On the multivariate spatial median for clustered data

The authors consider the multivariate one-sample location problem with clustered data from a nonparametric viewpoint. They propose the spatial median and its affine equivariant version as companion estimators to the affine invariant sign test of Larocque (2003). They extend the asymptotics of the proposed estimators to cluster dependent data and explore the limiting as well as finite-sample efficiencies for multivariate Student distributions. They demonstrate that the efficiency of the spatial median suffers less from intracluster correlation than the mean vector. They use data on the well-being of pupils in Finnish schools to illustrate their work.     

Confidence intervals for the difference between two median survival times for clustered survival data

Clustered survival data arise often in clinical trial design, where the correlated subunits from the same cluster are randomized to different treatment groups. Under such design, we consider the problem of constructing confidence interval for the difference of two median survival time given the covariates. We use Cox gamma frailty model to account for the within-cluster correlation. Based on the conditional confidence intervals, we can identify the possible range of covariates over which the two groups would provide different median survival times. The associated coverage probability and the expected length of the proposed interval are investigated via a simulation study. The implementation of the confidence intervals is illustrated using a real data set.     

A weighted Jackknife method for clustered data

We propose a weighted delete-one-cluster Jackknife based framework for few clusters with severe cluster-level heterogeneity. The proposed method estimates the mean for a condition by a weighted sum of estimates from each of the Jackknife procedures. Influence from a heterogeneous cluster can be weighted appropriately, and the conditional mean can be estimated with higher precision. An algorithm for estimating the variance of the proposed estimator is also provided, followed by the cluster permutation test for the condition effect assessment. Our simulation studies demonstrate that the proposed framework has good operating characteristics.     

Resampling method to estimate intra-cluster correlation for clustered binary data

ABSTRACT

Various methods have been proposed to estimate intra-cluster correlation coefficients (ICCs) for correlated binary data, and many are very sensitive to the type of design and underlying distributional assumptions. We proposed a new method to estimate ICC and its 95% confidence intervals based on resampling principles and U-statistics, where we resampled with replacement pairs of individuals from within and between clusters. We concluded from our simulation study that the resampling-based estimates approximate the population ICC more precisely than the analysis of variance and method of moments techniques for different event rates, varying number of clusters, and cluster sizes.     

Conservative confidence intervals for the intraclass correlation coefficient for clustered binary data

Asymptotic approaches are traditionally used to calculate confidence intervals for intraclass correlation coefficient in a clustered binary study. When sample size is small to medium, or correlation or response rate is near the boundary, asymptotic intervals often do not have satisfactory performance with regard to coverage. We propose using the importance sampling method to construct the profile confidence limits for the intraclass correlation coefficient. Importance sampling is a simulation based approach to reduce the variance of the estimated parameter. Four existing asymptotic limits are used as statistical quantities for sample space ordering in the importance sampling method. Simulation studies are performed to evaluate the performance of the proposed accurate intervals with regard to coverage and interval width. Simulation results indicate that the accurate intervals based on the asymptotic limits by Fleiss and Cuzick generally have shorter width than others in many cases, while the accurate intervals based on Zou and Donner asymptotic limits outperform others when correlation and response rate are close to their boundaries.     

A weighted polynomial regression method for local fitting of spatial data

Typically, parametric approaches to spatial problems require restrictive assumptions. On the other hand, in a wide variety of practical situations nonparametric bivariate smoothing techniques has been shown to be successfully employable for estimating small or large scale regularity factors, or even the signal content of spatial data taken as a whole.We propose a weighted local polynomial regression smoother suitable for fitting of spatial data. To account for spatial variability, we both insert a spatial contiguity index in the standard formulation, and construct a spatial-adaptive bandwidth selection rule. Our bandwidth selector depends on the Gearys local indicator of spatial association. As illustrative example, we provide a brief Monte Carlo study case on equally spaced data, the performances of our smoother and the standard polynomial regression procedure are compared.This note, though it is the result of a close collaboration, was specifically elaborated as follows: paragraphs 1 and 2 by T. Sclocco and the remainder by M. Di Marzio. The authors are grateful to the referees for constructive comments and suggestions.     

