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