# Detecting algorithmic bias in medical-AI models using conformal trees
**Authors**: Jeffrey Smith, Andre Holder, Rishikesan Kamaleswaran, Yao Xie
> School of Industrial and Systems Engineering,
Georgia Institute of Technology, Atlanta, GA, 30332, USAjsmith312@gatech.edu
> Department of Biomedical Informatics, Emory University School of Medicine, Atlanta, GA, 30303, USA
> Department of Surgery, Duke University School of Medicine, Durham, NC 27708, USA
> School of Industrial and Systems Engineering,
Georgia Institute of Technology, Atlanta, GA, 30332, USA
Abstract
With the growing prevalence of machine learning and artificial intelligence-based medical decision support systems, it is equally important to ensure that these systems provide patient outcomes in a fair and equitable fashion. This paper presents an innovative framework for detecting areas of algorithmic bias in medical-AI decision support systems. Our approach efficiently identifies potential biases in medical-AI models, specifically in the context of sepsis prediction, by employing the Classification and Regression Trees (CART) algorithm with conformity scores. We verify our methodology by conducting a series of synthetic data experiments, showcasing its ability to estimate areas of bias in controlled settings precisely. The effectiveness of the concept is further validated by experiments using electronic medical records from Grady Memorial Hospital in Atlanta, Georgia. These tests demonstrate the practical implementation of our strategy in a clinical environment, where it can function as a vital instrument for guaranteeing fairness and equity in AI-based medical decisions.
keywords: Algorithmic Bias, Subgroup Fairness,Trustworthy-AI
1 Introduction
Machine learning (ML) and artificial intelligence (AI) technologies are becoming increasingly prevalent in critical decision-making processes in industries such as finance [1, 2], education [3, 4, 5], and criminal justice [6, 7, 8]. As a result, the deployment of these technologies in such consequential domains has given rise to significant ethical considerations, particularly in terms of the influence of societal biases on model fairness. In medical applications, this bias has the potential to disproportionately affect particular patient subgroups and further amplify pre-existing disparities. The well documented exacerbation of existing disparities in healthcare data [9, 10, 11, 12, 13], underscores the urgency of identifying these biases to ensure fair and equitable ML applications in this domain, especially for diverse and often underrepresented patient sub-populations.
Broadly, fairness can be grouped into three categories: individual [14], group [14], and causality-based [15]. Group fairness, as opposed to causality-based fairness and individual fairness, which both necessitate domain expertise to establish a just causal framework and aim for equality solely among comparable individuals, operates without presumption of knowledge and pursues equality across groups often framed in terms of one-dimensional protected attributes such as race, gender, or socio-economic status.
While there has been much interest in group fairness measures [16], researchers have noted their limitations. According to research by Castelnovo et al. [17], simply excluding protected features from the decision-making process does not inherently guarantee demographic parity, which is achieved when both protected and unprotected groups have equal probability of being assigned to the positive predicted class. Achieving demographic parity may involve using different treatment strategies for different groups in order to mitigate the impact of correlations between variables, a strategy that may be considered inequitable or counter-intuitive. Dwork et al. [14] further expound on a “catalogue of evils” that highlight numerous ways the satisfaction of existing fairness definitions could prove ineffective in offering substantial fairness assurances.
Although a number of group fairness metrics have been developed recently [18, 19, 20, 14, 15, 16], Dwork and Ilvento [21] raise a notable issue that predictors may be adjusted in a way that they meet independent group fairness criteria, but their predictions contradict fairness at an interconnected subgroup level. This more nuanced case of group fairness spanning multiple subgroups is termed intersectional group fairness [22]. Within this context, intersectionality posits that the interaction between multiple dimensions of identity may result in distinct and varying degrees of prejudice directed towards different potential subgroups [23]. More abstractly, this problem may be connected to the concept of identifying “fairness gerrymandering,” [24] where a classifier’s results are deemed “fair” for each specific group (such as race, gender, insurance status, etc.), but significantly violate fairness when it comes to structured subgroups, such as specific combinations of protected features.
In the healthcare domain, medical-AI decision support systems frequently function as black-box models, oftentimes providing limited insight into the structure of their training data, if any, as well as no visibility into the parameters used in model development. Developing effective and fair prediction models in this context poses unique difficulties, such as the potential absence of patient demographic representation in the training data and, in some instances, the complete absence of demographic information. The distinct challenges of healthcare data coupled with the intersectional group fairness contradictions could result in both inaccurate diagnoses and suboptimal interventions for certain structured subgroups.
In this paper, we address the challenge of detecting “algorithmic bias” in medical-AI models. These models utilize discrete time intervals for data organization (i.e., the 1-hour epoch structure we use that is normalized to ICU admission). They also include outcome prediction, with a defined prediction horizon. In particular, we present a novel framework utilizing a well-studied statistical approach, namely Classification and Regression Trees (CART) decision trees to detect regions of bias generated by a medical-AI model via uncertainty quantification. Moreover, this framework allows researchers and clinicians to evaluate the reliability of a prediction model, for a patient considering their individual characteristics. This methodology can be used on the output of any arbitrary prediction model to evaluate the effectiveness of the model in making accurate predictions for a specific patient and to assess whether the model should be applied to that type of patient. Our goal can be summarized as follows:
Using data, we aim to detect “algorithmic bias”, via uncertainty quantification, generated by inferior algorithmic performance and directly identify structured subgroups, defined by various combinations of attributes, impacted by this bias.
The contributions of the work include:
- We present a model-agnostic framework to systematically and rigorously detect biased regions through the retrospective analysis of results generated by medical-AI prediction algorithms. This method addresses gaps in current fairness evaluation methods that requires one to preselect groups in which bias is tested and paves the way for safer and more trustworthy medical-AI applications.
- Empirically, we evaluate the effectiveness of our technique in recognizing biased regions by conducting case studies using both synthetic and real data. Our findings demonstrate our ability to identify biased regions and gain insights into the characteristics that define these regions.
2 Related Works
Group Fairness
Several studies have addressed the challenges of group fairness by developing predictors that ensure fairness across numerous subgroups via “fairness auditing.” Kearns et al. [24] propose a zero-sum game played between an “Auditor” and “Learner” to evaluate a predictor’s fairness by minimizing error while adhering to specified fairness constraints. Separately, Herbert-Johnson et al. [25] introduce a post-processing iterative boosting algorithm which combines all subgroups $c∈\mathcal{C}$ , where $\mathcal{C}$ represents a class of subgroups, until the model is $\alpha$ -calibration. Pastor, Alfaro, and Baralis [26] examine subgroup bias by exploring the feature space through data mining techniques.
Tree-based Failure Mode Analysis
Although decision trees may not be regarded as the most sophisticated method for failure mode analysis, they have the significant advantage of yielding results that are easily interpretable by humans. Consequently, decision trees have become increasingly prominent as a method for failure mode analysis. Chen et al. [27] train decision trees to diagnose failures in large-scale data systems by classifying system requests as successful or failed. Singla et al. [28] apply decision trees to identify and explain failure modes of deep neural networks, focusing on robustly extracted features. They evaluate performance using metrics such as Average Leaf Error Rate (ALER) and Base Error Rate (BER) to identify high-error clusters of labeled images. Nushi, Kamar, and Horvitz [29] employ decision trees as part of their hybrid human-machine failure analysis approach, Pandora, which similarly identifies failure clusters in high-error conditions.
In contrast to these works, our approach detects “algorithmic bias” within structured subgroups beyond binary classification contexts. It avoids computationally intensive exhaustive searches of all possible attribute combinations, integrates statistical rigor in the determination of bias, and does not explicitly rely on common fairness metrics which require the pre-selection of protected features.
3 Preliminaries
3.1 Classification and Regression Trees (CART)
Decision trees are a versatile and intuitive machine learning (ML) algorithm used for both classification and regression tasks, embodying a tree-link model of decisions and their possible consequences. The CART model [30], is a non-parametric ML decision tree methodology that is well suited for the prediction of dependent variables through the utilization of both categorical and continuous predictors. CART models offer a versatile approach to defining the conditional distribution of a response variable $y$ based on a set of predictor values $x$ [31].
In the classification setting, we are given the training data ( $\mathbf{X,Y}$ ), containing $n$ observations ( $\mathbf{x}_{i},y_{i}$ ), $i=1,...,n$ , each with $p$ features $\mathbf{x}_{i}∈\mathbb{R}^{p}$ and a class label $y_{i}∈\{1,...,K\}$ indicating which of $K$ possible labels is assigned to this given point. In the regression setting our output variable is a continuous response variable $y_{i}∈\mathbb{R}$ . Decision tree methods seek to recursively partition the dataset (feature space) into a number of hierarchically disjoint subsets with the aim of achieving progressively more homogeneous distributions of the response variable $y$ within each subset. An example of a decision tree is shown in Fig. 1.
<details>
<summary>x1.png Details</summary>
### Visual Description
# Technical Document Extraction: Decision Tree and Feature Space Partitioning
This image illustrates the relationship between a **Decision Tree** (right) and the corresponding **Recursive Partitioning of Feature Space** (left). It demonstrates how a hierarchical set of binary rules divides a 2D input space into distinct regions.
---
## 1. Component Isolation
The image is divided into two primary technical diagrams:
* **Left Region (Feature Space):** A rectangular plot representing a two-dimensional coordinate system defined by variables $x_1$ and $x_2$.
* **Right Region (Decision Tree):** A binary tree structure showing the logical flow of decisions leading to terminal nodes (leaves).
---
## 2. Right Region: Decision Tree Analysis
The diagram depicts a binary tree with three levels of nodes and specific decision criteria on the edges.
### Node Hierarchy
* **Root Node:** $t_0$
* **Internal Nodes (Level 1):** $t_1$ (left branch), $t_2$ (right branch)
* **Terminal Nodes / Leaves (Level 2):** $t_3, t_4, t_5, t_6$
### Decision Logic (Edges)
The tree splits based on thresholds $b_1, b_2,$ and $b_3$:
1. **First Split (at $t_0$):**
* Left Branch: $x_2 \leq b_1$ (leads to node $t_1$)
* Right Branch: $x_2 > b_1$ (leads to node $t_2$)
2. **Second Split (at $t_1$):**
* Left Branch: $x_1 \leq b_2$ (leads to leaf $t_3$)
* Right Branch: $x_1 > b_2$ (leads to leaf $t_4$)
3. **Third Split (at $t_2$):**
* Left Branch: $x_1 \leq b_3$ (leads to leaf $t_5$)
* Right Branch: $x_1 > b_3$ (leads to leaf $t_6$)
---
## 3. Left Region: Feature Space Partitioning
The rectangular space is defined by horizontal axis $x_1$ and vertical axis $x_2$. The space is partitioned into four distinct rectangular regions corresponding to the leaf nodes of the tree.
### Spatial Grounding and Mapping
By cross-referencing the tree logic with the visual partitions:
* **Horizontal Partition ($b_1$):** A horizontal line divides the space into a lower section ($x_2 \leq b_1$) and an upper section ($x_2 > b_1$).
* **Lower Section ($x_2 \leq b_1$):**
* Divided by a vertical line at $x_1 = b_2$.
* **Region $t_3$:** Located at the bottom-left ($x_1 \leq b_2$ AND $x_2 \leq b_1$).
* **Region $t_4$:** Located at the bottom-right ($x_1 > b_2$ AND $x_2 \leq b_1$).
* **Upper Section ($x_2 > b_1$):**
* Divided by a vertical line at $x_1 = b_3$.
* **Region $t_5$:** Located at the top-left ($x_1 \leq b_3$ AND $x_2 > b_1$).
* **Region $t_6$:** Located at the top-right ($x_1 > b_3$ AND $x_2 > b_1$).
---
## 4. Summary of Extracted Data Labels
| Label | Type | Description |
| :--- | :--- | :--- |
| $x_1$ | Axis/Variable | Horizontal feature dimension. |
| $x_2$ | Axis/Variable | Vertical feature dimension. |
| $t_0$ | Node | Root of the decision tree. |
| $t_1, t_2$ | Node | Intermediate decision points. |
| $t_3, t_4, t_5, t_6$ | Leaf/Region | Final classification or regression outcomes/regions. |
| $b_1, b_2, b_3$ | Constant | Threshold values for splitting the data. |
| $\leq, >$ | Operator | Comparison operators defining the decision boundaries. |
</details>
Figure 1: Example of an optimal axis-aligned decision tree with a depth of $K=2$ with $p=2$ dimensions. Splits occur along specific features in the form $x_{j}=b$ for $j=1,2$ .
Beginning from the root node, an optimal feature and split point are identified based on an appropriate optimization metric. The feature, split-point pair defines the partition splitting the feature space, and this procedure is repeated for every sub-feature space that is created. These partitions will ultimately result in the binary tree structure consisting of interconnected root, branch, and leaf nodes.
- Root nodes encapsulate the entire dataset, forming the foundational layer of the decision tree.
- Branch nodes are points in the dataset characterized by features and split points that serve as points of division for partitioning the feature space. Each of these branches extend to subsequent child nodes.
- Leaf nodes are the final nodes in the tree, classifying or predicting data points based on their localized patterns.
CART models take a top-down approach and can be used for both classification and regression problems, as the name implies. Partitions are determined by using a specified loss function to evaluate the quality of a potential split and are based on both the features and values, that provide optimal splits. The splitting criteria determine the optimal splits. In the classification setting, the criteria are often determined by the label impurity of data points within a partition. The splitting criteria for regression-based CART models focuses on minimizing the variance of data points in partitioned regions. CART models, as applied to both tasks, have two main stages: the decision tree’s generation and subsequent pruning. We now transition to a more granular discussion on CART’s implementation for both classification and regression problems.
Classification Trees
The CART method, in the context of classification tasks, is a powerful tool for categorizing outcomes into distinct classes based on input features. The objective is to partition the feature space into regions that maximize the the uniformity of the response variable’s classes within in each subsequent node during the partitioning process. This process begins at the root node and splits the feature space recursively based on a set of decision rules that maximally separate the classes.
When we consider splitting a classification tree, $T$ , at any node $t$ , we evaluate potential splits based on how well they separate the different classes of the response variable. For a given variable $X$ , a split point $s$ is chosen to divide node $t$ into left ( $t_{L}$ ) and right ( $t_{R}$ ) child nodes. This division is based on whether the values of $X$ are less than or equal to $s$ or greater than $s$ , formally defined as $t_{L}=\{{\textbf{X}∈ t:X≤ s}\}$ and $t_{R}=\{{\textbf{X}∈ t:X>s}\}$ . The effectiveness of a split is measured using the impurity metric of Information Gain, which gauges the value of the insight a feature offers about a response variable. In practical applications, this measure is determined using Entropy or the Gini index.
- Entropy functions as a metric of disorder or unpredictability. It measures the impurity or randomness of a node, especially in binary classification problems. Mathematically, it is expressed as:
$$
E=-\sum_{i=1}^{K}p_{i}\log_{2}p_{i},
$$
where $p_{i}$ is the probability of an instance belonging to the $i^{th}$ class.
- Gini index serves as an alternate measure of node impurity. Considered a computationally efficient alternative to entropy, it is formulated as follows:
$$
E=\sum_{i=1}^{K}p_{i}(1-p_{i}),
$$
where, yet again, $p_{i}$ is the probability of an instance belonging to the $i^{th}$ class.
- Information Gain is a metric calculated by observing the impurity of a node before and after a split and is formulated as:
$$
\text{IG}=E_{{\rm parent}}-\sum_{i=1}^{K}w_{i}E_{{\rm child}_{i}},
$$
where $w_{i}$ is the relative weight of the child node with respect to the parent node.
The algorithm uses these splitting criteria to divide the feature space into sub-regions recursively, terminating when any of the specified stopping criteria are satisfied. After the dividing procedure finishes, each region gets assigned a class label $1,...,K$ . This assigned class label will predict the classification of any points inside the region. Typically, the assigned class will be the most common class among the points in the region.
Regression Trees
Regression trees exhibit notable performance in the prediction of continuous output variables. The key aspect of their approach involves partitioning the feature space in such a way that the variation of the target variable is minimized within each segment of the space, referred to as nodes. To elaborate, when a regression tree, denoted as $T$ , undergoes a split at a node $t$ , we consider a potential division point, or split point $s$ , for a variable $X$ . This split point categorizes the data into left ( $t_{L}$ ) and right ( $t_{R}$ ) child nodes based on the condition whether $X≤ s$ or $X>s$ . These nodes are formally represented as $t_{L}=\{{\textbf{X}∈ t:X≤ s}\}$ and $t_{R}=\{{\textbf{X}∈ t:X>s}\}$ . The criterion for assessing the quality of a split in regression trees revolves around the variance within a node, given by
$$
\widehat{\Delta}(t)=\widehat{\rm VAR}(y|\textbf{X}\in t)=\frac{1}{n(t)}\sum_{%
\textbf{x}_{i}\in t}\left(y_{i}-\bar{y}_{t}\right)^{2},
$$
where $\bar{y}_{t}$ is the mean value of the target variable for the data points within node $t$ and $n(t)$ represents the count of these data points. The variance within the child nodes, left ( $t_{L}$ ) and right ( $t_{R}$ ), is similarly calculated. The decision to split a parent node $t$ into child nodes is based on the split that yields the highest decrease in variance, defined as
$$
\widehat{\Delta}(s,t)=\widehat{\Delta}(t)-(\widehat{W}(t_{L})\widehat{\Delta}(%
t_{L})+(\widehat{W}(t_{R})\widehat{\Delta}(t_{R})),
$$
where $\widehat{W}(t_{L})=n(t_{L})/n(t)$ and $\widehat{W}(t_{R})=n(t_{R})/n(t)$ denote the proportions of data points in $t$ allocated to $t_{L}$ and $t_{R}$ , respectively.
