ($<$210-16$$) | $0.092$ ($<$210-16$$) |  $-0.027$ ($0.120$) | $-0.045$ ($0.001$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
CheXpert CLF | Pleural Effusion | ✗ | $1.590$ ($<$210-16$$) |  $0.025$ ($0.060$) | $-0.067$ ($510-4$) | $0.036$ ($0.012$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
Cardiomegaly | ✗ | $1.613$ ($<$210-16$$) | $-0.090$ ($210-16$) | $0.454$ ($<$210-16$$) | $-0.081$ ($210-9$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)
No Finding | ✗ | $-2.356$ ($<$210-16$$) | $0.039$ ($0.002$) |  $-0.037$ ($0.057$) | $-0.071$ ($<$810-7$$)
✓ |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$) |  $0$ ($1$)

Table 2: The average coefficients (with p-values in brackets) for the impact
of protected features on predictions for Pleural Effusion, Cardiomegaly, and
No Finding. For predicting a scan’s race, the prevalent label White is chosen
as the reference level. Cells containing p-values $<$ 0.05 are colored red
marking significant influence.

Dataset: | MIMIC | CheXpert
---|---|---
Embedding: | CFM | CheSS | CLF | CheSS | CLF
Orthogonalized? | ✗ | ✓ | $\Delta$ | ✗ | ✓ | $\Delta$ | ✗ | ✓ | $\Delta$ | ✗ | ✓ | $\Delta$ | ✗ | ✓ | $\Delta$
Enl. Cardiomed. | 0.728 $\pm$ 0.009 | 0.721 $\pm$ 0.018 | -0.97 % | 0.636 $\pm$ 0.003 | 0.643 $\pm$ 0.007 | +1.09 % | 0.601 $\pm$ 0.002 | 0.593 $\pm$ 0.004 | -1.35 % | 0.621 $\pm$ 0.001 | 0.636 $\pm$ 0.003 | +2.36 % | 0.634 $\pm$ 0.000 | 0.639 $\pm$ 0.001 | +0.78 %
Cardiomegaly | 0.780 $\pm$ 0.002 | 0.775 $\pm$ 0.003 | -0.65 % | 0.750 $\pm$ 0.001 | 0.751 $\pm$ 0.001 | +0.13 % | 0.737 $\pm$ 0.000 | 0.736 $\pm$ 0.001 | -0.14 % | 0.789 $\pm$ 0.000 | 0.791 $\pm$ 0.001 | +0.25 % | 0.799 $\pm$ 0.000 | 0.793 $\pm$ 0.000 | -0.76 %
Lung Opacity | 0.696 $\pm$ 0.003 | 0.684 $\pm$ 0.005 | -1.75 % | 0.626 $\pm$ 0.002 | 0.627 $\pm$ 0.004 | +0.16 % | 0.623 $\pm$ 0.001 | 0.612 $\pm$ 0.002 | -1.80 % | 0.685 $\pm$ 0.000 | 0.684 $\pm$ 0.000 | -0.15 % | 0.695 $\pm$ 0.000 | 0.690 $\pm$ 0.000 | -0.72 %
Lung Lesion | 0.731 $\pm$ 0.006 | 0.718 $\pm$ 0.007 | -1.81 % | 0.623 $\pm$ 0.003 | 0.630 $\pm$ 0.004 | +1.11 % | 0.576 $\pm$ 0.009 | 0.591 $\pm$ 0.014 | +2.54 % | 0.672 $\pm$ 0.002 | 0.700 $\pm$ 0.002 | +4.00 % | 0.701 $\pm$ 0.001 | 0.707 $\pm$ 0.002 | +0.85 %
Edema | 0.843 $\pm$ 0.001 | 0.837 $\pm$ 0.002 | -0.72 % | 0.804 $\pm$ 0.000 | 0.798 $\pm$ 0.000 | -0.75 % | 0.791 $\pm$ 0.000 | 0.783 $\pm$ 0.001 | -1.02 % | 0.791 $\pm$ 0.000 | 0.789 $\pm$ 0.000 | -0.25 % | 0.788 $\pm$ 0.000 | 0.783 $\pm$ 0.000 | -0.64 %
