Statistical Methods in Medical Research

Mediation analysis quantifies how an exposure affects an outcome through an intermediate variable. We extend mediation analysis to capture the cumulative effects of longitudinal predictors on longitudinal outcomes. Our proposed model examines how mediators transmit the effects of the current and previous exposure on the current outcome. We construct a least-squared estimator for cumulative indirect effect (CIE) and used three approaches (exact form, delta method, and bootstrap procedure) to estimate its standard error (SE). The estimator of CIE is unbiased with no unmeasured confounding and independent model errors between mediator model and outcome model at all time points, as shown in statistical inference and in simulations. While three SE estimates are numerically similar, bootstrap procedure is recommended due to its simplicity in implementation. We apply this method to Framingham Heart Study offspring cohort to assess if DNA methylation mediates the association of alcohol consumption with systolic blood pressure over two time points. We identify two CpGs (cg05130679 and cg05465916) as mediators and construct a composite DNA methylation score from 11 CpGs, which mediates for 39% of the cumulative effect. In conclusion, we propose an unbiased estimator for CIE. Future studies will investigate the missingness in mediators and outcomes.

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.

We present a novel sequential multiple assignment randomized trial (SMART) design that integrates response-adaptive randomization with tailoring functions (RA-TF-SMART). We develop percentile-based and Z-score RA-TFs that incorporate both within-patient and between-patient adaptation to map continuous outcomes to randomization probabilities. We apply Q-learning, tree-based reinforcement learning, and G-estimation to estimate dynamic treatment regimens (DTRs). We compare our RA-TF-SMART designs to balanced randomized SMARTs (BR-SMARTs), tailoring function SMARTs (TF-SMARTs), and generalized outcome-adaptive SMARTs (GO-SMARTs). This study addresses limitations in SMART methodology by presenting designs where randomization probability does not require dichotomization of continuous outcomes and utilizes both individual patient outcomes and accumulated treatment efficacy data from prior participants. RA-TF-SMARTs offer a flexible framework that maximizes benefit for trial participants while maintaining statistical validity for post-trial DTR estimation.

Cancer therapy balances between two competing objectives - treatment efficacy against the tumour and the risk of treatment related severe adverse events, including patient death. Most existing optimal control theory (OCT) formulations rely on optimising heuristic cost functionals that lack direct clinical interpretability. In clinical practice treatment efficacy and patient tolerability are primarily assessed through survival metrics and adverse event rates. Here we introduce the Continuous Lifetime Payoff (CLP), a novel OCT objective functional that directly links treatment decisions to patient survival. It explicitly incorporates tumour dynamics, tumour eradication, and patient mortality from tumour progression, drug-related toxicity and age. We fit age-related mortality from life tables and infer parameters from simulated survival data. The CLP provides a clinically grounded framework for optimising chemotherapy regimens.

Abstract Purpose This study introduces a novel approach to address unmeasured confounding in terminal event studies using the prior event rate ratio (PERR) method. The proposed approach PERR_{proxy} used a proxy event to replace the original terminal event in the pre-exposure period, enabling the application of PERR in terminal event settings. Additionally, we also applied difference in difference (DID) regression, which is conceptually analogous to PERR to estimate the standard errors and confidence intervals of PERR_{proxy}. Methods We conducted numeric simulations to evaluate the validity of PERR_{proxy} approach and assessed its performance under varying levels of unmeasured confounding effects, baseline hazard ratios, and the correlation between the proxy and terminal events. To demonstrate its practical applicability, we also performed an empirical analysis to investigate the impact of severe hospitalized COVID-19 on circulatory system disease mortality using the PERR_{proxy}. Results In simulation studies, PERR_{proxy} effectively reduced the unmeasured confounding effects compared to the conventional methods. The performance of PERR_{proxy} was influenced by the strength of unmeasured confounding, baseline hazard ratios, and the correlation between the proxy and terminal outcomes. In addition, difference in difference (DID) regression had much faster computational speed for estimating standard errors and confidence intervals compared to bootstrap. In the empirical analysis, PERR_{proxy} identified that severe hospitalized COVID-19 as a significant risk factor for the circulatory system disease mortality and reduced the unmeasured confounding effects. Conclusions The PERR_{proxy} approach extends the applicability of the original PERR method to terminal event studies, offering a promising solution for addressing unmeasured confounding. Additionally, the DID regression framework provides a computationally efficient alternative for parameter estimation in PERR-based studies. However, careful consideration is still required in PERR_{proxy} for proxy events selection and other underlying assumptions of the PERR method to ensure valid results. Keywords: prior event rate ratio, unmeasured confounding, proxy event, terminal event study, observational study, electronic health records

