Inverse Cluster Size Research Articles

Adjusting for informative cluster size in pseudo-value-based regression approaches with clustered time to event data.

Informative cluster size (ICS) arises in situations with clustered data where a latent relationship exists between the number of participants in a cluster and the outcome measures. Although this phenomenon has been sporadically reported in the statistical literature for nearly two decades now, further exploration is needed in certain statistical methodologies to avoid potentially misleading inferences. For inference about population quantities without covariates, inverse cluster size reweightings are often employed to adjust for ICS. Further, to study the effect of covariates on disease progression described by a multistate model, the pseudo-value regression technique has gained popularity in time-to-event data analysis. We seek to answer the question: "How to apply pseudo-value regression to clustered time-to-event data when cluster size is informative?" ICS adjustment by the reweighting method can be performed in two steps; estimation of marginal functions of the multistate model and fitting the estimating equations based on pseudo-value responses, leading to four possible strategies. We present theoretical arguments and thorough simulation experiments to ascertain the correct strategy for adjusting for ICS. A further extension of our methodology is implemented to include informativeness induced by the intracluster group size. We demonstrate the methods in two real-world applications: (i) to determine predictors of tooth survival in a periodontal study and (ii) to identify indicators of ambulatory recovery in spinal cord injury patients who participated in locomotor-training rehabilitation.

Two weights make a wrong: Cluster randomized trials with variable cluster sizes and heterogeneous treatment effects

In cluster randomized trials (CRTs), the hierarchical nesting of participants (level 1) within clusters (level 2) leads to two conceptual populations: clusters and participants. When cluster sizes vary and the goal is to generalize to a hypothetical population of clusters, the unit average treatment effect (UATE), which averages equally at the cluster level rather than equally at the participant level, is a common estimand of interest. From an analytic perspective, when a generalized estimating equations (GEE) framework is used to obtain averaged treatment effect estimates for CRTs with variable cluster sizes, it is natural to specify an inverse cluster size weighted analysis so that each cluster contributes equally and to adopt an exchangeable working correlation matrix to account for within-cluster correlation. However, such an approach essentially uses two distinct weights in the analysis (i.e. both cluster size weights and covariance weights) and, in this article, we caution that it will lead to biased and/or inefficient treatment effect estimates for the UATE estimand. That is, two weights “make a wrong” or lead to poor estimation characteristics. These findings are based on theoretical derivations, corroborated via a simulation study, and illustrated using data from a CRT of a colorectal cancer screening program. We show that, an analysis with both an independence working correlation matrix and weighting by inverse cluster size is the only approach that always provides valid results for estimation of the UATE in CRTs with variable cluster sizes.

Variance estimation in tests of clustered categorical data with informative cluster size.

In the analysis of clustered data, inverse cluster size weighting has been shown to be resistant to the potentially biasing effects of informative cluster size, where the number of observations within a cluster is associated with the outcome variable of interest. The method of inverse cluster size reweighting has been implemented to establish clustered data analogues of common tests for independent data, but the method has yet to be extended to tests of categorical data. Many variance estimators have been implemented across established cluster-weighted tests, but potential effects of differing methods on test performance has not previously been explored. Here, we develop cluster-weighted estimators of marginal proportions that remain unbiased under informativeness, and derive analogues of three popular tests for clustered categorical data, the one-sample proportion, goodness of fit, and independence chi square tests. We construct these tests using several variance estimators and show substantial differences in the performance of cluster-weighted tests based on variance estimation technique, with variance estimators constructed under the null hypothesis maintaining size closest to nominal. We illustrate the proposed tests through an application to a data set of functional measures from patients with spinal cord injuries participating in a rehabilitation program.

Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems

Great progress has been made in recent years towards understanding the properties of disordered electronic systems. In part, this is made possible by recent advances in quantum effective medium methods which enable the study of disorder and electron-electronic interactions on equal footing. They include dynamical mean-field theory and the Coherent Potential Approximation, and their cluster extension, the dynamical cluster approximation. Despite their successes, these methods do not enable the first-principles study of the strongly disordered regime, including the effects of electronic localization. The main focus of this review is the recently developed typical medium dynamical cluster approximation for disordered electronic systems. This method has been constructed to capture disorder-induced localization and is based on a mapping of a lattice onto a quantum cluster embedded in an effective typical medium, which is determined self-consistently. Unlike the average effective medium-based methods mentioned above, typical medium-based methods properly capture the states localized by disorder. The typical medium dynamical cluster approximation not only provides the proper order parameter for Anderson localized states, but it can also incorporate the full complexity of Density-Functional Theory (DFT)-derived potentials into the analysis, including the effect of multiple bands, non-local disorder, and electron-electron interactions. After a brief historical review of other numerical methods for disordered systems, we discuss coarse-graining as a unifying principle for the development of translationally invariant quantum cluster methods. Together, the Coherent Potential Approximation, the Dynamical Mean-Field Theory and the Dynamical Cluster Approximation may be viewed as a single class of approximations with a much-needed small parameter of the inverse cluster size which may be used to control the approximation. We then present an overview of various recent applications of the typical medium dynamical cluster approximation to a variety of models and systems, including single and multiband Anderson model, and models with local and off-diagonal disorder. We then present the application of the method to realistic systems in the framework of the DFT and demonstrate that the resulting method can provide a systematic first-principles method validated by experiment and capable of making experimentally relevant predictions. We also discuss the application of the typical medium dynamical cluster approximation to systems with disorder and electron-electron interactions. Most significantly, we show that in the limits of strong disorder and weak interactions treated perturbatively, that the phenomena of 3D localization, including a mobility edge, remains intact. However, the metal-insulator transition is pushed to larger disorder values by the local interactions. We also study the limits of strong disorder and strong interactions capable of producing moment formation and screening, with a non-perturbative local approximation. Here, we find that the Anderson localization quantum phase transition is accompanied by a quantum-critical fan in the energy-disorder phase diagram.

Robust estimation of marginal regression parameters in clustered data.

We develop robust methods for analyzing clustered data where estimation of marginal regression parameters is of interest. Inverse cluster size reweighting in the objective function to be minimized is incorporated to handle the issue of informative cluster size. Performance of the resulting estimators is studied by simulation. Large sample inference and variance estimation is carried out. The methodology is illustrated using a periodontal disease dataset.

Trend of the magnetic anisotropy for individual Mn dopants near the (1 1 0) GaAs surface

Using a microscopic finite-cluster tight-binding model, we investigate the trend of the magnetic anisotropy energy as a function of the cluster size for an individual Mn impurity positioned in the vicinity of the (1 1 0) GaAs surface. We present results of calculations for large cluster sizes containing approximately 104 atoms, which have not been investigated so far. Our calculations demonstrate that the anisotropy energy of a Mn dopant in bulk GaAs, found to be non-zero in previous tight-binding calculations, is purely a finite size effect that vanishes with inverse cluster size. In contrast to this, we find that the splitting of the three in-gap Mn acceptor energy levels converges to a finite value in the limit of the infinite cluster size. For a Mn in bulk GaAs this feature is related to the nature of the mean-field treatment of the coupling between the impurity and its nearest neighbor atoms. We also calculate the trend of the anisotropy energy in the sublayers as the Mn dopant is moved away from the surface towards the center of the cluster. Here the use of large cluster sizes allows us to position the impurity in deeper sublayers below the surface, compared to previous calculations. In particular, we show that the anisotropy energy increases up to the fifth sublayer and then decreases as the impurity is moved further away from the surface, approaching its bulk value. The present study provides important insights for experimental control and manipulation of the electronic and magnetic properties of individual Mn dopants at the semiconductor surface by means of advanced scanning tunneling microscopy techniques.

Frequency and Relative Prevalence of Calcium Blips and Puffs in a Model of Small IP3R Clusters

In this work, we model the local calcium release from clusters with a few inositol 1,4,5-trisphosphate receptor (IP3R) channels, focusing on the stochastic process in which an open channel either triggers other channels to open (as a puff) or fails to cause any channel to open (as a blip). We show that there are linear relations for the interevent interval (including blips and puffs) and the first event latency against the inverse cluster size. However, nonlinearity is found for the interpuff interval and the first puff latency against the inverse cluster size. Furthermore, the simulations indicate that the blip fraction among all release events and the blip frequency are increasing with larger basal [Ca2+], with blips in turn giving a growing contribution to basal [Ca2+]. This result suggests that blips are not just lapses to trigger puffs, but they may also possess a biological function to contribute to the initiation of calcium waves by a preceding increase of basal [Ca2+] in cells that have small IP3R clusters.