Consolidation | 0.748 $\pm$ 0.008 | 0.742 $\pm$ 0.009 | -0.81 % | 0.648 $\pm$ 0.001 | 0.650 $\pm$ 0.005 | +0.31 % | 0.638 $\pm$ 0.003 | 0.640 $\pm$ 0.002 | +0.31 % | 0.669 $\pm$ 0.001 | 0.683 $\pm$ 0.001 | +2.05 % | 0.689 $\pm$ 0.001 | 0.692 $\pm$ 0.002 | +0.43 %
Pneumonia | 0.703 $\pm$ 0.005 | 0.704 $\pm$ 0.004 | +0.14 % | 0.586 $\pm$ 0.005 | 0.597 $\pm$ 0.009 | +1.84 % | 0.589 $\pm$ 0.003 | 0.607 $\pm$ 0.002 | +2.97 % | 0.610 $\pm$ 0.002 | 0.652 $\pm$ 0.003 | +6.44 % | 0.652 $\pm$ 0.001 | 0.659 $\pm$ 0.004 | +1.06 %
Atelectasis | 0.746 $\pm$ 0.002 | 0.734 $\pm$ 0.004 | -1.63 % | 0.702 $\pm$ 0.000 | 0.696 $\pm$ 0.001 | -0.86 % | 0.685 $\pm$ 0.001 | 0.671 $\pm$ 0.001 | -2.09 % | 0.631 $\pm$ 0.000 | 0.636 $\pm$ 0.001 | +0.79 % | 0.632 $\pm$ 0.000 | 0.633 $\pm$ 0.001 | +0.16 %
Pneumothorax | 0.843 $\pm$ 0.005 | 0.830 $\pm$ 0.007 | -1.57 % | 0.649 $\pm$ 0.002 | 0.645 $\pm$ 0.005 | -0.62 % | 0.634 $\pm$ 0.003 | 0.638 $\pm$ 0.004 | +0.63 % | 0.732 $\pm$ 0.001 | 0.749 $\pm$ 0.001 | +2.27 % | 0.730 $\pm$ 0.001 | 0.740 $\pm$ 0.001 | +1.35 %
Pleural Effusion | 0.870 $\pm$ 0.001 | 0.859 $\pm$ 0.002 | -1.28 % | 0.802 $\pm$ 0.000 | 0.792 $\pm$ 0.001 | -1.26 % | 0.797 $\pm$ 0.000 | 0.781 $\pm$ 0.000 | -2.05 % | 0.792 $\pm$ 0.000 | 0.798 $\pm$ 0.000 | +0.75 % | 0.804 $\pm$ 0.000 | 0.801 $\pm$ 0.000 | -0.37 %
Pleural Other | 0.894 $\pm$ 0.009 | 0.874 $\pm$ 0.021 | -2.29 % | 0.711 $\pm$ 0.005 | 0.746 $\pm$ 0.011 | +4.69 % | 0.684 $\pm$ 0.006 | 0.693 $\pm$ 0.009 | +1.30 % | 0.723 $\pm$ 0.001 | 0.756 $\pm$ 0.004 | +4.37 % | 0.718 $\pm$ 0.001 | 0.718 $\pm$ 0.004 | +0.00 %
Fracture | 0.752 $\pm$ 0.007 | 0.739 $\pm$ 0.013 | -1.76 % | 0.643 $\pm$ 0.005 | 0.648 $\pm$ 0.009 | +0.77 % | 0.642 $\pm$ 0.004 | 0.652 $\pm$ 0.012 | +1.53 % | 0.668 $\pm$ 0.001 | 0.682 $\pm$ 0.003 | +2.05 % | 0.667 $\pm$ 0.001 | 0.673 $\pm$ 0.003 | +0.89 %
Support Devices | 0.909 $\pm$ 0.001 | 0.905 $\pm$ 0.001 | -0.44 % | 0.801 $\pm$ 0.000 | 0.801 $\pm$ 0.001 | +0.00 % | 0.767 $\pm$ 0.001 | 0.763 $\pm$ 0.001 | -0.52 % | 0.731 $\pm$ 0.000 | 0.748 $\pm$ 0.000 | +2.27 % | 0.711 $\pm$ 0.000 | 0.721 $\pm$ 0.000 | +1.39 %
No Finding | 0.801 $\pm$ 0.003 | 0.770 $\pm$ 0.005 | -4.03 % | 0.746 $\pm$ 0.001 | 0.725 $\pm$ 0.002 | -2.90 % | 0.747 $\pm$ 0.000 | 0.728 $\pm$ 0.001 | -2.61 % | 0.833 $\pm$ 0.000 | 0.824 $\pm$ 0.001 | -1.09 % | 0.854 $\pm$ 0.000 | 0.844 $\pm$ 0.000 | -1.18 %
Total | 0.789 $\pm$ 0.005 | 0.778 $\pm$ 0.009 | -1.39 % | 0.695 $\pm$ 0.003 | 0.696 $\pm$ 0.005 | -0.14 % | 0.679 $\pm$ 0.003 | 0.678 $\pm$ 0.006 | -0.14 % | 0.710 $\pm$ 0.001 | 0.723 $\pm$ 0.002 | +1.83 % | 0.720 $\pm$ 0.001 | 0.721 $\pm$ 0.002 | +0.13 %

