Unequal Cluster Sizes Research Articles

Explaining the variation in the attained power of a stepped-wedge trial with unequal cluster sizes

BackgroundIn a cross-sectional stepped-wedge trial with unequal cluster sizes, attained power in the trial depends on the realized allocation of the clusters. This attained power may differ from the expected power calculated using standard formulae by averaging the attained powers over all allocations the randomization algorithm can generate. We investigated the effect of design factors and allocation characteristics on attained power and developed models to predict attained power based on allocation characteristics.MethodBased on data simulated and analyzed using linear mixed-effects models, we evaluated the distribution of attained powers under different scenarios with varying intraclass correlation coefficient (ICC) of the responses, coefficient of variation (CV) of the cluster sizes, number of cluster-size groups, distributions of group sizes, and number of clusters. We explored the relationship between attained power and two allocation characteristics: the individual-level correlation between treatment status and time period, and the absolute treatment group imbalance. When computational time was excessive due to a scenario having a large number of possible allocations, we developed regression models to predict attained power using the treatment-vs-time period correlation and absolute treatment group imbalance as predictors.ResultsThe risk of attained power falling more than 5% below the expected or nominal power decreased as the ICC or number of clusters increased and as the CV decreased. Attained power was strongly affected by the treatment-vs-time period correlation. The absolute treatment group imbalance had much less impact on attained power. The attained power for any allocation was predicted accurately using a logistic regression model with the treatment-vs-time period correlation and the absolute treatment group imbalance as predictors.ConclusionIn a stepped-wedge trial with unequal cluster sizes, the risk that randomization yields an allocation with inadequate attained power depends on the ICC, the CV of the cluster sizes, and number of clusters. To reduce the computational burden of simulating attained power for allocations, the attained power can be predicted via regression modeling. Trial designers can reduce the risk of low attained power by restricting the randomization algorithm to avoid allocations with large treatment-vs-time period correlations.

Noise robust hybrid algorithm for segmenting image with unequal cluster sizes based on chaotic crow search and improved fuzzy c-means

Adding spatial penalty to fuzzy C-means (FCM) model is an important way to reduce the influence of noise in image segmentation. However, these improved algorithms easily cause segmentation failures when the image has the characteristics of unequal cluster sizes. Besides, they often fall into local optimal solutions if the initial cluster centers are improper. This paper presents a noise robust hybrid algorithm for segmenting image with unequal cluster sizes based on chaotic crow search algorithm and improved fuzzy c-means to overcome the above defects. Firstly, each size of clusters is integrated into the objective function of noise detecting fuzzy c-means algorithm (NDFCM), which can reduces the contribution of larger clusters to objective function and then the new membership degree and cluster centers are deduced. Secondly, a new expression called compactness, representing the pixel distribution of each cluster, is introduced into the iteration process of clustering. Thirdly, we use two- paths to seek the optimal solutions in each step of iteration: one path is produced by the chaotic crow search algorithm and the other is originated by gradient method. Furthermore, the better solutions of the two-paths go to next generation until the end of the iteration. Finally, the experiments on the synthetic and non–destructive testing (NDT) images show that the proposed algorithm behaves well in noise robustness and segmentation performance.

On Inference When Using State Corporate Laws for Identification

A popular research design identifies the effects of corporate governance by (changes in) state laws, clustering standard errors by state of incorporation. Using Monte-Carlo simulations, this paper shows that conventional statistical tests based on these standard errors dramatically overreject: in a typical design, randomly generated “placebo laws” are “significant” at the 1/5/10% level 9/21/30% of the time. This poor coverage is due to the extremely unequal cluster sizes, especially Delaware's concentration of half of all incorporations. Fixes recommended in the literature fail, including degrees-of-freedom corrections and the cluster wild bootstrap. The paper proposes a permutation test for valid inference.

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.

An Energy Efficient Routing Protocol Based on Layers and Unequal Clusters in Underwater Wireless Sensor Networks

Underwater Wireless Sensor Networks (UWSNs) have drawn tremendous attentions from all fields because of their wide application. Underwater wireless sensor networks are similar to terrestrial Wireless Sensor Networks (WSNs), however, due to different working environment and communication medium, UWSNs have many unique characteristics such as high bit error rate, long end-to-end delay and low bandwidth. These characteristics of UWSNs lead to many problems such as retransmission, high energy consumption and low reliability. To solve these problems, many routing protocols for UWSNs are proposed. In this paper, a localization-free routing protocol, named energy efficient routing protocol based on layers and unequal clusters (EERBLC) is proposed. EERBLC protocol consists of three phases: layer and unequal cluster formation, transmission routing, maintenance and update of clusters. In the first phase, the monitoring area under the water is divided into layers, the nodes in the same layer are clustered. For balancing energy of the whole network and avoiding the “hotspot” problem, a novel unequal clustering method based on layers for UWSNs is proposed, in which a new calculation method of unequal cluster size is presented. Meanwhile, a new cluster head selection mechanism based on energy balance and degree is given. In the transmission phase, EERBLC protocol proposes a novel next forwarder selection method based on the forwarding ratio and the residual energy. In the third phase, Intra and inter cluster updating method is presented. The simulation results show that the EERBLC can effectively balance the energy consumption, prolong the network lifetime, and increase the amount of data transmission compared with DBR and EEDBR protocols.

