Biometrics

Mendelian randomization (MR) is widely used to draw causal conclusions in the presence of unmeasured confounding, but most MR analyses focus on average treatment effects and rely on strong assumptions. For precision medicine, the primary target is instead the individualized treatment effect (ITE); yet in MR, such effects are not point-identified under core IV assumptions, and valid inference is particularly challenging. We therefore propose a robust partial identification inference framework for ITE under MR allowing multiple instruments. Under minimal causal assumptions, we derive a sharp inference procedure for the intersection bounds of ITE by adopting a multiplier bootstrap procedure with data-adaptive bootstrap distribution shifting and heterogeneous variance adjustment. In theory, we prove that the proposed method achieves nominal coverage and asymptotic sharpness. Further, we extend the procedure to tolerate possible invalid IVs under a minimal proportion rule assumption by aggregating over instrument subsets while preserving coverage. Simulation studies demonstrate that the proposed methods attain nominal coverage and substantially shorter intervals than existing procedures. We illustrate the framework using data from the Alzheimers Disease Neuroimaging Initiative to assess heterogeneous causal effects of TREM2 expression on Alzheimers disease risk across education-defined subgroups.

Advances in spatial transcriptomics (ST) technologies enable systematic molecular characterization of tumor microenvironment, tumor gradients and gene regulatory networks. Cancer progression is known to vary along pathological gradients, yet existing network approaches for gene network inference typically ignore hierarchical spatial organization across the tumor. We develop a Bayesian multi-resolution spatial graphical regression (mSGR) framework to infer spatially varying gene networks from multi-resolution ST data. The proposed model allows precision matrices to vary across hierarchically structured spatial domains, capturing both local and global organization within the tumor. To identify spatially varying regulatory relationships, we introduce a spatially structured edge selection strategy that borrows strength across regions according to spatial proximity and pathological gradients, while Gaussian-process priors flexibly model spatial variation in edge strengths. Scalable inference is achieved through an augmented mean-field variational Bayes algorithm with node-wise parallel regressions, enabling efficient estimation in high-dimensional settings. Simulation studies demonstrate improved recovery of network structures compared with competing approaches. Applying mSGR to multi-resolution ST data from kidney cancer reveals stronger regulatory connectivity in transitional regions of epithelial-mesenchymal transition pathway and identifies hub genes along the tumor gradient, illustrating how spatially resolved network analysis can provide key insights into tumor microenvironment organization.

Recently, Hu et al. (2024) conducted a benchmarking study showing that most existing Mendelian randomization (MR) methods exhibit substantial bias and inflated type-I error rates in real data. They attributed these failures to two largely neglected sources of bias: winner's curse and polygenicity-induced bias. Although a few methods have been developed to address one or both of these issues, existing approaches either do not fully account for both biases or are restricted to the univariable setting. In this paper, we propose a multivariable Rao-Blackwellization that corrects winner's curse while accounting for polygenicity and sample structure in a unified framework. Unlike univariable Rao-Blackwellization, where instrument selection yields a truncated normal statistic amenable to a Mills-ratio correction, multivariable Rao-Blackwellization conditions on a noncentral $\chi^2$ statistic, for which no analogous correction is available. We derive closed-form conditional moments under this instrument selection model and use them to construct bias-corrected summary statistics that can be integrated into a wide range of existing MR methods. Simulations and real data analyses show that, when combined with methods such as MR-cML and MR-BEE, the proposed correction substantially improves type-I error control and yields more robust inference.

