Equal Cluster Sizes Research Articles

Quantitative evaluation of internal cluster validation indices using binary data sets

AbstractAimsDifferent clustering methods often classify the same data set differently. Selecting the “best” clustering solution from alternatives is possible with cluster validation indices. Because of the large variety of cluster validation indices (CVIs), choosing the most suitable index concerning the data set and clustering algorithms is challenging. We aim to assess different internal clustering validation indices.MethodsArtificial binary data sets with equal‐ and unequal‐sized well‐separated a priori clusters were simulated and three levels of noise were then added. Twenty replications of each of the six types of data sets (two group sizes × three levels of noise) were created and analyzed by three clustering algorithms with Jaccard dissimilarity. Twenty‐seven clustering validation indices are evaluated including both geometric and non‐geometric indices.ResultsAlthough, in theory, all CVIs could differentiate between good and wrong classifications, only a few perform as expected with noisy data. Tau and silhouette widths proved to be the best geometric CVIs both for equal and unequal cluster sizes. Among non‐geometric indices, crispness and OptimClass performed best.ConclusionWe recommend using these best‐performing CVIs. We suggest plotting the CVI value against the number of clusters because the lack of a sharp peak means that the position of the maximum is uncertain.

Evaluating the Efficiency of Restricted Pseudo Likelihood Estimation in Balanced and Unbalanced Clustered Binary Data Models

Clustered binary data analysis is a common task in various fields, such as social sciences and epidemiology. Restricted Pseudo Likelihood Estimation (PLE) is a widely used approach for analyzing clustered binary data, providing flexibility in handling complex dependencies within clusters. This study aims to evaluate the efficiency of Restricted Pseudo Likelihood Estimation in balanced and unbalanced clustered binary data models. Using simulated data, we compare the performance of PLE in balanced and unbalanced clustered binary data scenarios. We consider various factors such as the number of clusters, cluster sizes, and intra-cluster correlation. The preferred class of models for clustered binary data is the Hierarchical Generalized Linear Model (HGLM). This article compares the performance of a restricted pseudo-likelihood estimation method of the Hierarchical Generalized Linear Model (HGLM) with equal and unequal cluster sizes. Through comprehensive simulation experiments, we assess the accuracy and precision of PLE estimates in terms of parameter estimation, standard errors, and hypothesis testing. Our findings provide insights into the efficiency of Restricted Pseudo Likelihood Estimation (RPLE) in balanced and unbalanced clustered binary data models. The results highlight the advantages and limitations of PLE in different scenarios, aiding researchers in selecting appropriate modeling approaches for their specific data characteristics. The results can guide researchers in making informed decisions regarding the selection and application of PLE in their own studies, ultimately enhancing the validity and reliability of statistical analyses in the presence of clustered binary data.

The impact of endoplasmic reticulum morphology on IRE1 protein clustering.

The Unfolded Protein Response (UPR) is a signaling network that responds to an increased load of unfolded proteins within the Endoplasmic Reticulum (ER), referred to as ER stress. ER stress activates the transmembrane signaling protein IRE1, which subsequently forms oligomers and clusters with signaling activity that modifies target gene expression to reestablish ER homeostasis. Experiments have shown that IRE1 clusters can have complex shapes that form along the ER tube structure, including wrapping around an ER tube, and suggest that IRE1 protein clusters tend to be found on narrower ER tubes. However, the influence of ER morphology on the dynamics of these IRE1 clusters remains largely unexplored. We quantitatively modeled the dynamics of IRE1 protein clusters on a tubular surface, using a kinetic Monte Carlo algorithm that treats the IRE1 proteins as particles undergoing stochastic diffusion. We show that cluster shape exhibits a phase transition from approximately round clusters to clusters that wrap around the ER tube as a cluster becomes sufficiently large or on sufficiently narrow tubes. These wrapped clusters, which easily form on narrow tubes, evaporate much slower compared to circular clusters of equal size that form on larger tubes. The threshold concentration between cluster growth and decay is also lower for wrapped clusters compared to circular clusters, exhibiting greater stability of wrapped clusters under varying cellular concentrations of IRE1. This modeled behavior aligns with experimental observations and suggests cluster wrapping around the ER tube is important to understanding IRE1 cluster dynamics and corresponding signaling. UPR malfunction is implicated in the development of many diseases including neurodegeneration and cancer. By furthering understanding of IRE1 protein clustering, we gain insight into how ER morphology controls the IRE1 behavior and impacts UPR signaling

