True Correlation Structure Research Articles

Modeling Longitudinal Outcomes in a Small Matched-Pair Sample Motivated by Cardiovascular Data: A Simulation Study

Introduction: In cardiovascular research, matched pairs with longitudinal outcomes are used to assess time and treatment effects on relevant cardiovascular parameters of interest. Generalized estimating equations (GEE) with independent working correlation are commonly employed for unbiased estimation, but their consistency depends on large sample properties. This study investigates the validity of independent GEE for small sample matched pair data under various working correlations, comparing the results with quasi-least squares (QLS) estimates through simulation. Methods: We simulate a hospital cohort with longitudinal outcomes for two exposure groups and individual random effects. After obtaining the propensity score-matched sample, we performed different simulation scenarios across various mortality rates and the random effect. We compare results from GEE and QLS methods under independence, exchangeable, and AR1 working correlation structures. Results: The independence structure often yields a wider range of relative biases and higher standard errors when there is considerable drop-out. Conversely, the exchangeable structure appears as the true correlation structure, providing more accurate and reliable estimates. No significant discrepancies are observed between the results generated by GEE and QLS methods in this study, and the performance of both approaches is significantly influenced by mortality rates and the standard deviation of the random intercept. Conclusion: Our findings suggest that GEE with an independent working correlation structure is misspecified due to incorrect correlation assumptions and is less efficient. Therefore, correctly specifying the working correlation structure is important for small sample data.

Near-boundary error reduction with an optimized weighted Dirichlet–Neumann boundary condition for stochastic PDE-based Gaussian random field generators

Random field generation through the solution of stochastic partial differential equations is a computationally inexpensive method of introducing spatial variability into numerical analyses. This is particularly important in systems where material heterogeneity has influence over the response to certain stimuli. Whilst it is a convenient method, spurious values arise in the near boundary of the domain due to the non-exact nature of the specific boundary condition applied. This change in the correlation structure can amplify or dampen the system response in the near-boundary region depending on the chosen boundary condition, and can lead to inconsistencies in the overall behaviour of the system. In this study, a weighted Dirichlet–Neumann boundary condition is proposed as a way of controlling the resulting structure in the near-boundary region. The condition relies on a weighting parameter which scales the application to have a more dominant Dirichlet or Neumann component, giving a closer approximation to the true correlation structure of the Matérn autocorrelation function on which the formulation is based on. Two weighting coefficients are proposed and optimal values of the weighting parameter are provided. Through parametric investigation, the weighted Dirichlet–Neumann approach is shown to yield more consistent correlation structures than the common boundary conditions applied in the current literature. We also propose a relationship between the weighting parameter and the desired length-scale parameter of the field such that the optimal value can be chosen for a given problem.

Selection Criterion of Working Correlation Structure for Spatially Correlated Data

To obtain regression parameter estimates in generalized estimation equation modeling, whether in longitudinal or spatially correlated data, it is necessary to specify the structure of the working correlation matrix. The regression parameter estimates can be affected by the choice of this matrix. Within spatial statistics, the correlation matrix also influences how spatial variability is modeled. Therefore, this study proposes a new method for selecting a working matrix, based on conditioning the variance-covariance matrix naive. The method performance is evaluated by an extensive simulation study, using the marginal distributions of normal, Poisson, and gamma for spatially correlated data. The correlation structure specification is based on semivariogram models, using the Wendland, Matérn, and spherical model families. The results reveal that regarding the hit rates of the true spatial correlation structure of simulated data, the proposed criterion resulted in better performance than competing criteria: quasi-likelihood under the independence model criterion QIC, correlation information criterion CIC, and the Rotnizky–Jewell criterion RJC. The application of an appropriate spatial correlation structure selection was shown using the first-semester average rainfall data of 2021 in the state of Pernambuco, Brazil.

Robust Sourcing Under Multilevel Supply Risks: Analysis of Random Yield and Capacity

We consider the optimal sourcing problem when the available suppliers are subject to ambiguously correlated supply risks. This problem is motivated by the increasing severity of supply risks and difficulty evaluating common sources of vulnerability in upstream supply chains, which are problems reported by many surveys of goods-producing firms. We propose a distributionally robust model that accommodates (i) multiple levels of supply disruption, not just full delivery or no delivery, and (ii) can use data-driven estimates of the underlying correlation to develop sourcing strategies in situations where the true correlation structure is ambiguous. Using this framework, we provide analytical results regarding the form of a worst-case supply distribution and show that taking such a worst-case perspective is appealing due to severe consequences associated with supply chain risks. Moreover, we show how our distributionally robust model may be used to offer guidance to firms considering whether to exert additional effort in attempt to better understanding the prevailing correlation structure. Extensive computational experiments further demonstrate the performance of our distributionally robust approach and show how supplier characteristics and the type of supply uncertainty affect the optimal sourcing decision. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods and Analysis. Funding: K. Pan was supported in part by the Research Grants Council of Hong Kong [Grant 15501920] and in part by the National Natural Science Foundation of China [Grant 72001185]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1254 ) or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at http://dx.doi.org/10.5281/zenodo.7129356 .