A simple approach to analyzing clustered longitudinal data

When modeling correlated binary data in the presence of informative cluster sizes, generalized estimating equations with either resampling or inverse-weighting, are often used to correct for estimation bias. However, existing methods for the clustered longitudinal setting assume constant cluster sizes over time. We present a subject-weighted generalized estimating equations scheme that provides valid parameter estimation for the clustered longitudinal setting while allowing cluster sizes to change over time. We compare, via simulation, the performance of existing methods to our subject-weighted approach. The subject-weighted approach was the only method that showed negligible bias, with excellent coverage, for all model parameters.     

Bootstrapping Spatial Median for Location Problems

In multivariate location problems, the sample mean is most widely used, having various advantages. It is, however, very sensitive to outlying observations and inefficient for data from heavy tailed distributions. In this situation, the spatial median is more robust than the sample mean and could be a reasonable alternative. We reviewed several spatial median based testing methods for multivariate location and compared their significance level and power through Monte Carlo simulations. The results show that bootstrap method is efficient for the estimation of the covariance matrix of the sample spatial median. We also proposed bootstrap simultaneous confidence intervals based on the spatial median for multiple comparisons in the multi-sample case.     

A new alternative to the standard F test for clustered data

The data collection process and the inherent population structure are the main causes for clustered data. The observations in a given cluster are correlated, and the magnitude of such correlation is often measured by the intra-cluster correlation coefficient. The intra-cluster correlation can lead to an inflated size of the standard F test in a linear model. In this paper, we propose a solution to this problem. Unlike previous adjustments, our method does not require estimation of the intra-class correlation, which is problematic especially when the number of clusters is small. Our simulation results show that the new method outperforms the existing methods.     

One-sample location tests for multilevel data

In this paper, we consider testing the location parameter with multilevel (or hierarchical) data. A general family of weighted test statistics is introduced. This family includes extensions to the case of multilevel data of familiar procedures like the t, the sign and the Wilcoxon signed-rank tests. Under mild assumptions, the test statistics have a null limiting normal distribution which facilitates their use. An investigation of the relative merits of selected members of the family of tests is achieved theoretically by deriving their asymptotic relative efficiency (ARE) and empirically via a simulation study. It is shown that the performance of a test depends on the clusters configurations and on the intracluster correlations. Explicit formulas for optimal weights and a discussion of the impact of omitting a level are provided for 2 and 3-level data. It is shown that using appropriate weights can greatly improve the performance of the tests. Finally, the use of the new tests is illustrated with a real data example.     

Variance reduction for kernel estimators in clustered/longitudinal data analysis

We develop a variance reduction method for the seemingly unrelated (SUR) kernel estimator of Wang (2003). We show that the quadratic interpolation method introduced in Cheng et al. (2007) works for the SUR kernel estimator. For a given point of estimation, Cheng et al. (2007) define a variance reduced local linear estimate as a linear combination of classical estimates at three nearby points. We develop an analogous variance reduction method for SUR kernel estimators in clustered/longitudinal models and perform simulation studies which demonstrate the efficacy of our variance reduction method in finite sample settings.     

Continuous covariate frailty models for censored and truncated clustered data

Using some logarithmic and integral transformation we transform a continuous covariate frailty model into a polynomial regression model with a random effect. The responses of this mixed model can be ‘estimated’ via conditional hazard function estimation. The random error in this model does not have zero mean and its variance is not constant along the covariate and, consequently, these two quantities have to be estimated. Since the asymptotic expression for the bias is complicated, the two-large-bandwidth trick is proposed to estimate the bias. The proposed transformation is very useful for clustered incomplete data subject to left truncation and right censoring (and for complex clustered data in general). Indeed, in this case no standard software is available to fit the frailty model, whereas for the transformed model standard software for mixed models can be used for estimating the unknown parameters in the original frailty model. A small simulation study illustrates the good behavior of the proposed method. This method is applied to a bladder cancer data set.     

