The motivation for the tool stems from a persistent gap in how statistical software presents regression. According to the authors, mainstream packages such as SPSS, Minitab, Tableau, and JMP present regression results through disconnected numerical tables and static plots, obscuring the geometric relationships that are essential for deep conceptual understanding. A systematic survey of ten widely used systems conducted by the developers found that none of them provides geometric error decomposition or sampling-based interval exploration, while coordinated diagnostic views appear only in limited or partial form in a small number of systems. RegVis was built specifically to close those three gaps within a single, installation-free web application.

The first and perhaps most striking feature is the geometric decomposition of variance. For every data point, RegVis draws a red translucent square whose side length equals the absolute residual, the distance between the observed value and the fitted value predicted by the regression line. It also draws a blue translucent square whose side length equals the distance between the observed value and the mean of the response. The aggregate area of all red squares corresponds to the sum of squared errors, SSE, while the aggregate area of all blue squares corresponds to the total sum of squares, TSS. The identity R² = 1 − SSE/TSS, usually presented as an abstract algebraic formula, thus becomes a visible contrast between two fields of colored area. When a user drags a point or adds an outlier, the squares animate smoothly, and the shift in the ratio of red to blue area makes the change in explained variance perceptually apparent rather than merely calculable.

The developers are careful to note that this area-based rendering is intended as a conceptual aid rather than a precise measurement instrument. Research in graphical perception going back to the classic work of Cleveland and McGill shows that area encodings are less accurate than position or length for exact quantitative comparison. RegVis embraces this limitation deliberately: the squares are meant to convey a holistic, intuitive sense of how total variance splits into explained and unexplained components, while the exact numeric value of R² remains available in a live readout at all times. Users can toggle each layer independently and limit the maximum number of displayed squares to reduce visual clutter, supporting a progressive disclosure approach that starts with a simple scatterplot and builds toward the full decomposition.

The second major innovation tackles one of the most misunderstood concepts in statistics: the confidence interval. Studies have shown that even professional researchers harbor robust misinterpretations of what a confidence interval actually means. RegVis addresses this with an interactive sampling demonstration. Clicking the Sample button draws a random subsample of a user-configured proportion from the current dataset, fits a least-squares line to that subset, and overlays the result as a persistent semi-transparent grey trace. Each click produces a fresh, independent sample, and the accumulating cloud of grey lines gives learners a visceral sense of sampling variability. The displayed confidence band, computed once from the complete dataset, serves as the reference against which the repeated sample lines can be judged.

Because the dataset is treated as a finite population, the behavior of the containment rate follows finite-population sampling theory rather than the textbook infinite-population ideal. A grey trace counts as contained only if it lies within the confidence band at every value of x. As the sampling fraction increases, sampling variance shrinks through the finite-population correction, and containment rises monotonically toward certainty at 100 percent, where the subsample coincides with the population. To make this relationship observable rather than merely asserted, a Simulate function draws 100 independent samples at each of eleven sampling fractions from 0 to 100 percent, fits a line to each, and plots the resulting containment rate against the sampling fraction. In one illustrative workflow described in the paper, a 50-point dataset at the 95 percent confidence level produced a curve rising from roughly 10 percent containment at a 10 percent sampling fraction to full containment at 100 percent.

Underlying both features is a coordinated three-view architecture designed around Shneiderman’s direct-manipulation principle. The primary scatterplot displays the data, the red regression line, the toggleable error squares, adjustable confidence and prediction intervals, and a live regression equation with R² and sample size. A residuals-versus-fitted-values panel reveals heteroscedasticity as cone-shaped patterns, non-linearity as systematic curves, and outliers as extreme vertical displacements. A Q-Q plot maps empirical residual quantiles against theoretical normal quantiles, exposing heavy tails, skewness, and individual aberrant points. Hovering over any observation in any view triggers synchronized highlighting across all three panels, allowing users to trace a single data point through data space, error space, and distributional space simultaneously. Clicking anywhere adds a point; clicking an existing point removes it; and every such action triggers the full pipeline of data update, statistical recalculation, and rendering across all three views.