Instrumental variable (IV) methods are widely used in health and social sciences to estimate causal treatment effects among compliers. In certain research settings, the instrument-treatment association (first stage) and the instrument-outcome association (reduced form) are each estimated from a different dataset. Two-Sample Instrumental Variables (TSIV), proposed by Angrist and Krueger (1992), addresses this by combining first-stage and reduced-form estimates from separate data sources into a single causal effect estimate. However, TSIV identification requires that instrument compliance behavior be consistent across the two samples, a condition that is rarely verified in practice. We show mathematically and empirically that when compliance differs between samples, the raw TSIV estimator does not converge to the true Local Average Treatment Effect (LATE) and instead attenuates toward a predictably biased limit proportional to the ratio of first-stage compliance rates between the two samples. To address this, we formalize a framework for estimating LATE with TSIV under two key assumptions: (1) Covariate Overlap, requiring that the two samples share sufficient common support in their covariate distributions, and (2) Compliance Transportability, requiring that compliance behavior is identical across populations after conditioning on observed covariates. We consider a setting in which a health policy instrument and outcomes are recorded in administrative claims while treatment and covariates are collected in a survey. We use a C-statistic derived from pooled covariates to detect population mismatch and an Inverse Probability Weighting (IPW) correction that reweights the first-stage sample to approximate the administrative covariate distribution. In Monte Carlo simulations across eight scenarios calibrated to a survey-Medicaid setting, IPW-TSIV reduces bias in estimating the LATE, achieving 88% reduction in the primary scenario, 82% under severe selection, and 79% when state-level expansion policy drives compliance heterogeneity. We further validate this framework using the Oregon Health Insurance Experiment, where partitioning the public-use lottery data (N = 24,646) into two non-overlapping samples with substantively meaningful compliance heterogeneity yields a verifiable benchmark against the true causal effect. IPW-TSIV reduces mean absolute bias by 71.6% relative to the oracle S2-specific LATE across 10 independent replications (C-statistic = 0.78), outperforms naive TSIV in all 10 splits, and reduces mean bias relative to the full-data LATE from +0.016 to +0.008. This framework provides applied researchers with actionable diagnostic thresholds to detect sample mismatch, validate transportability assumptions, and determine when structural TSIV estimation is reliable.

In meta-analyses of continuous outcomes, the sample mean and standard deviation (SD) are essential for synthesizing effect sizes across studies. However, clinical studies frequently report alternative summary statistics, such as the median, quartiles, and range. To enable inclusion of such studies, various methods have been proposed to estimate the sample mean and SD from these reported summaries. We propose the Bayesian Order Statistics-based Estimator (BOSE), which leverages the joint likelihood of observed order statistics together with weakly informative priors to obtain the full posterior distribution for the mean and SD without relying on computationally intensive iterative procedures such as Markov chain Monte Carlo algorithms. Our numerical studies demonstrate that BOSE performs competitively with existing approaches in estimating the mean, while achieving superior performance for estimating the SD across all evaluated scenarios, particularly in small-sample settings. Under non-normal distributions including skewed, heavy-tailed, and bimodal settings with mild or moderate deviations from normality, BOSE remains robust and stable, whereas methods specifically designed for skewed distributions may become unstable or even inapplicable. Beyond point estimation, BOSE naturally provides empirically validated posterior credible intervals, enabling researchers to formally quantify uncertainty for study-level estimates and make reliable, evidence-based decisions in meta-analytic research synthesis. A publicly accessible web application implementing BOSE and competing methods is also provided to facilitate practical use in meta-analytic research.

Joint modelling of longitudinal data using non-linear mixed effects models and time-to-event outcomes provides a suitable framework to account for informative censoring when estimating biomarker dynamics and quantifying event risk using covariates and longitudinal trajectories. Their usefulness in clinical research depends on data collection design, particularly to precisely estimate the association (link) parameter between longitudinal and survival processes. However, optimal design strategies have so far been addressed separately for longitudinal and survival endpoints and remain unexplored for joint models. We propose two Fisher Information Matrix (FIM) computation methods for joint models, relying on Monte-Carlo integration over observations combined with either Markov Chains Monte-Carlo or Adaptive Gaussian Quadrature to integrate random effects. Their accuracy is assessed against clinical trial simulations in an oncological example based on the HORIZON III study with a tumour-growth-survival model including discrete and continuous covariates. We apply these methods to quantify the impact of follow-up duration, sampling richness, sample size, and covariate distribution on parameter uncertainty and test power. In our example, longitudinal-parameter uncertainty is barely affected by follow-up duration or sampling richness, whereas survival-parameter uncertainty decreases substantially from 1-year to 2-year follow-up. The number of subjects needed (NSN) to achieve <15\% uncertainty on the link parameter is comparable for a 2-year rich design and a 3-year sparse design. Optimal covariate distributions are stable across designs and systematically improve test power, outperforming longer and richer but non-optimised designs. These FIM-based methods accurately predict uncertainty and test powers, enabling design evaluation and NSN computation for joint-model-based clinical studies.