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.

Physical activity (PA) plays an important role in maintaining and improving health. Daily steps have been a key PA measure that is easily accessible with common wearable devices. However, methods are lacking to recommend a personalized optimal distribution of daily steps over a period of time for the best of certain health biomarkers. In this paper, we fill this void based on the data from the All of Us Research Program which includes months of step counts as well as repeated measurements of key health biomarkers. We develop a new offline reinforcement learning (RL) algorithm to learn personalized and optimal PA distributions associated with cardiometabolic risk, where the action is a function representing the daily step distribution over a period of time. Simulation studies demonstrate the advantage of the proposed approach over existing continuous-action RL methods. The learned optimal policy from the All of Us data generally suggests people take more daily steps and also follow a more consistent pattern of PA over time while offering tailored recommendations for subgroups in blood glucose level, body mass index, blood pressure, age, and sex.

Machine learning models for causal inference aim to adjust for confounding factors that are associated with both an exposure and an outcome, creating a spurious biased association. But, these methods are rarely empirically evaluated to assess their success in mitigating such bias. Recent advances in knowledge representation, including both foundation models and knowledge graphs, could enrich these models, but rigorous evaluations are needed in order to assess their potential. Here, we ask whether enriching existing causal inference models with knowledge representations from foundation models can improve confounding control. Rather than using semi-simulated data to address this question, we focus on examples of real confounding: we emulate target randomized active comparator trials that are subject to confounding by indication. Our results can guide researchers aiming to develop or apply methods for discovering causal effects from observational data.

Background: In prospective cohort studies, where an exposure is collected repeatedly, interest often lies in determining whether the timing of that exposure has a differential effect on a later outcome. The Structured Life Course Modeling Approach (SLCMA), where users select between temporal hypotheses of exposure specified a priori, provides one way to analyse such longitudinal data. However, few studies using SLCMA consider the effect of time-varying covariates (TVC) which may impact associations. Methods: We present a modified version of the SLCMA - called direct and mediated effects (DME)-SLCMA - which corrects for TVC. We first develop the DME-SLCMA method, test it through simulation, and apply it to psychosocial data from the Drakenstein Child Health Study (DCHS, n=336) to investigate relationships between maternal psychopathology, TVC of socioeconomic status, and offspring depressive symptoms. Results: We found that, on average, offspring depressive symptoms score increased by 3.9% (95% CI: 1.0%-6.9%, p = 0.039) for each unit of maternal psychopathology (SRQ) at 48 months whilst adjusting for time-varying socioeconomic status (at 18, 30, 42 and 54 months). Our simulations identified several realistic scenarios where selections ignoring TVC - with TVC mediated exposure effects present - were prone to be incorrect, including our DCHS example. Conclusion: DME-SLCMA is a robust new approach for life course modelling in the presence of time-varying covariates. We recommend adjusting for TVC whenever possible, and, when not possible, our simulation study identified that scenarios where mediated effects are comparable, or greater, in magnitude to direct effects are most prone to confounding.