Relative efficiency of unequal cluster sizes in stepped wedge and other trial designs under longitudinal or cross-sectional sampling.

Background: A cluster trial with unequal cluster sizes often has lower precision than one with equal clusters, with a corresponding inflation of the design effect. For parallel group trials, adjustments to the design effect are available under sampling models with a single intracluster correlation. Design effects for equal clusters under more complex scenarios have appeared recently (including stepped wedge trials under cross‐sectional or longitudinal sampling). We investigate the impact of unequal cluster size in these more general settings. Results: Assuming a linear mixed model with an exchangeable correlation structure that incorporates cluster and subject autocorrelation, we compute the relative efficiency (RE) of a trial with clusters of unequal size under a size‐stratified randomization scheme, as compared to an equal cluster trial with the same total number of observations. If there are no within‐cluster time effects, the RE exceeds that for a parallel trial. In general, the RE is a weighted average of the RE for a parallel trial and the RE for a crossover trial in the same clusters. Existing approximations for parallel designs are extended to the general setting. Increasing the cluster size by the factor (1 + CV2), where CV is the coefficient of variation of cluster size, leads to conservative sample sizes, as in a popular method for parallel trials. Conclusion: Methods to assess experimental precision for single‐period parallel trials with unequal cluster sizes can be extended to stepped wedge and other complete layouts under longitudinal or cross‐sectional sampling. In practice, the loss of precision due to unequal cluster sizes is unlikely to exceed 12%.

Energy‐efficient medium access control and routing protocol for multihop wireless sensor networks

This work presents a combined energy-efficient medium access control (MAC) and routing protocol for large-scale wireless sensor networks that aims to minimise energy consumption and prolong the network lifetime. The proposed communication framework employs the following measures to enhance the network energy efficiency. Firstly, it provides an adaptive intra-cluster schedule to arbitrate media access of sensor nodes within a cluster, minimising idle listening on sensor nodes, leading to improved energy performance. Secondly, it proposes an on-demand source cross-layer routing protocol ensuring selection of best routes based on energy level and channel quality indicator for the multihop inter-cluster data transmission. Lastly, an unequal cluster size technique based on cluster head residual energy and distance away from the base station is utilised. This technique balances the energy among clusters and avoids early network partitioning. This work further presents the analytical performance model for energy consumption and delay of the proposed communication framework. The performance measures used for evaluation are energy consumption, delay, and network lifetime. The results indicate that combining routing and MAC schemes conserves energy better than utilising MAC scheme alone.

Relative efficiency of unequal versus equal cluster sizes in cluster randomized trials using generalized estimating equation models.

There is growing interest in conducting cluster randomized trials (CRTs). For simplicity in sample size calculation, the cluster sizes are assumed to be identical across all clusters. However, equal cluster sizes are not guaranteed in practice. Therefore, the relative efficiency (RE) of unequal versus equal cluster sizes has been investigated when testing the treatment effect. One of the most important approaches to analyze a set of correlated data is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which the "working correlation structure" is introduced and the association pattern depends on a vector of association parameters denoted by ρ. In this paper, we utilize GEE models to test the treatment effect in a two-group comparison for continuous, binary, or count data in CRTs. The variances of the estimator of the treatment effect are derived for the different types of outcome. RE is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal cluster sizes. We discuss a commonly used structure in CRTs-exchangeable, and derive the simpler formula of RE with continuous, binary, and count outcomes. Finally, REs are investigated for several scenarios of cluster size distributions through simulation studies. We propose an adjusted sample size due to efficiency loss. Additionally, we also propose an optimal sample size estimation based on the GEE models under a fixed budget for known and unknown association parameter (ρ) in the working correlation structure within the cluster.

Unequal cluster sizes in stepped-wedge cluster randomised trials: a systematic review

ObjectivesTo investigate the extent to which cluster sizes vary in stepped-wedge cluster randomised trials (SW-CRT) and whether any variability is accounted for during the sample size calculation and analysis of...

Sample size estimation for cluster randomized controlled trials

Cluster randomized controlled trials (cRCTs) are commonly used by clinical researchers. The advantages of cRCTs include preventing treatment contamination, enhancing administrative efficiency, convenience, external validity, ethical considerations, and likelihood of increased compliance by participants. However, when designing a cRCT, clinical researchers are faced with challenges, such as cluster units that may not have an equal number of participants within each. In this Technical Note, we discuss approaches for estimating the sample size, while taking into account unequal cluster sizes, and strategies for optimizing the design of cluster trials.

