Multi-response Mendelian randomization: Identification of shared and distinct exposures for multimorbidity and multiple related disease outcomes.

The existing framework of Mendelian randomization (MR) infers the causal effect of one or multiple exposures on one single outcome. It is not designed to jointly model multiple outcomes, as would be necessary to detect causes of more than one outcome and would be relevant to model multimorbidity or other related disease outcomes. Here, we introduce multi-response Mendelian randomization (MR2), an MR method specifically designed for multiple outcomes to identify exposures that cause more than one outcome or, conversely, exposures that exert their effect on distinct responses. MR2 uses a sparse Bayesian Gaussian copula regression framework to detect causal effects while estimating the residual correlation between summary-level outcomes, i.e., the correlation that cannot be explained by the exposures, and vice versa. We show both theoretically and in a comprehensive simulation study how unmeasured shared pleiotropy induces residual correlation between outcomes irrespective of sample overlap. We also reveal how non-genetic factors that affect more than one outcome contribute to their correlation. We demonstrate that by accounting for residual correlation, MR2 has higher power to detect shared exposures causing more than one outcome. It also provides more accurate causal effect estimates than existing methods that ignore the dependence between related responses. Finally, we illustrate how MR2 detects shared and distinct causal exposures for five cardiovascular diseases in two applications considering cardiometabolic and lipidomic exposures and uncovers residual correlation between summary-level outcomes reflecting known relationships between cardiovascular diseases.

Bayesian joint modeling of multivariate longitudinal and survival outcomes using Gaussian copulas.

There is an increasing interest in the use of joint models for the analysis of longitudinal and survival data. While random effects models have been extensively studied, these models can be hard to implement and the fixed effect regression parameters must be interpreted conditional on the random effects. Copulas provide a useful alternative framework for joint modeling. One advantage of using copulas is that practitioners can directly specify marginal models for the outcomes of interest. We develop a joint model using a Gaussian copula to characterize the association between multivariate longitudinal and survival outcomes. Rather than using an unstructured correlation matrix in the copula model to characterize dependence structure as is common, we propose a novel decomposition that allows practitioners to impose structure (e.g., auto-regressive) which provides efficiency gains in small to moderate sample sizes and reduces computational complexity. We develop a Markov chain Monte Carlo model fitting procedure for estimation. We illustrate the method's value using a simulation study and present a real data analysis of longitudinal quality of life and disease-free survival data from an International Breast Cancer Study Group trial.

Asymmetric relationships and their effects on coexistence.

Species coexistence attracts wide interest in ecology. Modern coexistence theory (MCT) identifies coexistence mechanisms, one of which, storage effects, hinges on relationships between fluctuations in environmental and competitive pressures. However, such relationships are typically measured using covariance, which does not account for the possibility that environment and competition may be more related to each other when they are strong than when weak, or vice versa. Recent work showed that such 'asymmetric tail associations' (ATAs) are common between ecological variables, and are important for extinction risk, ecosystem stability, and other phenomena. We extend MCT, decomposing storage effects to show the influence of ATAs. Analysis of a simple model and an empirical example using diatoms illustrate that ATA influences can be comparable in magnitude to other mechanisms of coexistence and that ATAs can make the difference between species coexistence and competitive exclusion. ATA influences may be an important new mechanism of coexistence.

Testing for similarity of multivariate mixed outcomes using generalized joint regression models with application to efficacy-toxicity responses.

A common problem in clinical trials is to test whether the effect of an explanatory variable on a response of interest is similar between two groups, for example, patient or treatment groups. In this regard, similarity is defined as equivalence up to a pre-specified threshold that denotes an acceptable deviation between the two groups. This issue is typically tackled by assessing if the explanatory variable's effect on the response is similar. This assessment is based on, for example, confidence intervals of differences or a suitable distance between two parametric regression models. Typically, these approaches build on the assumption of a univariate continuous or binary outcome variable. However, multivariate outcomes, especially beyond the case of bivariate binary responses, remain underexplored. This paper introduces an approach based on a generalized joint regression framework exploiting the Gaussian copula. Compared to existing methods, our approach accommodates various outcome variable scales, such as continuous, binary, categorical, and ordinal, including mixed outcomes in multi-dimensional spaces. We demonstrate the validity of this approach through a simulation study and an efficacy-toxicity case study, hence highlighting its practical relevance.