Table 3: Prediction performance original versus orthogonalized data on the
MIMIC and CheXpert datasets. The table shows the mean and standard deviation
of the AUC over 10 randomly initialized runs. Additionally, $\Delta$ depicts
the percentual change from the original to the corrected embedding.

|  |  | Age | Sex | Race [White] | Race [Black] | Race [Asian]
---|---|---|---|---|---|---|---
| Emb. | Orthogonalized? | MAE | $\bm{R}^{\bm{2}}$ | AUC | Sens. | Spec. | AUC | Sens. | Spec. | AUC | Sens. | Spec. | AUC | Sens. | Spec.
MIMIC | CFM | ✗ | 8.040 $\pm$1.456 | 0.314 $\pm$0.241 | 0.979 $\pm$0.000 | 0.933 $\pm$0.025 | 0.902 $\pm$0.039 | 0.848 $\pm$0.004 | 0.940 $\pm$0.016 | 0.497 $\pm$0.049 | 0.871 $\pm$0.004 | 0.521 $\pm$0.055 | 0.944 $\pm$0.015 | 0.870 $\pm$0.005 | 0.179 $\pm$0.028 | 0.991 $\pm$0.005
✓ | 10.969 $\pm$0.433 | -0.294 $\pm$0.087 | 0.507 $\pm$0.031 | 0.775 $\pm$0.077 | 0.224 $\pm$0.100 | 0.501 $\pm$0.045 | 1.000 $\pm$0.000 | 0.000 $\pm$0.000 | 0.509 $\pm$0.042 | 0.000 $\pm$0.000 | 1.000 $\pm$0.000 | 0.464 $\pm$0.076 | 0.000 $\pm$0.000 | 1.000 $\pm$0.000
CheSS | ✗ | 7.893 $\pm$ 0.066 | 0.331 $\pm$ 0.010 | 0.945 $\pm$ 0.001 | 0.907 $\pm$ 0.011 | 0.820 $\pm$ 0.016 | 0.761 $\pm$ 0.004 | 0.975 $\pm$ 0.008 | 0.158 $\pm$ 0.039 | 0.768 $\pm$ 0.003 | 0.143 $\pm$ 0.039 | 0.968 $\pm$ 0.010 | 0.733 $\pm$ 0.004 | 0.013 $\pm$ 0.008 | 0.997 $\pm$ 0.002
✓ | 9.908 $\pm$ 0.082 | -0.083 $\pm$ 0.016 | 0.482 $\pm$ 0.037 | 0.982 $\pm$ 0.024 | 0.013 $\pm$ 0.025 | 0.489 $\pm$ 0.060 | 1.000 $\pm$ 0.000 | 0.000 $\pm$ 0.000 | 0.478 $\pm$ 0.068 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000 | 0.514 $\pm$ 0.048 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000
CLF | ✗ | 8.816 $\pm$ 0.063 | 0.161 $\pm$ 0.011 | 0.832 $\pm$ 0.000 | 0.804 $\pm$ 0.015 | 0.702 $\pm$ 0.016 | 0.631 $\pm$ 0.005 | 0.988 $\pm$ 0.004 | 0.048 $\pm$ 0.012 | 0.652 $\pm$ 0.005 | 0.050 $\pm$ 0.012 | 0.987 $\pm$ 0.005 | 0.663 $\pm$ 0.007 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000
✓ | 9.941 $\pm$ 0.095 | -0.093 $\pm$ 0.025 | 0.456 $\pm$ 0.039 | 0.994 $\pm$ 0.007 | 0.007 $\pm$ 0.006 | 0.500 $\pm$ 0.048 | 1.000 $\pm$ 0.000 | 0.000 $\pm$ 0.000 | 0.498 $\pm$ 0.056 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000 | 0.493 $\pm$ 0.084 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000
CheXpert | CheSS | ✗ | 9.333 $\pm$ 0.103 | 0.529 $\pm$ 0.009 | 0.946 $\pm$ 0.000 | 0.912 $\pm$ 0.008 | 0.821 $\pm$ 0.018 | 0.780 $\pm$ 0.001 | 0.984 $\pm$ 0.007 | 0.136 $\pm$ 0.038 | 0.762 $\pm$ 0.001 | 0.018 $\pm$ 0.006 | 0.999 $\pm$ 0.001 | 0.816 $\pm$ 0.001 | 0.173 $\pm$ 0.047 | 0.983 $\pm$ 0.007
✓ | 13.875 $\pm$ 0.035 | -0.009 $\pm$ 0.005 | 0.499 $\pm$ 0.010 | 1.000 $\pm$ 0.000 | 0.000 $\pm$ 0.000 | 0.501 $\pm$ 0.007 | 1.000 $\pm$ 0.000 | 0.000 $\pm$ 0.000 | 0.498 $\pm$ 0.015 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000 | 0.500 $\pm$ 0.007 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000
CLF | ✗ | 10.693 $\pm$ 0.047 | 0.385 $\pm$ 0.005 | 0.868 $\pm$ 0.000 | 0.854 $\pm$ 0.008 | 0.696 $\pm$ 0.014 | 0.721 $\pm$ 0.001 | 0.993 $\pm$ 0.003 | 0.040 $\pm$ 0.012 | 0.688 $\pm$ 0.002 | 0.003 $\pm$ 0.002 | 1.000 $\pm$ 0.000 | 0.764 $\pm$ 0.000 | 0.049 $\pm$ 0.016 | 0.993 $\pm$ 0.003
✓ | 13.866 $\pm$ 0.035 | -0.009 $\pm$ 0.005 | 0.496 $\pm$ 0.010 | 1.000 $\pm$ 0.000 | 0.000 $\pm$ 0.000 | 0.501 $\pm$ 0.008 | 1.000 $\pm$ 0.000 | 0.000 $\pm$ 0.000 | 0.497 $\pm$ 0.021 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000 | 0.503 $\pm$ 0.009 | 0.000 $\pm$ 0.000 | 1.000 $\pm$ 0.000