An examination of a method for marginal inference when the cluster size is informative

Generalised Estimating Equations (GEE) are a popular method to fit marginal models to clustered data. When the cluster size, i.e. the total number of members in the cluster, is informative then inference may be for the population of typical cluster members (i.e. inference for a typical member of a typical cluster) or the population of all cluster members. Applying the GEE with independence working correlation provides inference for the population of all members, and with additional weighting by the inverse cluster size gives inference for the population of typical members. In an earlier volume an adaptation of GEE termed modified within-cluster resampling (MWCR) was proposed to give unbiased inference for the population of typical members with increased efficiency by recognising the correlation between measurements. We describe how bias can arise when MWCR is used. This potential for bias was not clear when the method was proposed. We present conditions, concerning the data structure and the choice of the working correlation, which if satisfied will allow consistent estimation from MWCR. We illustrate Statistica Sinica: Preprint doi:10.5705/ss.2011.252

Marginal association measures for clustered data.

The use of correlation coefficients in measuring the association between two continuous variables is common, but regular methods of calculating correlations have not been extended to the clustered data framework. For clustered data in which observations within a cluster may be correlated, regular inferential procedures for calculating marginal association between two variables can be biased. This is particularly true for data in which the number of observations in a given cluster is informative for the association being measured. In this paper, we apply the principle of inverse cluster size reweighting to develop estimators of marginal correlation that remain valid in the clustered data framework when cluster size is informative for the correlation being measured. These correlations are derived as analogs to regular correlation estimators for continuous, independent data, namely, Pearson's ρ and Kendall's τ. We present the results of a simple simulation study demonstrating the appropriateness of our proposed estimators and the inherent bias of other inferential procedures for clustered data. We illustrate their use through an application to data from patients with incomplete spinal cord injury in the U.S.A.

Fitting marginal accelerated failure time models to clustered survival data with potentially informative cluster size

Methods for analyzing clustered survival data are gaining popularity in biomedical research. Naive attempts to fitting marginal models to such data may lead to biased estimators and misleading inference when the size of a cluster is statistically correlated with some cluster specific latent factors or one or more cluster level covariates. A simple adjustment to correct for potentially informative cluster size is achieved through inverse cluster size reweighting. We give a methodology that incorporates this technique in fitting an accelerated failure time marginal model to clustered survival data. Furthermore, right censoring is handled by inverse probability of censoring reweighting through the use of a flexible model for the censoring hazard. The resulting methodology is examined through a thorough simulation study. Also an illustrative example using a real dataset is provided that examines the effects of age at enrollment and smoking on tooth survival.

Nonequilibrium cluster perturbation theory

The cluster perturbation theory (CPT) is one of the simplest but systematic quantum cluster approaches to lattice models of strongly correlated electrons with local interactions. By treating the inter-cluster potential, in addition to the interactions, as a perturbation, it is shown that the CPT can be reformulated as an all-order re-summation of diagrams within standard weak-coupling perturbation theory where vertex corrections are neglected. This reformulation is shown to allow for a straightforward generalization of the CPT to the general non-equilibrium case using contour-ordered Green's functions. Solving the resulting generalized CPT equation on the discretized Keldysh-Matsubara time contour, the transient dynamics of an essentially arbitrary initial pure or mixed state can be traced. In this way, the time-dependent expectation values of one-particle observables can be obtained within an approximation that neglects spatial correlations beyond the extension of the reference cluster. The necessary computational effort is very moderate. A detailed discussion and simple test calculations are presented to demonstrate the strengths and the shortcomings of the proposed approach. The non-equilibrium CPT is systematic and is controlled in principle by the inverse cluster size. It interpolates between the non-interacting and the atomic or decoupled-cluster limit which are recovered exactly and is found to predict the correct dynamics at very short times in a general non-trivial case. The effects of initial-state correlations on the subsequent dynamics and the necessity to extend the Keldysh contour by the imaginary Matsubara branch are analyzed carefully and demonstrated numerically. It is furthermore shown that the approach can describe the dissipation of spin and charge to an uncorrelated bath with an essentially arbitrary number of degrees of freedom.

Giant magnetoresistance and Co-cluster structure in phase-separated Co-Cu granular alloys.

We examine the effect of Co-cluster morphology on giant magnetoresistance (MR) in phase-separated Co-Cu films. The Co clusters were characterized through grazing incidence, anomalous, small-angle x-ray scattering. With thermal annealing the Co cluster diameter increases from 21 to \ensuremath{\sim}250 \AA{} with a concomitant drop from \ensuremath{\sim}35% to 1% in the 4.2 K MR. The MR scales approximately as the inverse cluster size. Comparison with theory indicates that interfacial spin-dependent electron scattering is the dominant scattering mechanism underlying giant MR for cluster diameters up to at least 250 \AA{}.