## Violin Plot: Predicted Causal Effect (ATE) vs. Dataset Size under Different Scenarios ### Overview The image presents six violin plots arranged in a 2x3 grid. Each plot visualizes the distribution of predicted causal effects (ATE) for different dataset sizes under varying scenarios: Biased, Direct-Effect, Indirect-Effect, Fair Observable, Fair Unobservable, and Fair Additive Noise. The x-axis represents dataset size, categorized into ranges, while the y-axis represents the predicted causal effect (ATE). The violin plots show the distribution of the predicted causal effect for each dataset size range. ### Components/Axes * **Y-axis:** "Pred. Causal Effect (ATE)" with a scale from -0.2 to 0.2, marked at -0.2, -0.1, 0.0, 0.1, and 0.2. * **X-axis:** "Dataset Size" categorized into five ranges: 98-250, 250-630, 630-1583, 1583-3981, and 3981-9998. * **Violin Plots:** Each violin plot is filled with a light purple color and outlined in black. Each violin plot contains a box plot with a black box and whiskers. * **Titles:** Each plot has a title indicating the scenario: 1. Biased 2. Direct-Effect 3. Indirect-Effect 4. Fair Observable 5. Fair Unobservable 6. Fair Additive Noise ### Detailed Analysis **Plot 1: Biased** * The violin plots show a decreasing spread as the dataset size increases. * The median (black box) is close to 0 for all dataset sizes. * The distribution is wider for smaller dataset sizes (98-250 and 250-630) and becomes narrower for larger dataset sizes (1583-3981 and 3981-9998). **Plot 2: Direct-Effect** * Similar to the "Biased" scenario, the spread of the violin plots decreases with increasing dataset size. * The median is close to 0 for all dataset sizes. * The distribution is wider for smaller dataset sizes and narrower for larger dataset sizes. **Plot 3: Indirect-Effect** * The spread of the violin plots decreases with increasing dataset size. * The median is close to 0 for all dataset sizes. * The distribution is wider for smaller dataset sizes and narrower for larger dataset sizes. **Plot 4: Fair Observable** * The spread of the violin plots decreases with increasing dataset size. * The median is close to 0 for all dataset sizes. * The distribution is wider for smaller dataset sizes and narrower for larger dataset sizes. **Plot 5: Fair Unobservable** * The spread of the violin plots decreases with increasing dataset size. * The median is close to 0 for all dataset sizes. * The distribution is wider for smaller dataset sizes and narrower for larger dataset sizes. **Plot 6: Fair Additive Noise** * The spread of the violin plots decreases with increasing dataset size. * The median is close to 0 for all dataset sizes. * The distribution is wider for smaller dataset sizes and narrower for larger dataset sizes. ### Key Observations * In all six scenarios, the spread of the predicted causal effect (ATE) decreases as the dataset size increases. This suggests that larger datasets lead to more precise estimates of the causal effect. * The medians of the distributions are generally close to 0 across all dataset sizes and scenarios, indicating that the average predicted causal effect is near zero. * The "Biased" scenario shows a wider distribution for smaller dataset sizes compared to the "Fair" scenarios, suggesting that bias can lead to more variable estimates, especially with limited data. ### Interpretation The plots demonstrate the impact of dataset size on the precision of predicted causal effects under different scenarios. The consistent trend of decreasing spread with increasing dataset size highlights the importance of having sufficient data for reliable causal inference. The scenarios with "Fair" conditions generally exhibit narrower distributions, suggesting that addressing biases and confounding factors can improve the accuracy and stability of causal effect estimates. The "Biased" scenario shows that even with increasing dataset size, the initial bias can still lead to more variable estimates compared to the "Fair" scenarios. The plots suggest that increasing dataset size can mitigate the impact of noise and unobserved confounders, leading to more precise causal effect estimates.