The process of developing the tree $T$ is iterative, identifying the variable and split point that maximizes variance reduction. Similar to its classification counterpart, the recursive partitioning of the feature space aims at reducing variance with the ultimate goal of accurately estimating the conditional mean response $\mu(x)$ , in the tree’s terminal nodes. The predicted response for data points in node $t$ is the mean target variable value, $\bar{y}_{t}$ , for those points.
Without limitations, the tree generation process of the CART algorithm will continue until each data point is represented by a single leaf node. This is often not recommended as fully growing a tree to maturity introduces the risk of overfitting. To counter this, the tree development process includes constraints such as minimal sample split, maximum tree depth, and cost-complexity pruning to fine-tune the tree’s structure and fit.
3.2 Conformal Prediction
Conformal prediction is a statistical framework where the aim is to quantify uncertainty in the predictions made by some arbitrary prediction algorithm by converting point-predictions into set-valued functions with coverage guarantees. Consider a training set $\{(X_{i},Y_{i})\}^{n}_{i=1}$ and a test point $\{X_{n+1},Y_{n+1}\}$ sampled i.i.d. from some unknown distribution $P$ . Using $\{(X_{i},Y_{i})\}^{n}_{i=1}\cup\{X_{n+1}\}$ as input, conformal prediction produces a set-valued function, denoted by $\hat{C}(\dot{)}$ , that satisfies the guarantee $\mathbb{P}(Y_{n+1}∈\hat{C}(X_{n+1}))≥ 1-\alpha$ , where $\alpha∈(0,1)$ is a nominal error level.
4 Conformal tree based method for algorithm bias detection
Given a pre-trained prediction algorithm $\mathcal{A}$ , our objective is two-fold. First, can we detect the presence of bias in the predictions made by the algorithm? Second, if bias is detected, can we precisely identify the region $\mathcal{S}$ within the $p$ -dimensional feature space where the algorithm exhibits suboptimal performance, a region we term the “algorithmic bias” region. In this context, $p$ denotes the number of features which can be categorical and/or continuous valued.
We assume that the true region $\mathcal{S}$ is defined by a subset of key variables (features) $j∈ S$ . For real-valued features, this is represented as $X_{j}∈[L_{j},U_{j}]$ , where $L_{j}$ and $U_{j}$ represent some lower and upper bounds, respectively. For categorical value features, $X_{j}∈ C_{j}$ , $j∈ S$ . For example, if $p=10$ and $S=\{1,3\}$ , the algorithmic bias region might be defined by age $X_{1}∈[35,50]$ and gender $X_{3}=\{{\rm Female}\}$ .
This formulation implies that the subset of variables in the set $S$ will be the most critical in causing the bias, defining the algorithmic bias region $\mathcal{S}$ . For instance, in our example, age and gender are the two most important features in defining the algorithmic bias region $\mathcal{S}$ . Fig. 2 depicts the concept, where green dots signify superior performance, blue dots indicate worse performance, and the algorithmic bias region is delineated by a dashed-line box inside the feature space for $X∈\mathbb{R}^{p}$ .
<details>
<summary>x2.png Details</summary>
### Visual Description
# Technical Document Extraction: Prediction Algorithm Performance
## 1. Document Metadata
* **Title:** Prediction Algorithm Performance
* **Image Type:** 2D Scatter Plot with Classification Boundary
* **Primary Language:** English
* **Coordinate System:** Cartesian ($x_0, x_1$)
## 2. Component Isolation
### A. Header
* **Text:** "Prediction Algorithm Performance"
* **Placement:** Centered at the top of the image.
### B. Main Chart Area (Axes and Data)
* **X-Axis Label ($x_0$):** Located at the bottom center.
* **X-Axis Scale:** Numerical markers at 0, 2, 4, 6, 8, and 10.
* **Y-Axis Label ($x_1$):** Located at the left center, rotated 90 degrees.
* **Y-Axis Scale:** Numerical markers at 0, 2, 4, 6, 8, and 10.
* **Background:** Light gray grid-less field.
### C. Data Series and Classification
The plot contains approximately 150-170 individual data points categorized by color and spatial location.
* **Series 1: Green Circles (Negative/Background Class)**
* **Visual Trend:** These points are distributed across the entire $10 \times 10$ plane, except for a specific rectangular region in the mid-right quadrant.
* **Spatial Distribution:** High density in the ranges $x_0 \in [0, 5]$ and $x_0 \in [9, 10]$, and $x_1 \in [0, 3]$ and $x_1 \in [8, 10]$.
* **Outliers/Anomalies:** There is one green point located inside the dashed boundary at approximately $[x_0 \approx 7, x_1 \approx 0.8]$ and another near $[x_0 \approx 3, x_1 \approx 7]$.
* **Series 2: Blue Circles (Positive/Target Class)**
* **Visual Trend:** These points are clustered tightly within a specific rectangular region.
* **Spatial Distribution:** Concentrated within the bounds of $x_0 \in [5, 9]$ and $x_1 \in [3, 8]$.
### D. Decision Boundary (Dashed Box)
* **Type:** Rectangular dashed line.
* **Coordinates:**
* **Bottom-Left Corner:** $[5.0, 3.0]$
* **Top-Right Corner:** $[9.0, 8.0]$
* **Function:** This represents the algorithm's prediction zone. Points inside the box are predicted as the "Blue" class; points outside are predicted as the "Green" class.
## 3. Data Analysis and Observations
### Classification Accuracy
* **True Positives (Blue points inside the box):** The majority of blue points (approx. 45-50 points) are correctly contained within the dashed boundary.
* **False Positives (Green points inside the box):** There are no green points visible inside the dashed boundary, suggesting high precision for the "Blue" class.
* **False Negatives (Blue points outside the box):** There are no blue points visible outside the dashed boundary, suggesting high recall for the "Blue" class.
* **True Negatives (Green points outside the box):** The vast majority of green points are correctly located outside the dashed boundary.
### Summary of Spatial Logic
The algorithm has defined a rectangular decision rule:
$IF (5.0 \leq x_0 \leq 9.0) \text{ AND } (3.0 \leq x_1 \leq 8.0) \text{ THEN Class = Blue, ELSE Class = Green.}$
Based on the visual evidence, this algorithm achieves near-perfect separation of the two classes in this specific feature space.
</details>
Figure 2: Illustration of the algorithmic bias region $\mathcal{S}$ in the feature space, where the algorithm $\mathcal{A}$ exhibits suboptimal performance.
Without knowing the true algorithmic bias region, $\mathcal{S}$ , of the algorithm $\mathcal{A}$ , as represented using blue dots in Fig. 2, we want to estimate it using test data. We can evaluate the performance of the algorithm on a collection of test samples $x_{i}∈\mathbb{R}^{p}$ , $i=1,...,n$ . The response associated with each test sample is $y_{i}∈\mathbb{R}$ . Based on this, we can evaluate the algorithm performance using residuals.
$$
\epsilon_{i}=y_{i}-f(x_{i}),\quad i=1,\ldots,n.
$$
We note that alternative measures of algorithm performance, such as conformity scores, may replace residuals.
Our goal is to estimate the region $\widehat{\mathcal{S}}$ using $\{\epsilon_{i}\}_{i=1}^{n}$ as follows:
$$
\widehat{\mathcal{S}}=\{X_{j}\in[L_{j},U_{j}]\mbox{ or }X_{j}\in C_{j},j\in%
\widehat{S}\}, \tag{1}
$$
where $S$ , $L_{j}$ , $U_{j}$ , and $C_{j}$ are parameters to be determined.
Continuing with our previous example, if we estimate $\widehat{S}=\{1,5\}$ , this implies that we have correctly predicted the first important feature and incorrectly predicted the second. If $\widehat{S}=S$ , then we estimate the correct subset variables used to define the algorithmic bias region. Once $S$ is estimated, the other parameters can be easier to decide.
We hypothesize that the residuals within the bias region are larger. Thus, we formulate our problem as follows.
$$
\begin{split}\max_{\widehat{S}}\frac{1}{n(\widehat{S})}\sum_{x_{i}\in\widehat{%
S}}|\epsilon_{i}|,\end{split} \tag{2}
$$
where $\widehat{S}$ is defined in (1), and $n(\widehat{S})$ represents the number of data points $\widehat{S}$ .
We apply decision trees, specifically Classification And Regression Trees (CART), as proposed by Breiman et al. [30], to solve (2). The CART algorithm recursively partitions the feature space until some stopping criteria are achieved and provides a piecewise constant approximation of the response function, here representing algorithm performance. The effectiveness of our methodology relies on the compactness of the estimated value $\widehat{S}$ to the true value $S$ .
Bias Testing
Due to limited samples, bias estimation will have uncertainty, which we take into account in the bias detection through a conformal prediction procedure. This procedure provides a confidence interval for the estimated accuracy for each region. The confidence intervals are formed as follows. For each node in the decision tree, we can compute the confidence interval using the residuals $\epsilon_{i}$ of samples that fall into the region at a user-specified level $\alpha$ , such that if bias exists, we detect it with at least probability $1-\alpha$ . Confidence intervals are computed via quantiles. Formally defining $\text{Quantile}(\alpha;X):=∈f\{x:\alpha≤\mathbb{P}(X≤ x)\}$ , we obtain our lower and upper bounds via
$$
\hat{q}_{l}=\text{Quantile}\left(\frac{\alpha}{2};\sum^{n}_{i=1}{\epsilon_{i}}%
\right),\quad\hat{q}_{u}=\text{Quantile}\left(1-\frac{\alpha}{2};\sum^{n}_{i=1%
}{\epsilon_{i}}\right)
$$
respectively, and confidence intervals via
$$
\hat{C}_{j}(x)=\left[\hat{f}_{j}(x)+\hat{q}_{l_{j}},\hat{f}_{j}(x)+\hat{q}_{u_%
{j}}\right] \tag{3}
$$
where $\hat{f}_{j}(x)$ is the point prediction in the $j^{\text{th}}$ node of the decision tree.
To detect bias, we iterate over each terminal node, comparing the upper bound of the selected terminal node’s confidence intervals with the lower bound of the remaining terminal nodes. When the confidence intervals mutually overlap, we can claim no detection, meaning that we believe that the node does not have sufficient statistical evidence to indicate that a particular group suffers from significantly larger bias. If the upper bound of the selected terminal node is less than or equal to the lower bound of the other terminal nodes, we consider that node to have bias at significance level $\alpha$ . Alternatively stated, we are able to detect “algorithmic bias” with probability $1-\alpha$ . Fig. 3 provides a visual example of the implementation of these confidence intervals in the bias detection procedure. This bias detection method serves to audit the performance of any given pre-trained prediction algorithm $\mathcal{A}$ and is thus model agnostic.
<details>
<summary>x3.png Details</summary>
### Visual Description
# Technical Document Extraction: Decision Tree and Feature Space Partitioning
This image illustrates the relationship between a partitioned 2D feature space, a corresponding decision tree structure, and the resulting leaf node predictions.
## 1. Component Isolation
The image is divided into three primary functional regions:
* **Left Region (Feature Space Plot):** A 2D scatter plot showing data points and recursive partitioning.
* **Top-Right Region (Decision Tree Diagram):** A hierarchical tree structure representing the logic used to partition the feature space.
* **Bottom-Right Region (Prediction Plot):** A visualization of the values or intervals associated with the terminal (leaf) nodes of the tree.
---
## 2. Feature Space Plot (Left)
This region represents a two-dimensional feature space defined by axes $x_1$ and $x_2$.
### Axis Labels
* **X-axis:** $x_1$ (Horizontal)
* **Y-axis:** $x_2$ (Vertical)
### Data Points
The plot contains a high density of scatter points in two distinct colors:
* **Green Points:** Predominant throughout the space.
* **Yellow/Gold Points:** Interspersed among the green points, appearing as a secondary class or value.
* **Note:** Some points appear with lower opacity (faded), possibly indicating historical data or points outside a specific weight threshold.
### Partitioning (Spatial Grounding)
The space is divided into three distinct rectangular regions by solid black lines:
1. **Right Vertical Partition:** A vertical line at a specific $x_1$ value separates the rightmost section of the plot.
2. **Left Horizontal Partition:** A horizontal line at a specific $x_2$ value divides the remaining left area into an upper-left and a lower-left quadrant.
---
## 3. Decision Tree Diagram (Top-Right)
This diagram maps the logic of the partitions shown in the Feature Space Plot.
### Nodes and Hierarchy
* **Root Node ($t_0$):** The top-level node representing the first split.
* **Left Branch:** Leads to leaf node **$t_1$**.
* **Right Branch:** Leads to internal node **$t_2$**.
* **Internal Node ($t_2$):** Represents the second split.
* **Left Branch:** Leads to leaf node **$t_3$**.
* **Right Branch:** Leads to leaf node **$t_4$**.
### Terminal Nodes (Leaf Nodes)
The leaf nodes are identified as:
* **$t_1$**: Corresponds to the right-hand vertical partition in the feature space.
* **$t_3$**: Corresponds to one of the horizontal sub-partitions on the left.
* **$t_4$**: Corresponds to the other horizontal sub-partition on the left.
Below each leaf node ($t_1, t_3, t_4$), there is a horizontal bracket symbol `[—]` indicating a range or interval associated with that node.
---
## 4. Prediction Plot (Bottom-Right)
This chart visualizes the output or "prediction" associated with each leaf node.
### Axis and Markers
* **X-axis Labels:** $t_1$, $t_3$, $t_4$ (corresponding to the tree's leaf nodes).
* **Y-axis:** Unlabeled vertical axis representing a continuous value.
* **Baseline:** A horizontal dashed line runs across the plot, representing a global mean or a reference threshold.
### Data Series: Prediction Intervals
For each leaf node, a vertical error bar (I-beam) is plotted:
* **$t_1$ Trend:** The interval is centered slightly above the dashed baseline.
* **$t_3$ Trend:** The interval is significantly larger and centered below the dashed baseline.
* **$t_4$ Trend:** The interval is centered slightly above the dashed baseline, similar to $t_1$ but with a slightly different vertical span.
---
## 5. Summary of Logic and Flow
1. **Input:** Data points are distributed in a 2D space ($x_1, x_2$).
2. **Process:** A decision tree performs recursive binary splitting.
* The first split at $t_0$ isolates the right-hand region ($t_1$).
* The second split at $t_2$ divides the remaining left-hand region into $t_3$ and $t_4$.
3. **Output:** Each resulting region (leaf node) is assigned a predictive value and an uncertainty interval, as shown in the bottom-right graph. $t_3$ shows the highest variance/interval range compared to $t_1$ and $t_4$.
</details>
(a)
<details>
<summary>x4.png Details</summary>
### Visual Description
# Technical Document Extraction: Decision Tree Partitioning and Node Analysis
This image illustrates the relationship between a 2D feature space partition, a corresponding decision tree structure, and a statistical representation of the leaf nodes.
## 1. Component Isolation
The image is divided into three primary functional regions:
* **Left Region (Feature Space):** A 2D scatter plot showing data points and rectangular partitions.
* **Top-Right Region (Decision Tree):** A hierarchical tree diagram representing the logic used to create the partitions.
* **Bottom-Right Region (Node Statistics):** A chart showing intervals or error bars associated with the terminal leaf nodes.
---
## 2. Feature Space Analysis (Left Region)
This is a 2D coordinate system representing a feature space defined by two variables.
* **Axis Labels:**
* **X-axis:** $x_1$ (horizontal)
* **Y-axis:** $x_2$ (vertical)
* **Data Points:** The space is populated with numerous circular data points.
* **Green Points:** The majority class/type, distributed throughout the space.
* **Yellow Points:** Sparsely distributed throughout the space.
* **Blue Points:** Concentrated specifically in the top-right quadrant of the right-hand partition.
* **Transparency:** Some points appear semi-transparent, possibly indicating density or a third dimension/weight.
* **Partitions:** The space is divided by black boundary lines:
1. A primary vertical split at a specific value of $x_1$.
2. In the right-hand section ($x_1 > \text{split value}$), a horizontal split at a specific value of $x_2$.
* **Spatial Logic:**
* **Left Partition:** Contains mostly green and yellow points.
* **Top-Right Partition:** Contains a high concentration of blue points.
* **Bottom-Right Partition:** Contains mostly green and yellow points.
---
## 3. Decision Tree Diagram (Top-Right Region)
This diagram maps the recursive splitting of the feature space.
* **Nodes:** Represented by circles containing labels.
* **Root Node ($t_0$):** The starting point of the tree.
* **Internal Node ($t_2$):** A decision point following the first split.
* **Leaf Nodes ($t_1, t_3, t_4$):** Terminal nodes representing the final partitioned regions.
* **Flow/Structure:**
* $t_0$ splits into $t_1$ (left branch) and $t_2$ (right branch).
* $t_2$ splits into $t_3$ (left branch) and $t_4$ (right branch).
* **Mapping to Feature Space:**
* **$t_1$** corresponds to the large rectangular region on the left of the $x_1$ split.
* **$t_3$** corresponds to the bottom-right rectangular region.
* **$t_4$** corresponds to the top-right rectangular region (where the blue points are clustered).
* **Symbols:** Beneath each leaf node ($t_1, t_3, t_4$) is a horizontal bracket symbol indicating a range or interval associated with that node.
---
## 4. Node Statistics Chart (Bottom-Right Region)
This chart visualizes data properties for each terminal leaf node.
* **X-axis Labels:** $t_1, t_3, t_4$ (corresponding to the leaf nodes of the tree).
* **Y-axis:** Unlabeled vertical axis representing a value or magnitude.
* **Components:**
* **Dashed Horizontal Line:** Represents a baseline or threshold value across all nodes.
* **Vertical Intervals (Error Bars/Ranges):**
* **$t_1$:** A vertical interval positioned significantly **above** the dashed baseline.
* **$t_3$:** A vertical interval positioned **below** the dashed baseline.
* **$t_4$:** A vertical interval that **straddles** the dashed baseline (mostly above it).
