The problem the paper addresses is deceptively simple. Statisticians routinely fit regression models with an intercept, allowing the line to pass through the centroid of the data cloud. But many scientific questions demand a no-intercept, or homogeneous, model, in which the response must vanish when the predictor vanishes. Metabolic rate, for instance, should arguably fall to zero as body mass falls to zero. The default strategy for switching between these specifications has always been to refit the model from scratch, which implicitly assumes the analyst can return to the full design matrix and response vector. In legacy studies, privacy-restricted medical research, and federated learning systems, that assumption often fails.

Selvitella’s central theorem shows that for any quadratic-loss regression, including ordinary least squares, ridge regression, and weighted least squares, the no-intercept estimator can be written in closed form as a function of the with-intercept coefficients, the empirical means of the variables, and the Gram matrices that summarize the data’s geometry. The key identity decomposes the uncentered cross-products used by the no-intercept model into centered cross-products plus a mean-product term. Because the with-intercept solution already satisfies a centered normal equation, the no-intercept solution can be recovered by a single matrix inversion that depends only on the design, the weighting, and any penalty structure, never on the response itself.

Just as importantly, uncertainty propagates through the same bridge. The no-intercept estimator turns out to be an affine function of the with-intercept estimator, so standard covariance calculus delivers its variance directly. Under Gaussian errors, the transformed estimator exactly reproduces the finite-sample distribution of the classical no-intercept solution, and under weak moment conditions it inherits the usual asymptotic normality through Slutsky’s theorem. In other words, confidence intervals and hypothesis tests for the through-the-origin model can be reconstructed from published summaries alone, an algebraically exact result rather than an approximation.

The first major application is federated learning, where data are distributed across many clients, such as hospitals or mobile devices, that never share raw observations. If a consortium has already trained a with-intercept model through the popular FedAvg protocol and then decides it needs the no-intercept version, the conventional answer is a second full federated run, costing on the order of the number of rounds times the sample size times the predictor dimension in local computation, plus repeated communication rounds. The new theorem shows the server can instead aggregate one round of local sufficient statistics, each client transmitting only its sample count, moment vectors, and Gram matrix, and then compute the no-intercept estimator with a cost that scales with the predictor dimension alone and is independent of the number of clients, observations, or communication rounds.

Numerical experiments in the paper back the theory with striking precision. In simulations with ten clients and five thousand observations, the transformed estimator agreed with the centralized no-intercept ordinary least squares solution at floating-point precision, around ten to the minus fifteen. The direct federated no-intercept baseline, by contrast, drifted from the centralized answer as client data became heterogeneous, with relative errors growing from roughly four ten-thousandths in the balanced case to a full ten percent under severe non-IID conditions, a manifestation of the well-known client-drift problem. Timing measurements showed the transformation running between one thousand and twenty-three thousand times faster than a second federated training run, and the agreement held at machine precision even in high-dimensional settings with up to five hundred predictors.

The paper is careful about scope. The savings are relative to a second iterative federated optimization, not to one-shot protocols in which clients transmit exact Gram matrices from the outset, and the method is not by itself a formal privacy guarantee. Aggregate summaries can still leak information when a client holds few observations or unusual covariate patterns, so the author notes that secure aggregation or calibrated noise would need to be layered on top where formal differential privacy is required. There is also a communication trade-off: the one-shot summary protocol costs on the order of the predictor dimension squared per client, so it beats a second federated run only when the number of additional rounds exceeds roughly a quarter of the predictor dimension, a break-even the paper quantifies explicitly.

The second implication is more conceptually surprising: a regression reversal phenomenon. In simple linear regression, the slope estimated with an intercept and the slope estimated through the origin can have strictly opposite signs. Selvitella proves an exact necessary and sufficient condition, namely that the centered cross-product and the uncentered cross-product have opposite signs, which happens precisely when the mean-product term dominates the centered signal. The reversal is driven by the intercept constraint and by leverage, not by classical confounding, yet it is structurally analogous to Simpson’s paradox: in both cases an unadjusted association can point opposite to an adjusted one, with the omitted constant term playing the role of a lurking group-level offset. Because the condition can be checked from summary statistics alone, practitioners can detect potential sign flips before committing to a refit.