## Violin Plots: Predicted Causal Effect (ATE) vs. Dataset Size ### Overview The image presents six violin plots arranged in a 2x3 grid. Each plot visualizes the relationship between "Dataset Size" (x-axis) and "Pred. Causal Effect (ATE)" (y-axis) under different algorithmic conditions: Biased, Direct-Effect, Indirect-Effect, Fair Observable, Fair Unobservable, and Fair Additive Noise. Each violin plot also includes a black line representing the median and a gray triangle marking the mean. ### Components/Axes * **X-axis Label:** "Dataset Size" with markers: 98-250, 250-630, 630-1583, 1583-3981, 3981-9998. * **Y-axis Label:** "Pred. Causal Effect (ATE)" ranging from approximately -0.2 to 0.2. * **Plot Titles:** 1. Biased, 2. Direct-Effect, 3. Indirect-Effect, 4. Fair Observable, 5. Fair Unobservable, 6. Fair Additive Noise. These are positioned at the top-center of each respective plot. * **Violin Plots:** Each plot displays the distribution of the predicted causal effect for a given dataset size and algorithmic condition. * **Median Line:** A black line within each violin plot indicates the median value. * **Mean Triangle:** A gray triangle within each violin plot indicates the mean value. ### Detailed Analysis **1. Biased:** * Trend: The violin plots show a slight upward trend in the median and mean as dataset size increases. * Data Points (approximate): * 98-250: Median ≈ -0.02, Mean ≈ -0.03 * 250-630: Median ≈ 0.00, Mean ≈ 0.01 * 630-1583: Median ≈ 0.02, Mean ≈ 0.03 * 1583-3981: Median ≈ 0.04, Mean ≈ 0.05 * 3981-9998: Median ≈ 0.06, Mean ≈ 0.07 **2. Direct-Effect:** * Trend: The violin plots show a clear upward trend in both the median and mean as dataset size increases. The distribution also appears to narrow with increasing dataset size. * Data Points (approximate): * 98-250: Median ≈ -0.05, Mean ≈ -0.06 * 250-630: Median ≈ 0.00, Mean ≈ 0.01 * 630-1583: Median ≈ 0.05, Mean ≈ 0.06 * 1583-3981: Median ≈ 0.10, Mean ≈ 0.11 * 3981-9998: Median ≈ 0.15, Mean ≈ 0.16 **3. Indirect-Effect:** * Trend: The violin plots show a relatively flat trend with some variability. The median and mean remain close to zero across dataset sizes. * Data Points (approximate): * 98-250: Median ≈ 0.00, Mean ≈ 0.01 * 250-630: Median ≈ 0.00, Mean ≈ 0.00 * 630-1583: Median ≈ 0.00, Mean ≈ -0.01 * 1583-3981: Median ≈ 0.00, Mean ≈ 0.00 * 3981-9998: Median ≈ 0.00, Mean ≈ 0.01 **4. Fair Observable:** * Trend: Similar to the "Biased" plot, there's a slight upward trend in the median and mean as dataset size increases. * Data Points (approximate): * 98-250: Median ≈ -0.02, Mean ≈ -0.03 * 250-630: Median ≈ 0.00, Mean ≈ 0.01 * 630-1583: Median ≈ 0.02, Mean ≈ 0.03 * 1583-3981: Median ≈ 0.04, Mean ≈ 0.05 * 3981-9998: Median ≈ 0.06, Mean ≈ 0.07 **5. Fair Unobservable:** * Trend: The violin plots show a clear upward trend in both the median and mean as dataset size increases. The distribution also appears to narrow with increasing dataset size. * Data Points (approximate): * 98-250: Median ≈ -0.05, Mean ≈ -0.06 * 250-630: Median ≈ 0.00, Mean ≈ 0.01 * 630-1583: Median ≈ 0.05, Mean ≈ 0.06 * 1583-3981: Median ≈ 0.10, Mean ≈ 0.11 * 3981-9998: Median ≈ 0.15, Mean ≈ 0.16 **6. Fair Additive Noise:** * Trend: The violin plots show a relatively flat trend with some variability. The median and mean remain close to zero across dataset sizes. * Data Points (approximate): * 98-250: Median ≈ 0.00, Mean ≈ 0.01 * 250-630: Median ≈ 0.00, Mean ≈ 0.00 * 630-1583: Median ≈ 0.00, Mean ≈ -0.01 * 1583-3981: Median ≈ 0.00, Mean ≈ 0.00 * 3981-9998: Median ≈ 0.00, Mean ≈ 0.01 ### Key Observations * The "Direct-Effect" and "Fair Unobservable" plots exhibit the most pronounced positive correlation between dataset size and predicted causal effect. * The "Indirect-Effect" and "Fair Additive Noise" plots show minimal change in the predicted causal effect across different dataset sizes. * The "Biased" and "Fair Observable" plots show a moderate positive correlation. * The distributions in the "Direct-Effect" and "Fair Unobservable" plots become narrower with increasing dataset size, suggesting greater certainty in the predicted causal effect. ### Interpretation The plots demonstrate how different algorithmic conditions influence the relationship between dataset size and the accuracy of predicted causal effects. The "Direct-Effect" and "Fair Unobservable" conditions benefit significantly from