---
## 5. Summary of Logic and Trends
1. **Partitioning Trend:** The data is first split on $x_1$ (creating node $t_1$ and $t_2$). The right-hand side ($t_2$) is then further refined by a split on $x_2$ (creating nodes $t_3$ and $t_4$).
2. **Data Correlation:** The concentration of blue points in the top-right of the feature space is captured by leaf node $t_4$.
3. **Statistical Outcome:** The chart in the bottom right indicates that the data in region $t_1$ has a higher mean/value than the baseline, region $t_3$ has a lower value, and region $t_4$ is centered near or slightly above the baseline.
</details>
(b)
Figure 3: The plots present 2D examples of (a) the determination of no bias, and (b) the determination of bias when using the conformal prediction procedure within our bias detection framework.
Bias Detection Framework
Let $D$ represent a dataset of patients, modeled as a tuple $(X,y)$ , where $X∈\mathbb{R}^{m× p}$ denotes a $p$ -dimensional feature matrix for $m$ patients, and $y∈[0,1]$ represents the performance metric corresponding to the prediction outcome for each patient in the pre-trained prediction algorithm $\mathcal{A}(X)$ . In this context, $X$ includes both categorical and continuous variables that capture the features of each patient, while $y$ evaluates the performance of the algorithm’s predictions on a scale from 0 (worst performance) to 1 (best performance).
Let $\alpha^{*}$ be the user-specified bias detection threshold, $K$ be the number of epochs, and $\Omega$ denote the hyperparameter space for the decision tree model. Our first objective is to identify a robust set of hyperparameters. For each epoch $k=1,2,...,K$ , we randomly shuffle the rows of the dataset $D$ and conduct a five-fold cross-validated grid search over the hyperparameter space $\Omega$ , yielding the optimized set of hyperparameters $\Omega_{k}$ .
Next, we fit our decision tree model $\Phi_{k}(D,\Omega_{k})$ to the data. For each fitted decision tree $\Phi_{k}$ , we test for the presence of bias at different nominal error levels $\alpha_{i}∈\{0.1,0.2,...,0.9,1.0\}$ using our conformal prediction procedure. If bias is detected at any nominal error level $\alpha_{i}≤\alpha^{*}$ , we conclude that bias is present at the user-specified threshold $\alpha^{*}$ , otherwise the framework reports no bias. We outline Algorithm 1 below.
Input: Dataset $D=(X,y)$ , Pre-trained prediction algorithm $\mathcal{A}(X)$ , User-specified detection threshold $\alpha^{*}$ , Number of epochs $K$ , Hyperparameter space $\Omega$
Output: Bias detection result (Yes/No)
for $k=1$ to $K$ do
Randomly shuffle the rows of dataset $D$ ;
Perform 5-fold cross-validated grid search over $\Omega$ to find optimized hyperparameters $\Omega_{k}$ ;
Fit decision tree model $\Phi_{k}(D,\Omega_{k})$ ;
for each nominal error level $\alpha_{i}∈\{0.1,0.2,...,1.0\}$ do
Apply conformal prediction procedure to test for bias at $\alpha_{i}$ ;
end for
end for
if Bias is detected such that $\alpha_{i}≤\alpha^{*}$ for any $\alpha_{i}$ then
Report Bias Detected;
end if
else
Report No Bias Detected;
end if
Algorithm 1 Bias Detection
5 Data
In this section, we describe the dataset used in our real-world case study. We begin with a discussion of the sepsis definition and follow with the data pre-processing steps implemented prior to model development.
5.1 Sepsis Definition
We adopted the revised Sepsis-3 definition as proposed by Singer et al. [32], which defines sepsis as a life-threatening organ failure induced by a dysregulated host response to infection. We implement the suspicion of infection criteria by identifying instances where the delivery of antibiotics in conjunction with orders for bacterial blood cultures occurred within a predetermined period. It is then determined that organ dysfunction has occurred when there is at least a two-point increase in the Sequential Organ Failure Assessment (SOFA) score during a specified period of time. The SOFA score is a numerical representation of the degradation of six organ systems (respiratory, coagulatory, liver, cardiovascular, renal, and neurologic) [33]. This definition was utilized to identify patients meeting the sepsis criteria and to ascertain the most likely onset time of sepsis.
5.2 Cohorts
5.2.1 Grady Memorial Hospital
Electronic health record (EHR) data was collected from 73,484 adult patients admitted to the intensive care unit (ICU) at Grady Memorial Hospital in Atlanta, Georgia from 2016 - 2020. This data included a total of 119,733 individual patient visits, referred to as “encounters”, where, 18,464 (15.42%) visits resulted in the retrospective diagnosis of sepsis. For our study, we excluded patients with less than 24 hours of continuous data, as well as, patients diagnosed with sepsis within the first six hours, reducing our dataset to 10,274 patient encounters involving 9,827 unique patients. Among these, 1,770 (17.23%) visits were retrospectively diagnosed with sepsis during their ICU stay. The general demographic and clinical characteristics of the analyzed cohort of patients are summarized in Table 1.
Table 1: Baseline characteristics of Grady patients grouped by cohort.
| n | | 10274 | 8504 | 1770 | |
| --- | --- | --- | --- | --- | --- |
| Age, median [Q1,Q3] | | 53.0 [36.0,65.0] | 53.0 [36.0,64.0] | 54.0 [36.0,66.0] | 0.248 |
| Gender, n (%) | Female | 3429 (33.4) | 2909 (34.2) | 520 (29.4) | $<$ 0.001 |
| Male | 6845 (66.6) | 5595 (65.8) | 1250 (70.6) | | |
| Race, n (%) | Asian | 125 (1.2) | 99 (1.2) | 26 (1.5) | $<$ 0.001 |
| Black | 6711 (65.3) | 5631 (66.2) | 1080 (61.0) | | |
| Hispanic | 479 (4.7) | 387 (4.6) | 92 (5.2) | | |
| Other | 305 (3.0) | 233 (2.7) | 72 (4.1) | | |
| White | 2654 (25.8) | 2154 (25.3) | 500 (28.2) | | |
| ICU Length of stay (LOS), mean (SD) | | 6.8 (9.4) | 4.3 (3.5) | 19.1 (16.5) | $<$ 0.001 |
| LOS in hospital, mean (SD) | | 14.7 (19.7) | 10.5 (10.5) | 34.7 (35.2) | $<$ 0.001 |
5.2.2 Emory University Hospital
EHR data were collected from 580,172 adult patients admitted to the Emory University Hospital ICU in Atlanta, Georgia between 2013 and 2021. Of these visits, 67,200 (11.58%) resulted in the retrospective diagnosis of sepsis. Following the same cohort generation procedure used for the Grady dataset, the Emory dataset was reduced to 69,232 patient encounters, of which 5,704 (8.24%) were retrospectively diagnosed with sepsis during their ICU stay. The demographic and clinical characteristics of the Emory patient cohort are summarized in Table 2.
Table 2: Baseline characteristics of Emory patients grouped by cohort.
| n | | 69232 | 63528 | 5704 | |
| --- | --- | --- | --- | --- | --- |
| Age, median [Q1,Q3] | | 63.0 [51.0,73.0] | 63.0 [51.0,73.0] | 63.0 [52.0,72.0] | 0.476 |
| Gender, n (%) | Female | 32141 (46.4) | 29596 (46.6) | 2545 (44.6) | 0.004 |
| Male | 37091 (53.6) | 33932 (53.4) | 3159 (55.4) | | |
| Race, n (%) | Asian | 1949 (2.8) | 1798 (2.8) | 151 (2.6) | <0.001 |
| Black | 27280 (39.4) | 24824 (39.1) | 2456 (43.1) | | |
| Multiple | 300 (0.4) | 270 (0.4) | 30 (0.5) | | |
| Other | 3751 (5.4) | 3344 (5.3) | 407 (7.1) | | |
| White | 35952 (51.9) | 33291 (52.4) | 2661 (46.7) | | |
| ICU Length of stay (LOS), mean (SD) | | 6.3 (10.8) | 4.7 (8.1) | 16.1 (17.8) | <0.001 |
| LOS in hospital, mean (SD) | | 12.6 (15.2) | 10.5 (11.7) | 25.9 (24.9) | <0.001 |
6 Sepsis Prediction Model
In developing the sepsis prediction model, we reference the model development procedure described in Yang et al. [34], which is one of the best-performing algorithms for sepsis detection. We detail the model development process in Appendix B.
7 Synthetic Data Experiments
In this section, we conduct experiments utilizing three synthetic data simulations using multidimensional uniform distributions. The objective of these simulations is to methodically assess the effectiveness of the conformal tree procedure in the context of detecting algorithmic bias regions. The first experiment evaluates the sensitivity of our bias detection approach when no bias exists. The final two experiments assess the effectiveness of the CART algorithm in the context of detecting algorithmic bias regions. This comparison is carried out by evaluating the coverage ratio, which serves as our primary performance criterion. This metric has been designed to effectively analyze and encompass the potential presence of an algorithmic bias region that may emerge within the feature space.
7.1 Performance Metrics
We introduce a refined performance metric, namely the coverage ratio, designed to account for the presence of distinct region(s) characterized by algorithmic bias within the feature space.
Coverage Ratio in n-Dimensional Space
The Coverage Ratio ( $\mathit{CVR}$ ) in $n$ -dimensional space provides a measure of how well the estimated region approximates the true region in higher-dimensional space. The metric quantifies the relationship between the hypervolumes of the true and estimated regions compared to the overlapping hypervolume covered by both regions. When $n=2$ or $n=3$ , $\mathit{CVR}$ is comparable to measuring the ratio of overlap between the area or volume of two sets, respectively. This metric is extended to higher-dimensional spaces as follows:
Given a dataset $\mathcal{D}⊂\mathbb{R}^{n}$ , consider two n-dimensional bounded regions defined by sets $\mathcal{S}$ (true region) and $\mathcal{\hat{S}}$ (estimated region). Let $|S|$ and $|\hat{S}|$ represent the hypervolumes of the true and estimated regions, respectively, in the $n$ -dimensional space, and let $|S\cap\hat{S}|$ denote the hypervolume of overlap common to both regions. Mathematically, we define $\mathit{CVR}$ as:
$$
\mathit{CVR}=\frac{1}{2}\left(\frac{|S\cap\hat{S}|}{|S|}+\frac{|S\cap\hat{S}|}%
{|\hat{S}|}\right). \tag{4}
$$
7.2 Experiments
In our first experiment, we evaluated the sensitivity of our method using synthetically generated datasets, without explicitly defining biased regions. We conducted 500 replications for each of the following sample sizes: $n_{s}=[500,750,1000,2000,$ $3000,6000,8000]$ , across dimensions $p∈[2,3,4,5]$ . The feature vectors $x_{i}$ for $i=1,2,...,5$ were drawn from a uniform distribution over the range $[-10,10]$ , and the corresponding $y$ values were generated from a uniform distribution $Y\sim U(0,1)$ .
We initialized the experiment by setting a significance level $\alpha=0.2$ , aiming to detect bias with a confidence level of $1-\alpha=0.80$ . For each simulation run, we applied bootstrap aggregation (bagging) with five estimators, using majority voting to determine the presence of bias. The effectiveness of our bias detection framework was evaluated based on the false discovery rate.
In the subsequent experiments, we introduced a single implicit bias region across a variety of sample sizes and dimensions. We conducted 100 replications for each of the following sample sizes: $n_{s}=[150,200,300,400,500,750,1000,2000]$ , across dimensions $p∈[2,3,4]$ . Similarly to the first experiment, the features $x_{i}$ for $i=1,2,...,4$ were sampled from a uniform distribution over the range $[-10,10]$ .
To simulate an algorithmic bias region, we generated the corresponding $y$ values from a uniform distribution within the range [0.8, 1.0]. A central point, denoted $c_{i}$ , was randomly selected within the feature space. Data points located within a defined distance from this central point were modified so that their corresponding $y$ values followed a uniform distribution within the interval $[0.3,0.6]$ . This region of reduced output values represents a potential area of algorithmic bias within the feature space.
The objective of the second experiment was to examine the relationship between the data sample topology and the performance of our bias detection framework when applied to a predefined algorithmic bias region. For each sample size, $n_{s}$ , a single algorithmic bias region was established and consistently maintained across all replications as the benchmark (true region). The experiment focused on evaluating the positional variability of data points, where new data points were randomly generated in each replication.
The primary goal of our third experiment was to assess how the location of the algorithmic bias region affects the performance of our detection framework. To isolate this effect, the topology of the feature space remained fixed across all replications, allowing us to focus on how variations in the bias region’s location influence model performance. We evaluated the effectiveness of our bias detection framework using the Coverage Ratio ( $\mathit{CVR}$ ) performance metric, which measures the alignment between the estimated region produced by the model and the predefined true bias region.
7.3 Results
Our simulations were designed with two primary objectives: first, to assess the framework’s ability to detect bias in scenarios where no bias is present, and second, to explore the complex relationships between algorithmic bias regions and the topologies of the feature space. Table LABEL:tab:falsediscovery_rate presents the false discovery rates observed in the first experiment, where we tested the framework’s sensitivity to bias detection in the absence of bias. The table shows results across various sample sizes ( $n_{s}$ ) and feature space dimensionalities ( $p$ ), where the findings indicate that false discovery rates decrease as sample sizes increase, with similar trends observed across different values of $p$ .
Table 3: False discovery rates across sample sizes and feature space dimensionalities.
| 2 3 4 | 0.0100 0.0000 0.0000 | 0.0040 0.0025 0.0060 | 0.0160 0.0075 0.0095 | 0.0120 0.0000 0.0149 | 0.0060 0.0000 0.0050 | 0.0080 0.0000 0.0000 | 0.0100 0.0000 0.0000 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 5 | 0.0080 | 0.0040 | 0.0060 | 0.0087 | 0.0100 | 0.0050 | 0.0000 |
Fig. 4 provides a visual representation of the ability of our approach to accurately estimate the borders of regions characterized by algorithmic bias. The true region(s) are delineated and filled in blue, whereas the estimated region(s) consist of points located inside the red dashed lines. Figs. 4a and 4b illustrate examples of the ability to identify bias regions in simulated output in the context of two and three-dimensional scenarios respectively.
<details>
<summary>x5.png Details</summary>
### Visual Description
# Technical Document Extraction: 2D Spatial Estimation Plot
## 1. Component Isolation
The image is a 2D scatter plot representing a spatial distribution of data points with overlaid estimation regions and a color-coded intensity scale.
* **Header/Legend Region:** Located at the top right of the main axes.
* **Main Chart Region:** A square coordinate system ranging from -10 to 10 on both axes.
* **Colorbar/Scale Region:** A vertical bar located to the right of the main chart.
---
## 2. Metadata and Labels
* **X-axis Label ($x_0$):** Horizontal axis, ranging from -10.0 to 10.0.
* **Y-axis Label ($x_1$):** Vertical axis, ranging from -10.0 to 10.0.
* **Legend (Top Right):**
* **Blue Solid Line:** $S$ (True) - Represents the ground truth boundary.
* **Red Solid Line:** $\hat{S}$ (Est) - Represents the estimated boundary.
* **Colorbar (Right):** A continuous vertical scale ranging from approximately **0.3** (dark purple) to **1.0** (bright yellow).
---
## 3. Data Series and Visual Trends
### A. Scatter Points (Data Distribution)
* **Trend:** The points are distributed quasi-randomly across the entire $20 \times 20$ unit field.
* **Color Mapping:**
* Points outside the central region are primarily colored in the **green-to-yellow** spectrum (values $\approx 0.7$ to $1.0$ on the colorbar).
* Points located within the central bounded region are colored in the **dark purple-to-blue** spectrum (values $\approx 0.3$ to $0.5$ on the colorbar).
* **Observation:** There is a clear correlation between spatial location and the value represented by the color; the central "box" contains significantly lower values than the surrounding field.
### B. Boundary Regions (The "Boxes")
There are two overlapping rectangular regions centered near the origin:
1. **True Boundary ($S$):**
* **Visual:** A light blue/lavender shaded rectangle with a faint blue dashed border.
* **Spatial Grounding:** Extends approximately from **-3.5 to 0.5** on the $x_0$ axis and **-1.0 to 3.0** on the $x_1$ axis.
2. **Estimated Boundary ($\hat{S}$):**
* **Visual:** A thick, red dashed line forming a rectangle.
* **Spatial Grounding:** Extends approximately from **-3.0 to 0.3** on the $x_0$ axis and **-0.8 to 3.0** on the $x_1$ axis.
* **Trend:** The estimated boundary ($\hat{S}$) is slightly smaller than the true boundary ($S$) but captures the majority of the low-value (purple) data points.
---
## 4. Detailed Data Extraction
### Axis Markers
| Axis | Min Value | Max Value | Major Increments |
| :--- | :--- | :--- | :--- |
| **$x_0$ (Horizontal)** | -10.0 | 10.0 | 2.5 |
| **$x_1$ (Vertical)** | -10.0 | 10.0 | 2.5 |
### Region Coordinates (Approximate)
| Feature | $x_0$ Range | $x_1$ Range | Visual Style |
| :--- | :--- | :--- | :--- |
| **True Region ($S$)** | [-3.5, 0.5] | [-1.0, 3.0] | Light Blue Fill, Blue Dash |
| **Estimated Region ($\hat{S}$)** | [-3.0, 0.3] | [-0.8, 3.0] | Red Dash |
---
## 5. Summary of Information
This chart visualizes a classification or estimation task. The goal appears to be identifying a specific spatial region (the "True" box) where data values (represented by color) are significantly lower than the background.
* **Background Values:** High (Yellow/Green, $\approx 0.85$).
* **Target Region Values:** Low (Purple/Blue, $\approx 0.4$).
* **Estimation Performance:** The red dashed box ($\hat{S}$) provides a high-accuracy estimation of the true region ($S$), with a slight under-estimation on the left and bottom edges.