High-dimensional sparse vine copula regression with application to genomic prediction.

High-dimensional data sets are often available in genome-enabled predictions. Such data sets include nonlinear relationships with complex dependence structures. For such situations, vine copula-based (quantile) regression is an important tool. However, the current vine copula-based regression approaches do not scale up to high and ultra-high dimensions. To perform high-dimensional sparse vine copula-based regression, we propose 2 methods. First, we show their superiority regarding computational complexity over the existing methods. Second, we define relevant, irrelevant, and redundant explanatory variables for quantile regression. Then, we show our method's power in selecting relevant variables and prediction accuracy in high-dimensional sparse data sets via simulation studies. Next, we apply the proposed methods to the high-dimensional real data, aiming at the genomic prediction of maize traits. Some data processing and feature extraction steps for the real data are further discussed. Finally, we show the advantage of our methods over linear models and quantile regression forests in simulation studies and real data applications.

Principal stratification for quantile causal effects under partial compliance.

Within the principal stratification framework in causal inference, the majority of the literature has focused on binary compliance with an intervention and modelling means. Yet in some research areas, compliance is partial, and research questions-and hence analyses-are concerned with causal effects on (possibly high) quantiles rather than on shifts in average outcomes. Modelling partial compliance is challenging because it can suffer from lack of identifiability. We develop an approach to estimate quantile causal effects within a principal stratification framework, where principal strata are defined by the bivariate vector of (partial) compliance to the two levels of a binary intervention. We propose a conditional copula approach to impute the missing potential compliance and estimate the principal quantile treatment effect surface at high quantiles, allowing the copula association parameter to vary with the covariates. A bootstrap procedure is used to estimate the parameter to account for inflation due to imputation of missing compliance. Moreover, we describe precise assumptions on which the proposed approach is based, and investigate the finite sample behavior of our method by a simulation study. The proposed approach is used to study the 90th principal quantile treatment effect of executive stay-at-home orders on mitigating the risk of COVID-19 transmission in the United States.

Identifying the relation between food groups and biological ageing: a data-driven approach.

BACKGROUND: Heterogeneity in ageing rates drives the need for research into lifestyle secrets of successful agers. Biological age, predicted by epigenetic clocks, has been shown to be a more reliable measure of ageing than chronological age. Dietary habits are known to affect the ageing process. However, much remains to be learnt about specific dietary habits that may directly affect the biological process of ageing. OBJECTIVE: To identify food groups that are directly related to biological ageing, using Copula Graphical Models. METHODS: We performed a preregistered analysis of 3,990 postmenopausal women from the Women's Health Initiative, based in North America. Biological age acceleration was calculated by the epigenetic clock PhenoAge using whole-blood DNA methylation. Copula Graphical Modelling, a powerful data-driven exploratory tool, was used to examine relations between food groups and biological ageing whilst adjusting for an extensive amount of confounders. Two food group-age acceleration networks were established: one based on the MyPyramid food grouping system and another based on item-level food group data. RESULTS: Intake of eggs, organ meat, sausages, cheese, legumes, starchy vegetables, added sugar and lunch meat was associated with biological age acceleration, whereas intake of peaches/nectarines/plums, poultry, nuts, discretionary oil and solid fat was associated with decelerated ageing. CONCLUSION: We identified several associations between specific food groups and biological ageing. These findings pave the way for subsequent studies to ascertain causality and magnitude of these relationships, thereby improving the understanding of biological mechanisms underlying the interplay between food groups and biological ageing.