larger datasets, showing a clear positive trend in the predicted causal effect. This suggests that these algorithms are able to more accurately estimate the causal effect as more data becomes available. Conversely, the "Indirect-Effect" and "Fair Additive Noise" conditions are largely unaffected by dataset size, indicating that the predicted causal effect is relatively stable regardless of the amount of data. The "Biased" and "Fair Observable" conditions show a moderate improvement with larger datasets, but not as pronounced as the "Direct-Effect" and "Fair Unobservable" conditions. The narrowing of the distributions in the "Direct-Effect" and "Fair Unobservable" plots with increasing dataset size suggests that larger datasets lead to more precise estimates of the causal effect, reducing uncertainty. This highlights the importance of data quantity in achieving reliable causal inference, particularly when using algorithms that are sensitive to dataset size. The differences between the plots underscore the impact of algorithmic design choices on the robustness and accuracy of causal effect estimation.

## Violin Plot Grid: Predicted Causal Effect (ATE) by Dataset Size and Method ### Overview The image displays a 2x3 grid of six violin plots. Each subplot visualizes the distribution of the **Predicted Average Treatment Effect (ATE)** for a different causal estimation method across five increasing dataset size categories. The plots compare how the precision and bias of the ATE estimates change with more data for each method. ### Components/Axes * **Overall Layout:** Six subplots arranged in two rows and three columns. * **Subplot Titles (Methods):** 1. **Biased** (Top Left) 2. **Direct-Effect** (Top Center) 3. **Indirect-Effect** (Top Right) 4. **Fair Observable** (Bottom Left) 5. **Fair Unobservable** (Bottom Center) 6. **Fair Additive Noise** (Bottom Right) * **Y-Axis (Common to all subplots):** Labeled **"Pred. Causal Effect (ATE)"**. The scale ranges from -0.2 to 0.2, with major grid lines at intervals of 0.1. * **X-Axis (Common to all subplots):** Labeled **"Dataset Size"**. It contains five categorical bins representing ranges of dataset sizes: * `98-250` * `250-630` * `630-1583` * `1583-3981` * `3981-9998` * **Plot Elements:** Each category on the x-axis has a corresponding **violin plot**. The violin shows the probability density of the data at different values, with a wider section indicating a higher frequency of data points. Inside each violin is a miniature **box plot** (black bar with white median line and whiskers), summarizing the median, interquartile range, and range of the distribution. ### Detailed Analysis The analysis is segmented by subplot (method). For each, the visual trend of the distributions as dataset size increases is described, followed by approximate data points. **1. Biased** * **Trend:** The distributions are centered near zero but show significant spread, especially for smaller datasets. The variance (spread of the violin) decreases noticeably as dataset size increases. The median (white line) remains close to zero across all sizes. * **Data Points (Approximate Median & Spread):** * `98-250`: Median ~0.0, wide spread from ~-0.15 to +0.25. * `250-630`: Median ~0.0, spread narrows (~-0.1 to +0.15). * `630-1583`: Median ~0.0, spread continues to narrow. * `1583-3981`: Median ~0.0, relatively tight distribution. * `3981-9998`: Median ~0.0, tightest distribution, but with a long tail extending to ~-0.15. **2. Direct-Effect** * **Trend:** Similar to "Biased," distributions are centered near zero and variance decreases with more data. The initial spread for the smallest dataset appears slightly larger than in the "Biased" plot. * **Data Points (Approximate Median & Spread):** * `98-250`: Median ~0.0, very wide spread from ~-0.15 to +0.2. * `250-630`: Median ~0.0, spread narrows. * `630-1583`: Median ~0.0, spread narrows further. * `1583-3981`: Median ~0.0, tight distribution. * `3981-9998`: Median ~0.0, very tight distribution. **3. Indirect-Effect** * **Trend:** Distributions are centered near zero. The variance reduction with increasing dataset size is very pronounced. The smallest dataset shows a particularly wide and tall distribution. * **Data Points (Approximate Median & Spread):** * `98-250`: Median ~0.0, extremely wide and tall distribution (high density around zero but large