That pipeline is fast. The developers instrumented each architectural layer with performance timers and ran 30 repetitions at six dataset sizes across Microsoft Edge, Firefox, and Chrome on a 12-core Windows desktop. The computation layer, which recalculates least-squares coefficients, residuals, SSE, TSS, R², and interval bounds from scratch after every change, stayed below 1.2 milliseconds on average across all tested conditions. End-to-end latency rose gradually from under 2.3 milliseconds at ten points to a maximum of 17.93 milliseconds at 1,000 points in Firefox, comfortably below the roughly 100-millisecond threshold at which interactions are typically perceived as instantaneous. Rendering dominates the total, contributing an order of magnitude more time than computation even at the largest tested size, and the authors note that performance was not evaluated beyond 1,000 points. A benchmark script ships with the repository so that others can verify or extend the measurements.

The system is implemented entirely in client-side JavaScript using D3.js version 7, with HTML5 and CSS3, and requires no server-side installation; it can be deployed as a static file on any web host or opened directly from a local filesystem. The choice of D3.js over higher-level charting libraries was deliberate, since the custom geometric error squares, the sampling accumulation overlay, and the synchronized cross-view highlighting all demand low-level control over SVG elements that pre-built statistical libraries do not expose. The software is released under the MIT license on GitHub, includes a guided five-step tutorial covering dataset construction, variance decomposition, interval sampling, and diagnostic coordination, and supports CSV import and export with basic validation for reproducibility.

An exploratory usability assessment with eighteen expert participants, each holding at least one graduate-level statistics course, yielded a mean System Usability Scale score of 79.2 with a standard deviation of 12.8, a figure the authors describe as indicating high perceived usability, though they caution that the study lacked a controlled baseline and formal hypothesis testing. The developers also disclose that Ahmad is the founder of SpimeLab Inc., which hosts the platform, while stating that the reported results were developed independently of commercial considerations. Looking forward, the design principles of geometric correspondence, observable sampling, and coordinated multi-view interaction are described as general enough to extend to multiple regression, generalized linear models, ANOVA, and cross-validation. For now, RegVis positions itself as the first comprehensive visual environment in which regression mathematics is not merely calculable but directly observable, one clickable point at a time.

Subject of Research: Interactive web-based visualization of linear regression analysis, variance decomposition, and confidence interval sampling for statistics education

Article Title: RegVis: An interactive web-based visualization system for linear regression analysis and diagnostics

Article References: Ahmad, U., & Uzair, H. (2026). RegVis: An interactive web-based visualization system for linear regression analysis and diagnostics. SoftwareX, 36, Article 103045. https://doi.org/10.1016/j.softx.2026.103045

Image Credits: AI Generated

DOI: 10.1016/j.softx.2026.103045

Keywords: linear regression, data visualization, statistics education, confidence intervals, D3.js, variance decomposition, open-source software, residual diagnostics, interactive learning, R-squared, sampling variability, web application

The standard approach to testing the liberation hypothesis has been regression-based. Researchers build statistical models of sentencing outcomes, include measures of case severity, and test whether the effect of race or sex on punishment grows stronger as severity weakens, typically through interaction terms. If the coefficient linking defendant race to sentence length is larger in low-severity cases than in high-severity cases, the finding is taken as evidence of liberation. Holmes and Arrigo do not dispute that these coefficients answer a real question. They simply point out that it is not the question Kalven and Zeisel asked. The original theory was about what explains disagreement, not merely about how large a particular effect happens to be in one setting compared with another.

The distinction the authors draw is between effects and contributions. A regression coefficient can tell us, for example, that the odds of a judge-jury disagreement are twice as high when a defendant appears sympathetic than when the defendant does not. What it cannot tell us is how much of the overall variation in disagreements is actually accounted for by sympathy compared with the strength of the evidence. Two variables can have identical coefficients yet contribute very different shares of explained variance, depending on how much they vary and how they correlate with everything else in the model. Effect size and explanatory power are related but distinct quantities, and a theory about the sources of judicial discretion, the authors contend, is fundamentally a theory about explanatory power.

To capture that second quantity, Holmes and Arrigo turn to variance decomposition, a family of techniques borrowed from economics, psychology, and statistics that partition a model’s explained variance among its predictors. The study employs two such methods. Dominance analysis, adapted for logistic regression, determines the relative importance of predictors in the decision to incarcerate by comparing each variable’s contribution across all possible reorderings of the model. Shapley value decomposition, rooted in cooperative game theory, assigns each predictor its average marginal contribution to the model’s R-squared across every possible subset of predictors. These approaches are computationally demanding. With 26 variables, a complete Shapley decomposition would require estimating more than 67 million submodels, a burden the authors managed by computing individual contributions for key independent variables while grouping control variables.