Bifurcations of a Neural Network Model with Symmetry

We analyze a family of clustered excitatory-inhibitory neural networks and the underlying bifurcation structures that arise because of permutation symmetries in the network as the global coupling strength $g$ is varied. We primarily consider two network topologies: an all-to-all connected network which excludes self-connections, and a network in which the excitatory cells are broken into clusters of equal size. Although in both cases the bifurcation structure is determined by symmetries in the system, the behavior of the two systems is qualitatively different. In the all-to-all connected network, the system undergoes Hopf bifurcations leading to periodic orbit solutions; notably, for large $g$, there is a single, stable periodic orbit solution and no stable fixed points. By contrast, in the clustered network, there are no Hopf bifurcations, and there is a family of stable fixed points for large $g$.

Simultaneous confidence bands for nonparametric regression with equal cluster size

Repeated/clustered data are becoming increasingly popular in many research studies. While there is a considerable literature in the estimation of the nonparametric component functions, the problem has been addressed by very few in statistical inference such as simultaneous confidence bands (SCBs). Based on the efficient kernel estimation incorporating within‐cluster error correlation, we provide asymptotic Wald‐type‐based SCBs for the nonparametric function when the repeated measures have the equal cluster size. The performance of the SCBs is evaluated by simulation studies which support the asymptotic theory. The proposed method is also applied to a dataset on the Junior School Project of test scores.

Design and analysis of cluster randomized trials with time-to-event outcomes under the additive hazards mixed model.

A primary focus of current methods for cluster randomized trials (CRTs) has been for continuous, binary, and count outcomes, with relatively less attention given to right-censored, time-to-event outcomes. In this article, we detail considerations for sample size requirement and statistical inference in CRTs with time-to-event outcomes when the intervention effect parameter is specified through the additive hazards mixed model (AHMM), which includes a frailty term to explicitly account for the dependency between the failure times. First, we discuss improved inference for the treatment effect parameter via bias-corrected sandwich variance estimators and randomization-based test under AHMM, addressing potential small-sample biases in CRTs. Next, we derive a new sample size formula for AHMM analysis of CRTs accommodating both equal and unequal cluster sizes. When the cluster sizes vary, our sample size formula depends on the mean and coefficient of variation of cluster sizes, based on which we articulate the impact of cluster size variation in CRTs with time-to-event outcomes. Furthermore, we obtain the insight that the classical variance inflation factor for CRTs with a non-censored outcome can in fact apply to CRTs with a time-to-event outcome, providing that an appropriate definition of the intraclass correlation coefficient is considered under AHMM. Simulation studies are carried out to illustrate key design and analysis considerations in CRTs with a small to moderate number of clusters. The proposed sample size procedure and analytical methods are further illustrated using the context of the STrategies to Reduce Injuries and Develop Confidence in Elders CRT.