Metatranscriptomic (MTX) sequencing enables profiling of gene expression across microbial communities, providing a framework for linking genetic potential with functional activity. However, standard pipelines report normalized abundances rather than raw counts, limiting the use of count-based RNA-seq methods, while Gaussian-based alternatives rely on transformations and assumptions that are often poorly suited to MTX data. We propose a new modeling framework for differential expression analysis of MTX data, built on a scale mixture of exponential distributions, that incorporates DNA abundance to adjust for genomic potential, accommodates subject-specific random effects, treats zeros as left-censored, and employs a mixture prior to handle extreme sparsity. Applied to the IBDMDB multi-omics cohort, differential expression results vary substantially across models, including among Gaussian approaches with different pseudocount choices. Our approach identifies a distinct subset of candidate genes not detected by existing Gaussian methods; these may provide useful leads toward a novel understanding of transcriptomic patterns associated with dysbiosis in inflammatory bowel disease. Estimated dysbiosis effect directions are consistent between our model and Gaussian-based approaches, while effect sizes from our model tend to be larger in absolute value.

We address the problem of fitting a collection of location-specific models under a spatial smoothness assumption. Existing approaches penalize roughness in the model parameters directly, an assumption that breaks down when smoothness is a function of parameters and auxiliary covariates rather than the parameters themselves. Our framework, the Auxiliary-Transformed Location-Aware Smoothing (ATLAS) penalty, generalizes spatial smoothness by penalizing roughness in transformations of model parameters using auxiliary information. As a concrete case study, we develop a spatially smooth deconvolution model for spatial transcriptomics that estimates tumor mixing coefficients from thousands of spots distributed on a single tissue slide. To handle the computational challenges posed by the nonlinear likelihood, nonsmooth nonconvex penalty, and spatially coupled estimation, we propose an alternating direction method of multipliers (ADMM) algorithm. Through simulation studies, we demonstrate that our framework provides substantially better spatial domain detection than approaches that smooth model parameters directly, with particularly strong gains when auxiliary covariates carry calibrated spatial structure.

Meta-analysis usually reduces each study to an effect estimate with a standard error and pools these by inverse-variance weighting: fixed effect (FE), random effects (RE), or unrestricted weighted least squares (UWLS). We propose information-geometric meta-integration (IGMI), representing each study by its sampling distribution, the Gaussian N(theta_i, Sigma_i), and pooling studies as a weighted Frechet mean (barycenter) under Bures-Wasserstein (BW), Fisher-Rao, or Wasserstein-Fisher-Rao (WFR) geometry. In the scalar fixed-variance case the BW barycenter mean is exactly the FE estimate; the minimized Frechet functional reproduces the Higgins-Thompson I^2 and DerSimonian-Laird tau^2 heterogeneity statistics; and a Frechet-scatter pivot reproduces the Hartung-Knapp-Sidik-Jonkman interval at m = 1 and yields an exact Hotelling F(m, K-m) region for m outcomes under proportional total covariances. WFR adds a robust outlier-resistant pool: as its length scale delta grows without bound it converges monotonically to BW, whereas finite delta gives a redescending M-estimator with rejection point exactly pi*delta. Simulations show calibrated multivariate coverage at small K, where Wald intervals undercover, and strong resistance of the equal-weight WFR pool to contamination. In 2,445 Cochrane meta-analyses, WFR most often wins leave-one-out predictive scoring. In 835 bivariate meta-analyses, the closed-form BW barycenter matches REML multivariate meta-analysis predictively and is exactly invariant to the unreported within-study correlation, unlike the likelihood estimate.

Longitudinal scRNA-seq experiments offer a powerful approach for dissecting temporal gene expression dynamics in individual cell types. However, few methods have been developed specifically to address the unique statistical challenges of repeated measures in scRNA-seq data. Here, we introduce a novel method, REBEL (Repeated measures Empirical Bayes differential Expression analysis using Linear mixed models), for analyzing cell type-specific differential expression in repeated measures scRNA-seq experiments. Using simulation studies, we demonstrate that, relative to conventional repeated measures analysis methods and other scRNA-seq approaches, REBEL controls the false discovery rate and exhibits competitive power across a range of simulation scenarios. We further validate REBEL by analyzing a longitudinal scRNA-seq dataset from patients with B-cell lymphoma receiving chimeric antigen receptor (CAR)-T cell therapy. REBEL is implemented as an R package, available at https://github.com/ewynn610/REBEL.