An analysis of the dual burden of childhood stunting and wasting in Myanmar: a copula geoadditive modelling approach.

OBJECTIVE: To analyze the spatial variation and risk factors of the dual burden of childhood stunting and wasting in Myanmar. DESIGN: Analysis was carried out on nationally representative data obtained from the Myan-mar Demographic and Health Survey conducted during 2015-2016. Childhood stunting and wasting are used as proxies of chronic and acute childhood undernutrition. A child with standardized height-for-age Z score (HAZ) below -2 is categorized as stunted while that with a weight-for-height Z score (WHZ) below -2 as wasted. SETTING: A nationally representative sample of households from the 15 states and regions of Myanmar. PARTICIPANTS: Children under the age of five (n 4162). RESULTS: Overall marginal prevalence of childhood stunting and wasting were 28.9% (95% CI 27.5, 30.2) and 7.3% (95% CI 6.5, 8.0) while their concurrent prevalence was 1.6% (95% CI 1.2, 2.0). The study revealed mild positive association between stunting and wasting across Myanmar. Both stunting and wasting had significant spatial variation across the country with eastern regions having higher burden of stunting while southern regions having higher prevalence of wasting. Child age and maternal weight-for-height Z score had significant non- linear association with both stunting and wasting while child gender, ethnicity and household wealth quintile had significant association with stunting. CONCLUSION: The study provides data-driven evidence about the association between stunting and wasting and their spatial variation across Myanmar. The resulting insights can aid in the formulation and implementation of targeted, region-specific interventions towards improving the state of childhood under-nutrition in Myanmar.

Modelling alternately recurring events using subject specific hazard estimation approach.

The motivation for this paper is to account for subject specific variations in a Cox proportional hazard model for alternating recurrent events. This is done through two sets of frailty components, whose marginal distributions are bound together by a copula function. The likelihood function involves unobservable variables, which requires the use of the EM algorithm. This leads to intractable integrals, which after some approximations, are solved using computationally intensive techniques. The results are applied to a real-life data. A simulation study is also carried out to check for consistency.

DOD-Combo: Bayesian dose finding design in combination trials with meta-analytic-predictive prior.

Combination therapy, a treatment modality that involves multiple treatment agents, has become imperative for improving treatment effectiveness and addressing resistance in the field of oncology. However, determining the most effective dose for these combinations, particularly when dealing with intricate drug interactions and diverse toxicity patterns, presents a substantial challenge. This paper introduces a novel Bayesian dose-finding design for combination therapies with information borrowing, named the DOD-Combo design. Leveraging historical single-agent trials and the meta-analytic-predictive (MAP) power prior, our approach utilizes a copula-type model to connect individual drug priors with joint toxicity probabilities in combination treatments. The MAP power prior allows the integration of information from multiple historical trials, constructing informative priors for each agent. Extensive simulations confirm our method's superior performance compared to combination designs with no information borrowing. By adaptively incorporating historical data, our approach reduces sample sizes and enhances efficiency in selecting the maximum tolerated dose (MTD), effectively addressing the intricate challenges presented by combination trials.

Generation of realistic virtual adult populations using a model-based copula approach.

Incorporating realistic sets of patient-associated covariates, i.e., virtual populations, in pharmacometric simulation workflows is essential to obtain realistic model predictions. Current covariate simulation strategies often omit or simplify dependency structures between covariates. Copula models are multivariate distribution functions suitable to capture dependency structures between covariates with improved performance compared to standard approaches. We aimed to develop and evaluate a copula model for generation of adult virtual populations for 12 patient-associated covariates commonly used in pharmacometric simulations, using the publicly available NHANES database, including sex, race-ethnicity, body weight, albumin, and several biochemical variables related to organ function. A multivariate (vine) copula was constructed from bivariate relationships in a stepwise fashion. Covariate distributions were well captured for the overall and subgroup populations. Based on the developed copula model, a web application was developed. The developed copula model and associated web application can be used to generate realistic adult virtual populations, ultimately to support model-based clinical trial design or dose optimization strategies.