Energy Efficiency Maximization for UAV-Assisted Full-Duplex NOMA System: User Clustering and Resource Allocation

In this paper, the problem of energy efficiency maximization of an unmanned-aerial-vehicle (UAV)-assisted full-duplex non-orthogonal-multiple-access (FD-NOMA) system is investigated. In order to reduce the interference among users in the considered system, we first propose a novel and less complex adaptive-geometric distribution (AGD) which adaptively maintain equal cluster size and allocate resource optimally among the users. Later, we formulate an optimization problem of energy efficiency maximization subject to the transmit power budget at the UAV for downlink (DL) users and for each uplink (UL) user and UAV placement. To transform the non-convex problem into convex one, we decouple the original optimization problem into different sub-problems in which the problems of power allocation and UAV placement are alternatively solved using Dinkelbach’s algorithm with successive convex approximation (SCA) and particle swarm optimization (PSO), respectively. Later, the unified solution for optimal power allocation and UAV placement is proposed based on block-coordinate descent (BCD) algorithm. Extensive simulation results validate the superiority of the proposed AGD clustering based proposed algorithm for UAV-assisted FD-NOMA system over the conventional clustering and other orthogonal schemes.

Power analysis for cluster randomized trials with continuous coprimary endpoints.

Pragmatic trials evaluating health care interventions often adopt cluster randomization due to scientific or logistical considerations. Systematic reviews have shown that coprimary endpoints are not uncommon in pragmatic trials but are seldom recognized in sample size or power calculations. While methods for power analysis based on K ( ) binary coprimary endpoints are available for cluster randomized trials (CRTs), to our knowledge, methods for continuous coprimary endpoints are not yet available. Assuming a multivariate linear mixed model (MLMM) that accounts for multiple types of intraclass correlation coefficients among the observations in each cluster, we derive the closed-form joint distribution of K treatment effect estimators to facilitate sample size and power determination with different types of null hypotheses under equal cluster sizes. We characterize the relationship between the power of each test and different types of correlation parameters. We further relax the equal cluster size assumption and approximate the joint distribution of the K treatment effect estimators through the mean and coefficient of variation of cluster sizes. Our simulation studies with a finite number of clusters indicate that the predicted power by our method agrees well with the empirical power, when the parameters in the MLMM are estimated via the expectation-maximization algorithm. An application to a real CRT is presented to illustrate the proposedmethod.

A New Estimator for Standard Errors with Few Unbalanced Clusters

In linear regression analysis, the estimator of the variance of the estimator of the regression coefficients should take into account the clustered nature of the data, if present, since using the standard textbook formula will in that case lead to a severe downward bias in the standard errors. This idea of a cluster-robust variance estimator (CRVE) generalizes to clusters the classical heteroskedasticity-robust estimator. Its justification is asymptotic in the number of clusters. Although an improvement, a considerable bias could remain when the number of clusters is low, the more so when regressors are correlated within cluster. In order to address these issues, two improved methods were proposed; one method, which we call CR2VE, was based on biased reduced linearization, while the other, CR3VE, can be seen as a jackknife estimator. The latter is unbiased under very strict conditions, in particular equal cluster size. To relax this condition, we introduce in this paper CR3VE-λ, a generalization of CR3VE where the cluster size is allowed to vary freely between clusters. We illustrate the performance of CR3VE-λ through simulations and we show that, especially when cluster sizes vary widely, it can outperform the other commonly used estimators.

Clustered Wireless Sensor Network Assisted the Design of Intelligent Art System

Based on the principle of cluster wireless sensor network, this article introduces typical routing protocols in wireless sensors, and wireless sensor network protocol in detail analyzes their advantages and disadvantages and addresses their shortcomings. First, in the clustering network, a uniform clustering protocol with multiple hops in the circular network is proposed. The circular network is divided into rings of equal width, and clusters of equal size are set on different rings. Secondly, the ordinary nodes on each layer of the ring send the collected data to the auxiliary intelligent nodes in the cluster in a single-hop manner, and the auxiliary intelligent nodes located on the outer ring transfer the data to the auxiliary intelligent nodes located on the adjacent inner ring. Finally, on the basis of studying the clustering network protocol, this paper proposes a new clustering routing algorithm, a multihop adaptive clustering routing algorithm. The simulation results show that the algorithm can effectively extend the life of the network, save network energy consumption, and achieve network load balance. At the same time, the initial energy of the auxiliary intelligent node is set according to the energy consumption of the ordinary node and the relative distance between the auxiliary intelligent node and the base station on each layer of the ring. The theoretical and simulation results prove that, compared with the clustered network and auxiliary intelligent nodes, the clustered network can extend the life of the network.