</details>
(a)
<details>
<summary>x6.png Details</summary>
### Visual Description
# Technical Document Extraction: 3D Spatial Estimation Plot
## 1. Component Isolation
The image is a 3D scatter plot visualizing a dataset within a coordinate system, featuring two overlapping bounding boxes and a color-coded intensity scale.
* **Header/Legend Region:** Top right corner.
* **Main Chart Region:** Central 3D coordinate space.
* **Colorbar Region:** Far right vertical axis.
---
## 2. Metadata and Legend Extraction
**Spatial Grounding [x, y]:** The legend is located in the upper right quadrant of the image frame.
| Label | Visual Representation | Description |
| :--- | :--- | :--- |
| **$S$ (True)** | Solid Blue Line | Represents the ground truth spatial volume (True set). |
| **$\hat{S}$ (Est)** | Solid Red Line | Represents the estimated spatial volume (Estimated set). |
---
## 3. Axis and Coordinate System
The plot uses a Cartesian 3D coordinate system with the following scales:
* **$x_0$ Axis (Bottom Left):** Ranges from **-10.0 to 10.0** with increments of 2.5.
* **$x_1$ Axis (Bottom Right):** Ranges from **-10.0 to 10.0** with increments of 2.5.
* **$x_2$ Axis (Vertical):** Ranges from **-10.0 to 10.0** with increments of 2.5.
---
## 4. Data Series and Trends
### A. Scatter Plot (Data Points)
* **Distribution:** Points are distributed throughout the 3D volume, primarily concentrated between -10 and 10 on all axes.
* **Color Encoding:** Points are colored based on a value scale (likely probability or density).
* **Trend:** Points outside the bounding boxes generally appear in the yellow-to-green range (higher values on the colorbar). Points inside the blue/red bounding boxes transition into darker purple/blue shades (lower values on the colorbar).
* **Colorbar Scale:**
* **Range:** 0.4 to 0.9.
* **Gradient:** Dark Purple (0.4) $\rightarrow$ Teal (0.6) $\rightarrow$ Green (0.8) $\rightarrow$ Yellow (0.9).
### B. Bounding Volumes (The Cuboids)
* **$S$ (True) - Blue Box:**
* **Trend:** A semi-transparent blue cuboid located in the positive $x_0$, positive $x_1$, and negative $x_2$ octant.
* **Approximate Bounds:**
* $x_0$: ~[2.5, 10.0]
* $x_1$: ~[0.0, 7.5]
* $x_2$: ~[-7.5, 2.5]
* **$\hat{S}$ (Est) - Red Box:**
* **Trend:** A red wireframe/outline cuboid that closely overlaps the blue box.
* **Observation:** The red box appears slightly larger or shifted along the $x_1$ and $x_2$ axes compared to the blue box, indicating a high-accuracy estimation with minor marginal error.
---
## 5. Technical Summary
This visualization depicts a 3D point cloud where a specific subset of data (characterized by lower values, ~0.4 - 0.5) is being isolated. The **True Volume ($S$)** defines the actual region containing these low-value points. The **Estimated Volume ($\hat{S}$)** shows the result of a model's attempt to bound that same region. The high degree of overlap between the blue solid volume and the red outline suggests the estimation algorithm is performing with high precision.
</details>
(b)
Figure 4: Examples of the experimental results in 2(a) and 3(b) dimensional space.
We provide a summary of the results achieved by our approach, as depicted in Fig. 5, and confirm the efficacy of our bias detection framework in accurately detecting algorithmic bias regions. To provide precise details, Fig. 5 shows the mean performance of each experiment at the various sample size test points for multiple $n$ -dimensional cases. The figures incorporate 95% confidence intervals for both experiments. These results indicate that our method can efficiently detect the presence of algorithmic bias layered in the feature space.
<details>
<summary>extracted/5962136/images/lineplot_CVR_2D.png Details</summary>
### Visual Description
# Technical Data Extraction: Coverage Ratio vs. $n_s$
## 1. Document Overview
This image is a line graph illustrating the relationship between a variable $n_s$ and the "Coverage Ratio" for two distinct experimental groups. The chart includes error bars for each data point, representing variability or confidence intervals.
## 2. Component Isolation
### Header/Title
* **Content:** None present.
### Main Chart Area
* **Y-Axis Label:** Coverage Ratio
* **Y-Axis Scale:** Linear, ranging from approximately 0.88 to 1.00. Major tick marks are labeled at: `0.90`, `0.92`, `0.94`, `0.96`, `0.98`, `1.00`.
* **X-Axis Label:** $n_s$
* **X-Axis Scale:** Categorical/Ordinal (non-linear spacing). Labels are: `500`, `750`, `1000`, `2000`, `3000`, `6000`, `8000`.
* **Data Series:** Two lines with circular markers and vertical error bars.
### Legend
* **Location:** Bottom right [x ≈ 0.85, y ≈ 0.20].
* **Title:** Experiment
* **Series 1:** Blue dot (●) labeled "**1**"
* **Series 2:** Orange dot (●) labeled "**2**"
---
## 3. Trend Verification and Data Extraction
### Experiment 1 (Blue Line)
* **Visual Trend:** The series starts at a moderate value, drops significantly at $n_s=750$, and then exhibits a consistent upward slope, plateauing as it approaches a Coverage Ratio of 1.00.
* **Data Points (Approximate):**
* **500:** ~0.941 (High error bar variance)
* **750:** ~0.918 (Local minimum)
* **1000:** ~0.971
* **2000:** ~0.978
* **3000:** ~0.992
* **6000:** ~0.997
* **8000:** ~0.998 (Highest value)
### Experiment 2 (Orange Line)
* **Visual Trend:** The series starts at the lowest recorded value and shows a strong, continuous upward slope. It overtakes Experiment 1 at $n_s=750$ and $n_s=2000$, eventually converging with Experiment 1 at the highest $n_s$ values.
* **Data Points (Approximate):**
* **500:** ~0.897 (Global minimum)
* **750:** ~0.950
* **1000:** ~0.963
* **2000:** ~0.985
* **3000:** ~0.994
* **6000:** ~0.995
* **8000:** ~0.996
---
## 4. Data Table Reconstruction
| $n_s$ (X-axis) | Experiment 1 (Blue) Coverage Ratio | Experiment 2 (Orange) Coverage Ratio |
| :--- | :--- | :--- |
| **500** | ~0.941 | ~0.897 |
| **750** | ~0.918 | ~0.950 |
| **1000** | ~0.971 | ~0.963 |
| **2000** | ~0.978 | ~0.985 |
| **3000** | ~0.992 | ~0.994 |
| **6000** | ~0.997 | ~0.995 |
| **8000** | ~0.998 | ~0.996 |
---
## 5. Key Observations
* **Convergence:** Both experiments converge toward a Coverage Ratio of 1.00 as $n_s$ increases beyond 3000.
* **Anomalies:** Experiment 1 shows a non-monotonic behavior between $n_s=500$ and $n_s=1000$, whereas Experiment 2 is strictly increasing.
* **Precision:** Error bars generally decrease in size as $n_s$ increases, suggesting higher stability/certainty at larger sample sizes or parameters.
</details>
(a)
<details>
<summary>extracted/5962136/images/lineplot_CVR_3D.png Details</summary>
### Visual Description
# Technical Data Extraction: Coverage Ratio vs. $n_s$
## 1. Image Overview
This image is a line graph with error bars comparing two experimental datasets. It illustrates the relationship between a sample size or parameter denoted as $n_s$ and a "Coverage Ratio."
## 2. Component Isolation
### Header/Title
* No explicit title is present within the image frame.
### Main Chart Area
* **X-Axis Label:** $n_s$
* **X-Axis Markers (Categorical/Log-spaced):** 500, 750, 1000, 2000, 3000, 6000, 8000. Note: The spacing on the x-axis is not linear; it appears to be treated as categorical or semi-logarithmic.
* **Y-Axis Label:** Coverage Ratio
* **Y-Axis Markers:** 0.84, 0.86, 0.88, 0.90, 0.92, 0.94, 0.96, 0.98, 1.00.
### Legend
* **Location:** Bottom right [approximate coordinates: x=0.85, y=0.20].
* **Title:** Experiment
* **Series 1:** Blue circle ($\bullet$) - Experiment 1
* **Series 2:** Orange circle ($\bullet$) - Experiment 2
---
## 3. Trend Verification and Data Extraction
### Trend Analysis
* **Experiment 1 (Blue):** Shows a steep upward slope from $n_s = 500$ to $n_s = 2000$, after which the growth rate decelerates significantly, approaching an asymptote near 1.00.
* **Experiment 2 (Orange):** Starts at a higher coverage ratio than Experiment 1 at $n_s = 500$. It follows a similar upward trajectory but with a less steep initial slope. It is surpassed by Experiment 1 at $n_s = 2000$, then converges toward the same asymptote near 1.00.
### Data Table (Estimated Values)
Values are extracted based on visual alignment with axis markers. Error bars indicate standard deviation or confidence intervals.
| $n_s$ (X-axis) | Experiment 1 (Blue) | Experiment 2 (Orange) |
| :--- | :--- | :--- |
| **500** | ~0.852 | ~0.883 |
| **750** | ~0.916 | ~0.947 |
| **1000** | ~0.969 | ~0.970 |
| **2000** | ~0.990 | ~0.986 |
| **3000** | ~0.992 | ~0.991 |
| **6000** | ~0.996 | ~0.995 |
| **8000** | ~0.997 | ~0.996 |
---
## 4. Detailed Observations
* **Crossover Point:** A significant crossover occurs between $n_s = 1000$ and $n_s = 2000$. At $n_s = 1000$, the two experiments are nearly identical. By $n_s = 2000$, Experiment 1 (Blue) achieves a higher coverage ratio than Experiment 2 (Orange).
* **Error Bars:**
* Experiment 1 (Blue) has larger error bars at lower $n_s$ values (500, 750), suggesting higher variance in results for smaller sample sizes.
* Experiment 2 (Orange) has relatively consistent error bar sizes until $n_s = 2000$.
* For both series, error bars become extremely small as $n_s$ reaches 6000 and 8000, indicating high precision and convergence as the coverage ratio nears 1.00.
* **Saturation:** Both experiments show "saturation" behavior, where increasing $n_s$ beyond 3000 yields diminishing returns in the Coverage Ratio, as both are already >0.99.
</details>
(b)
<details>
<summary>extracted/5962136/images/lineplot_CVR_4D.png Details</summary>
### Visual Description
# Technical Data Extraction: Coverage Ratio Analysis
## 1. Image Overview
This image is a line graph comparing the performance of two experimental conditions (labeled "1" and "2") across varying sample sizes ($n_s$). The chart includes error bars representing variability or confidence intervals for each data point.
## 2. Component Isolation
### Header/Metadata
* **Language:** English
* **Primary Metric (Y-axis):** Coverage Ratio
* **Independent Variable (X-axis):** $n_s$ (Sample Size)
### Main Chart Area
* **X-axis Scale:** Categorical/Discrete numerical values: 5000, 7500, 10000, 20000, 30000, 60000, 80000.
* **Y-axis Scale:** Linear scale ranging from 0.84 to 1.00 with increments of 0.02.
* **Data Series 1 (Blue):** Represents "Experiment 1".
* **Data Series 2 (Orange):** Represents "Experiment 2".
### Legend (Spatial Grounding: Bottom Right [~720, 880])
* **Title:** Experiment
* **Blue Circle:** 1
* **Orange Circle:** 2
---
## 3. Data Extraction and Trend Analysis
### Series 1: Experiment 1 (Blue Line)
* **Trend Verification:** This line is nearly perfectly horizontal, maintaining a value extremely close to 1.00 across all values of $n_s$. It shows negligible variance and no significant response to the increase in sample size.
* **Data Points (Approximate):**
* $n_s$ 5000: ~1.00
* $n_s$ 7500: ~1.00
* $n_s$ 10000: ~1.00
* $n_s$ 20000: ~0.998 (slightest dip)
* $n_s$ 30000: ~1.00
* $n_s$ 60000: ~1.00
* $n_s$ 80000: ~1.00
### Series 2: Experiment 2 (Orange Line)
* **Trend Verification:** This line exhibits significant volatility and a general downward trend as $n_s$ increases. It starts relatively high (~0.93), fluctuates between $n_s$ 7500 and 60000, and then drops sharply at the final data point ($n_s$ 80000). Error bars are prominent, indicating lower precision compared to Experiment 1.
* **Data Points and Error Estimates:**
* **$n_s$ 5000:** ~0.932 (Error range: ~0.920 to 0.945)
* **$n_s$ 7500:** ~0.937 (Error range: ~0.925 to 0.948)
* **$n_s$ 10000:** ~0.920 (Error range: ~0.908 to 0.933)
* **$n_s$ 20000:** ~0.936 (Error range: ~0.922 to 0.948)
* **$n_s$ 30000:** ~0.905 (Error range: ~0.892 to 0.918)
* **$n_s$ 60000:** ~0.922 (Error range: ~0.910 to 0.935)
* **$n_s$ 80000:** ~0.854 (Error range: ~0.840 to 0.868)
---
## 4. Summary Table of Extracted Data
| $n_s$ (Sample Size) | Experiment 1 (Blue) | Experiment 2 (Orange) |
| :--- | :--- | :--- |
| 5000 | ~1.00 | ~0.932 |
| 7500 | ~1.00 | ~0.937 |
| 10000 | ~1.00 | ~0.920 |
| 20000 | ~0.998 | ~0.936 |
| 30000 | ~1.00 | ~0.905 |
| 60000 | ~1.00 | ~0.922 |
| 80000 | ~1.00 | ~0.854 |
## 5. Technical Observations
* **Performance Gap:** Experiment 1 consistently outperforms Experiment 2 in terms of Coverage Ratio across the entire tested range.
* **Stability:** Experiment 1 is highly stable, whereas Experiment 2 shows sensitivity to the sample size, particularly at the highest value (80000), where performance degrades significantly.
* **Uncertainty:** The presence of error bars on Experiment 2 suggests a stochastic element or higher variance in results that is not present (or is negligible) in Experiment 1.
</details>
(c)
<details>
<summary>extracted/5962136/images/lineplot_CVR_5D.png Details</summary>
### Visual Description
# Technical Data Extraction: Coverage Ratio Analysis
## 1. Image Classification and Overview
This image is a line graph with error bars comparing two experimental datasets. It plots the **Coverage Ratio** (y-axis) against a variable denoted as **$n_s$** (x-axis).
## 2. Component Isolation
### Header/Title
* No explicit title is present in the image.
### Main Chart Area
* **X-Axis Label:** $n_s$
* **X-Axis Markers (Categorical/Discrete):** 5000, 7500, 10000, 20000, 30000, 60000, 80000. Note: The spacing between markers is uniform, indicating $n_s$ is treated as a categorical variable or a non-linear scale.
* **Y-Axis Label:** Coverage Ratio
* **Y-Axis Markers (Linear):** 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00.
### Legend (Spatial Grounding: Bottom Right [x $\approx$ 0.8, y $\approx$ 0.2])
* **Title:** Experiment
* **Series 1 (Blue Circle):** Labeled "1"
* **Series 2 (Orange Circle):** Labeled "2"
---
## 3. Data Series Analysis
### Series 1 (Blue Line)
* **Trend Verification:** The blue line starts high, drops sharply at $n_s=7500$, recovers to near-maximum levels, dips slightly at $n_s=30000$, and plateaus at 1.00 for the final two data points. It consistently maintains a higher coverage ratio than Series 2.
* **Error Bars:** Very small vertical error bars, indicating high precision/low variance.
| $n_s$ | Coverage Ratio (Approx.) |
| :--- | :--- |
| 5000 | 0.997 |
| 7500 | 0.968 |
| 10000 | 0.995 |
| 20000 | 0.998 |
| 30000 | 0.987 |
| 60000 | 1.000 |
| 80000 | 1.000 |
### Series 2 (Orange Line)
* **Trend Verification:** The orange line exhibits significant volatility. It starts at a moderate level, rises slightly, drops sharply to its lowest point at $n_s=20000$, recovers significantly at $n_s=30000$, and then follows a downward trend for the remainder of the scale.
* **Error Bars:** Substantial vertical error bars, indicating lower precision/higher variance compared to Series 1.
| $n_s$ | Coverage Ratio (Approx.) |
| :--- | :--- |
| 5000 | 0.955 |
| 7500 | 0.962 |
| 10000 | 0.978 |
| 20000 | 0.932 |
| 30000 | 0.976 |
| 60000 | 0.956 |
| 80000 | 0.938 |
---
## 4. Comparative Summary
* **Performance:** Experiment 1 (Blue) outperforms Experiment 2 (Orange) at every measured interval of $n_s$.
* **Stability:** Experiment 1 reaches a perfect coverage ratio (1.00) at higher values of $n_s$ (60000+), whereas Experiment 2 shows a decline in performance in the same range.
* **Reliability:** The error bars suggest that the results for Experiment 1 are much more consistent across trials than those for Experiment 2.
</details>
(d)
Figure 5: The plots show the mean coverage ratio for multiple $n$ -dimensional test points: 2D(a), 3D(b), 4D(c), and 5D(d).
8 Real-Data Experiment
In the second phase of our empirical study, we evaluate the effectiveness of the sepsis prediction model and aim to identify any potential algorithmic biases. During this assessment, the test dataset is used to sequentially process the continuous data of each patient via the prediction model. We further refine the test data by only applying the model to patients whose EHR data includes at least one occurrence of sepsis. Implementing this approach results in an hourly forecast for every occurrence of a patient’s data. Subsequently, we compute the performance of the classification model for every individual patient. Here, we selected model accuracy as the performance measure, implying it is the variable we are using to identify algorithmic bias. Next, we combine the accuracy of each patient’s performance measure with their corresponding demographic data, which includes a range of factors such as gender, race, age, insurance type, and the existence and number of pre-existing comorbidities. One-hot encoding is used to transform non-numeric features into a numeric representation. Lastly, we define a threshold significance level $\alpha^{*}=0.20$ , meaning we want to detect “algorithmic bias” with a confidence level of at least $80\%$ , and define our hyper-parameter space $\Omega$ , as outlined in Table 4.