Table 4: Regression/Classification performance for deriving protected features
from an embedding vector with mean and standard deviation over 10 randomly
initialized runs. The displayed metrics include mean absolute error (MAE),
$R^{2}$ for age regression as well as AUC, sensitivity (sens.), and
specificity (spec.) for classification.

Figure 1:  Geometric visualization of the orthogonalization method. The column
space of the protected features ($\text{col}(\bm{X})$) contains all the
possible vectors that can be formed by taking linear combinations of the
respective features, i.e. the hypothesis space of a linear model. For an
embedding vector $\bm{e}\in\bm{E}$, the orthogonalization is equivalent to the
residual between $\bm{e}$ and its projection onto $\text{col}(\bm{X})$. With
$\mathcal{P}_{X}^{\bot}\bm{e}$ being perpendicular to $\text{col}(\bm{X})$,
the influence of protected features in $\bm{e}$ is neutralized.

| PCA components by sex |  PCA components by Pleural Effusion
---|---|---
| Original | Orthogonalized | Original | Orthogonalized
MIMIC CFM |  |  |  |
MIMIC CheSS |  |  |  |
MIMIC CLF |  |  |  |
CheXpert CheSS |  |  |  |
CheXpert CLF |  |  |  |
Figure 2: Marginalized distributions per sex (left two columns) and label
Pleural Effusion (right two columns) over the PCA reduction of the original
versus the orthogonalized corrected embedding.

(a) Classification of sex.

(b) Classification of race label White.

(c) Classification of race label Black.

(d) Classification of race label Asian.

(e) Regression of age.

Figure 3: Regression/Classification performance for deriving protected
features from an embedding vector with mean and standard deviation over 10
randomly initialized runs. The displayed metrics include mean absolute error
(MAE) in years for age regression as well as AUC for classification of sex and
race.

(a) Change in AUC for Pleural Effusion.

(b) Change in AUC for Cardiomegaly.

Figure 4: Difference from overall AUC per subgroup based on downstream
classifiers for original and orthogonalized embeddings. The closer a bar is to
zero, the less disparity from the mean AUC exists.

SUPPLEMENTAL MATERIAL

##  Appendix A Influence of Protected Features on Model Prediction

###  A.1 Pathology: Pleural Effusion

(a) Coefficients for Pleural Effusion.

(b) p-values associated with the respective coefficients.

Figure 5: Distribution of derived coefficients and p-values for 10 downstream
models per embedding and protected feature category on the label Pleural
Effusion.

###  A.2 Pathology: Cardiomegaly

(a) Coefficients for Cardiomegaly.

(b) p-values associated with the respective coefficients.

Figure 6: Distribution of derived coefficients and p-values for 10 downstream
models per embedding and protected feature category on the label Cardiomegaly.

###  A.3 Pathology: No Finding

(a) Coefficients for No Finding.

(b) p-values associated with the respective coefficients.

Figure 7: Distribution of derived coefficients and p-values for 10 downstream
models per embedding and protected feature category on the label No Finding.