Sample size calculation in hierarchical factorial trials with unequal cluster sizes.

Motivated by a suicide prevention trial with hierarchical treatment allocation (cluster-level and individual-level treatments), we address the sample size requirements for testing the treatment effects as well as their interaction. We assume a linear mixed model, within which two types of treatment effect estimands (controlled effect and marginal effect) are defined. For each null hypothesis corresponding to an estimand, we derive sample size formulas based on large-sample z-approximation, and provide finite-sample modifications based on a t-approximation. We relax the equal cluster size assumption and express the sample size formulas as functions of the mean and coefficient of variation of cluster sizes. We show that the sample size requirement for testing the controlled effect of the cluster-level treatment is more sensitive to cluster size variability than that for testing the controlled effect of the individual-level treatment; the same observation holds for testing the marginal effects. In addition, we show that the sample size for testing the interaction effect is proportional to that for testing the controlled or the marginal effect of the individual-level treatment. We conduct extensive simulations to validate the proposed sample size formulas, and find the empirical power agrees well with the predicted power for each test. Furthermore, the t-approximations often provide better control of type I error rate with a small number of clusters. Finally, we illustrate our sample size formulas to design the motivating suicide prevention factorial trial. The proposed methods are implemented in the R package H2x2Factorial.

HLBC: A Hierarchical Layer-Balanced Clustering Scheme for Energy Efficient Wireless Sensor Networks

Battery lifetime maximization of Wireless Sensor Network (WSN) nodes has been an area of prime research interest, and clustering of WSN nodes for realizing the goal of network lifetime maximization has been extensively addressed in the literature. Various clustering schemes for partitioning WSN nodes into equal, unequal or random cluster sizes have been prevalent, each having their own set of pros and cons. This paper proposes a Hierarchical Layer-Balanced Clustering (HLBC) scheme that partitions the WSN into a number of hierarchical layers of varied sizes while maintaining equal sized clusters at every layer. The proposed HLBC architecture is presented along with the proposed Frame Forwarding Algorithm (FFA) and the proposed TDMA Schedule for eliminating the synchronization problem. Extensive simulation experiments have been carried out under varying network scenarios to assess the performance of the proposed scheme, and the results demonstrate its efficacy. In particular, the proposed HLBC was found to outperform state of the art approaches LEACH, UCR and EMUC by 77.71%, 54.23% and 27.93%, respectively, in terms of Network Lifetime.

Choice of Strata Boundaries for Allocation Proportional to Stratum Cluster Totals in Stratified Cluster Sampling

In survey planning, sometimes, there arises situation to use cluster sampling because of nature of spatial relationship between elements of population or physical feature of land over which elements are dispersed or unavailability of reliable list of elements. At the same time, there requires technique and strategy for ensuring precision of the sample in representing the parent population. Although several theoretical cum practical works have been done in cluster sampling, stratified sampling and stratified cluster sampling, so far, the problem of stratified cluster sampling for a study variable based on an auxiliary variable, which is required in practice, has never been approached. For the first time, this paper deals with the problem of optimum stratification of population of clusters in cluster sampling with clusters of equal size of a characteristic y under study based on highly correlated concomitant variable x for allocation proportional to stratum cluster totals under a super population model. Equations giving optimum strata boundaries (OSB) for dividing population, in which sampling unit of the population is a cluster, are obtained by minimising sampling variance of the estimator of population mean. As the equations are implicit in nature, a few methods of finding approximately optimum strata boundaries (AOSB) are deduced from the equations giving OSB. In deriving the equations, mathematical tools of calculus and algebra are used in addition to statistical methods of finding conditional expectation of variance. All the proposed methods of stratification are empirically examined by illustrating in live data, population of villages in Lunglei and Serchhip districts of Mizoram State, India, and found to perform efficiently in stratifying the population. The proposed methods may provide practically feasible solution in planning socio-economic survey.