Model misspecification in stepped wedge trials: Random effects for time or treatment.

Mixed models are commonly used to analyze stepped wedge trials (SWTs) to account for clustering and repeated measures on clusters. One critical issue researchers face is whether to include a random time effect or a random treatment effect. When the wrong model is chosen, inference on the treatment effect may be invalid. We explore asymptotic and finite-sample convergence of variance component estimates when the model is misspecified and how misspecification affects the estimated variance of the treatment effect. For asymptotic results, we rely on analytical solutions rather than simulation studies, which allow us to succinctly describe the convergence of misspecified estimates, even though there are multiple roots for each misspecified model. We found that both direction and magnitude of the bias associated with model-based standard errors depends on the study design and magnitude of the true variance components. We identify some scenarios in which choosing the wrong random effect has a large impact on model-based inference. However, many trends depend on trial design and assumptions about the true correlation structure, so we provide tools for researchers to investigate specific scenarios of interest. We use data from an SWT on disinvesting from weekend services in hospital wards to demonstrate how these results can be applied as a sensitivity analysis, which quantifies the impact of misspecification under a variety of settings and directly compares the potential consequences of different modeling choices. Our results will provide guidance for prespecified model choices and supplement sensitivity analyses to inform confidence in the validity of results.

Power considerations for generalized estimating equations analyses of four-level cluster randomized trials.

In this article, we develop methods for sample size and power calculations in four-level intervention studies when intervention assignment is carried out at any level, with a particular focus on cluster randomized trials (CRTs). CRTs involving four levels are becoming popular in healthcare research, where the effects are measured, for example, from evaluations (level 1) within participants (level 2) in divisions (level 3) that are nested in clusters (level 4). In such multilevel CRTs, we consider three types of intraclass correlations between different evaluations to account for such clustering: that of the same participant, that of different participants from the same division, and that of different participants from different divisions in the same cluster. Assuming arbitrary link and variance functions, with the proposed correlation structure as the true correlation structure, closed-form sample size formulas for randomization carried out at any level (including individually randomized trials within a four-level clustered structure) are derived based on the generalized estimating equations approach using the model-based variance and using the sandwich variance with an independence working correlation matrix. We demonstrate that empirical power corresponds well with that predicted by the proposed method for as few as eight clusters, when data are analyzed using the matrix-adjusted estimating equations for the correlation parameters with a bias-corrected sandwich variance estimator, under both balanced and unbalanceddesigns.

Power and sample size calculations for cluster randomized trials with binary outcomes when intracluster correlation coefficients vary by treatment arm.

Generalized estimating equations are commonly used to fit logistic regression models to clustered binary data from cluster randomized trials. A commonly used correlation structure assumes that the intracluster correlation coefficient does not vary by treatment arm or other covariates, but the consequences of this assumption are understudied. We aim to evaluate the effect of allowing variation of the intracluster correlation coefficient by treatment or other covariates on the efficiency of analysis and show how to account for such variation in sample size calculations. We develop formulae for the asymptotic variance of the estimated difference in outcome between treatment arms obtained when the true exchangeable correlation structure depends on the treatment arm and the working correlation structure used in the generalized estimating equations analysis is: (i) correctly specified, (ii) independent, or (iii) exchangeable with no dependence on treatment arm. These formulae require a known distribution of cluster sizes; we also develop simplifications for the case when cluster sizes do not vary and approximations that can be used when the first two moments of the cluster size distribution are known. We then extend the results to settings with adjustment for a second binary cluster-level covariate. We provide formulae to calculate the required sample size for cluster randomized trials using these variances. We show that the asymptotic variance of the estimated difference in outcome between treatment arms using these three working correlation structures is the same if all clusters have the same size, and this asymptotic variance is approximately the same when intracluster correlation coefficient values are small. We illustrate these results using data from a recent cluster randomized trial for infectious disease prevention in which the clusters are groups of households and modest in size (mean 9.6 individuals), with intracluster correlation coefficient values of 0.078 in the control arm and 0.057 in an intervention arm. In this application, we found a negligible difference between the variances calculated using structures (i) and (iii) and only a small increase (typically ) for the independent correlation structure (ii), and hence minimal effect on power or sample size requirements. The impact may be larger in other applications if there is greater variation in the ICC between treatment arms or with an additional covariate. The common approach of fitting generalized estimating equations with an exchangeable working correlation structure with a common intracluster correlation coefficient across arms likely does not substantially reduce the power or efficiency of the analysis in the setting of a large number of small or modest-sized clusters, even if the intracluster correlation coefficient varies by treatment arm. Our formulae, however, allow formal evaluation of this and may identify situations in which variation in intracluster correlation coefficient by treatment arm or another binary covariate may have a more substantial impact on power and hence sample size requirements.