range). * `250-630`: Median ~0.0, spread reduces dramatically. * `630-1583`: Median ~0.0, tight distribution. * `1583-3981`: Median ~0.0, very tight distribution. * `3981-9998`: Median ~0.0, extremely tight distribution. **4. Fair Observable** * **Trend:** Distributions are centered near zero. Variance decreases with dataset size. The shape and spread appear very similar to the "Biased" method. * **Data Points (Approximate Median & Spread):** * `98-250`: Median ~0.0, wide spread. * `250-630`: Median ~0.0, spread narrows. * `630-1583`: Median ~0.0, spread narrows further. * `1583-3981`: Median ~0.0, tight distribution. * `3981-9998`: Median ~0.0, tight distribution. **5. Fair Unobservable** * **Trend:** This method shows a distinct pattern. While the median remains near zero, the **variance does not decrease consistently** with dataset size. The distributions for the two largest dataset sizes (`1583-3981` and `3981-9998`) appear wider than those for the middle sizes, suggesting instability or increased uncertainty with more data for this method. * **Data Points (Approximate Median & Spread):** * `98-250`: Median ~0.0, wide spread. * `250-630`: Median ~0.0, spread narrows. * `630-1583`: Median ~0.0, relatively tight. * `1583-3981`: Median ~0.0, spread increases again. * `3981-9998`: Median ~0.0, spread remains wide. **6. Fair Additive Noise** * **Trend:** Distributions are centered near zero. Variance decreases with dataset size, but the rate of decrease appears slower compared to methods like "Indirect-Effect." The distributions remain relatively wide even for larger datasets. * **Data Points (Approximate Median & Spread):** * `98-250`: Median ~0.0, very wide spread. * `250-630`: Median ~0.0, spread narrows. * `630-1583`: Median ~0.0, spread narrows further. * `1583-3981`: Median ~0.0, moderately wide distribution. * `3981-9998`: Median ~0.0, moderately wide distribution. ### Key Observations 1. **Universal Trend:** For five of the six methods (all except "Fair Unobservable"), the variance (uncertainty) of the predicted ATE decreases as the dataset size increases. This is the expected behavior of consistent estimators. 2. **Bias:** All methods appear to be **unbiased** on average, as the median of every distribution is centered at or very near 0.0 on the y-axis. 3. **Method Comparison:** * The **"Indirect-Effect"** method shows the most dramatic reduction in variance, achieving the tightest distributions for large datasets. * The **"Fair Unobservable"** method is an outlier. Its variance does not monotonically decrease and is notably high for the largest datasets, indicating potential issues with this estimation approach under the tested conditions. * The **"Biased"** and **"Fair Observable"** methods show very similar performance profiles. * The **"Fair Additive Noise"** method retains higher variance than others at large dataset sizes. ### Interpretation This visualization is a comparative performance analysis of different causal inference methods. The **Predicted ATE** is the estimated average effect of a treatment or intervention. The plots reveal how the **precision** (inverse of variance) of these estimates improves with more data. * **What the data suggests:** Most methods become more precise with larger datasets, which validates their statistical consistency. The "Indirect-Effect" method appears most efficient in this test. The anomalous behavior of "Fair Unobservable" suggests that incorporating unobservable confounders in a fairness-aware model may introduce instability or require a different modeling approach that doesn't scale well with data size in this scenario. * **How elements relate:** The x-axis (Dataset Size) is the independent variable. The y-axis (Predicted ATE) is the dependent variable whose distribution is measured. The subplot titles (Methods) are the different models or algorithms being tested. The violin shape directly visualizes the uncertainty in the causal estimate for each method at each data scale. * **Notable anomalies:** The primary anomaly is the **"Fair Unobservable"** method's failure to reduce variance with the largest datasets. This could indicate overfitting, model misspecification, or a fundamental challenge in estimating causal effects when accounting for unobservable factors in a fairness context. The long tail in the largest dataset for the "Biased" method is a minor secondary anomaly.