Methods for dealing with unequal cluster sizes in cluster randomized trials: A scoping review.

In a cluster-randomized trial (CRT), the number of participants enrolled often varies across clusters. This variation should be considered during both trial design and data analysis to ensure statistical performance goals are achieved. Most methodological literature on the CRT design has assumed equal cluster sizes. This scoping review focuses on methodology for unequal cluster size CRTs. EMBASE, Medline, Google Scholar, MathSciNet and Web of Science databases were searched to identify English-language articles reporting on methodology for unequal cluster size CRTs published until March 2021. We extracted data on the focus of the paper (power calculation, Type I error etc.), the type of CRT, the type and the range of parameter values investigated (number of clusters, mean cluster size, cluster size coefficient of variation, intra-cluster correlation coefficient, etc.), and the main conclusions. Seventy-nine of 5032 identified papers met the inclusion criteria. Papers primarily focused on the parallel-arm CRT (p-CRT, n = 60, 76%) and the stepped-wedge CRT (n = 14, 18%). Roughly 75% of the papers addressed trial design issues (sample size/power calculation) while 25% focused on analysis considerations (Type I error, bias, etc.). The ranges of parameter values explored varied substantially across different studies. Methods for accounting for unequal cluster sizes in the p-CRT have been investigated extensively for Gaussian and binary outcomes. Synthesizing the findings of these works is difficult as the magnitude of impact of the unequal cluster sizes varies substantially across the combinations and ranges of input parameters. Limited investigations have been done for other combinations of a CRT design by outcome type, particularly methodology involving binary outcomes-the most commonly used type of primary outcome in trials. The paucity of methodological papers outside of the p-CRT with Gaussian or binary outcomes highlights the need for further methodological development to fill the gaps.

Cutoff for Exact Recovery of Gaussian Mixture Models

We determine the information-theoretic cutoff value on separation of cluster centers for exact recovery of cluster labels in a K-component Gaussian mixture model with equal cluster sizes. Moreover, we show that a semidefinite programming (SDP) relaxation of the K-means clustering method achieves such sharp threshold for exact recovery without assuming the symmetry of cluster centers.

Emergence and stability of periodic two-cluster states for ensembles of excitable units.

We study dynamics in ensembles of identical excitable units with global repulsive interaction. Starting from active rotators with additional higher order Fourier modes in on-site dynamics, we observe, at sufficiently strong repulsive coupling, large-scale collective oscillations in which the elements form two separate clusters. Transitions from quiescence to clustered oscillations are caused by global bifurcations involving the unstable clustered steady states. For clusters of equal size, the scenarios evolve either through simultaneous formation of two heteroclinic trajectories or through two simultaneous saddle-node bifurcations on invariant circles. If the sizes of clusters differ, two global bifurcations are separated in the parameter space. Stability of clusters with respect to splitting perturbations depends on the kind of higher order corrections to on-site dynamics; we show that for periodic oscillations of two equal clusters the Watanabe-Strogatz integrability marks a change of stability. By extending our studies to ensembles of voltage-coupled Morris-Lecar neurons, we demonstrate that similar bifurcations and switches in stability occur also for more elaborate models in higher dimensions.

Testing for Factor Loading Differences in Mixture Simultaneous Factor Analysis: A Monte Carlo Simulation-Based Perspective

ABSTRACT Factor analysis is ubiquitously applied in behavioral sciences for capturing covariances of observed variables by latent variables (factors). When factor-analyzing data from many groups of subjects, mixture simultaneous factor analysis (MSFA) determines which groups have the same factor model by clustering them based on their factor loadings, factor (co)variances and residual variances. Two Monte Carlo simulations are performed to investigate the power and type I error of Wald tests for factor loading differences in MSFA, as affected by characteristics of the data (sample size in terms of number and size of groups), factor models (item communality levels, sizes and types of loading differences) and clustering (cluster size, classification error and uncertainty). The results were better in case of equal cluster sizes, strongly overdetermined factors, high communalities, and larger primary loading differences.