Robust approach for variable selection with high dimensional longitudinal data analysis.

This article proposes a new robust smooth-threshold estimating equation to select important variables and automatically estimate parameters for high dimensional longitudinal data. A novel working correlation matrix is proposed to capture correlations within the same subject. The proposed procedure works well when the number of covariates increases as the number of subjects n increases. The proposed estimates are competitive with the estimates obtained with the true correlation structure, especially when the data are contaminated. Moreover, the proposed method is robust against outliers in the response variables and/or covariates. Furthermore, the oracle properties for robust smooth-threshold estimating equations under "large n, diverging " are established under some regularity conditions. Extensive simulation studies and a yeast cell cycle data are used to evaluate the performance of the proposed method, and results show that the proposed method is competitive with existing robust variable selection procedures.

Power calculation for analyses of cross-sectional stepped-wedge cluster randomized trials with binary outcomes via generalized estimating equations.

Power calculation for stepped-wedge cluster randomized trials (SW-CRTs) presents unique challenges, beyond those of standard parallel cluster randomized trials, due to the need to consider temporal within cluster correlations and background period effects. To date, power calculation methods specific to SW-CRTs have primarily been developed under a linear model. When the outcome is binary, the use of a linear model corresponds to assessing a prevalence difference; yet trial analysis often employs a nonlinear link function. We propose power calculation methods for cross-sectional SW-CRTs under a logistic model fitted by generalized estimating equations. Firstly, under an exchangeable correlation structure, we show the power based on a logistic model is lower than that from assuming a linear model in the absence of period effects. We then evaluate the impact of background prevalence changes over time on power. To allow the correlation among outcomes in the same cluster to change over time and with treatment status, we generalize the methods to more complex correlation structures. Our simulation studies demonstrate that the proposed power calculation methods perform well with the model-based variance under the true correlation structure and reveal that a working independence structure can result in substantial efficiency loss, while a working exchangeable structure performs well even when the underlying correlation structure deviates from exchangeable. An extension to our methods accounts for variable cluster sizes and reveals that unequal cluster sizes have a modest impact on power. We illustrate the approaches by application to a quality of care improvement trial for acute coronary syndrome.

Network analysis of earthquake ground motion spatial correlation: a case study with the San Jacinto seismic nodal array

SUMMARYThe spatial correlation of earthquake ground motion intensity can be measured from strong motion data; however, the data used in past studies is sparsely sampled in space, and only the interstation distance was considered as a correlation variable. These limitations mean that we have only weak constraints on the true correlation structure of ground motion and that potentially important aspects of spatial correlation are unconstrained. In this study, we combine a large-N seismic array and graph analytics to explore this issue at a local scale using small local and regional earthquakes. Our result suggests site conditions, and how they interact with the incident seismic wavefield, strongly condition the spatial correlation of ground motion. Future progress in characterizing ground motion spatial variability will require dense wavefield measurements, either through nodal deployments, or perhaps distributed acoustic sensing measurements, of seismic wavefields. Aftershock sequences of major earthquakes would provide particularly data-rich targets of opportunity.

Choosing the frequency of follow-up in longitudinal studies: Is more necessarily better?

Background Follow-up frequency is an important design parameter in longitudinal studies. We quantified the impact of reducing follow-up frequency on the precision of estimated regression parameters, and investigated the impact of incorrectly assuming an exchangeable correlation structure on estimates of the loss of precision resulting from reduced follow-up. Methods We estimated the loss in precision on deleting every second observation from three longitudinal cohorts: patients with Childhood Systemic Lupus Erythematosus (cSLE), the Canadian Haemophilia Prophylaxis Study (CHPS), and patients with Juvenile Dermatomyositis (JDM). We compared these results with those from a theoretical formula assuming an exchangeable correlation structure. Results The increase in sample size needed to compensate for halving follow-up frequency was 9%, 6% and 28% for the cSLE, CHPS and JDM cohorts respectively. Under the assumption of an exchangeable correlation, the estimated increases in sample size were 22%, 11% and 10% respectively. Conclusions Reducing follow-up frequency can result in minimal loss of information, as seen in the CHPS cohort. While using a theoretical formula based on an exchangeable correlation structure is convenient, it can be inaccurate when the true correlation structure is not exchangeable.