Fast, closed-form, and efficient estimators for hierarchical models with AR(1) covariance and unequal cluster sizes

ABSTRACTThis article is concerned with statistically and computationally efficient estimation in a hierarchical data setting with unequal cluster sizes and an AR(1) covariance structure. Maximum likelihood estimation for AR(1) requires numerical iteration when cluster sizes are unequal. A near optimal non-iterative procedure is proposed. Pseudo-likelihood and split-sample methods are used, resulting in computing weights to combine cluster size specific parameter estimates. Results show that the method is statistically nearly as efficient as maximum likelihood, but shows great savings in computation time.

An imbalance in cluster sizes does not lead to notable loss of power in cross-sectional, stepped-wedge cluster randomised trials with a continuous outcome

BackgroundThe current methodology for sample size calculations for stepped-wedge cluster randomised trials (SW-CRTs) is based on the assumption of equal cluster sizes. However, as is often the case in cluster randomised trials (CRTs), the clusters in SW-CRTs are likely to vary in size, which in other designs of CRT leads to a reduction in power. The effect of an imbalance in cluster size on the power of SW-CRTs has not previously been reported, nor what an appropriate adjustment to the sample size calculation should be to allow for any imbalance. We aimed to assess the impact of an imbalance in cluster size on the power of a cross-sectional SW-CRT and recommend a method for calculating the sample size of a SW-CRT when there is an imbalance in cluster size.MethodsThe effect of varying degrees of imbalance in cluster size on the power of SW-CRTs was investigated using simulations. The sample size was calculated using both the standard method and two proposed adjusted design effects (DEs), based on those suggested for CRTs with unequal cluster sizes. The data were analysed using generalised estimating equations with an exchangeable correlation matrix and robust standard errors.ResultsAn imbalance in cluster size was not found to have a notable effect on the power of SW-CRTs. The two proposed adjusted DEs resulted in trials that were generally considerably over-powered.ConclusionsWe recommend that the standard method of sample size calculation for SW-CRTs be used, provided that the assumptions of the method hold. However, it would be beneficial to investigate, through simulation, what effect the maximum likely amount of inequality in cluster sizes would be on the power of the trial and whether any inflation of the sample size would be required.

Gaussian model-based partitioning using iterated local search.

The emergence of Gaussian model-based partitioning as a viable alternative to K-means clustering fosters a need for discrete optimization methods that can be efficiently implemented using model-based criteria. A variety of alternative partitioning criteria have been proposed for more general data conditions that permit elliptical clusters, different spatial orientations for the clusters, and unequal cluster sizes. Unfortunately, many of these partitioning criteria are computationally demanding, which makes the multiple-restart (multistart) approach commonly used for K-means partitioning less effective as a heuristic solution strategy. As an alternative, we propose an approach based on iterated local search (ILS), which has proved effective in previous combinatorial data analysis contexts. We compared multistart, ILS and hybrid multistart-ILS procedures for minimizing a very general model-based criterion that assumes no restrictions on cluster size or within-group covariance structure. This comparison, which used 23 data sets from the classification literature, revealed that the ILS and hybrid heuristics generally provided better criterion function values than the multistart approach when all three methods were constrained to the same 10-min time limit. In many instances, these differences in criterion function values reflected profound differences in the partitions obtained.

Estimating Standardized Effect Sizes for Two- and Three-Level Partially Nested Data

ABSTRACTAlthough previous research has discussed an effect size estimator for partially nested cluster randomized designs, the existing estimator (a) is not efficient when used with primary data, (b) can be biased when the homogeneity of variance assumption is violated, and (c) has not yet been empirically evaluated for its finite sample properties. The present paper addresses these limitations by proposing an alternative maximum likelihood estimator for obtaining standardized mean difference effect size and the corresponding sampling variance for partially nested data, as well as the variants that do not make an assumption of homogeneity of variance. The typical estimator, denoted as d (dW with pooled SD and dC with control arm SD), requires input of summary statistics such as observed means, variances, and the intraclass correlation, and is useful for meta-analyses and secondary data analyses; the newly proposed estimator ( and ) takes parameter estimates from a correctly specified multilevel model as input and is mainly of interest to researchers doing primary research. The simulation results showed that the two methods (d and ) produced unbiased point and variance estimates for effect size. As expected, in general, was more efficient than d with unequal cluster sizes, especially with large average cluster size and large intraclass correlation. Furthermore, under heterogeneous variances, demonstrated a greater relative efficiency with small sample size for the unclustered control arm. Real data examples, one from a youth preventive program and one from an eating disorder intervention, were used to demonstrate the methods presented. In addition, we extend the discussion to a scenario with a three-level treatment arm and an unclustered control arm, and illustrate the procedures for effect size estimation using a hypothetical example of multiple therapy groups of clients clustered within therapists.