Background. Self-exciting (Hawkes) point processes are a natural model for the temporal clustering of human physical activity (PA) recorded by accelerometers, yet they have seldom been used in this setting---in part because the usual maximum-likelihood fitting is challenging due to potential estimation bias and convergence failures on these data. A moment-based alternative---estimating the Hawkes branching ratio from the dispersion index, the variance-to-mean ratio of event counts---is kernel-independent and computationally trivial, but it has not been evaluated for accelerometry or adapted to the intensity-marked recordings accelerometers provide. Methods. Treating each minute above a sedentary threshold as an event, we estimated the Hawkes branching ratio $n$ by maximum likelihood and, as a kernel-independent and far cheaper alternative, from the dispersion index. We compared four dispersion-based estimators---event-count-based, intensity-mark-weighted using the mark-moment ratio, and time-of-day (TOD) adjusted variants of each---against the marked and unmarked maximum-likelihood estimates. Estimators were evaluated for mutual agreement, goodness of fit, and finite-window results in two National Health and Nutrition Examination Survey (NHANES) accelerometry cohorts (hip-worn, $n=2{,}560$; wrist-worn, $n=3{,}132$). We related the resulting temporal clustering measures to all-cause mortality using survey-weighted Cox models, adjusting for PA frequency, Peak30 (the average of the 30 highest PA values), and demographic covariates. Results. Event-count-based dispersion estimates agreed strongly with maximum-likelihood branching ratios ($r\approx0.74$ in both cohorts); the intensity-marked variant incorporating PA intensity variability agreed less well. Marked and unmarked Hawkes models yielded similar excitation and decay parameters, suggesting PA intensity added little clustering information beyond event timing. In the survival analysis, temporal clustering was associated with all-cause mortality independently of PA frequency and Peak30; the direction of association differed between the hip- and wrist-worn cohorts. Conclusions. A scalable dispersion-index estimator recovers the Hawkes branching ratio and matches maximum-likelihood estimates without requiring kernel specification or iterative optimization. It offers a practical tool for quantifying temporal clustering in accelerometry, enabling decomposition of temporal PA patterns into its exogenous initiation and endogenous persistence. Such temporal patterns carry health-relevant information beyond PA intensity and volume. Keywords: dispersion index; Hawkes process; branching ratio; temporal clustering; point process estimation; accelerometry; mortality

Objective To evaluate the feasibility, safety, patient acceptance, and preliminary clinical relevance of automated self-operated transvaginal ultrasound for ovarian stimulation monitoring. Design Prospective observational pilot study. Subjects Ten women undergoing ovarian stimulation for in vitro fertilization or fertility preservation at a single high-volume private IVF center. Exposure Participants performed investigational self-operated transvaginal ultrasound examinations immediately following standard monitoring visits. Patients inserted and stabilized the ultrasound probe while ovarian and endometrial imaging was acquired through controlled motorized probe rotation without real-time anatomical guidance. Main Outcome Measure(s) The primary outcome was feasibility, defined as the generation of evaluable imaging datasets suitable for ovarian stimulation monitoring. Secondary outcomes included bilateral ovarian visualization, procedural safety, patient-reported outcomes, follicular assessment, and agreement of endometrial thickness measurements with standard transvaginal ultrasound. Result(s) Nineteen investigational scan attempts were performed, yielding 18 evaluable datasets (94.7%). Bilateral ovarian visualization was achieved in 16 of 18 evaluable examinations (88.9%), whereas partial ovarian visualization occurred in 2 examinations (11.1%). No adverse events, adverse device effects, vaginal injury, bleeding, or infection were observed. Patient-reported outcomes demonstrated high procedural acceptability, with all participants expressing willingness to reuse the system. Compared with standard transvaginal ultrasound monitoring, investigational self-operated acquisition significantly improved overall examination experience (Wilcoxon p=0.002). Investigational imaging demonstrated clinically relevant agreement with standard transvaginal ultrasound for follicular categorization and endometrial assessment. Counts of follicles [≥]14 mm correlated strongly with mature oocyte recovery for both investigational and standard ultrasound measurements (Spearman {rho}=0.83 and {rho}=0.80, respectively). Endometrial thickness measurements also demonstrated strong correlation between modalities (Spearman {rho}=0.91). Conclusion(s) This prospective pilot study demonstrates the feasibility of automated self-operated transvaginal ultrasound during ovarian stimulation monitoring. Investigational imaging generated clinically relevant monitoring information without observed safety concerns and was associated with high patient acceptance. These findings support further investigation of patient-operated acquisition strategies and standardized imaging workflows in reproductive medicine.