A copula-based method of risk prediction for autonomous underwater gliders in dynamic environments.

Autonomous underwater gliders (AUGs) are effective platforms for oceanic research and environmental monitoring. However, complex underwater environments with uncertainties could pose the risk of vehicle loss during their missions. It is therefore essential to conduct risk prediction to assist decision making for safer operations. The main limitation of current studies for AUGs is the lack of a tailored method for risk analysis considering both dynamic environments and potential functional failures of the vehicle. Hence, this study proposed a copula-based approach for evaluating the risk of AUG loss in dynamic underwater environments. The developed copula Bayesian network (CBN) integrated copula functions into a traditional Bayesian belief network (BBN), aiming to handle nonlinear dependencies among environmental variables and inherent technical failures. Specifically, potential risk factors with causal effects were captured using the BBN. A Gaussian copula was then employed to measure correlated dependencies among identified risk factors. Furthermore, the dependence analysis and CBN inference were performed to assess the risk level of vehicle loss given various environmental observations. The effectiveness of the proposed method was demonstrated in a case study, which considered deploying a Slocum G1 Glider in a real water region. Risk mitigation measures were provided based on key findings. This study potentially contributes a tailored tool of risk prediction for AUGs in dynamic environments, which can enhance the safety performance of AUGs and assist in risk mitigation for decision makers.

Survival Analysis Without Sharing of Individual Patient Data by Using a Gaussian Copula.

Cox regression and Kaplan-Meier estimations are often needed in clinical research and this requires access to individual patient data (IPD). However, IPD cannot always be shared because of privacy or proprietary restrictions, which complicates the making of such estimations. We propose a method that generates pseudodata replacing the IPD by only sharing non-disclosive aggregates such as IPD marginal moments and a correlation matrix. Such aggregates are collected by a central computer and input as parameters to a Gaussian copula (GC) that generates the pseudodata. Survival inferences are computed on the pseudodata as if it were the IPD. Using practical examples we demonstrate the utility of the method, via the amount of IPD inferential content recoverable by the GC. We compare GC to a summary-based meta-analysis and an IPD bootstrap distributed across several centers. Other pseudodata approaches are also considered. In the empirical results, GC approximates the utility of the IPD bootstrap although it might yield more conservative inferences and it might have limitations in subgroup analyses. Overall, GC avoids many legal problems related to IPD privacy or property while enabling approximation of common IPD survival analyses otherwise difficult to conduct. Sharing more IPD aggregates than is currently practiced could facilitate "second purpose"-research and relax concerns regarding IPD access.

Uncertainty analysis and economic value prediction of water environmental capacity based on Copula and Bayesian model: A case study of Yitong River, China.

Water environmental capacity (WEC) is an indicator of environment management. The uncertainty analysis of WEC is more closely aligned with the actual conditions of the water body. It is crucial for accurately formulating pollution total emissions control schemes. However, the current WEC uncertainty analysis method ignored the connection between water quality and discharge, and required a large amount of monitoring data. This study analyzed the uncertainty of the WEC and predicted its economic value based on Copula and Bayesian model for the Yitong River in China. The Copula model was employed to calculate joint probabilities of water quality and discharge. And the posterior distribution of WEC with limited data was obtained by the Bayesian formula. The results showed that the WEC-COD in the Yitong River was 9009.67 t/a, while NH3-N had no residual WEC. Wanjinta Highway Bridge-Kaoshan Town reach had the most serious pollution. In order to make it have WEC, the reduction of COD and NH3-N was 5330.47 t and 3017.87 t. The economic value of WEC-COD was 5.97 × 107 CNY, and the treatment cost was 2.04 × 108 CNY to make NH3-N have residual WEC. The economic value distribution of WEC was extremely uneven, which could be utilized by adjusting the sewage outlet. In addition, since the treated water was discharged into the Sihua Bridge-Wanjinta Highway Bridge reach, the WEC-COD and the economic value were 19,488.51 t/a and 8.24 × 107 CNY. Increasing the flow of rivers could effectively improve WEC and economic value. This study provided an evaluation tool for guiding river water environment management.