Table 4: CART bias detection hyper-parameter tuning grid
| “criterion” | [“squared_error”, “absolute_error”] |
| --- | --- |
| “splitter” | [“best"] |
| “ccp_alpha” | [0.0, 0.0001, 0.0005, 0.001] |
| “max_depth” | [3,4] |
| “min_samples_leaf” | [10, 30, 50, 60, 100] |
| “min_samples_split” | [10, 30, 50, 60, 100] |
| “max_features” | [None,“log2”,“sqrt”] |
8.1 Results
8.1.1 Grady Memorial Hospital
The final results of our bias detection framework for Grady Memorial Hospital are shown in Fig. 6. Our findings indicate that bias was detected for patients located in Node 7, with a significance level of $\alpha^{*}=0.20$ . Fig. 6a illustrates the complete decision tree generated by the sepsis prediction model for the test dataset. Each node contains the feature split-point pair selected by the model at that node, the number of instances in the node, the predicted response variable $\hat{y}$ for the samples, the standard deviation within the node, and the conformal prediction set based on the significance level $\alpha^{*}$ .
Fig. 6b visualizes the confidence intervals for each node’s conformal predictions, providing a detailed view of prediction uncertainty across the tree. Fig. 6c displays the optimized significance levels $\alpha^{*}$ across all leaf nodes, as summarized in Table 5. Notably, the optimized confidence level for Node 7 is 0.9, which translates to an optimized significance level of $\alpha_{7}^{*}=0.10$ .
Fig. 6d provides a simplified representation of the key attributes that define this suboptimal path. Based on our bias detection analysis, we conclude that the sepsis prediction model $\mathcal{A}$ may underperform for the subgroup characterized as “ventilated patients, younger than 45 years old, residing more than 3.35 miles from Grady Hospital.” This summary not only highlights the algorithmic bias detected but also provides valuable insight into the demographic and clinical attributes associated with suboptimal model performance.
<details>
<summary>x7.png Details</summary>
### Visual Description
# Technical Document: Decision Tree Analysis
## 1. Overview
This image displays a **Decision Tree** diagram, likely representing a regression or classification model. The tree consists of 11 nodes (numbered #0 through #10), organized into four hierarchical levels. Each node contains statistical data including sample size, mean value, standard deviation, and an 80% Confidence Interval (CI).
The nodes are color-coded based on their "value" metric:
* **Darker Orange:** Higher values (e.g., Node #3 at 0.86).
* **Light Orange/Peach:** Moderate values (e.g., Node #0 at 0.4).
* **White/Light Grey:** Lower values (e.g., Node #7 at 0.21).
---
## 2. Component Breakdown and Flow
### Level 0: Root Node
* **Node #0**
* **Condition:** `on_vent > 0.5`
* **Samples:** 446
* **Value:** 0.4
* **Std Dev (σ):** 0.36
* **80% CI:** [0.07, 0.91]
* **Flow:** If **False**, proceed to Node #1 (Left). If **True**, proceed to Node #4 (Right).
---
### Level 1: Primary Splits
* **Node #1 (False branch of Node #0)**
* **Condition:** `age > 62.5`
* **Samples:** 65
* **Value:** 0.81
* **Std Dev (σ):** 0.24
* **80% CI:** [0.52, 1.0]
* **Flow:** Splits into Node #2 and Node #3.
* **Node #4 (True branch of Node #0)**
* **Condition:** `age > 44.5`
* **Samples:** 381
* **Value:** 0.33
* **Std Dev (σ):** 0.33
* **80% CI:** [0.0, 0.89]
* **Flow:** Splits into Node #5 and Node #8.
---
### Level 2: Secondary Splits
* **Node #2 (Leaf Node)**
* **Samples:** 32
* **Value:** 0.75
* **Std Dev (σ):** 0.31
* **80% CI:** [0.22, 0.99]
* **Node #3 (Leaf Node)**
* **Samples:** 33
* **Value:** 0.86
* **Std Dev (σ):** 0.14
* **80% CI:** [0.7, 0.98]
* **Node #5**
* **Condition:** `dist_to_grady > 3.35`
* **Samples:** 130
* **Value:** 0.23
* **Std Dev (σ):** 0.27
* **80% CI:** [0.03, 0.67]
* **Flow:** Splits into Node #6 and Node #7.
* **Node #8**
* **Condition:** `age > 78.5`
* **Samples:** 251
* **Value:** 0.38
* **Std Dev (σ):** 0.35
* **80% CI:** [0.05, 0.97]
* **Flow:** Splits into Node #9 and Node #10.
---
### Level 3: Tertiary Leaf Nodes
* **Node #6**
* **Samples:** 10
* **Value:** 0.42
* **Std Dev (σ):** 0.45
* **80% CI:** [0.03, 0.94]
* **Node #7**
* **Samples:** 120
* **Value:** 0.21
* **Std Dev (σ):** 0.25
* **80% CI:** [0.02, 0.61]
* **Node #9**
* **Samples:** 232
* **Value:** 0.36
* **Std Dev (σ):** 0.34
* **80% CI:** [0.03, 0.96]
* **Node #10**
* **Samples:** 19
* **Value:** 0.64
* **Std Dev (σ):** 0.37
* **80% CI:** [0.09, 0.98]
---
## 3. Data Summary Table
| Node ID | Condition | Samples | Value | Std Dev (σ) | 80% CI |
| :--- | :--- | :--- | :--- | :--- | :--- |
| #0 | on_vent > 0.5 | 446 | 0.4 | 0.36 | [0.07, 0.91] |
| #1 | age > 62.5 | 65 | 0.81 | 0.24 | [0.52, 1.0] |
| #2 | (Leaf) | 32 | 0.75 | 0.31 | [0.22, 0.99] |
| #3 | (Leaf) | 33 | 0.86 | 0.14 | [0.7, 0.98] |
| #4 | age > 44.5 | 381 | 0.33 | 0.33 | [0.0, 0.89] |
| #5 | dist_to_grady > 3.35 | 130 | 0.23 | 0.27 | [0.03, 0.67] |
| #6 | (Leaf) | 10 | 0.42 | 0.45 | [0.03, 0.94] |
| #7 | (Leaf) | 120 | 0.21 | 0.25 | [0.02, 0.61] |
| #8 | age > 78.5 | 251 | 0.38 | 0.35 | [0.05, 0.97] |
| #9 | (Leaf) | 232 | 0.36 | 0.34 | [0.03, 0.96] |
| #10 | (Leaf) | 19 | 0.64 | 0.37 | [0.09, 0.98] |
---
## 4. Key Trends and Observations
1. **Primary Driver:** The initial split is based on `on_vent`. When `on_vent` is False (Node #1), the predicted "value" is significantly higher (0.81) compared to when it is True (Node #4, value 0.33).
2. **Age Influence:** Age is a recurring decision factor (Nodes #1, #4, and #8). In the `on_vent` True branch, a higher age (> 78.5) leads to a higher value (0.64 in Node #10) compared to younger groups.
3. **Sample Distribution:** The majority of the data (381/446 samples) follows the `on_vent > 0.5 = True` path.
4. **Confidence:** Node #3 shows the highest precision with the lowest standard deviation (0.14) and the narrowest 80% CI [0.7, 0.98].
</details>
(a)
<details>
<summary>x8.png Details</summary>
### Visual Description
# Technical Document Extraction: Prediction Sets (80% CI)
## 1. Document Metadata
* **Title:** Prediction Sets (80% CI)
* **Primary Language:** English
* **Chart Type:** Error Bar Plot (Point estimates with Confidence Intervals)
* **X-Axis Label:** Node
* **Y-Axis Label:** Accuracy
* **Legend Title:** Nodes:
* **Legend Categories:**
* **Branch:** Represented by Blue dots and Blue error bars.
* **Terminal:** Represented by Orange dots and Red error bars.
## 2. Spatial Grounding and Component Isolation
* **Header:** Contains the title "Prediction Sets (80% CI)".
* **Main Chart Area:** Occupies the central region, plotting Accuracy (0.0 to 1.0) against Node index (0 to 10).
* **Legend Location:** Positioned on the right-hand side, vertically centered relative to the plot area.
* **Data Series Identification:**
* **Blue Series (Branch):** Nodes 0, 1, 4, 5, 8.
* **Orange/Red Series (Terminal):** Nodes 2, 3, 6, 7, 9, 10.
## 3. Data Extraction and Trend Analysis
### Trend Verification
* **Branch Nodes:** The accuracy for branch nodes is highly volatile, ranging from approximately 0.35 to 0.75. The confidence intervals for these nodes are generally very wide, often spanning nearly the entire 0.0 to 1.0 range (e.g., Nodes 0, 4, 8).
* **Terminal Nodes:** These nodes show a moderate range of accuracy (approx. 0.3 to 0.85). Node 3 stands out with the highest accuracy and the narrowest confidence interval in the entire set. Other terminal nodes (6, 9, 10) show very wide confidence intervals similar to the branch nodes.
### Data Table (Estimated Values)
The following table transcribes the visual data points. Values are estimated based on the axis scales.
| Node | Type | Accuracy (Point) | CI Lower Bound | CI Upper Bound |
| :--- | :--- | :--- | :--- | :--- |
| 0 | Branch | ~0.49 | ~0.07 | ~0.91 |
| 1 | Branch | ~0.76 | ~0.52 | ~1.00 |
| 2 | Terminal | ~0.61 | ~0.22 | ~0.99 |
| 3 | Terminal | ~0.84 | ~0.70 | ~0.98 |
| 4 | Branch | ~0.45 | ~0.00 | ~0.89 |
| 5 | Branch | ~0.35 | ~0.03 | ~0.67 |
| 6 | Terminal | ~0.49 | ~0.03 | ~0.94 |
| 7 | Terminal | ~0.32 | ~0.02 | ~0.61 |
| 8 | Branch | ~0.51 | ~0.05 | ~0.97 |
| 9 | Terminal | ~0.50 | ~0.03 | ~0.96 |
| 10 | Terminal | ~0.54 | ~0.09 | ~0.98 |
## 4. Key Observations
* **Highest Accuracy:** Node 3 (Terminal) at ~0.84.
* **Lowest Accuracy:** Node 7 (Terminal) at ~0.32.
* **Precision:** Node 3 has the most precise prediction (smallest error bar).
* **Uncertainty:** Nodes 0, 4, 8 (Branch) and 2, 6, 9, 10 (Terminal) exhibit extreme uncertainty, with 80% confidence intervals covering roughly 80-90% of the possible accuracy range.
* **Boundary Hits:** Node 1 (Branch) upper CI reaches the 1.0 limit. Node 4 (Branch) lower CI reaches the 0.0 limit.
</details>
(b)
<details>
<summary>x9.png Details</summary>
### Visual Description
# Technical Document Extraction: Bias Detection Threshold Chart
## 1. Document Metadata
* **Title:** Bias Detection Threshold
* **Chart Type:** Vertical Bar Chart with a Threshold Line
* **Language:** English
## 2. Component Isolation
### A. Header
* **Text:** "Bias Detection Threshold" (Centered at the top).
### B. Main Chart Area (Axes and Data)
* **Y-Axis Label:** "Max Confidence Level (1-α)"
* **Y-Axis Scale:** Linear, ranging from 0.0 to 1.0 with major tick marks every 0.2 units (0.0, 0.2, 0.4, 0.6, 0.8, 1.0).
* **X-Axis Label:** "Node"
* **X-Axis Categories:** Non-sequential integers representing specific nodes: 2, 3, 6, 7, 9, 10.
* **Threshold Line:** A horizontal dashed red line is positioned at $y = 0.8$.
### C. Legend/Visual Indicators
* **Data Series:** Blue bars representing the confidence level for each node.
* **Reference Line:** Red dashed line representing the "Bias Detection Threshold."
## 3. Data Extraction and Trend Verification
### Trend Analysis
The data shows a non-linear distribution across the selected nodes. The confidence levels are low for the initial nodes (2 and 3), rise significantly to a peak at Node 7, and then decline through Node 10. Only one node (Node 7) exceeds the defined threshold of 0.8.
### Data Table Reconstruction
The following table represents the approximate values extracted from the visual height of the blue bars relative to the Y-axis markers.
| Node (X-Axis) | Max Confidence Level (Y-Axis) | Status Relative to Threshold (0.8) |
| :--- | :--- | :--- |
| 2 | ~0.10 | Below |
| 3 | ~0.10 | Below |
| 6 | ~0.40 | Below |
| 7 | ~0.90 | **Above** |
| 9 | ~0.70 | Below |
| 10 | ~0.20 | Below |
## 4. Key Findings and Observations
* **Threshold Breach:** Node 7 is the only component that surpasses the Bias Detection Threshold, reaching a confidence level of approximately 0.9.
* **Near-Threshold:** Node 9 shows a high confidence level (~0.7) but remains below the 0.8 detection limit.
* **Baseline Nodes:** Nodes 2 and 3 show the lowest confidence levels, both sitting at approximately 0.1.
* **Gaps in Data:** The X-axis is not continuous (missing nodes 1, 4, 5, and 8), suggesting this chart focuses on a specific subset of nodes under investigation.
</details>
(c)
<details>
<summary>x10.png Details</summary>
### Visual Description
# Technical Document Extraction: Decision Logic Flowchart
## 1. Image Overview
The image depicts a linear logic flow or a decision path consisting of three sequential nodes connected by arrows. Each node contains a specific condition or variable state enclosed in large parentheses. The background is a light gray, and the text and symbols are black.
## 2. Component Isolation and Transcription
The diagram is structured as a horizontal sequence from left to right.
| Sequence | Component Type | Text/Content | Technical Interpretation |
| :--- | :--- | :--- | :--- |
| **Node 1** | Input/Condition | `Ventilated = 1` | Boolean or binary flag indicating the subject is ventilated. |
| **Transition** | Connector | `→` | Logical progression to the next criteria. |
| **Node 2** | Filter/Condition | `Age ≤ 44` | Numerical constraint: Age is less than or equal to 44. |
| **Transition** | Connector | `→` | Logical progression to the final criteria. |
| **Node 3** | Filter/Condition | `Distance to Grady > 3.35 mi` | Spatial constraint: Distance to a location named "Grady" is greater than 3.35 miles. |
## 3. Logic Flow and Data Analysis
### Sequential Trend
The logic follows an "AND" gate progression or a specific branch of a decision tree. For a case to satisfy this specific path, all three conditions must be met in the order presented.
1. **Initial State:** The process begins with the requirement that the subject is "Ventilated" (Value = 1).
2. **Demographic Filter:** If the first condition is met, the logic proceeds to check if the subject's age is 44 years or younger.
3. **Geographic Filter:** If the age condition is met, the final check determines if the distance to "Grady" (likely referring to Grady Memorial Hospital or a specific landmark) exceeds 3.35 miles.
### Summary of Extracted Data Points
* **Variable 1:** `Ventilated` (Binary: 1)
* **Variable 2:** `Age` (Threshold: ≤ 44)
* **Variable 3:** `Distance to Grady` (Threshold: > 3.35; Unit: miles)
## 4. Final Transcription
The complete logical string represented in the image is:
**[ Ventilated = 1 ] → [ Age ≤ 44 ] → [ Distance to Grady > 3.35 mi ]**
</details>
(d)
Figure 6: Grady bias detection model results. 6a displays the complete decision tree, where the intensity of node shading corresponds to the magnitude of the point prediction—darker nodes indicate higher point prediction values, while lighter nodes indicate lower point prediction values. 6b shows the predicted confidence intervals $\hat{\mathcal{C}}_{j}$ for each branch (blue) and terminal (red) node at significance level $\alpha$ . 6c presents the maximum bias detection confidence level $1-\alpha^{*}_{j}$ for the $j^{\text{th}}$ terminal node. 6d provides a simplified representation of the nodes along that route.
Table 5: Optimized significance level $\alpha$ per node.
| 2 3 6 | 1.0 1.0 0.60 | 0.00 0.00 0.40 |
| --- | --- | --- |
| 7 | 0.10 | 0.90 |
| 9 | 0.30 | 0.70 |
| 10 | 0.80 | 0.20 |
<details>
<summary>x11.png Details</summary>
### Visual Description
# Technical Document Extraction: Accuracy Distribution Analysis
## 1. Document Metadata
* **Title:** Accuracy Distribution by Race, Gender, and Bias
* **Language:** English
* **Chart Type:** Grouped Box Plot (Facet Grid)
* **Primary Variables:**
* **Accuracy** (Y-axis, continuous 0.0 to 1.0)
* **Race** (X-axis, categorical: Black, White, Other Race, Hispanic, Asian)
* **Gender** (Legend/Color: Male, Female)
* **Bias Group** (Facet/Sub-plot: False, True)
---
## 2. Component Isolation
### Header
* **Main Title:** Accuracy Distribution by Race, Gender, and Bias
* **Sub-titles:**
* Left Plot: `Bias Group = False`
* Right Plot: `Bias Group = True`
### Legend (Spatial Placement: Top Right of the right-hand plot)
* **Title:** Gender
* **Light Blue (#B9CFE0):** Male
* **Dark Blue (#4E84B0):** Female
### Axis Definitions
* **Y-Axis (Left of both plots):** Label: "Accuracy". Markers: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0.
* **X-Axis (Bottom of both plots):** Label: "Race". Categories: Black, White, Other Race, Hispanic, Asian.