The empirical setting is the federal sentencing system, using data from the United States Sentencing Commission. The authors exploit a structural feature of the federal guidelines: the sentencing table’s zones, labeled A through D, which impose progressively tighter constraints on judicial discretion as case severity rises. Zone A cases, the least serious, permit probation and give judges wide latitude, with incarceration imposed roughly 43 percent of the time. Zone D cases are so severe that imprisonment is essentially mandatory, with incarceration rates near 96 percent. By estimating separate models within each zone, the researchers create a natural severity spectrum across which to compare both the effects and the contributions of extralegal factors, sidestepping the interpretive distortions that arise when predicted probabilities are transformed through the nonlinear sigmoid function at very different base rates.

The results deliver the study’s central punchline. When the authors ran conventional regression tests, asking whether the effect of race, ethnicity, sex, or citizenship on sentencing outcomes was moderated by case severity, the pattern expected under the liberation hypothesis was largely absent. But when they decomposed the variance, asking whether the contribution of extralegal factors to explained variation in sentencing grew as severity declined, evidence consistent with liberation emerged. In other words, the two inquiries, which many researchers have treated as interchangeable, produce different answers, and only the variance-based inquiry finds support for the theory. A variable’s coefficient can stay flat across the severity spectrum even as its share of explained variance shifts substantially, because contributions depend on the full covariance structure of the model, not on any single slope.

The technical choices behind this finding deserve attention in their own right. The authors restricted their sentence-length models to cases receiving prison time, noting that coding non-prison sentences as zero months would create heterogeneous masses at zero across zones, conflating the incarceration decision with the length decision and corrupting comparisons between zones. They also declined to include a Heckman selection correction in the length models because the correction term was substantively collinear with legally relevant variables such as pre-sentence detention, a problem documented in prior methodological work. All models were interpreted on a common log-odds scale rather than as predicted probabilities, precisely because the sigmoid transformation would compress marginal effects differently in low-incarceration Zone A than in near-certain-incarceration Zone D, undermining interpretive consistency.

What does this mean for our understanding of American courts? If the liberation hypothesis survives only under the contribution framing, then the mechanism Kalven and Zeisel described, in which doubts about evidence free decision-makers to follow sentiment, is best understood as a claim about what drives variation in punishment, not about how much any demographic characteristic changes the average sentence. That reframing has practical consequences. A small coefficient for race in a low-severity zone could still reflect a meaningful share of the discretion exercised in those cases, while a statistically significant interaction term might reflect little explained variance at all. Policymakers seeking to rein in unwarranted disparity would therefore want to know not just whether disparities exist in lenient cases, but how much of the discretionary variation those disparities actually occupy.

The study also carries a broader methodological message for criminology and the social sciences generally. Regression coefficients have long been the default currency of quantitative testing, and interaction terms the default tool for moderation. Holmes and Arrigo join a growing literature, including recent work applying variance decomposition to inequality of opportunity, arguing that relative importance analysis should supplement rather than replace conventional regression. Their demonstration that a classic criminological theory changes its empirical verdict depending on which question is asked is a vivid case study in why the choice of statistical lens is itself a theoretical commitment. As sentencing researchers continue to debate when and whether race, sex, and citizenship shape punishment, the liberation hypothesis may need to be re-litigated, this time in the currency of explained variance.

Subject of Research: Testing the liberation hypothesis in federal sentencing using variance decomposition methods

Article Title: Using Variance Decomposition to Test Criminal Justice Theory: A Redeliberation of the Liberation Hypothesis

Article References: Holmes, B., & Arrigo, S. R. (2026). Using Variance Decomposition to Test Criminal Justice Theory: A Redeliberation of the Liberation Hypothesis. American Journal of Criminal Justice. https://doi.org/10.1007/s12103-026-09950-y

Image Credits: AI Generated

DOI: 10.1007/s12103-026-09950-y

Keywords: liberation hypothesis, sentencing, variance decomposition, Shapley values, dominance analysis, federal courts, racial disparity, judicial discretion, criminology, United States Sentencing Commission, case severity, extralegal factors