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Implements missing-data sensitivity analyses for confirmatory clinical trials including MMRM under MAR (with Kenward-Roger correction), reference-based multiple imputation (J2R, CR, CIR, LMCF per Carpenter-Roger 2013), Permutt delta-adjustment / tipping-point analysis, pattern-mixture identifying restrictions (CCMV, NCMV, ACMV), and the Cro vs Bartlett variance debate. Use when handling missing primary or secondary endpoint data in regulatory submissions following NRC 2010 and ICH E9(R1).

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Performs time-to-event analysis for clinical trials including Cox proportional hazards regression with PH diagnostics, restricted mean survival time (RMST) under non-PH, competing risks via Fine-Gray vs cause-specific Cox, weighted log-rank and MaxCombo for non-proportional hazards, recurrent events (Andersen-Gill, PWP, WLW), and interval-censored data. Use when analyzing time-to-event endpoints (OS, PFS, DOR, TTR, TTNT) in oncology or other clinical trials.

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Hardware-agnostic quantum ML framework with automatic differentiation. Use when training quantum circuits via gradients, building hybrid quantum-classical models, or needing device portability across IBM/Google/Rigetti/IonQ. Best for variational algorithms (VQE, QAOA), quantum neural networks, and integration with PyTorch/JAX/TensorFlow. For hardware-specific optimizations use qiskit (IBM) or cirq (Google); for open quantum systems use qutip.

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Performs logistic regression for clinical trial outcomes (binary, ordinal, multinomial) with marginal-vs-conditional estimand reporting per FDA 2023 covariate adjustment guidance, g-computation/standardisation for marginal effects, modified Poisson for RR, Brant test for proportional odds, Firth penalty for separation, and Hauck-Donner detection. Use when modeling binary or ordinal endpoints in confirmatory or exploratory clinical trials.

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Implements multiplicity control for confirmatory clinical trials using graphical procedures (Bretz-Maurer-Hommel), gatekeeping (parallel, serial, mixed), Hochberg/Hommel/Holm with PRDS, and the closed-testing principle (Marcus-Peritz-Gabriel; Goeman 2021 admissibility). Covers FDA Multiple Endpoints Final Guidance (October 2022), graphical procedures via R gMCP, primary + key-secondary + subgroup hierarchies, and FWER vs FDR distinction. Use when designing the multiplicity strategy for confirmatory trials with multiple primary or key secondary endpoints.

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Performs subgroup and heterogeneous treatment effect (HTE) analyses for clinical trials. Covers Mantel-Haenszel pooling, Breslow-Day, interaction tests in regression, RERI for additive interaction, modern data-adaptive HTE methods (STEPP, SIDES, causal forests, X/R-learners), Bayesian shrinkage (Dixon-Simon, MAP, EXNEX), graphical multiplicity (Bretz-Maurer), and credibility frameworks (Sun BMJ, EMA 2019). Use when analyzing treatment effects across patient subgroups for regulatory submissions or precision-medicine claims.

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Solve quantitative problems in biophysics — pharmacokinetics (PK volume of distribution, clearance, half-life), epidemiology (R0, attack rate), toxicology (LD50, NOAEL), population genetics (Hardy-Weinberg, Fst), enzyme kinetics (Michaelis-Menten), thermodynamics. Use for first-principles quantitative biology calculations, dose calculations, exposure assessment, and biophysical-property estimation.

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Build publication-quality figures in R with ggplot2 using the grammar of graphics (data + aesthetics + geometries + scales + facets + themes) with CVD-safe palettes, cairo_pdf TrueType embedding, programmatic aes via tidy evaluation, and the theme_classic publication baseline. Use when producing static figures in R for papers, presentations, or reports.

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Designs Bayesian clinical trials including Phase I dose-finding (BOIN, CRM, EWOC, mTPI-2), meta-analytic-predictive (MAP) priors with robust mixtures for external data borrowing, EXNEX for basket trials, hierarchical models for safety AE (Berry-Berry), Bayesian platform trials (I-SPY 2, GBM AGILE, REMAP-CAP), and posterior probability stopping rules. Covers FDA Bayesian Devices Guidance (2010), FDA Bayesian Methodology in Drugs Draft (January 2026), BOIN Fit-for-Purpose qualification (December 2021), and Project Optimus dose-optimisation. Use when designing dose-finding studies, platform trials, or sensitivity analyses with informative priors.

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Computes and interprets treatment effect measures (OR, RR, RD, HR, NNT) with calibrated confidence intervals (Wilson, Newcombe, Miettinen-Nurminen, MOVER, profile likelihood, Bender NNT) and reports marginal vs conditional estimands per FDA 2023 covariate adjustment guidance. Use when reporting treatment effects in confirmatory trials, comparing effect sizes across studies, or constructing forest plots.

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Sparse Identification of Nonlinear Dynamics (SINDy) — discover governing equations from time-series data. Builds sparse dynamical system models dx/dt = f(x) from measurements using PySINDy. Use when you have trajectory data and want to find the underlying ODE.