A new framework for water quality forecasting coupling causal inference, time-frequency analysis and uncertainty quantification.

Accurate forecasting of water quality variables in river systems is crucial for relevant administrators to identify potential water quality degradation issues and take countermeasures promptly. However, pure data-driven forecasting models are often insufficient to deal with the highly varying periodicity of water quality in today's more complex environment. This study presents a new holistic framework for time-series forecasting of water quality parameters by combining advanced deep learning algorithms (i.e., Long Short-Term Memory (LSTM) and Informer) with causal inference, time-frequency analysis, and uncertainty quantification. The framework was demonstrated for total nitrogen (TN) forecasting in the largest artificial lakes in Asia (i.e., the Danjiangkou Reservoir, China) with six-year monitoring data from January 2017 to June 2022. The results showed that the pre-processing techniques based on causal inference and wavelet decomposition can significantly improve the performance of deep learning algorithms. Compared to the individual LSTM and Informer models, wavelet-coupled approaches diminished well the apparent forecasting errors of TN concentrations, with 24.39%, 32.68%, and 41.26% reduction at most in the average, standard deviation, and maximum values of the errors, respectively. In addition, a post-processing algorithm based on the Copula function and Bayesian theory was designed to quantify the uncertainty of predictions. With the help of this algorithm, each deterministic prediction of our model can correspond to a range of possible outputs. The 95% forecast confidence interval covered almost all the observations, which proves a measure of the reliability and robustness of the predictions. This study provides rich scientific references for applying advanced data-driven methods in time-series forecasting tasks and a practical methodological framework for water resources management and similar projects.

Hybrid life-cycle and hierarchical archimedean copula analyses for identifying pathways of greenhouse gas mitigation in domestic sewage treatment systems.

Electricity consumption and anaerobic reactions cause direct and indirect greenhouse gas (GHG) emissions within domestic sewage treatment systems (DSTSs). GHG emissions in DSTSs were influenced by the sewage quantity and the efficacy of treatment technologies. To address combined effects of these variables, this study presented an approach for identifying pathways for GHG mitigation within the DSTSs of cities under climate change and socio-economic development, through combining life cycle analysis (LCA) and the Hierarchical Archimedean copula (HAC) methods. The approach was innovative in the following aspects: 1) quantifying the GHG emissions of the DSTSs; 2) identifying the correlations among temperature changes, socioeconomic development, and domestic sewage quantity, and 3) predicting the future fluctuations in GHG emissions from the DSTSs. The effectiveness of the proposed approach was validated through its application to an urban agglomeration in the Pearl River Delta (PRD), China. To identify the potentials of GHG mitigation in the DSTSs, two pathways (i.e., general and optimized) were proposed according to the different technical choices for establishing facilities from 2021 to 2030. The results indicated that GHG emissions from the DSTS in the PRD were [3.01, 4.96] Mt CO2eq in 2021, with substantial contributions from Shenzhen and Guangzhou. Moreover, GHG emissions from the sewage treatment facilities based on Anaerobic-Anoxic-Axic (AAO) technology were higher than those based on other technologies. Under the optimized pathway, GHG emissions, contributed by the technologies of Continuous Cycle Aeration System (CASS) and Oxidation Ditch (OD), were the lowest. Through the results of correlation analysis, the impact of socioeconomic development on domestic sewage quantities was more significant than that of climate change. Domestic sewage quantities in the cities of the PRD would increase by 4.10%-28.38%, 17.14%-26.01%, and 18.15%-26.50% from 2022 to 2030 under three Representative Concentration Pathways (RCPs) 2.6, 4.5, and 8.5. These findings demonstrated that the capacities of domestic sewage treatment facilities in most cities of the PRD should be substantially improved from 0.12 to 2.99 times between 2022 and 2030. Under the optimized pathway, the future GHG emissions of the CASS method would be the lowest, followed by the OD method.