A Revisit to the Asymmetric Stochastic Volatility Models' Correlation Structures

The asymmetric stochastic volatility (ASV) models extend the stochastic volatility model (SV) by modeling the correlation between the asset return and its volatility. We prove by simulation studies that fitting the ASV models may infer erroneous estimations of the correlation coefficients. Even if the true return-volatility correlation structure is different from the ASV models' specifications, the ASV models constantly provide significant correlation coefficients far from zero. The misidentification of the correlation parameter can be a significant issue considering that in-conclusive empirical evidence regarding the asymmetry has been documented, and the ASV model can only represent the general asymmetry according to its return-volatility correlation specification. The incorporation of ex-post volatility helps the ASV model to identify the true return-volatility correlation. Empirical evidence on major U.S. equity market indexes verifies that ASV models with ex-post volatility obtain significantly different estimations of the correlation parameters compared to the benchmark ASV models. Moreover, the inclusion of the ex-post volatility also significantly improves the ASV models in return density forecasting. It is necessary to include the ex-post volatility when estimating the ASV model.

Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations

The generalized estimating equations (GEE) is one of the statistical approaches for the analysis of longitudinal data with correlated response. A working correlation structure for the repeated measures of the outcome variable of a subject needs to be specified by this method and the GEE estimator for the regression parameter will be the most efficient if the working correlation matrix is correctly specified. Hence it is desirable to choose a working correlation matrix that is the closest to the underlying structure among a set of working correlation structures. The quasi-likelihood Information criteria (QIC) was proposed for the selection of the working correlation structure and the best subset of explanatory variables in GEE. However, its success rate in selecting the true correlation structure has been established to be about 29.4%. Likewise, past studies have shown that its bias increases with the number of parameters. By considering longitudinal data with binary response, we establish numerically through simulations the consistency property of QIC in selecting the true working correlation structure and the conditions for its consistency. Further, we propose a modified QIC that penalizes for the number of parameter estimates in the original QIC and numerically establish that the penalization enhances the consistency of QIC in selecting the true working correlation structure. The results indicate that QIC selects the true correlation structure with probability approaching one if only parsimonious structures are considered otherwise the selection rates are less than 50% regardless of the increase in the sample size, measurements per subject and level of correlation. Further, we established that the probability of selecting the true correlation structure <i>R₀</i> almost surely converges to one when we penalize for the number of correlation parameters estimated.

Simultaneous estimation and inference for multiple response variables

Multivariate data arise frequently in biomedical and health studies where multiple response variables are collected across subjects. Unlike a univariate procedure fitting each response separately, a multivariate regression model provides a unique opportunity in studying the joint evolution of various response variables. In this paper, we propose two estimation procedures that improve estimation efficiency for the regression parameter by accommodating correlations among the response variables. The proposed procedures do not require knowledge of the true correlation structure nor does it estimate the parameters associated with the correlation. Theoretical and simulation results confirm that the proposed estimators are more efficient than the one obtained from the univariate approach. We further propose simple and powerful inference procedures for a goodness-of-fit test that possess the chi-squared asymptotic properties. Extensive simulation studies suggest that the proposed tests are more powerful than the Wald test based on the univariate procedure. The proposed methods are also illustrated through the mother’s stress and children’s morbidity study.

Reservoir history matching using constrained ensemble Kalman filtering

The high heterogeneity of petroleum reservoirs, represented by their spatially varying rock properties (porosity and permeability), greatly dictates the quantity of recoverable oil. In this work, the estimation of the spatial permeability distribution, which is crucial for predicting the future performance of a reservoir, is carried out through a history matching technique based on constrained ensemble Kalman filtering (EnKF). The main contribution in this work is the novel implementation of hard and soft constraints in the recursive EnKF estimation methodology for petroleum reservoirs. Hard data is obtained from the actual values of the reservoir parameters at discrete locations obtained by core sampling and well logging, while the soft data considered is obtained from correlograms, which characterize the spatial correlation of the rock properties in a reservoir. In each time update, the parameter estimates obtained from the unconstrained EnKF are modified by one of two novel algorithms. In the first, the correlation matrix obtained after the unconstrained EnKF update is transformed to honour the true correlation structure from the correlogram by applying a projection‐based method. The second algorithm involves the use of a technique for soft constrained covariance localization. We observe that the soft data constrained localization method results in the best estimates of the permeability and also reduces the computational time significantly. We quantify the improvement in estimation performance of each of the constrained methods over unconstrained estimation. The method, while developed for estimation in petroleum reservoirs, is also generally applicable to systems with spatial heterogeneity and underlying spatial correlations.