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Discover governing equations from data using PySR (evolutionary symbolic regression). Physics-constrained search with dimensional analysis, custom operators, and complexity-accuracy tradeoffs. Use when you need an interpretable equation, not a black-box model.

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Estimates the minimum biological replicates (or cells/events) for a target power at a target FDR in genomics experiments using ssizeRNA, PROPER, powsimR for scRNA-seq, and pilot-data dispersion estimation from DESeq2/edgeR. Covers the biological-versus-technical replication distinction (technical replicates do not add degrees of freedom for biological inference), replicate-number-versus-sequencing-depth budgeting, scRNA-seq sample-versus-cell allocation under a pseudobulk model, and the critique that "n=3" is a publication convention rather than a power calculation. Use when budgeting a sequencing experiment, writing the sample-size justification in a grant, estimating replicates from pilot data, allocating a fixed budget between samples and depth, or planning scRNA-seq cohort size. For clinical-trial sample size see clinical-biostatistics/power-and-sample-size; for the power-given-n direction see experimental-design/power-analysis.

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Computes sample size and power for clinical trials including continuous, binary, and time-to-event endpoints; superiority, non-inferiority, and equivalence designs; FDA 2016 non-inferiority margin selection with M1/M2 framework; Schoenfeld 1981 and Lakatos 1988 for survival; Schuirmann TOST and 80-125% bioequivalence; minimum clinically important difference (MCID) vs δ distinction. Use when justifying trial size in protocol or SAP per CONSORT 2025 item 7.

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Architectures and techniques for neural operators on discontinuous PDE solutions (shocks, contact discontinuities, steep gradients). Covers local-global spectral design (ShockFNO), reflection padding for non-periodic BCs, resolution scaling for shock width, and frequency-band error diagnostics. Use for low-viscosity Burgers, compressible Euler, Riemann problems, or any PDE where standard FNO produces Gibbs oscillations.

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Frequency-domain analysis — FFT, power spectral density (Welch/periodogram), spectrograms, wavelet transforms, and coherence. Use for any signal with periodic, quasi-periodic, or transient frequency content in physics data.

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Simulate wave propagation — acoustic, electromagnetic, elastic, and quantum waves. FDTD, spectral methods, and absorbing boundary conditions for 1D/2D/3D wave equations with sources, scattering, and dispersion.

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Google quantum computing framework. Use when targeting Google Quantum AI hardware, designing noise-aware circuits, or running quantum characterization experiments. Best for Google hardware, noise modeling, and low-level circuit design. For IBM hardware use qiskit; for quantum ML with autodiff use pennylane; for physics simulations use qutip.

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Quantum physics simulation library for open quantum systems. Use when studying master equations, Lindblad dynamics, decoherence, quantum optics, or cavity QED. Best for physics research, open system dynamics, and educational simulations. NOT for circuit-based quantum computing—use qiskit, cirq, or pennylane for quantum algorithms and hardware execution.

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Designs adaptive clinical trials including group-sequential (O'Brien-Fleming, Pocock, Lan-DeMets spending), sample-size re-estimation (blinded Friede-Kieser, unblinded Cui-Hung-Wang, Mehta-Pocock promising zone), seamless Phase 2/3 with treatment-arm selection, population enrichment, and response-adaptive randomisation. Covers FDA 2019 Final Adaptive Designs Guidance, FDA 2022 Master Protocols, and ICH E20 Step 2b/3 draft (June 2025, NOT final). Use when planning interim analyses, sample-size re-estimation, or master/platform-trial designs.

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Tests associations between categorical variables in clinical data using chi-square, Fisher's exact, Boschloo, Cochran-Mantel-Haenszel, and modern McNemar variants with calibrated confidence intervals (Wilson, Newcombe, Miettinen-Nurminen). Use when analyzing categorical outcomes, paired binary endpoints, or testing treatment-outcome independence in confirmatory or exploratory clinical trials.

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Reads, validates, and prepares CDISC SDTM and ADaM clinical trial data for analysis. Covers SDTM domain joins (DM, AE, EX, VS, LB, DS), ADaM architecture (ADSL, BDS, OCCDS, ADTTE) with traceability, treatment-emergent AE conventions, baseline derivation, SUPPQUAL/NSV handling, Define-XML 2.1, and Pinnacle 21 / CORE validation. Use when working with clinical trial datasets in CDISC SDTM/ADaM format, preparing analysis-ready data, or validating for regulatory submission.

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Monte Carlo simulation for statistical mechanics — Ising model, Metropolis-Hastings, Wolff cluster algorithm, observables (magnetization, susceptibility, specific heat), finite-size scaling, and critical phenomena analysis.

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Prepares statistical reports for clinical trials following CONSORT 2025, SPIRIT 2025, ICH E9(R1) estimands, and FDA 2023 covariate adjustment guidance. Covers Table 1 generation, analysis populations (ITT/FAS/PP/Safety), the 5 ICH E9(R1) intercurrent-event strategies, MMRM under MAR (mmrm), reference-based MI (rbmi J2R/CR/CIR), Permutt tipping-point sensitivity, and Rubin's-rules vs frequentist variance debate. Use when preparing regulatory submissions, defining estimands, or implementing missing-data sensitivity analyses.

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