---
## 3. Data Extraction and Trend Analysis
### Sub-plot 1: Bias Group = False
**General Trend:** Accuracy is significantly higher across all racial groups compared to the "Bias Group = True" condition. Medians generally fall between 0.2 and 0.5.
| Race | Gender | Median Accuracy (Approx) | Interquartile Range (IQR) | Whiskers/Outliers |
| :--- | :--- | :--- | :--- | :--- |
| **Black** | Male | 0.46 | ~0.12 to 0.91 | 0.0 to 1.0 |
| | Female | 0.42 | ~0.14 to 0.89 | 0.03 to 1.0 |
| **White** | Male | 0.28 | ~0.08 to 0.90 | 0.0 to 0.99 |
| | Female | 0.42 | ~0.08 to 0.93 | 0.01 to 1.0 |
| **Other Race**| Male | 0.37 | ~0.20 to 0.53 | 0.03 to 0.70 |
| | Female | 0.49 | ~0.16 to 0.83 | 0.03 to 0.97 |
| **Hispanic** | Male | 0.22 | ~0.09 to 0.56 | 0.02 to 0.99 |
| | Female | 0.50 | ~0.22 to 0.62 | 0.02 to 0.96 |
| **Asian** | Male | 0.38 | ~0.19 to 0.63 | 0.06 to 0.93 |
| | Female | ~0.84 | N/A (Single line) | No whiskers shown |
### Sub-plot 2: Bias Group = True
**General Trend:** Accuracy drops drastically. Most medians fall below 0.2. There is a high concentration of outliers in the "Black" category and a significant collapse in performance for "Hispanic" and "Asian" groups.
| Race | Gender | Median Accuracy (Approx) | Interquartile Range (IQR) | Whiskers/Outliers |
| :--- | :--- | :--- | :--- | :--- |
| **Black** | Male | 0.08 | ~0.04 to 0.25 | Whiskers to ~0.55; Outliers at ~0.56, 0.61, 0.62, 0.65, 1.0 |
| | Female | 0.26 | ~0.10 to 0.42 | 0.04 to 0.52; Outlier at ~0.91 |
| **White** | Male | 0.10 | ~0.03 to 0.42 | 0.01 to 0.98 |
| | Female | 0.19 | ~0.10 to 0.21 | 0.02 to 0.23; Outlier at 1.0 |
| **Other Race**| Male | 0.20 | ~0.04 to 0.38 | 0.01 to 0.49 |
| | Female | 0.35 | ~0.22 to 0.49 | 0.08 to 0.63 |
| **Hispanic** | Male | 0.06 | ~0.03 to 0.08 | 0.02 to 0.08; Outlier at ~0.17 |
| | Female | 0.08 | ~0.05 to 0.09 | 0.04 to 0.10 |
| **Asian** | Male | ~0.05 | N/A (Single line) | No whiskers shown |
| | Female | N/A | No data shown | No data shown |
---
## 4. Key Observations & Logic Check
1. **Bias Impact:** The introduction of "Bias Group = True" causes a visible downward shift in the entire distribution of accuracy scores across almost all demographics.
2. **Gender Disparity:** In the "Bias Group = True" condition, Females generally maintain a higher median accuracy than Males in the Black, White, and Other Race categories.
3. **Data Sparsity:** The "Asian" category in both plots and the "Hispanic" category in the "Bias Group = True" plot show very narrow IQRs or single lines, suggesting a much smaller sample size for these specific intersections.
4. **Outlier Presence:** The "Black Male" and "Black Female" categories under "Bias Group = True" show significant upward outliers, indicating that while the majority of the group has low accuracy, a small subset achieves very high accuracy (up to 1.0).
</details>
Figure 7: Analysis of bias detection model results. This plot displays the distribution of accuracy scores grouped by Race, Gender, and Bias, highlighting differences in model performance across different sub-groups.
Additionally, Fig. 7 visualizes the distribution of accuracy scores defined by race, gender, and bias group. Each subplot represents a different bias category, with individual boxes for each combination of race and gender. This plot illustrates notable differences in the accuracy scores between patient subgroups based on bias group identification. Furthermore, it highlights gender-based differences within the biased group, showing that, on average, this model performs worse for men.
8.1.2 Emory University Hospital
The final results of our bias detection framework for the Emory University Hospital cohort are presented in Fig. 8. Although Node 8 in Fig. 8a represents the group of patients with the worst model performance, the confidence intervals shown in Fig. 8b exhibit overlap across all terminal nodes. This overlap suggests that there is not enough evidence to indicate bias in the model’s performance for this cohort at significance level $\alpha^{*}=0.20$ . Furthermore, Fig. 8c illustrates the optimized significance level $\alpha^{*}$ across all leaf nodes, indicating that bias would only be detected at $\alpha=0.60$ , corresponding to a confidence level of 0.40.
<details>
<summary>x12.png Details</summary>
### Visual Description
# Technical Document Extraction: Bias Detection Analysis
This document provides a comprehensive extraction of the data and structural components from the provided image, which depicts a statistical analysis of bias detection across different nodes in a decision tree or similar hierarchical model.
## 1. Component Isolation
The image is divided into three primary functional regions:
* **Header:** Contains the root node identifier.
* **Main Chart (Left):** A bar chart showing confidence levels across specific nodes.
* **Decision Tree Fragment (Right):** A diagram showing the relationship and statistical properties of nodes 10, 11, and 12.
* **Footer:** Contains confidence interval (CI) data for specific nodes.
---
## 2. Header Information
* **Top Label:** `node #0` (contained in a light orange rounded rectangle).
---
## 3. Main Chart: Bias Detection Threshold
This is a bar chart measuring confidence levels against a predefined threshold.
### Metadata
* **Title:** Bias Detection Threshold
* **Y-Axis Label:** Max Confidence Level (1-α)
* **Y-Axis Scale:** 0.0 to 1.0 (increments of 0.2)
* **X-Axis Label:** Node
* **X-Axis Categories:** 3, 4, 5, 8, 9, 11, 12
* **Threshold Line:** A horizontal red dashed line is positioned at **y = 0.8**.
### Data Points (Bar Heights)
| Node | Visual Height (Approx. Value) | Trend Description |
| :--- | :--- | :--- |
| 3 | 0.2 | Low confidence. |
| 4 | 0.1 | Decrease from node 3. |
| 5 | 0.0 | No bar visible; value is zero or null. |
| 8 | 0.6 | Significant increase; highest bar in the set. |
| 9 | 0.5 | Slight decrease from node 8. |
| 11 | 0.5 | Equal to node 9. |
| 12 | 0.4 | Slight decrease from node 11. |
**Trend Observation:** All nodes remain below the red dashed "Bias Detection Threshold" of 0.8, indicating that none of the nodes have reached the critical threshold for bias detection based on the "Max Confidence Level."
---
## 4. Decision Tree Diagram (Right Side)
This section details the flow and statistical breakdown of a specific branch.
### Node #10 (Parent)
* **Condition:** `White > 0.5`
* **Samples:** 1062
* **Value:** 0.34
* **Std Dev (σ):** 0.32
* **80% CI:** [0.02, 0.88]
### Node #11 (Left Child of 10)
* **Samples:** 565
* **Value:** 0.32
* **Std Dev (σ):** 0.32
* **80% CI:** [0.02, 0.86]
### Node #12 (Right Child of 10)
* **Samples:** 497
* **Value:** 0.37
* **Std Dev (σ):** 0.33
* **80% CI:** [0.02, 0.89]
---
## 5. Footer: Confidence Intervals
The footer contains four distinct labels corresponding to the 80% Confidence Intervals for various nodes.
| Node Reference (Implicit) | Text Content | Color Coding |
| :--- | :--- | :--- |
| Node 3 (Approx) | `80% CI = [0.09, 0.95]` | Dark Orange |
| Node 4 (Approx) | `80% CI = [0.13, 0.93]` | Dark Orange |
| Node 8/9 (Approx) | `80% CI = [0.01, 0.6]` | White/Grey |
| Node 11/12 (Approx) | `80% CI = [0.02, 0.75]` | White/Grey |
---
## 6. Summary of Findings
The visualization indicates a bias detection process where multiple nodes (3, 4, 8, 9, 11, 12) are evaluated. While Node 8 shows the highest confidence level (~0.6), it still falls short of the 0.8 threshold required for a positive bias detection. The tree fragment shows that splitting Node 10 (based on the "White" attribute) results in two sub-groups (Nodes 11 and 12) with similar standard deviations but slightly different mean values (0.32 vs 0.37).
</details>
(a)
<details>
<summary>x13.png Details</summary>
### Visual Description
# Technical Document Extraction: Prediction Sets (80% CI)
## 1. Document Metadata
* **Title:** Prediction Sets (80% CI)
* **Chart Type:** Error Bar Plot (Point estimates with Confidence Intervals)
* **Primary Language:** English
## 2. Component Isolation
### Header
* **Title Text:** "Prediction Sets (80% CI)"
* **Context:** The chart visualizes accuracy metrics across different nodes, likely within a decision tree or hierarchical model, showing the mean accuracy and the 80% Confidence Interval (CI).
### Main Chart Area (Axes and Scales)
* **Y-Axis Label:** Accuracy
* **Y-Axis Scale:** 0.0 to 1.0 (increments of 0.2 marked: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0).
* **X-Axis Label:** Node
* **X-Axis Scale:** Discrete integer values from 0 to 12.
### Legend (Spatial Grounding: [x=0.95, y=0.50])
The legend is located on the center-right exterior of the plot.
* **Legend Title:** Nodes:
* **Category 1:** "Branch" (Represented by a Blue dot with Blue error bars).
* **Category 2:** "Terminal" (Represented by an Orange dot with Red/Brown error bars).
---
## 3. Data Extraction and Trend Verification
### Trend Analysis
* **Branch Nodes (Blue):** These nodes (0, 1, 2, 6, 7, 10) show a fluctuating accuracy. There is an initial slight upward trend (0-2), a significant drop at nodes 6 and 7, and a recovery at node 10.
* **Terminal Nodes (Orange/Red):** These nodes (3, 4, 5, 8, 9, 11, 12) appear in clusters. The first cluster (3-5) shows very stable, high accuracy (~0.53). The second cluster (8-9) shows a sharp dip followed by an increase. The final cluster (11-12) shows a slight upward trend.
### Data Table (Estimated Values)
The following table reconstructs the visual data points. Note: Accuracy values are point estimates (dots), and CI represents the vertical bars.
| Node | Node Type | Color (Point/Bar) | Estimated Accuracy (Mean) | Estimated 80% CI Range |
| :--- | :--- | :--- | :--- | :--- |
| 0 | Branch | Blue / Blue | 0.46 | [0.06, 0.86] |
| 1 | Branch | Blue / Blue | 0.52 | [0.12, 0.92] |
| 2 | Branch | Blue / Blue | 0.53 | [0.11, 0.94] |
| 3 | Terminal | Orange / Red | 0.52 | [0.09, 0.95] |
| 4 | Terminal | Orange / Red | 0.53 | [0.13, 0.93] |
| 5 | Terminal | Orange / Red | 0.54 | [0.11, 0.96] |
| 6 | Branch | Blue / Blue | 0.43 | [0.02, 0.84] |
| 7 | Branch | Blue / Blue | 0.36 | [0.03, 0.69] |
| 8 | Terminal | Orange / Red | 0.31 | [0.01, 0.60] |
| 9 | Terminal | Orange / Red | 0.39 | [0.02, 0.75] |
| 10 | Branch | Blue / Blue | 0.45 | [0.02, 0.88] |
| 11 | Terminal | Orange / Red | 0.44 | [0.02, 0.86] |
| 12 | Terminal | Orange / Red | 0.46 | [0.02, 0.89] |
---
## 4. Technical Observations
* **Variance:** All nodes exhibit extremely wide confidence intervals, spanning approximately 0.6 to 0.8 units of accuracy. This indicates high uncertainty in the prediction sets for every node.
* **Comparative Performance:** Branch nodes 1 and 2, and Terminal nodes 3, 4, and 5 represent the peak accuracy performance (exceeding 0.5).
* **Lowest Performance:** Node 8 (Terminal) represents the lowest mean accuracy at approximately 0.31.
* **Visual Encoding:** The distinction between "Branch" and "Terminal" is made through both the point color (Blue vs. Orange) and the error bar color (Blue vs. Red/Brown), allowing for clear categorization of the model's structural components.
</details>
(b)
<details>
<summary>x14.png Details</summary>
### Visual Description
# Technical Data Extraction: Bias Detection Threshold Chart
## 1. Document Overview
This image is a bar chart titled **"Bias Detection Threshold"**. It visualizes the confidence levels across various nodes relative to a predefined threshold.
## 2. Component Isolation
### Header
- **Title:** Bias Detection Threshold (Centered at top)
### Main Chart Area
- **Y-Axis Label:** Max Confidence Level (1-α)
- **X-Axis Label:** Node
- **Y-Axis Scale:** 0.0 to 1.0 (increments of 0.2)
- **X-Axis Categories:** 3, 4, 5, 8, 9, 11, 12
- **Threshold Line:** A horizontal dashed red line positioned at y = 0.8.
### Legend/Key
- There is no explicit text legend.
- **Visual Key:**
- **Blue Bars:** Represent the "Max Confidence Level" for each specific node.
- **Red Dashed Line:** Represents the "Bias Detection Threshold".
---
## 3. Data Extraction & Trend Verification
### Threshold Analysis
- **Value:** 0.8
- **Trend:** Constant across all nodes.
- **Observation:** All data points (bars) fall below this threshold, indicating that no node has reached the level required for bias detection according to this specific metric.
### Node Data Table
The following table reconstructs the values represented by the blue bars.
| Node (X-Axis) | Max Confidence Level (Y-Axis) | Status Relative to Threshold (0.8) |
| :--- | :--- | :--- |
| 3 | 0.2 | Below |
| 4 | 0.1 | Below |
| 5 | 0.0 | Below |
| 8 | 0.6 | Below |
| 9 | 0.5 | Below |
| 11 | 0.5 | Below |
| 12 | 0.4 | Below |
### Visual Trend Description
1. **Initial Low Phase:** Nodes 3 and 4 show very low confidence levels (0.2 and 0.1 respectively). Node 5 shows a null or zero value.
2. **Significant Increase:** There is a sharp upward trend moving from Node 5 to Node 8, where the confidence level peaks for this dataset at 0.6.
3. **Plateau and Slight Decline:** Following the peak at Node 8, the values stabilize at 0.5 for Nodes 9 and 11, before dipping slightly to 0.4 at Node 12.
4. **Overall Conclusion:** The data shows a "hump" distribution centered around Node 8, but remains consistently below the critical 0.8 threshold line.
---
## 4. Technical Summary
The chart evaluates seven specific nodes (3, 4, 5, 8, 9, 11, 12) against a **Bias Detection Threshold of 0.8**.
- **Maximum Value Recorded:** 0.6 (Node 8).
- **Minimum Value Recorded:** 0.0 (Node 5).
- **Compliance:** 100% of the nodes are below the bias detection threshold, as no blue bar intersects or exceeds the red dashed line.
</details>
(c)
Figure 8: Emory bias detection model results. 8a displays the complete decision tree, 8b shows the predicted confidence intervals $\hat{\mathcal{C}}_{j}$ for each branch (blue) and terminal (red) node at significance level $\alpha$ . 8c presents the maximum bias detection confidence level $1-\alpha^{*}_{j}$ for the $j^{\text{th}}$ terminal node.
9 Conclusion
This paper introduces a novel approach to detecting and analyzing regions of algorithmic bias in medical-AI decision support systems. Our framework leverages the Classification and Regression Trees (CART) method, enhanced with conformal prediction intervals, to provide a robust mechanism for detecting and addressing potential biases in AI applications within the healthcare sector. We evaluated our technique through synthetic data experiments, demonstrating its capability to identify regions of bias, assuming such regions exist in the data. Furthermore, we extended our analysis to a real-world dataset by conducting an experiment using electronic health record (EHR) data obtained from Grady Memorial Hospital. The integration of conformal prediction intervals with the CART algorithm allows users to test a variety of confidence levels, thereby providing a flexible tool for determining the existence of algorithmic bias. By adjusting the confidence levels, users can explore the robustness of the bias detection across different thresholds, enhancing the reliability of the findings.
The increasing integration of machine learning and artificial intelligence in healthcare underscores the urgent need for tools, techniques, and procedures that ensure the fair and equitable use of these technologies. Our framework addresses this challenge by offering a practical solution for healthcare practitioners and AI developers to identify and mitigate algorithmic biases. This, in turn, promotes the development of medical ML/AI decision support systems that are both ethically sound and clinically effective.
Acknowledgement
This work is partially supported by an NSF CAREER CCF-1650913, NSF DMS-2134037, CMMI-2015787, CMMI-2112533, DMS-1938106, DMS-1830210, NIGMS K23GM137182-03S1, Emory Hospital, and the Coca-Cola Foundation.
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Appendix A Data pre-processing
These datasets include a diverse range of continuous physiological measurements, vital signs, laboratory results, and medical treatment information for each encounter. Data also incorporated demographic information from the patient, including age, sex, race, zip code, and insurance status, which we utilize in later stages of the study. We perform feature reduction by removing physiological features missing more than 75% of their records. This resulted in 39 continuous patient features remaining for analysis as denoted by Table 6. In addition, we included two administrative identifiers: procedure and ventilation status.
Table 6: Patient physiologic features selected for analysis
We impute missing data through a forward-filling approach. When a feature $x$ has a previously recorded value, $v$ , at time step $t_{p}<t$ , we set $x_{v}^{(t)}=x_{v}^{t_{p}}$ to forward-fill the missing value of $v$ at time step $t$ . If no prior recorded value exists, the missing value remains unprocessed. Lastly, to mitigate data leakage, we remove sepsis patient data following their first retrospectively identified sepsis hour.
A.1 Feature engineering
Following our initial data pre-processing, which resulted in 41 selected physiological patient features, we further develop three categories of variables in this section. These include 72 variables for indicating the informativeness of missing features, 89 time-series based features, and eight clinically relevant features for assessing sepsis. The final dataset, following all feature engineering steps, resulted in a total of 210 features.