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.

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.

Progestin-IUD use was more frequent among IUA cases (29.9%) than polyp (3.4%) or infertile (11.1%) comparison groups. Compared to infertile comparators, any prior progestin-IUD use was independently associated with IUA (aOR 3.12; 95% CI 2.01, 4.85). There was a duration-response pattern: use of 5 years or less was modestly associated with IUA case status (aOR 1.99; 1.09, 3.64), whereas use >5 years conferred more than a seven-fold increase (aOR 7.26; 3.27, 16.11). The association persisted among surgically naive women (aOR 3.98; 2.44, 6.48) and was concentrated in those who were nulliparous, where use beyond five years conferred an approximately twelve-fold increase in odds (aOR 12.74; 5.25, 30.92). Progestin-IUD use was less frequent in polyp controls relative to IUA and infertile comparators, suggesting a possible role for progestin exposure in preventing endometrial polyp formation. The case control design does not allow for estimation of absolute risk for an individual and cannot inform causation. Further prospective studies are needed to better assess the relationship between progestin-IUD's, particularly when used beyond five years, and adverse fertility outcomes.

Background Simon two-stage designs for binary endpoints and their time-to-event analogues, including the Kwak and Jung method, rely on a fixed null benchmark. Their Type I error control is valid only when that benchmark is correctly specified. In practice, historical benchmarks are often inconsistent due to small samples, population heterogeneity, changing eligibility criteria, and evolving standards of care. Even modest misspecifications can substantially inflate the Type I error rate, leading to costly advancement of ineffective treatments. Methods We propose the Interval-Null Robust (INR) two-stage design framework that accounts for uncertainty in the historical null benchmark. We define the null hypothesis as a plausible range of clinically uninteresting values: p[isin][p0L, p0U] for binary endpoints and {lambda}[isin][{lambda}0L, {lambda}0U] (or equivalent survival probabilities) for time-to-event endpoints. Type I error is controlled uniformly over the full null interval: sup{theta}[isin]{theta}0 Pr{theta}(Go) [≤] . Under the monotonicity of the Go probability, the supremum occurs at the least favorable null configuration - p0U and {lambda}0L - but the design is not reduced to a point-null formulation. The interval defines the uncertainty set for error control and is used in selecting among feasible designs through robust criteria such as worst-case regret or minimal average expected sample size. Results Across representative planning scenarios for both endpoint types, classic designs calibrated to a single benchmark exhibit substantial Type I error inflation when the true null parameter exceeds the assumed planning value. INR designs maintain the nominal Type I error rate across the full null interval, directly addressing this vulnerability to benchmark misspecification. The robustness-efficiency trade-off can be managed through design constraints and robust optimization criteria while preserving uniform Type I error control. Conclusions INR two-stage designs offer a transparent framework for addressing historical control uncertainty in single-arm Phase II trials. By replacing reliance on a fixed benchmark assumption with a more realistic interval of clinically plausible null values, INR design reduces the risk of false-positive Go-decisions caused by benchmark misspecification. INR applies to both binary and time-to-event endpoints and is implemented in the open-source INRDesign R package and accompanying interactive Shiny app.

Biomarker measurements can assist with disease diagnosis and the assessment of disease risks, with the most recent measurements usually used by disease-risk models. However, a growing number of studies suggest that in addition to a biomarkers value, its inherent variability, estimated from several measurements over many days or years in an individual, can convey independent prognostic information about disease risks. Variance estimates require an individuals biomarker data to have been measured a sufficient number of times, ideally across a long time period, and are usually only available in a hospital setting or clinical trial. Furthermore, a single biomarker measurement will involve a combination of measurement-error, natural short-term variation over a daily time-period, variation over time periods of weeks and months, and slower age-dependent changes over several years. This paper develops a statistical method that accounts for these latter concerns, and applies it to Clinical Practice Research Datalink (CPRD) data collected by UK General Practitioners. It studies the associations between cardiovascular health outcomes and the within-person variances of eight routinely measured biomarkers. This involved Sequential Monte Carlo modeling to convert an individuals biomarker measurements (collected over months or years), into estimates for the biomarkers mean, linear age-dependent slope, within-person variance, and a variance due to variation on a daily time period or measurement errors. The result is a proof-of-principle that UK primary care Electronic Health Records (from CPRD) can be effectively used for this purpose. After adjusting for mean biomarker values, clear associations were found between mortality or cardiovascular disease risks and within-person variances for 6 of 8 biomarkers.