Statistical prediction models for binary outcomes are becoming increasingly popular. One significant challenge is calibrating these models to suit the characteristics of a target population that is structurally different from the original population. Calibration is especially challenging when there is no training data available from the target population. To address this problem, we propose a novel calibration method, SimCal, which uses synthetic data generated from the model development data in conjunction with marginal statistics from the calibration cohort. We show that expert judgment modeling (EJM) may be used for calibration if cross-sectional data from the target population are available comprising expert judgments about the potential outcome and the covariates. We describe three alternative calibration approaches when calibration data are lacking: similarity-binning averaging (SBA), adaptive calibration of predictions (ACP), and Elkan calibration. In a simulation study, we compare SBA, ACP, Elkan calibration, and SimCal. R code for applying these methods is provided from the re-analysis of data on coronary artery disease. We illustrate all 5 calibration approaches with a real data set for predicting functional outcome after stroke and all approaches but EJM in the re-analysis of the Cleveland Clinic data. None of the approaches performed convincingly well in all situations. SimCal performed well when model parameters were correctly specified. EJM failed on the stroke data. Further research is urgently required for calibration in the absence of calibration data.

ObjectiveStructured extraction from clinical free-text depends on human annotators whose labels are susceptible to errors and knowledge-driven mistakes; exhaustive quality control is impractical at scale. We evaluate whether disagreement among multiple locally hosted large language models (LLMs) can prioritize human annotations for targeted review. MethodsMultiple LLMs independently extract the same set of structured variables annotated by a human reviewer. For each annotation, an agreement score counts the LLMs matching the human label. Using four locally hosted LLMs (Gemma 3 27B, DeepSeek-R1 70B, GPT-OSS 120B, Mistral Large 3), we evaluated this approach on 910 German-language colonoscopy reports describing endoscopic mucosal resection, with five structured variables per case (anatomical location, two diameters, resection technique, multiple polyps), yielding 4,550 annotations and a 377-case adjudication sample. A stratified sample oversampling low-agreement strata was adjudicated blinded by an experienced reviewer and analyzed with prevalence-adjusted estimates ResultsHuman error rates rose as LLM agreement fell, from 0% at scores 3-4 to 76% at score 0. The lowest-agreement stratum was only 6.5% of annotations yet concentrated an estimated 80% of errors. The multi-LLM disagreement score achieved a prevalence-adjusted AUC-ROC of 0.991 (95% CI 0.987-0.994) and AUC-PR of 0.893 (95% CI 0.851-0.929) for error detection. DiscussionMulti-LLM disagreement outperformed single models and provided graded operating points for risk-stratified review. ConclusionMulti-LLM disagreement provides a scalable quality-control signal for targeted review of the highest-yield cases. Because all models run locally, the framework is GDPR-compliant; its language- and task-agnostic design supports application across clinical domains.

Spatial transcriptomics allows the unprecedented examination of gene expression levels at the resolution of spatially-situated single cells in a high-throughput manner. As the technology is adopted more broadly, studies frequently collect data from multiple tissue samples, which leads to unique challenges that traditional spatial statistical methods are not equipped to handle. In particular, factors that differ across samples, such as different coordinate systems, different numbers and types of cells, different underlying tissue architectures, among others, preclude the application of traditional methods to our new setting. In this work, we propose a novel method, TESSERA, based on a spatial generalized linear model, for analyzing multi-sample spatial transcriptomics count data. Importantly, we provide a mathematical and computational framework for efficient and scalable model fitting and statistical inference to accompany the specification of our model. Our method for fitting the model enables the estimation of a common set of fixed effects across samples. This allows us to address a variety of differential expression questions, such as identification of which genes are differentially expressed between conditions (e.g., diseases, treatments), while accounting for spatial correlation between cells within a sample. We benchmark our proposed method on simulated data and apply it to a spatial transcriptomics dataset of human kidney samples. We find that our method provides a hitherto nonexistent extension to the multi-sample setting while remaining competitive with or outperforming existing algorithms in the single-sample setting.