Undesirable properties of the dimensionless normalisation for spatio-temporal variables

We used theory and empirical data to demonstrate three undesirable properties of the dimensionless normalisation technique for gait spatio-temporal parameters. Firstly, it may not fully remove the correlation between leg length and spatio-temporal parameters. Secondly, it induces spurious correlation among spatio-temporal parameters, which might obscure their true correlation structure. Thirdly, it induces spurious correlation with external covariates, which complicates further statistical modelling. Therefore, depending on the objectives, we propose alternatives. If the objective is to compare datasets but remove the confounding effect of leg length, residualisation may be an alternative, although the generalisability of the residualisation is less well established than dimensionless normalisation. If the objective is to build a regression model, the raw spatio-temporal parameters could be used with leg length, or a function of leg length, as an explicit regressor to avoid spurious correlation. If correlation is the objective, partial correlation can be used instead.

Sample size estimation for a two-group comparison of repeated count outcomes using GEE

ABSTRACTRandomized clinical trials with count measurements as the primary outcome are common in various medical areas such as seizure counts in epilepsy trials, or relapse counts in multiple sclerosis trials. Controlled clinical trials frequently use a conventional parallel-group design that assigns subjects randomly to one of two treatment groups and repeatedly evaluates them at baseline and intervals across a treatment period of a fixed duration. The primary interest is to compare the rates of change between treatment groups. Generalized estimating equations (GEEs) have been widely used to compare rates of change between treatment groups because of its robustness to misspecification of the true correlation structure. In this paper, we derive a sample size formula for comparing the rates of change between two groups in a repeatedly measured count outcome using GEE. The sample size formula incorporates general missing patterns such as independent missing and monotone missing, and general correlation structures such as AR(1) and compound symmetry (CS). The performance of the sample size formula is evaluated through simulation studies. Sample size estimation is illustrated by a clinical trial example from epilepsy.

Working correlation structure selection in GEE analysis

The method of generalized estimating equations models the association between the repeated observations on a subject. It is important to select an appropriate working correlation structure for the repeated measures per subject in order to enhance efficiency of estimation of the regression parameter. Some existing selection criteria choose the structure for which the covariance matrix estimator and the specified working covariance matrix are closest. So, we define a new criterion based on this idea for selecting a working correlation structure. Also, we compare our criterion with some existing criteria to identify the true correlation structure via simulations for Poisson, binomial and normal responses, and exchangeable or AR(1) intracluster correlation structure. We assume that for each subject, the number of observations remains the same. Furthermore, we also illustrate the performance of our criterion using two data sets. Finally, we conclude that our approach is a good selection criterion in most of the different considered settings.

Efficient quantile marginal regression for longitudinal data with dropouts.

In many biomedical studies independent variables may affect the conditional distribution of the response differently in the middle as opposed to the upper or lower tail. Quantile regression evaluates diverse covariate effects on the conditional distribution of the response with quantile-specific regression coefficients. In this paper, we develop an empirical likelihood inference procedure for longitudinal data that accommodates both the within-subject correlations and informative dropouts under missing at random mechanisms. We borrow the matrix expansion idea of the quadratic inference function and incorporate the within-subject correlations under an informative working correlation structure. The proposed procedure does not assume the exact knowledge of the true correlation structure nor does it estimate the parameters of the correlation structure. Theoretical results show that the resulting estimator is asymptotically normal and more efficient than one attained under a working independence correlation structure. We expand the proposed approach to account for informative dropouts under missing at random mechanisms. The methodology is illustrated by empirical studies and a real-life example of HIV data analysis.

Estimating and Identifying Unspecified Correlation Structure for Longitudinal Data

Identifying correlation structure is important to achieving estimation efficiency in analyzing longitudinal data, and is also crucial for drawing valid statistical inference for large-size clustered data. In this article, we propose a nonparametric method to estimate the correlation structure, which is applicable for discrete longitudinal data. We use eigenvector-based basis matrices to approximate the inverse of the empirical correlation matrix and determine the number of basis matrices via model selection. A penalized objective function based on the difference between the empirical and model approximation of the correlation matrices is adopted to select an informative structure for the correlation matrix. The eigenvector representation of the correlation estimation is capable of reducing the risk of model misspecification, and also provides useful information on the specific within-cluster correlation pattern of the data. We show that the proposed method possesses the oracle property and selects the true correlation structure consistently. The proposed method is illustrated through simulations and two data examples on air pollution and sonar signal studies .