The third implication reaches into biology, where only aggregate results are often available and the intercept carries real scientific weight. Reanalyzing a published study of basal metabolic rate in sixty-eight adult men, the paper reconstructs the through-the-origin fit entirely from reported means, standard deviations, and the correlation coefficient. The no-intercept slope comes out near 0.0975, nearly double the with-intercept slope of about 0.054, illustrating how a nonzero intercept can attenuate an estimated scaling relationship. Goodness-of-fit measures can also be reconstructed, though the paper cautions that the uncentered coefficient of determination for no-intercept models is not directly comparable to the conventional centered one; on a common centered scale, the no-intercept model explains less variance than the with-intercept fit, consistent with the original authors’ conclusion that simple linear models inadequately capture allometric scaling.

The broader message is that a piece of linear algebra most statisticians thought they already understood still holds surprises with practical teeth. By placing the affine-to-linear map within a unified framework covering penalized and weighted losses, and by propagating both point estimates and their distributions, the result turns intercept-specification changes from an expensive refitting problem into a cheap post-processing step. For meta-analysts working with decades of published summaries, for consortia bound by data-use agreements, and for engineers designing federated systems, the ability to move between model specifications using sufficient statistics alone offers a rare combination of mathematical exactness and immediate utility, and it suggests that other transformations across the regression landscape may be waiting for similar treatment.

Subject of Research: Summary-statistics-only affine-to-linear transformations linking with-intercept and no-intercept quadratic-loss regression, with applications to federated learning, regression reversal, and biological scaling

Article Title: Affine-to-linear transformations for quadratic-loss regression: computational implications to federated learning, reversal paradox, and applications

Article References: Selvitella, A. M. (2026). Affine-to-linear transformations for quadratic-loss regression: computational implications to federated learning, reversal paradox, and applications. International Journal of Data Science and Analytics, 22(1), Article 289. https://doi.org/10.1007/s41060-026-01268-6

Image Credits: AI Generated

DOI: 10.1007/s41060-026-01268-6

Keywords: regression, federated learning, sufficient statistics, ordinary least squares, ridge regression, weighted least squares, intercept, Simpson's paradox, metabolic scaling, covariance propagation, communication cost, data privacy

The method, described in the journal BMC Bioinformatics by Jack Kamm, Jake Yeung, and William F. Forrest of Genentech, reframes guide assignment as a problem in classical statistics. Rather than fitting an elaborate probabilistic model to the guide count matrix, the authors propose treating the matrix of guide RNA counts across cells as a contingency table. For every cell-guide pair, they construct a two-by-two table that contrasts the counts involving that cell and that guide with all other counts, and then apply Fisher’s Exact Test, a workhorse of categorical data analysis, to ask whether the cell and guide barcodes are statistically associated. A significant association signals that the guide is genuinely present in the cell rather than appearing by chance through sequencing noise.

The elegance of this formulation lies in what it provides automatically. Because Fisher’s Exact Test conditions on the margins of the contingency table, the resulting p-values implicitly normalize for both cell-specific and guide-specific size factors, the uneven sequencing depth that plagues single-cell data. Cells that received more total reads and guides that were more abundant in the library are handled without any explicit normalization step. The method also produces a p-value for every cell-guide pair, giving users a graded measure of confidence rather than a hard binary call, and enabling downstream analyses such as full precision-recall curves when benchmarking alternative cutoffs.

Two statistical subtleties demanded additional innovation. First, because the method performs one test for every cell-guide combination, large screens can involve tens of millions of hypothesis tests. Naive application of standard multiple testing corrections such as the Benjamini-Hochberg procedure would be inappropriate, because the tests are strongly correlated: a cell that tests positive for one guide is less likely to test positive for others, and counts involving the same guide or the same cell share information. The authors developed a multiple testing correction strategy that accounts for this correlation structure, preserving control of the false discovery rate while avoiding the excessive conservatism that would come from ignoring the dependencies.

The second subtlety is a form of confounding that can flip conclusions in exactly the wrong way. Ambient RNA contamination, sometimes called soup or background noise, introduces guide molecules into cells that never received them. If the contamination is not independent across cells and guides, a phenomenon analogous to Simpson’s paradox can arise: an association that appears significant in the aggregate data disappears, or even reverses, once the noise component is considered separately. Fishash addresses this by testing an adjusted odds ratio that replaces the margins involving the focal cell and guide with their noise-based counterparts. To estimate the unobserved noise counts, the method performs an iterative rank-1 Poisson matrix completion: entries that have already been confidently assigned are masked out, the remaining background structure is fit under a Poisson likelihood by alternating updates of guide and cell abundance parameters, and the imputed noise counts feed back into refined p-values. The procedure repeats until the assignments stabilize.