Spatially resolved transcriptomics enables the systematic characterization of spatial gene expression variation across tissue sections. Spatially variable genes within the same biological pathway often exhibit similar spatial expression patterns, reflecting shared biological functions and tissue organization. However, existing gene set enrichment analysis methods typically ignore this spatial dependence, which may reduce power to detect spatially organized pathways and limit the interpretability of pathway-level findings. To address this limitation, we propose spaGSE, a Bayesian hierarchical model for spatial pathway enrichment analysis that integrates genelevel summary statistics from spatial expression analysis with predefined gene set annotations. spaGSE models latent spatially variable gene signals through a Gaussian mixture framework and links spatial variation to gene set membership using logistic regression. To support robust and interpretable inference, we impose a spike-and-slab prior on the enrichment coefficient. Through simulation studies and analyses of four public SRT datasets, we show that spaGSE is scalable and achieves higher power while maintaining false positive rate control compared with existing approaches. In real-data applications, spaGSE identifies biologically relevant pathways with coordinated spatial organization across cancer and developmental tissues, demonstrating the value of incorporating spatial information into pathway-level inference for spatial transcriptomics.

Standard genome-wide association studies (GWASs) are vulnerable to confounding factors, including stratification, assortative mating, and dynastic effects. Family studies such as sibling-based GWAS (sib-GWAS) mitigate such confounding and are becoming the tool of choice for teasing apart direct genetic effects--causal effects of ones genotype on ones own phenotype-- from other factors. However, due in part to their smaller sample sizes, sib-GWAS allelic effect estimates are substantially more variable than standard (i.e., population-based) GWAS estimates. The quantification of this uncertainty is essential for many uses of sib-GWAS, including polygenic scoring, causal inference (e.g., Mendelian randomization), disentangling direct from indirect familial effects, and measuring assortative mating. Here, we investigate sources of uncertainty in sib-GWAS allelic effect estimators. We study their impacts on the biases of three uncertainty measurement methods, including two that are commonly used and a new resampling-based approach we propose. We find that heterogeneity in allelic effects or heteroskedasticity across families (e.g., due to variation in genetic backgrounds or environments) can bias existing methods, and that this bias is more severe for small samples and rare variants. In contrast, the resampling-based approach we propose is approximately unbiased under all scenarios we considered. We validate our theoretical predictions, as well as the importance of effect heterogeneity and heteroskedasticity, using simulations and empirical analysis in the UK Biobank. In sum, this study helps understand the sources of uncertainty in family-based genotype-phenotype association studies and provides a robust method to estimate uncertainty.

Community-level estimates of healthcare utilization are essential for identifying inequities, allocating resources, and evaluating place-based interventions. However, in the United States, no single data source adequately captures healthcare utilization within geographically defined populations. Population-based surveys often lack sufficient geographic resolution, insurance claims represent only covered populations, and electronic health records are limited to care delivered within participating health systems. Increasingly, researchers combine these fragmented data sources, yet limited guidance exists for conducting valid population-based descriptive analyses using incomplete and overlapping data. We review the strengths and limitations of major data sources used to characterize community healthcare utilization and propose an approach for conducting population-based descriptive analyses using fragmented data. Rather than focusing on the limitations of individual data sources, our approach begins by explicitly defining the target population and the ideal observational study that would answer the research question. Available data sources are then conceptualized as incomplete or imperfect realizations of that ideal, providing a structured approach to (a) identifying sources of selection bias, missingness, and measurement error, (b) articulating required assumptions, and (c) selecting appropriate analytic strategies. We illustrate our approach using colorectal cancer screening utilization among adults residing in Brooklyn, New York during 2022. By shifting attention from individual data sources to the target community and the assumptions required for valid inference, this approach provides a practical approach for strengthening descriptive analyses of community healthcare utilization and informing place-based public health research, policy, and practice.

Purpose: Nonresponse to routinely collected patient-reported outcome measures (PROMs) threatens the representativeness of aggregated data. We characterized patient-, provider-, and clinic-level factors associated with PROMIS Global-10 nonresponse in routine radiation oncology care. Methods: In this retrospective cohort study, all adults seen at five Mass General Brigham radiation oncology clinics over one year were included. The primary outcome was patient-level nonresponse, defined as never completing the portal-administered Global-10 versus completing it at least once. Using iterative mixed-effects logistic regression, we modeled patient-, provider-, and clinic-level factors. Results: Among 12,214 patients, 71 providers, and five clinics, patient- and appointment-level response rates were 35.4% and 10.9%, with patient-level response ranging nearly fivefold across clinics (12.8% to 66.2%). In Model 1, male sex, lower education, not working, and recent surgery had higher odds of nonresponse, and longer time since diagnosis lower odds. After provider- and clinic-level factors were added, patient sex, education, and employment became nonsignificant, whereas recent surgery (adjusted odds ratio [aOR] 1.97) and longer time since diagnosis (aOR 0.46 for >12 months) persisted. A provider's historical collection rate was protective but attenuated at the clinic level. There, a later program launch (aOR 0.29) and higher historical collection rate (aOR 0.79) correlated with lower nonresponse, whereas academic versus community setting did not. Conclusions: Nonresponse to routinely collected PROMs is a multilevel phenomenon driven substantially by clinic-level implementation factors, not patient characteristics alone. Because response rate is only a proxy for representativeness, PROMs programs and PRO-based performance measures should prioritize representative collection over volume.