## Violin Plots: Causal Effect Analysis by Dataset Size and Condition ### Overview The image presents six violin plots arranged in two rows and three columns, comparing predicted causal effects (ATE) across different dataset sizes and experimental conditions. Each plot visualizes the distribution of causal effect estimates, with black box plots indicating median, quartiles, and outliers. The x-axis represents dataset size ranges (98-250, 250-630, 630-1583, 1583-3981, 3981-9998), while the y-axis shows causal effect values from -0.2 to 0.2. ### Components/Axes - **X-axis (Dataset Size)**: Categorical ranges (98-250, 250-630, 630-1583, 1583-3981, 3981-9998) - **Y-axis (Pred. Causal Effect (ATE))**: Continuous scale from -0.2 to 0.2 - **Violin Plots**: Purple distributions with black box plots (median, quartiles, outliers) - **Titles**: Six conditions labeled 1-6 (Biased, Direct-Effect, Indirect-Effect, Fair Observable, Fair Unobservable, Fair Additive Noise) ### Detailed Analysis 1. **Biased (1)**: - Distributions centered near 0 with moderate spread. - Median values stable across dataset sizes (~-0.02 to 0.02). - Outliers present in smaller datasets (98-250). 2. **Direct-Effect (2)**: - Slight positive trend as dataset size increases. - Median shifts from ~0.01 (98-250) to ~0.05 (3981-9998). - Spread narrows with larger datasets. 3. **Indirect-Effect (3)**: - Slight negative trend with increasing dataset size. - Median decreases from ~0.03 (98-250) to ~-0.02 (3981-9998). - Distributions become tighter in larger datasets. 4. **Fair Observable (4)**: - Similar to Biased but with reduced spread. - Median values stable (~-0.01 to 0.01). - Fewer outliers across all dataset sizes. 5. **Fair Unobservable (5)**: - Increased variability in larger datasets. - Median values stable (~-0.01 to 0.01). - Wider distributions in 1583-3981 and 3981-9998 ranges. 6. **Fair Additive Noise (6)**: - Distributions widen significantly with dataset size. - Median values stable (~-0.01 to 0.01). - Outliers increase in frequency for 250-630 and larger datasets. ### Key Observations - **Trend Divergence**: Direct-Effect (positive trend) and Indirect-Effect (negative trend) show opposing directional biases. - **Noise Impact**: Fair Additive Noise (6) exhibits the largest spread, suggesting noise amplifies uncertainty. - **Dataset Size Effects**: Larger datasets (3981-9998) generally show tighter distributions except in Fair Unobservable and Fair Additive Noise. - **Condition-Specific Variability**: Fair Unobservable (5) and Fair Additive Noise (6) demonstrate higher sensitivity to dataset size changes. ### Interpretation The plots reveal how experimental conditions influence causal effect estimation: 1. **Biased vs. Fair Conditions**: Fair conditions (4-6) show reduced spread compared to Biased (1), indicating better estimation stability. 2. **Direct vs. Indirect Effects**: Direct-Effect (2) demonstrates a consistent positive bias, while Indirect-Effect (3) shows a negative bias, suggesting methodological differences in effect measurement. 3. **Noise Sensitivity**: Fair Additive Noise (6) highlights how noise introduces uncertainty, particularly in larger datasets where spread increases despite more data. 4. **Unobservable Factors**: Fair Unobservable (5) shows dataset size has diminishing returns for reducing variability, implying unobservable confounders persist even with more data. These patterns suggest that experimental design (e.g., noise control, observability) critically impacts causal inference reliability, with larger datasets offering limited benefits in certain conditions.