Feature informative missingness
The presence of missing data, a common occurrence in routinely collected health information, can provide significant insights, as the nature of the missing data itself can be informative [35]. The collection times for clinical laboratory and treatment information fluctuate among individuals and may vary throughout their treatment period, resulting in a significant number of missing entries in the physiological data, including instances where entire features are absent. This phenomenon of missing data, particularly prevalent in ICU settings, is not without pattern as it often reflects the clinical judgments made regarding a patient’s critical condition. We introduce two missing data indicator sequences for 36 specific variables, which include all lab values, ventilation status, systolic blood pressure, diastolic blood pressure, and mean arterial pressure, with the aim to harness the latent predictive value embedded within these missing data points. The Measurement Frequency (f1) sequence counts the number of measurements taken for a variable before the current time. The Measurement Time Interval (f2) sequence records the time interval from the most recent measurement to the current time. A value of $-1$ is assigned when there is no prior recorded measurement.
Table 7 illustrates the application of two missing data indicator sequences through an example of an eight-hour time series for temperature measurements. The first row displays the temperature readings over time. The second row shows the measurement frequency sequence, indicating the cumulative number of temperature measurements taken up to each point in time. The final row presents the measurement time interval sequence, highlighting the time elapsed since the last temperature measurement, with a notation of -1 when there is no previous measurement to reference.
Table 7: Example of feature informative missingness sequences
| f1 score-f2 score | 0 $-1$ | 1 0 | 2 0 | 2 1 | 2 2 | 3 0 | 3 1 | 4 0 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
Clinical empiric features
Historically, rule-based severity scoring systems for diseases like the Sequential Organ Failure Assessment (SOFA) [36], quick-SOFA (qSOFA) [32], and the National Early Warning System (NEWS) [37] have been used to define sepsis in clinical settings. However, these systems may not satisfy the critical need for timely detection of sepsis to initiate effective treatment [38]. We highlight the importance of several measurements to quantify abnormalities according to some scoring system. The qSOFA score is identified as “1” with Systolic BP (SBP) $≤$ 100 mm Hg and Respiration rate (Resp) $≥$ 22/min, otherwise “0”. The measurements of platelets, bilirubin, mean arterial pressure (MAP), and creatinine are scored respectively under the rules of SOFA score, while heart rate, temperature, and respiration rate are scored on the basis of the NEWS score.
Time series features
To capture the dynamic changes in patients’ data records, we calculate two types of time-series features as follows.
- Differential features: These are derived by computing the difference between the current value and the previous measurement of a given feature. This calculation highlights the immediate changes in patient conditions.
- Sliding-window-based statistical features: For this analysis, we focus on eight vital sign measurements: Best Mean Arterial Pressure (MAP), Heart Rate (HR), Oxygen Saturation (SpO2), Respiratory Rate, Temperature, Diastolic Blood Pressure (DBP), Systolic Blood Pressure (SBP), and Mean Arterial Pressure (MAP). We employ a fixed-length rolling six-hour sliding window to segment each record. This fixed rolling window increments in one-hour steps. In instances where the window is less than six hours, the sliding window includes all available data. Finally, we calculate key statistical features for each window, including maximum, minimum, mean, median, standard deviation, and differential standard deviation for each of the selected measurements.
Sepsis label lead time
This study aims to develop a prognostic model that can accurately predict the onset of sepsis up to six hours before it happens. To highlight the significance of identifying sepsis at an early stage, we have introduced a six-hour lead time on the sepsis indicator variable. This adjustment enables the model to specifically focus on and recognize probable sepsis cases before they completely develop, thereby improving the model’s ability to forecast outcomes in clinical settings.
Appendix B XGBoost Model
The sepsis prediction model developed for this analysis was centered on the implementation of XGBoost [39], a robust tree-based gradient boosting algorithm known for its high computational efficiency and exceptional performance in managing complex and large datasets. We constructed this model using the Bayesian optimization technique with a Tree-structured Parzen Estimator (TPE) [40] approach. We applied this method to optimize hyperparameters, which helped establish the learning process, complexity, and generalization capability of the model. Hyperparameters included but were not limited to, the following: max depth, learning rate, and alpha and lambda regularization terms.
The Bayesian optimization technique involved a series of 20 evaluations. In each iteration, we tune the hyperparameters with the aim of maximizing the accuracy of the prediction model. The final model is an ensemble based on the average five-fold cross-validation performance measured across this accuracy optimized loss function.
B.1 Training, validation, and test sets
In crafting our machine learning model, we incorporated a nuanced approach that integrates stratified cross-validation, temporal partitioning of data, and ensemble techniques to address the inherent challenges of predicting sepsis through the use of temporal dataset. This framework is specifically designed to evaluate models on future, unobserved data, thus closely simulating real-world clinical forecasting scenarios and enhancing the model’s external validity. Our stratification strategy ensures that each subset for training and validation is a representative sample of the entire dataset by addressing class imbalance across folds. We incorporate an ensemble methodology to leverage the collective insights from multiple models, with the aim to reduce variability and enhance the reliability across predictions.
To construct our training, validation, and testing datasets we initially divided the dataset temporally, creating two groups: one with patients admitted to the ICU prior to 2019, designated for training and validation purposes, and the other comprised of patients from 2019 onwards for testing. Within the pre-2019 dataset, we performed stratified five-fold cross-validation to further partition the data into five exhaustive and mutually exclusive subsets. We execute this stratification with respect to the sepsis label to guarantee that each fold contains a proportional distribution of cases, both septic and non-septic.
Within each of these five stratified folds we include all relevant continuous physiological data for each patient, reflecting the previously mentioned comprehensive feature engineering process that was undertaken. We further temporally partition this data, allocating the initial 24 hours of records following a patient’s admission to the ICU to the training set, and the subsequent records, up to the 168th hour, to the validation set. This 168-hour cap is strategically selected to reduce the potential impacts of data bias that might arise from complications affecting a patient’s health status beyond the initial week of their ICU stay. To address the imbalance between sepsis and non-sepsis hours, we also undertake a down sampling of the non-sepsis instances within each fold. Each fold thus generates a model trained on its designated training data subset and validated on its respective validation set. Collectively, these models form an ensemble, capitalizing on the variability and strengths of each model trained and validated on slightly different data segments. Fig. 9 depicts the complete data pre-processing and model development pipeline using the Grady dataset.
<details>
<summary>x15.png Details</summary>
### Visual Description
# Technical Document Extraction: Study Flowchart and Machine Learning Pipeline
This document provides a comprehensive extraction of the data and structural flow represented in the provided image, which details a clinical study's methodology for sepsis prediction using machine learning.
## 1. Study Population and Data Filtering (Header Region)
The top section of the flowchart describes the initial cohort selection and exclusion criteria.
| Step | Description | Population Size (n) |
| :--- | :--- | :--- |
| **Initial Cohort** | Adult (age $\ge$ 18) ICU patient visits identified between 2016 and 2020. | $n = 119,733$ |
| **Filtered Cohort** | ICU patient visits with $\ge$ 24 hours of EHR data and no sepsis diagnosis within first six hours of admission. | $n = 10,274$ |
**Associated Statistics (Right-side callout):**
* **Number of sepsis cases:** 1,770 (17.23%)
* **Number of non-sepsis cases:** 8,504 (82.77%)
---
## 2. Data Processing and Temporal Partitioning (Main Chart Region)
Following filtering, the data undergoes "Data pre-processing & Feature Engineering" before being split by time.
### Temporal Partitioning
The data is divided into two primary temporal blocks:
1. **ICU Admission: 2016 – 2018**
* **Count:** $n = 6,364$
* **Breakdown:** 1,195 Sepsis cases / 5,169 Non-sepsis cases.
2. **ICU Admission: 2019 – 2020**
* **Count:** $n = 3,910$
* **Breakdown:** 575 Sepsis cases / 3,335 Non-sepsis cases.
---
## 3. Machine Learning Pipeline and Validation (Lower Region)
The pipeline utilizes the 2016–2018 data for model development and the 2019–2020 data for final testing.
### Cross-Validation Strategy
The 2016–2018 cohort is processed through **Stratified 5-Fold Cross-Validated Splits**.
* **Temporal Splits (hours) Legend:** A visual box indicates a timeline from 0 to 168 hours. A shaded region represents the 0–24 hour window.
* The diagram shows five boxes, each with a different segment shaded, representing the iterative nature of the 5-fold cross-validation.
### Model Development Flow
1. **Data Split:** The cross-validation data is split into a **Training set** and a **Validation set**.
2. **Algorithm:** The training and validation sets feed into an **XGBoost Classifier & Bayesian Optimization** process.
3. **Aggregation:** The results lead to an **Ensemble Model**.
### Final Evaluation
* The **Testing set** (derived from the 2019–2020 temporal partition) bypasses the training/validation loop and is fed directly into the final stage.
* **Output Results:** The final stage where the Ensemble Model's performance is measured against the Testing set.
---
## 4. Component Summary and Flow Logic
* **Language:** The document is entirely in English.
* **Flow Direction:** Top-to-bottom linear flow for data acquisition, branching into a parallel structure for temporal partitioning, and converging at the final output results.
* **Logic Check:** The sum of the temporal partitions ($6,364 + 3,910 = 10,274$) matches the filtered cohort total exactly. The sum of sepsis cases ($1,195 + 575 = 1,770$) and non-sepsis cases ($5,169 + 3,335 = 8,504$) also matches the reported statistics exactly, confirming data integrity across the diagram.
</details>
Figure 9: Illustration of the data pre-processing and model development procedure of the Grady sepsis prediction model.
B.2 Model results
Table 8 provides a comparative summary of the performance of individual XGBoost models and the ensemble model across both cohorts—Grady Memorial Hospital and Emory University Hospital. The table presents a horizontal comparison, reporting the accuracy and area under the curve (AUC) for each model across each cross-validation fold.
Table 8: Performance of different models on local test set formed by ourselves
| | Grady | Emory | | |
| --- | --- | --- | --- | --- |
| XGBoost Models (Folds) | Accuracy | AUC | Accuracy | AUC |
| 1 | 0.840 | 0.728 | 0.637 | 0.643 |
| 2 | 0.843 | 0.732 | 0.640 | 0.648 |
| 3 | 0.791 | 0.712 | 0.670 | 0.629 |
| 4 | 0.790 | 0.711 | 0.602 | 0.643 |
| 5 | 0.790 | 0.712 | 0.691 | 0.647 |
| Average | 0.814 | 0.722 | 0.651 | 0.646 |
| Ensemble Model | 0.824 | 0.738 | 0.665 | 0.667 |
Fig. 10 provides a comprehensive visualization of the sepsis prediction models’ performance for both Grady and Emory cohorts, across multiple evaluation metrics. The first row represents results from the model trained on Grady data, while the second row corresponds to the model trained on Emory data. These results are further categorized by the training and testing phases of model development. Figs. 10a and 10e depict confusion matrices based on the respective training datasets. The receiver operator characteristics (ROC) curves, shown in Figs. 10b and 10f, evaluate the model’s ability to generalize to unseen test data. Figs. 10c and 10g present confusion matrices for the test datasets, highlighting each model’s predictive accuracy on unseen data. Finally, Figs. 10d and 10h display the ROC curves for the test data. Table 9 provides a detailed summary of the classification performance metrics across both cohorts, providing further insights into the accuracy, precision, recall, F1-score, and F2-score for each model.
<details>
<summary>x16.png Details</summary>
### Visual Description
# Technical Document Extraction: Sepsis Prediction Model (Training) Confusion Matrix
## 1. Document Overview
This image is a technical visualization of a **Confusion Matrix** for a machine learning model titled **"Sepsis Prediction Model (Training)"**. It evaluates the performance of a binary classifier by comparing predicted labels against ground truth (true labels).
## 2. Component Isolation
### Header
* **Title:** Sepsis Prediction Model (Training)
### Main Chart (Heatmap)
* **Y-Axis (True Label):** Categories are "sepsis" (top) and "no sepsis" (bottom).
* **X-Axis (Predicted Label):** Categories are "sepsis" (left) and "no sepsis" (right).
* **Cell Content:** Each of the four quadrants contains a classification acronym, an absolute integer count, and a percentage representing the row-wise normalized value.
### Legend (Color Bar)
* **Placement:** Located on the far right of the image.
* **Scale:** A continuous gradient of blue shades.
* **Markers:** 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8.
* **Function:** Represents the normalized ratio (0.2 to 0.8+) of the predictions within each class.
---
## 3. Data Table Reconstruction
The following table transcribes the data points contained within the heatmap cells:
| True Label \ Predicted Label | sepsis (Predicted) | no sepsis (Predicted) |
| :--- | :--- | :--- |
| **sepsis (True)** | **TP (True Positive)**<br>Count: 682<br>Percentage: 74.29% | **FN (False Negative)**<br>Count: 236<br>Percentage: 25.71% |
| **no sepsis (True)** | **FP (False Positive)**<br>Count: 151,326<br>Percentage: 19.34% | **TN (True Negative)**<br>Count: 631,214<br>Percentage: 80.66% |
---
## 4. Trend and Statistical Analysis
### Spatial Grounding & Color Verification
* **Top-Left (TP):** Dark blue color. Matches the ~0.74 mark on the legend. This indicates a high success rate in identifying actual sepsis cases.
* **Bottom-Right (TN):** Darkest blue color. Matches the >0.8 mark on the legend. This indicates the highest success rate in identifying non-sepsis cases.
* **Top-Right (FN):** Lightest blue/grey. Matches the ~0.25 mark on the legend.
* **Bottom-Left (FP):** Very light blue. Matches the ~0.19 mark on the legend.
### Performance Summary
1. **Sensitivity (Recall):** The model correctly identifies **74.29%** of actual sepsis cases (682 out of 918 total sepsis cases).
2. **Specificity:** The model correctly identifies **80.66%** of non-sepsis cases (631,214 out of 782,540 total non-sepsis cases).
3. **Class Imbalance:** There is a massive class imbalance in the training set. There are only **918** total "sepsis" instances compared to **782,540** "no sepsis" instances.
4. **Error Analysis:**
* The model missed 236 sepsis cases (False Negatives).
* The model incorrectly flagged 151,326 healthy cases as sepsis (False Positives). Due to the high volume of "no sepsis" data, the absolute number of False Positives is significantly higher than the True Positives, despite the percentage (19.34%) being relatively low.
## 5. Language Declaration
The text in this image is entirely in **English**. No other languages were detected.
</details>
(a)
<details>
<summary>x17.png Details</summary>
### Visual Description
# Technical Document Extraction: Receiver Operating Characteristic (ROC) Curve
## 1. Component Isolation
* **Header:** Contains the main title of the chart.
* **Main Chart Area:** A square plot area with a grid, containing two data series (a solid curve and a dashed diagonal line).
* **Axes:** Horizontal (X) and Vertical (Y) axes with numerical scales and labels.
* **Legend:** Located in the bottom-right quadrant of the plot area.
---
## 2. Metadata and Labels
* **Title:** Receiver Operating Characteristic Curve (Training)
* **Y-Axis Label:** TPR (True Positive Rate)
* **X-Axis Label:** FPR (False Positive Rate)
* **Language:** English
---
## 3. Axis Scales and Markers
Both axes represent probabilities/rates ranging from 0.0 to 1.0.
| Axis | Minimum | Maximum | Major Tick Intervals |
| :--- | :--- | :--- | :--- |
| **X-Axis (FPR)** | 0.0 | 1.0 | 0.2 (0.0, 0.2, 0.4, 0.6, 0.8, 1.0) |
| **Y-Axis (TPR)** | 0.0 | 1.0 | 0.2 (0.0, 0.2, 0.4, 0.6, 0.8, 1.0) |
---
## 4. Legend and Data Series Analysis
The legend is positioned at approximately **[x=0.85, y=0.1]** relative to the plot area.
### Series 1: ROC Curve
* **Color:** Dark Blue
* **Line Style:** Solid
* **Legend Label:** ROC curve (area = 0.85)
* **Trend Verification:** The line starts at the origin (0,0) and slopes upward with a steep initial gradient. As it moves toward the top-right, the slope decreases (concave down), indicating diminishing returns in True Positive Rate for every increase in False Positive Rate. It eventually reaches the point (1,1).
* **Key Data Points (Estimated):**
* (0.0, 0.0)
* (0.1, 0.5)
* (0.2, 0.75)
* (0.4, 0.9)
* (0.6, 0.97)
* (1.0, 1.0)
### Series 2: Baseline
* **Color:** Orange/Gold
* **Line Style:** Dashed
* **Legend Label:** Baseline
* **Trend Verification:** This is a perfectly linear diagonal line starting at (0,0) and ending at (1,1). It represents a "Random Classifier" with an Area Under the Curve (AUC) of 0.5.
* **Key Data Points:**
* Linear progression where FPR = TPR (e.g., 0.2, 0.2; 0.5, 0.5; 0.8, 0.8).
---
## 5. Technical Summary
This image displays the performance of a classification model on a training dataset.
* **Performance Metric:** The Area Under the Curve (AUC) is explicitly stated as **0.85**.
* **Interpretation:** Since the dark blue ROC curve stays significantly above the orange dashed baseline (AUC 0.5), the model demonstrates strong predictive power. An AUC of 0.85 suggests that there is an 85% chance that the model will be able to distinguish between positive and negative classes correctly.
* **Visual Characteristics:** The plot includes a light gray grid to facilitate the reading of specific coordinate values. The curve is relatively smooth, suggesting a continuous probability output from the classifier.
</details>
(b)
<details>
<summary>x18.png Details</summary>
### Visual Description
# Technical Document Extraction: Sepsis Prediction Model (Test) Confusion Matrix
## 1. Document Overview
This image is a **Confusion Matrix** representing the performance of a "Sepsis Prediction Model" on a test dataset. It visualizes the relationship between the actual ground truth (True label) and the model's classifications (Predicted label).
## 2. Component Isolation
### Header
* **Title:** Sepsis Prediction Model (Test)
### Main Chart (Heatmap)
The chart is a 2x2 grid. The color intensity represents the normalized proportion of predictions within each true class row.