Testing for Hardy-Weinberg equilibrium (HWE) is a fundamental component of genetic data analysis, widely used for quality control and model validation. Although HWE testing is well established for autosomal loci, inference on the X chromosome is more complex due to sex-specific genotype structures and potential sex differences in minor allele frequency (sdMAF). Existing tests differ in their assumptions about sdMAF and male sample inclusion, often leading to distinct but poorly characterized null hypotheses. We develop a general statistical framework for HWE inference using the robust allele-based regression model. By formulating HWE testing as an assessment of allele-level dependence, the framework directly parameterizes Hardy-Weinberg disequilibrium, unifies existing Pearson{chi} 2-based tests under explicit modeling assumptions, and clarifies their null hypotheses, degrees of freedom, and sensitivity to sdMAF. The framework also accommodates covariate and population-structure adjustment within a unified regression-based formulation. The proposed framework provides robust, interpretable, and flexible inference, establishing a unified statistical foundation for HWE testing across autosomal and X-chromosomal regions. Simulation studies and analysis of high-coverage 1000 Genomes Project data demonstrate that commonly used X-chromosome tests can exhibit inflated type I error or misleading inference when sdMAF is present.

Gene Set Enrichment Analysis (GSEA) is one of the most popular methods for transcriptomic analysis, yet its statistical power is limited when the biological pathways exhibit heterogeneous or non-concordant expression patterns. We propose an improved GSEA method, integral-based GSEA (inGSEA). inGSEA introduces a novel enrichment score based on the Anderson-Darling weighted integral statistic. The new enrichment score enhances detection power for complex signals, particularly sparse and bidirectional ones, while the Cauchy combination of integral and classic maximum statistics provides robustness across diverse expression patterns. Extensive numerical studies demonstrate that inGSEA achieves superior power and well-calibrated false discoveries. Application to real-world datasets reveals biologically relevant pathways missed by the standard GSEA. inGSEA reduces the computational burden of permutation testing by employing a generalized gamma distribution to approximate the null distribution. inGSEA is accessible as a user-friendly web-based software tool (https://amss-stat.github.io/inGSEA).

Conventional LD measures such as r2 perform poorly in the rare common regime, particularly in asymmetric configurations such as nested haplotype structure. Because r2 is symmetric and quadratic, it removes directional structure in two ways: squaring discards the sign, or phase, retained by the signed LD coefficient D, while symmetric normalization hides the asymmetry between the conditional probabilities P(A|B) and P(B|A). Although D recovers the phase, it is locus symmetric and unnormalized; its magnitude is hard to compare across frequency regimes and it does not by itself express which way the asymmetry runs. We therefore analyze the conditional-probability asymmetry {Delta} = P(A|B) - P(B|A), together with r2 and D, as distinct scalar functions on the haplotype simplex under the Fisher information metric. The conditional probabilities P(A|B) and P(B|A) are bounded in [0, 1], directly express carrier-set inclusion, and are more readily visualized than D. Moreover, their difference admits the exact decomposition {Delta} = M + C into a marginal frequency term M and an LD-coupled term C. Prior work has characterized either the mathematical behavior of LD normalizations across allele-frequency space or the Fisher geometry of the haplotype simplex, but not their connection. We bridge this gap by showing that the geometric structure of the simplex explains why LD measures disagree in the rare common regime and why symmetric normalizations such as r2 lose directional information. We show that the fixed-frequency leaf is intrinsically anisotropic, positively curved, and frequency-dependent under the Fisher metric. These geometric predictions are tested empirically , in phased 1000 Genomes data1 and a two locus Wright Fisher model, in a companion paper (Ichikawa, preprint); the present note develops the geometry itself. Keywords: linkage disequilibrium; Fisher information metric; haplotype simplex; rare variant; conditional-probability asymmetry; nested haplotype structure