Level Based Clustering Algorithm with Unequal Cluster Size: An Energy Efficient Method in Wireless Sensor Network

Level Based Clustering Algorithm with Unequal Cluster Size: An Energy Efficient Method in Wireless Sensor Network

New improved estimators for overdispersion in models with clustered multinomial data and unequal cluster sizes

It is usual to rely on the quasi-likelihood methods for deriving statistical methods applied to clustered multinomial data with no underlying distribution. Even though extensive literature can be encountered for these kind of data sets, there are few investigations to deal with unequal cluster sizes. This paper aims to contribute to fill this gap by proposing new estimators for the intracluster correlation coefficient.

Optimal sample sizes for group testing in two-stage sampling

Optimal sample sizes under a budget constraint for estimating a proportion in a two-stage sampling process have been derived using individual testing. However, when group testing is used, these optimal sample sizes are not appropriate. In this study, optimal sample sizes at the cluster and individual levels are derived for group testing. First, optimal allocations of clusters and individuals are obtained under the assumption of equal cluster sizes. Second, we obtain the relative efficiency (RE) of unequal versus equal cluster sizes when estimating the average population proportion, . By multiplying the sample of clusters obtained assuming equal cluster size by the inverse of the RE, we adjust the sample size required in the context of unequal cluster sizes. We also show the adjustments that need to be made to allocate clusters and individuals correctly in order to estimate the required budget and achieve a certain power or precision.

Improved Graph Clustering

Graph clustering involves the task of dividing nodes into clusters, so that the edge density is higher within clusters as opposed to across clusters. A natural, classic and popular statistical setting for evaluating solutions to this problem is the stochastic block model, also referred to as the planted partition model. In this paper we present a new algorithm--a convexified version of Maximum Likelihood--for graph clustering. We show that, in the classic stochastic block model setting, it outperforms existing methods by polynomial factors when the cluster size is allowed to have general scalings. In fact, it is within logarithmic factors of known lower bounds for spectral methods, and there is evidence suggesting that no polynomial time algorithm would do significantly better. We then show that this guarantee carries over to a more general extension of the stochastic block model. Our method can handle the settings of semi-random graphs, heterogeneous degree distributions, unequal cluster sizes, unaffiliated nodes, partially observed graphs and planted clique/coloring etc. In particular, our results provide the best exact recovery guarantees to date for the planted partition, planted k-disjoint-cliques and planted noisy coloring models with general cluster sizes; in other settings, we match the best existing results up to logarithmic factors.

Avoiding Energy Holes Problem using Load Balancing Approach in Wireless Sensor Network

Clustering wireless sensor network is an efficient way to reduce the energy consumption of individual nodes in a cluster. In clustering, multihop routing techniques increase the load of the Cluster head near the sink. This unbalanced load on the Cluster head increases its energy consumption, thereby Cluster heads die faster and create an energy hole problem. In this paper, we propose an Energy Balancing Cluster Head (EBCH) in wireless sensor network. At First, we balance the intra cluster load among the cluster heads, which results in nonuniform distribution of nodes over an unequal cluster size. The load received by the Cluster head in the cluster distributes their traffic towards direct and multihop transmission based on the load distribution ratio. Also, we balance the energy consumption among the cluster heads to design an optimum load distribution ratio. Simulation result shows that this approach guarantees to increase the network lifetime, thereby balancing cluster head energy.

Relative Efficiency of Unequal Versus Equal Cluster Sizes for the Nonparametric Weighted Sign Test Estimators in Clustered Binary Data

We consider analysis of clustered binary data from multiple observations for each subject in which any two observations from a subject are assumed to have a common correlation coefficient. In the weighted sign test on proportion in clustered binary data, three weighting schemes are considered: equal weights to observations, equal weights to clusters and the optimal weights that minimize the variance of the estimator. Since the distribution of cluster sizes may not be exactly specified before the trial starts, the sample size is usually determined using an average cluster size without taking into account any potential imbalance in cluster size even though cluster size usually varies among clusters. In this paper we investigate the relative efficiency (RE) of unequal versus equal cluster sizes for clustered binary data using the weighted sign test estimators. The REs are computed as a function of correlation among observations within each subject and the various cluster size distributions. The required sample size for unequal cluster sizes will not exceed the sample size for an equal cluster size multiplied by the maximum RE. It is concluded that the maximum RE for various cluster size distributions considered here does not exceed 1.50, 1.61 and 1.12 for equal weights to observations, equal weights to clusters and optimal weights, respectively. It suggests sampling 50%, 61% and 12% more clusters depending on the weighting schemes than the number of clusters computed using an average cluster size.