Recognizing that fair comparison of guide assignment methods has been hampered by a lack of realistic test data, the authors also introduced a simulation framework that generates synthetic guide counts under a detailed model of sequencing noise. The simulator draws on the contamination model popularized by the Cellbender software, incorporating latent cell-level factors such as ambient RNA fractions, capture efficiencies, and dropout rates, along with guide-level expression variation. Crucially, the simulations can vary the number of guides in the library and the multiplicity of infection, the average number of guides delivered per cell, allowing benchmarking across the parameter regimes that matter in practice. The code to reproduce all results is publicly available alongside the method itself.

In benchmarks on both simulated and real datasets, Fishash compared favorably with existing approaches in both accuracy and runtime. On simulations varying the number of guides, the method achieved the top median F1 score in the majority of settings, performing particularly well when the guide library was large or guide RNA expression was low. In simulations varying the multiplicity of infection, Fishash remained competitive at low multiplicity, where most cells carry at most one guide, though other methods such as crispat-NB and dcCLEANSER edged ahead at high multiplicity. The authors emphasize that Fishash was conservative in its error control: its observed precision stayed above its nominal lower bound across the sweep, whereas several competing Bayesian and frequentist methods failed to control the false discovery rate at least once across the scenarios tested.

The stress tests also revealed the method’s limits, which the authors report transparently. When extra overdispersion was injected into the noise counts by replacing the Poisson distribution with a Geometric distribution, one of the most extreme overdispersion regimes available, precision dropped for many methods and Fishash no longer controlled the false discovery rate at its nominal five percent level in every scenario. Even so, its F1 scores remained qualitatively similar to the original benchmarks, indicating a competitive balance between precision and recall even under unusually harsh noise. The authors note that the conditional low-rank Poisson structure assumed for background counts is common to many single-cell contamination models, including Cellbender, SoupX, DecontX, and scAR, so the simulation represents a deliberately challenging departure from the assumptions shared across the field.

Practical accessibility was a design priority. Fishash is distributed as an easy-to-use R package on GitHub, and because it relies on exact tests over contingency tables rather than iterative model fitting, it scales comfortably to screens with tens of thousands of cells and guides, where computationally expensive probabilistic approaches can become prohibitive. The software also outputs its test statistic for every entry of the cell-guide matrix, which allows users to visualize the distribution of scores, select alternative significance cutoffs informed by the data, and compute full precision-recall curves during method benchmarking. The work was carried out entirely within Genentech, with the authors crediting colleagues in the company’s AI Biology and Translation department for discussions that shaped the manuscript.

For the growing community running Perturb-seq experiments, the significance of the method is straightforward: guide assignment is the gatekeeper step that determines whether downstream perturbation effects are real or artifacts of misassignment, and errors made here propagate through every subsequent analysis. By combining a classical, well-understood statistical test with modern corrections for multiplicity, confounding, and background contamination, and by packaging the result in fast, open software, Fishash lowers both the computational and conceptual barriers to reliable pooled single-cell screening. As CRISPR screens continue to expand in scale and ambition, tools that trade heavy model fitting for careful, assumption-aware statistics are likely to play an increasingly central role in turning raw sequencing counts into trustworthy biological conclusions.

Subject of Research: A contingency table and Fisher's Exact Test based method for assigning guide RNAs to cells in single-cell pooled CRISPR screens (Perturb-seq)

Article Title: Fishash: a contingency table approach to Perturb-seq guide assignment

Article References: Kamm, J., Yeung, J., & Forrest, W. F. (2026). Fishash: a contingency table approach to Perturb-seq guide assignment. BMC Bioinformatics. https://doi.org/10.1186/s12859-026-06631-9

Image Credits: AI Generated

DOI: 10.1186/s12859-026-06631-9

Keywords: Perturb-seq, CRISPR screen, single-cell RNA-seq, guide assignment, Fisher's exact test, Simpson's paradox, multiple testing correction, contingency table, false discovery rate, ambient RNA contamination, functional genomics, BMC Bioinformatics