Relative efficiency of equal versus unequal cluster sizes in cluster randomized trials with a small number of clusters

ABSTRACT To calculate sample sizes in cluster randomized trials (CRTs), the cluster sizes are usually assumed to be identical across all clusters for simplicity. However, equal cluster sizes are not guaranteed in practice, especially when the number of clusters is limited. Therefore, it is important to understand the relative efficiency (RE) of equal versus unequal cluster sizes when designing CRTs with a limited number of clusters. In this paper, we are interested in the RE of two bias-corrected sandwich estimators of the treatment effect in the Generalized Estimating Equation (GEE) models for CRTs with a small number of clusters. Specifically, we derive the RE of two bias-corrected sandwich estimators for binary, continuous, or count data in CRTs under the assumption of an exchangeable working correlation structure. We consider different scenarios of cluster size distributions and investigate RE performance through simulation studies. We conclude that the number of clusters could be increased by as much as 42% to compensate for efficiency loss due to unequal cluster sizes. Finally, we propose an algorithm of increasing the number of clusters when the coefficient of variation of cluster sizes is known and unknown.

Power calculation for cross-sectional stepped wedge cluster randomized trials with variable cluster sizes.

Standard sample size calculation formulas for stepped wedge cluster randomized trials (SW-CRTs) assume that cluster sizes are equal. When cluster sizes vary substantially, ignoring this variation may lead to an under-powered study. We investigate the relative efficiency of a SW-CRT with varying cluster sizes to equal cluster sizes, and derive variance estimators for the intervention effect that account for this variation under a mixed effects model-a commonly used approach for analyzing data from cluster randomized trials. When cluster sizes vary, the power of a SW-CRT depends on the order in which clusters receive the intervention, which is determined through randomization. We first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power. We then obtain an "expected" power based on a first-order approximation to the variance formula, where the expectation is taken with respect to all possible randomization sequences. Finally, we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation, instead of exact cluster sizes, are known in the design stage. We evaluate our methods through simulations and illustrate that the average power of a SW-CRT decreases as the variation in cluster sizes increases, and the impact is largest when the number of clusters is small.

The impact of varying cluster size in cross-sectional stepped-wedge cluster randomised trials

BackgroundCluster randomised trials with unequal sized clusters often have lower precision than with clusters of equal size. To allow for this, sample sizes are inflated by a modified version of the design effect for clustering. These inflation factors are valid under the assumption that randomisation is stratified by cluster size. We investigate the impact of unequal cluster size when that constraint is relaxed, with particular focus on the stepped-wedge cluster randomised trial, where this is more difficult to achieve.MethodsAssuming a multi-level mixed effect model with exchangeable correlation structure for a cross-sectional design, we use simulation methods to compare the precision for a trial with clusters of unequal size to a trial with clusters of equal size (relative efficiency). For a range of scenarios we illustrate the impact of various design features (the cluster-mean correlation – a function of the intracluster correlation and the cluster size, the number of clusters, number of randomisation sequences) on the average and distribution of the relative efficiency.ResultsSimulations confirm that the average reduction in precision, due to varying cluster sizes, is smaller in a stepped-wedge trial compared to the parallel trial. However, the variance of the distribution of the relative efficiency is large; and is larger under the stepped-wedge design compared to the parallel design. This can result in large variations in actual power, depending on the allocation of clusters to sequences. Designs with larger variations in cluster sizes, smaller number of clusters and studies with smaller cluster-mean correlations (smaller cluster sizes or smaller intra-cluster correlation) are particularly at risk.ConclusionThe actual realised power in a stepped-wedge trial might be substantially higher or lower than that estimated. This is particularly important when there are a small number of clusters or the variability in cluster sizes is large. Constraining the randomisation on cluster size, where feasible, might mitigate this effect.