* **Y-Axis (True label):** Categories are "sepsis" (top) and "no sepsis" (bottom).
* **X-Axis (Predicted label):** Categories are "sepsis" (left) and "no sepsis" (right).
### Legend (Color Bar)
* **Location:** Right side of the chart.
* **Type:** Gradient scale (light blue to dark blue).
* **Range:** Approximately 0.2 to 0.8.
* **Function:** Indicates the normalized ratio of the row total. Darker blue indicates a higher percentage of correct or incorrect classifications relative to that specific true label.
## 3. Data Table Reconstruction
The following table transcribes the numerical data and labels found within the four quadrants of the matrix.
| True Label \ Predicted Label | sepsis (Predicted) | no sepsis (Predicted) | Row Totals (Calculated) |
| :--- | :--- | :--- | :--- |
| **sepsis (True)** | **TP (True Positive)**<br>Count: 286<br>Percentage: 64.13% | **FN (False Negative)**<br>Count: 160<br>Percentage: 35.87% | 446 |
| **no sepsis (True)** | **FP (False Positive)**<br>Count: 81,098<br>Percentage: 17.19% | **TN (True Negative)**<br>Count: 390,794<br>Percentage: 82.81% | 471,892 |
## 4. Trend and Data Analysis
### Trend Verification
* **True Sepsis Row:** The "TP" cell is a medium-dark blue, while the "FN" cell is a light blue. This indicates the model is more likely to correctly identify sepsis than to miss it when it is present.
* **True No Sepsis Row:** The "TN" cell is the darkest blue in the entire chart, while the "FP" cell is nearly white. This indicates a very high success rate in identifying the absence of sepsis, though the raw number of false positives is high due to class imbalance.
### Key Performance Metrics (Derived from Data)
* **Sensitivity (Recall):** 64.13% (The model correctly identifies roughly 2/3 of actual sepsis cases).
* **Specificity:** 82.81% (The model correctly identifies over 4/5 of non-sepsis cases).
* **Class Imbalance:** There is a massive disparity in the dataset size. There are only 446 "sepsis" instances compared to 471,892 "no sepsis" instances.
* **False Positive Impact:** While the FP rate is relatively low (17.19%), because the "no sepsis" population is so large, the model generated 81,098 false alarms, which significantly outweighs the 286 true positives.
</details>
(c)
<details>
<summary>x19.png Details</summary>
### Visual Description
# Technical Document Extraction: Receiver Operating Characteristic (ROC) Curve
## 1. Header Information
* **Title:** Receiver Operating Characteristic Curve (Test)
* **Language:** English
## 2. Chart Metadata and Axes
* **Chart Type:** ROC Curve (Line Graph)
* **X-Axis Label:** FPR (False Positive Rate)
* **X-Axis Scale:** 0.0 to 1.0 (increments of 0.2 marked, with grid lines every 0.2)
* **Y-Axis Label:** TPR (True Positive Rate)
* **Y-Axis Scale:** 0.0 to 1.0 (increments of 0.2 marked, with grid lines every 0.2)
* **Grid:** Present (major grid lines at 0.2 intervals for both axes)
## 3. Legend Analysis
* **Location:** Bottom-right corner
* **Series 1:**
* **Label:** ROC curve (area = 0.81)
* **Color/Style:** Solid dark blue line
* **Series 2:**
* **Label:** Baseline
* **Color/Style:** Dashed orange/gold line
## 4. Data Series and Trend Verification
### Series 1: ROC Curve (Solid Blue Line)
* **Trend Description:** The line originates at [0,0] and slopes steeply upward and to the right. It exhibits a concave downward shape, indicating that the True Positive Rate increases significantly faster than the False Positive Rate in the initial stages. The curve stays consistently above the baseline, eventually flattening out as it approaches the top-right corner [1,1].
* **Key Data Points (Estimated):**
* [0.0, 0.0]: Origin
* [0.1, 0.45]: Rapid initial ascent.
* [0.2, 0.70]: The "elbow" of the curve where the rate of TPR gain begins to slow relative to FPR.
* [0.4, 0.85]: Continued gradual ascent.
* [0.6, 0.93]: Approaching the upper limit.
* [0.8, 0.98]: Near-perfect TPR.
* [1.0, 1.0]: Termination point.
* **Performance Metric:** The Area Under the Curve (AUC) is explicitly stated as **0.81**, indicating a good predictive model performance.
### Series 2: Baseline (Dashed Orange Line)
* **Trend Description:** A perfectly linear diagonal line sloping upward at a 45-degree angle from the origin to the top-right corner.
* **Key Data Points:**
* [0.0, 0.0]
* [0.5, 0.5]
* [1.0, 1.0]
* **Significance:** This represents a "random guess" classifier with an AUC of 0.5.
## 5. Component Isolation Summary
* **Header:** Contains the descriptive title identifying the data as "Test" results.
* **Main Chart Area:** Displays the relationship between sensitivity (TPR) and 1-specificity (FPR). The blue curve's position significantly above the diagonal baseline confirms the model has discriminative power.
* **Footer/Legend:** Provides the quantitative evaluation (AUC = 0.81) and defines the visual encoding for the two plotted lines.
</details>
(d)
<details>
<summary>x20.png Details</summary>
### Visual Description
# Technical Document Extraction: Sepsis Prediction Model (Training) Confusion Matrix
## 1. Document Header
* **Title:** Sepsis Prediction Model (Training)
* **Language:** English
## 2. Component Isolation
### Region A: Axis Labels and Categories
* **Y-Axis (Vertical):** "True label"
* Categories: `sepsis` (Top), `no sepsis` (Bottom)
* **X-Axis (Horizontal):** "Predicted label"
* Categories: `sepsis` (Left), `no sepsis` (Right)
### Region B: Main Heatmap Data
The image is a 2x2 confusion matrix. Each cell contains a classification acronym, a raw count, and a percentage. The color intensity (blue) correlates with the percentage value within each row.
| True Label | Predicted: sepsis | Predicted: no sepsis | Row Totals (Calculated) |
| :--- | :--- | :--- | :--- |
| **sepsis** | **TP (True Positive)**<br>Count: 170,263<br>Percentage: 74.80% | **FN (False Negative)**<br>Count: 57,372<br>Percentage: 25.20% | 227,635 (100%) |
| **no sepsis** | **FP (False Positive)**<br>Count: 1,096,321<br>Percentage: 32.36% | **TN (True Negative)**<br>Count: 2,291,455<br>Percentage: 67.64% | 3,387,776 (100%) |
### Region C: Legend (Color Bar)
* **Spatial Placement:** Located on the far right of the image.
* **Type:** Continuous gradient scale.
* **Color Mapping:** Lightest blue/white represents lower values (~0.3), while dark navy blue represents higher values (~0.7+).
* **Markers:** 0.3, 0.4, 0.5, 0.6, 0.7.
## 3. Trend and Data Verification
### Spatial Grounding & Color Check
* **Top-Left (TP):** Darkest blue. Matches the >0.7 mark on the legend (Value: 74.80%).
* **Bottom-Right (TN):** Medium-Dark blue. Matches the ~0.67 mark on the legend (Value: 67.64%).
* **Bottom-Left (FP):** Light blue. Matches the ~0.32 mark on the legend (Value: 32.36%).
* **Top-Right (FN):** White/Very Light blue. Matches the <0.3 mark on the legend (Value: 25.20%).
### Visual Trend Analysis
1. **Diagonal Dominance:** The "True Positive" and "True Negative" cells (top-left to bottom-right diagonal) are significantly darker than the off-diagonal cells. This indicates the model has a higher rate of correct predictions than incorrect ones for both classes.
2. **Class Imbalance:** There is a massive disparity in raw counts. The "no sepsis" true labels (totaling over 3.3 million) far outnumber the "sepsis" true labels (totaling approximately 227k), indicating a highly imbalanced training dataset.
3. **Sensitivity vs. Specificity:**
* The model correctly identifies 74.80% of actual sepsis cases (Sensitivity/Recall).
* The model correctly identifies 67.64% of actual non-sepsis cases (Specificity).
## 4. Summary of Facts
* **Total Samples in Training Set:** 3,615,411
* **Correct Predictions (TP + TN):** 2,461,718
* **Incorrect Predictions (FP + FN):** 1,153,693
* **Primary Observation:** While the model captures nearly 75% of sepsis cases, it generates a high volume of False Positives (over 1 million), which is reflected in the relatively lower percentage of 67.64% for True Negatives.
</details>
(e)
<details>
<summary>x21.png Details</summary>
### Visual Description
# Technical Document Extraction: Receiver Operating Characteristic (ROC) Curve
## 1. Document Metadata
* **Title:** Receiver Operating Characteristic Curve (Training)
* **Type:** Statistical Chart / Performance Metric Plot
* **Language:** English
## 2. Component Isolation
### Header
* **Text:** "Receiver Operating Characteristic Curve (Training)"
* **Position:** Top center.
### Main Chart Area
* **X-Axis Label:** "FPR" (False Positive Rate)
* **Y-Axis Label:** "TPR" (True Positive Rate)
* **X-Axis Scale:** 0.0 to 1.0 (increments of 0.2 marked: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
* **Y-Axis Scale:** 0.0 to 1.0 (increments of 0.2 marked: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
* **Grid:** Light gray horizontal and vertical grid lines corresponding to the 0.2 increments.
* **Background:** Light lavender/gray shaded area.
### Legend
* **Spatial Placement:** Bottom-right quadrant (approximate [x=0.6, y=0.1] in normalized coordinates).
* **Entry 1:** Solid dark blue line — "ROC curve (area = 0.78)"
* **Entry 2:** Dashed orange line — "Baseline"
---
## 3. Data Series Analysis and Trend Verification
### Series 1: ROC Curve
* **Color/Style:** Solid dark blue line.
* **Visual Trend:** The line starts at the origin (0,0) and exhibits a steep upward slope initially. As the FPR increases, the slope gradually decreases (concave down), eventually terminating at the top-right corner (1,1). The curve stays consistently above the baseline.
* **Key Data Points (Estimated):**
| FPR | TPR |
| :--- | :--- |
| 0.0 | 0.0 |
| 0.1 | 0.4 |
| 0.2 | 0.6 |
| 0.4 | 0.8 |
| 0.6 | 0.9 |
| 0.8 | 0.96 |
| 1.0 | 1.0 |
* **Metric:** Area Under the Curve (AUC) = 0.78.
### Series 2: Baseline
* **Color/Style:** Dashed orange line.
* **Visual Trend:** A perfectly linear diagonal line sloping upward at a 45-degree angle from the origin (0,0) to the maximum point (1,1).
* **Significance:** Represents a random classifier (AUC = 0.50).
---
## 4. Summary of Information
This chart visualizes the performance of a binary classification model on a training dataset.
* **Performance:** The model performs significantly better than a random guess, as indicated by the blue ROC curve bowing toward the top-left corner of the plot.
* **Discriminative Power:** With an Area Under the Curve (AUC) of **0.78**, the model has a "fair" to "good" ability to distinguish between the positive and negative classes.
* **Trade-off:** At a False Positive Rate (FPR) of 0.2, the model achieves a True Positive Rate (TPR) of approximately 0.6. To reach a TPR of 0.8, the model must accept an FPR of approximately 0.4.
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<details>
<summary>x22.png Details</summary>
### Visual Description
# Technical Document Extraction: Sepsis Prediction Model Confusion Matrix
## 1. Header Information
* **Title:** Sepsis Prediction Model (Test)
* **Language:** English
## 2. Component Isolation
### A. Main Chart: Confusion Matrix
The image displays a 2x2 confusion matrix (heatmap) evaluating the performance of a sepsis prediction model on a test dataset.
* **Y-Axis (True Label):**
* Top Row: `sepsis`
* Bottom Row: `no sepsis`
* **X-Axis (Predicted Label):**
* Left Column: `sepsis`
* Right Column: `no sepsis`
### B. Legend (Color Bar)
* **Location:** Right side of the chart.
* **Type:** Sequential blue color scale.
* **Range:** 0.35 (lightest blue/white) to 0.65 (darkest blue).
* **Function:** Represents the normalized proportion (percentage) of the row-wise classification.
---
## 3. Data Table Reconstruction
The following table represents the raw counts and row-normalized percentages extracted from the matrix cells.
| True \ Predicted | sepsis (Predicted) | no sepsis (Predicted) | Row Totals (Calculated) |
| :--- | :--- | :--- | :--- |
| **sepsis (True)** | **TP (True Positive)**<br>Count: 1198<br>Percentage: 66.93% | **FN (False Negative)**<br>Count: 592<br>Percentage: 33.07% | 1,790 |
| **no sepsis (True)** | **FP (False Positive)**<br>Count: 676391<br>Percentage: 33.51% | **TN (True Negative)**<br>Count: 1342175<br>Percentage: 66.49% | 2,018,566 |
---
## 4. Trend and Statistical Analysis
### Spatial Grounding & Color Verification
* **Dark Blue Cells (Diagonal):** The cells for **TP** (Top-Left) and **TN** (Bottom-Right) are the darkest blue. According to the color bar, these correspond to values > 0.65. This indicates the model correctly identifies both classes approximately 66-67% of the time relative to their true occurrence.
* **Light Blue/White Cells (Off-Diagonal):** The cells for **FN** (Top-Right) and **FP** (Bottom-Left) are light blue/white. These correspond to values between 0.33 and 0.35 on the color bar.
### Key Performance Metrics (Derived)
* **Sensitivity (Recall):** 66.93% (The ability to correctly identify patients with sepsis).
* **Specificity:** 66.49% (The ability to correctly identify patients without sepsis).
* **Class Imbalance:** There is a massive disparity in the sample sizes. The "no sepsis" group (2,018,566) is significantly larger than the "sepsis" group (1,790), representing a highly imbalanced dataset typical of medical screening.
* **False Positive Volume:** While the percentage of False Positives is relatively low (33.51%), the absolute number is very high (676,391) due to the large "no sepsis" population.
## 5. Summary of Findings
The model demonstrates balanced performance in terms of percentages, achieving roughly **66-67% accuracy for both positive and negative classes**. However, in a practical clinical setting, the high number of False Positives (676,391) compared to True Positives (1,198) suggests a very low Precision (Positive Predictive Value), which would likely lead to significant "alarm fatigue" for clinicians.
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<details>
<summary>x23.png Details</summary>
### Visual Description
# Technical Document Extraction: Receiver Operating Characteristic (ROC) Curve
## 1. Document Metadata
* **Title:** Receiver Operating Characteristic Curve (Test)
* **Type:** Line Graph / Performance Metric Chart
* **Language:** English
## 2. Component Isolation
### Header
* **Text:** "Receiver Operating Characteristic Curve (Test)"
* **Position:** Top-centered.
### Main Chart Area (Coordinate System)
* **Y-Axis Label:** TPR (True Positive Rate)
* **Y-Axis Scale:** 0.0 to 1.0 (increments of 0.2 marked, with grid lines at every 0.2 interval).
* **X-Axis Label:** FPR (False Positive Rate)
* **X-Axis Scale:** 0.0 to 1.0 (increments of 0.2 marked, with grid lines at every 0.2 interval).
* **Background:** Light gray with a white grid.
### Legend
* **Spatial Placement:** Bottom-right quadrant [approx. x=0.55 to 0.95, y=0.05 to 0.20].
* **Entry 1:** Solid dark blue line — "ROC curve (area = 0.73)"
* **Entry 2:** Dashed orange line — "Baseline"
---
## 3. Data Series Analysis and Trend Verification
### Series 1: Baseline
* **Visual Description:** A dashed orange line starting at the origin (0,0) and extending diagonally at a 45-degree angle to the top-right corner (1,1).
* **Trend:** Linear upward slope with a constant rate of change.
* **Significance:** Represents a random classifier (AUC = 0.50).
### Series 2: ROC Curve
* **Visual Description:** A solid dark blue curve that originates at (0,0), bows upward toward the top-left corner, and terminates at (1,1).
* **Trend:** Concave downward. The curve rises sharply at low FPR values and gradually flattens as it approaches an FPR of 1.0.
* **Key Data Points (Estimated):**
| Point | FPR (X-Axis) | TPR (Y-Axis) |
| :--- | :--- | :--- |
| Origin | 0.0 | 0.0 |
| Point A | ~0.1 | ~0.3 |
| Point B | ~0.2 | ~0.5 |
| Point C | ~0.4 | ~0.7 |
| Point D | ~0.6 | ~0.85 |
| Endpoint | 1.0 | 1.0 |
* **Metric:** The Area Under the Curve (AUC) is explicitly stated as **0.73**.
---
## 4. Summary of Information
This chart illustrates the performance of a binary classification model on a test dataset. The model's performance (blue line) is significantly better than a random guess (orange dashed line), as evidenced by the curve's position above the diagonal baseline.
The **AUC of 0.73** indicates a "fair" to "good" level of discrimination, meaning there is a 73% probability that the model will rank a randomly chosen positive instance higher than a randomly chosen negative one. The steepness of the curve at the beginning suggests the model maintains a relatively high True Positive Rate while keeping the False Positive Rate low in the initial threshold settings.
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Figure 10: The plots present the sepsis prediction model’s performance measures. Plots (a) and (b) show the confusion matrix and ROC curve results of the model against the training data, respectively. Plots (c) and (d) provide similar measures for the test dataset.
| Grady Training Set Test Set | 0.807 0.828 | Accuracy 0.004 0.004 | Precision 0.743 0.641 | Recall 0.009 0.007 | F1-Score 0.022 0.017 | F2-Score |
| --- | --- | --- | --- | --- | --- | --- |
| Emory | | Accuracy | Precision | Recall | F1-Score | F2-Score |
| Training Set | 0.806 | 0.004 | 0.736 | 0.009 | 0.022 | |
| Test Set | 0.824 | 0.003 | 0.652 | 0.007 | 0.017 | |
Table 9: Grady and Emory sepsis prediction model classification performance metrics for training and test sets.