Copula-based analysis of dependent current status data with semiparametric linear transformation model.

This paper discusses regression analysis of current status data with dependent censoring, a problem that often occurs in many areas such as cross-sectional studies, epidemiological investigations and tumorigenicity experiments. Copula model-based methods are commonly employed to tackle this issue. However, these methods often face challenges in terms of model and parameter identification. The primary aim of this paper is to propose a copula-based analysis for dependent current status data, where the association parameter is left unspecified. Our method is based on a general class of semiparametric linear transformation models and parametric copulas. We demonstrate that the proposed semiparametric model is identifiable under certain regularity conditions from the distribution of the observed data. For inference, we develop a sieve maximum likelihood estimation method, using Bernstein polynomials to approximate the nonparametric functions involved. The asymptotic consistency and normality of the proposed estimators are established. Finally, to demonstrate the effectiveness and practical applicability of our method, we conduct an extensive simulation study and apply the proposed method to a real data example.

Do non-linearity and non-Gaussianity truly matter in streamflow forecasting? A comparative study between PAR(p) and vine copula for Brazilian streamflow time series.

This study evaluates the joint impact of non-linearity and non-Gaussianity on predictive performance in 23 Brazilian monthly streamflow time series from 1931 to 2022. We consider point and interval forecasting, employing a PAR(p) model and comparing it with the periodic vine copula model. Results indicate that the Gaussian hypothesis assumed by PAR(p) is unsuitable; gamma and log-normal distributions prove more appropriate and crucial for constructing accurate confidence intervals. This is primarily due to the assumption of the Gaussian distribution, which can lead to the generation of confidence intervals with negative values. Analyzing the estimated copula models, we observed a prevalence of the bivariate Normal copula, indicating that linear dynamic dependence is frequent, and the Rotated Gumbel 180°, which exhibits lower tail dependence. Overall, the temporal dynamics are predominantly shaped by combining these two types of effects. In point forecasting, both models show similar behavior in the estimation set, with slight advantages for the copula model. The copula model performs better during the out-of-sample analysis, particularly for certain power plants. In interval forecasting, the copula model exhibits pronounced superiority, offering a better estimation of quantiles. Consistently demonstrating proficiency in constructing reliable and accurate intervals, the copula model reveals a notable advantage over the PAR(p) model in interval forecasting.

Likelihood Inference for Factor Copula Models with Asymmetric Tail Dependence.

For multivariate non-Gaussian involving copulas, likelihood inference is dominated by the data in the middle, and fitted models might not be very good for joint tail inference, such as assessing the strength of tail dependence. When preliminary data and likelihood analysis suggest asymmetric tail dependence, a method is proposed to improve extreme value inferences based on the joint lower and upper tails. A prior that uses previous information on tail dependence can be used in combination with the likelihood. With the combination of the prior and the likelihood (which in practice has some degree of misspecification) to obtain a tilted log-likelihood, inferences with suitably transformed parameters can be based on Bayesian computing methods or with numerical optimization of the tilted log-likelihood to obtain the posterior mode and Hessian at this mode.

Vulnerability Analysis Method Based on Network and Copula Entropy.

With the deepening of the diversification and openness of financial systems, financial vulnerability, as an endogenous attribute of financial systems, becomes an important measurement of financial security. Based on a network analysis, we introduce a network curvature indicator improved by Copula entropy as an innovative metric of financial vulnerability. Compared with the previous network curvature analysis method, the CE-based curvature proposed in this paper can measure market vulnerability and systematic risk with significant advantages.