BackgroundPolygenic embryo screening (PES) has recently become available to in-vitro fertilization (IVF) patients, allowing them to evaluate the genetic risk of each of their embryos for polygenic conditions such as heart attack or diabetes. Initial modeling predicted that transferring the embryo with the lowest genetic risk for one or more diseases would substantially reduce prevalence in the next generation, with relative risk reductions up to 50%. However, these models assumed the availability of a prespecified number of embryos and that the embryo with the most favorable polygenic risk is born once transferred to the uterus. In reality, a large percentage of embryo transfers do not lead to live births, and IVF frequently results in no or only a single live birth. MethodsTo quantify the expected risk reduction in the context of IVF, we used two datasets: 6944 ovarian stimulation cycles from 4452 Italian infertility patients and 2138 stimulation cycles of egg donors. In both datasets, we simulated the hypothetical application of PES in these cycles by assigning patients and their embryos randomly drawn polygenic risk scores for a given disease, assuming that embryos have been transferred in increasing order of their risk, and tracing their birth outcomes. We then compared the risk of the embryo born after hypothetical PES to the risk of an embryo born without PES. We either considered only completed cycles or integrated over possible birth outcomes of non-transferred embryos in incomplete cycles. ResultsIn stimulation cycles in infertility patients in which all embryos have been transferred and at least one child was born, we estimate that PES will result in relative risk reductions of just {approx}1-3%. In an intention-to-screen analysis of all completed cycles (regardless of birth outcomes), relative risk reductions are under 0.5%. The risk reductions increase, as expected, with more euploid blastocysts and with younger maternal age. Including incomplete cycles (in which not all embryos have been transferred) increases risk reductions to {approx}2-5%, due to the availability of more euploid blastocysts and a higher live birth rate per transfer in these cycles. Pooling all embryos from all cycles of the same patient increases risk reductions to {approx}5-10%. Relative risk reductions in egg donor cycles reach {approx}20% even with a single stimulation cycle per donor. ConclusionsWith the exception of particularly good-prognosis patients or cycles, typical infertility patients would benefit little from PES. In fertile patients, as represented by egg donors, PES is predicted to achieve greater relative risk reductions. However, even though these reductions are still substantially lower than prior estimates that did not account for realistic live birth rates. Ethical, social, and clinical issues associated with offering PES in the general population should be prioritized in future research.

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.

Infectious disease transmission evolves as a dynamic process shaped by biological mechanisms, population behavior, and intervention policies, yet public health responses are often driven by lagging indicators. Accurate short- and long-term disease forecasting is essential for the timely deployment of intervention strategies, healthcare capacity planning, and uncertainty-aware, risk-informed decision-making. To address this challenge, three broad classes of forecasting models have traditionally been used: statistical, machine learning, and mechanistic approaches. However, each of these modeling paradigms faces fundamental limitations. In particular, traditional statistical models often lack the flexibility needed to capture complex disease dynamics, machine learning approaches require large, high-quality data streams, and mechanistic models are notoriously difficult to calibrate. To overcome these challenges, we propose a novel physics-informed machine learning (PIML) framework for forecasting infectious disease dynamics. Our approach simultaneously forecasts new case and hospitalization counts, along with other key epidemiological quantities such as the time-varying reproduction number. This is achieved through the design of a machine learning model and estimation strategy regularized by a system of differential equations that encode disease dynamics of the SIHR model, thereby bridging the gap between purely data-driven and mechanistic models. We demonstrate the proposed methodology through in-depth numerical studies and an application to COVID-19 data collected in the state of South Carolina.

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.