A working estimating equation for spatial count data

This paper proposes a working estimating equation which is computationally easy to use for spatial count data. The proposed estimating equation is a modification of quasi-likelihood estimating equations without the need of correctly specifying the covariance matrix. Under some regularity conditions, we show that the proposed estimator has consistency and asymptotic normality. A simulation comparison also indicates that the proposed method has competitive performance in dealing with over-dispersion data from a parameter-driven model.     

A generalized model of logistic regression for clustered data

This paper proposes a generalized logistic regression model that can account for the correlation among responses on subunits. The subunits may arise as data on multiple observations within an individual. This method generalizes earlier work by Rosner (1984 a,b) and others. Methodological generalizations include: (1) the use of the more general Polya-Eggenberger distribution instead of the beta-binomial distribution to model the correlation structure, so that cases with negative, positive, or zero intraclass correlation can be handled; (2) a stepwise approach; (3) linear and non-linear regression; and, (4) the inclusion of the case of a truncated distribution. The model can accommodate missing data and covariates on the unit and subunit level. The derivative-free simplex algorithm is used to estimate the parameters.

The model is applied to data describing the progression of obstruction in coronary disease where multiple arterial segments are studied for each patient. The correlation in response that may exist for these multiple segments is accounted for in the analyses while attempting to examine associations with individual-specific (e.g., history of diabetes) and segment-specific (e.g., initial percent stenosis) covariates. Analyses were performed on a data set describing 382 patients with unoperated coronary artery disease and two coronary angiograms separated by at least one month and on a data set describing 284 patients undergoing percutaneous transluminal coronary angioplasty and studied by coronary angiograms.     

A model for estimating point sources in spatial data

We introduce a point source model which may be useful for estimating point sources in spatial data. It may also be useful for modelling general spatial data, and providing a simple explanatory model for some data, whilst in other cases it may give a parsimonious representation. The model assumes that there are point sources (or sinks), usually at unknown positions, and that the mean value at a site depends on the distance from these sources. We discuss the general form of the model, and some methods for estimating the sources and the regression parameters. We demonstrate the methodology using a simulation study, and apply the model to two real data sets. Some possibilities for further research are outlined.     

A random‐effects model for clustered circular data

This article considers a circular regression model for clustered data, where both the cluster effects and the regression errors have von Mises distributions. It involves β, a vector of parameters for the fixed effects, and two concentration parameters for the error distribution. A measure of intra‐cluster circular correlation and a predictor for an unobserved cluster random effect are studied. Preliminary estimators for the vector β and the two concentration parameters are proposed, and their performance is compared with that of the maximum likelihood estimators in a simulation study. A numerical example investigating the factors impacting the orientation taken by a sand hopper when released is presented. The Canadian Journal of Statistics 47: 712–728; 2019 © 2019 Statistical Society of Canada     

A spatial scan statistic for survival data based on generalized life distribution

ABSTRACT

For many years, detection of clusters has been of great public health interest and widely studied. Several methods have been developed to detect clusters and their performance has been evaluated in various contexts. Spatial scan statistics are widely used for geographical cluster detection and inference. Different types of discrete or continuous data can be analyzed using spatial scan statistics for Bernoulli, Poisson, ordinal, exponential, and normal models. In this paper, we propose a scan statistic for survival data which is based on generalized life distribution model that provides three important life distributions, viz. Weibull, exponential, and Rayleigh. The proposed method is applied to the survival data of tuberculosis patients in Nainital district of Uttarakhand, India, for the year 2004–05. The Monte Carlo simulation studies reveal that the proposed method performs well for different survival distributions.     

Maximum likelihood estimation for generalized conditionally autoregressive models of spatial data

Conditionally autoregressive (CAR) models are often used to analyze a spatial process observed over a lattice or a set of irregular regions. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. To accommodate directional and inherent anisotropy variation, a new class of spatial models is proposed that adaptively determines neighbors based on a bivariate kernel using the distances and angles between the centroid of the regions. The newly proposed model generalizes the usual CAR model in a sense of accounting for adaptively determined weights. Maximum likelihood estimators are derived and simulation studies are presented for the sampling properties of the estimates on the new model, which is compared to the CAR model. Finally the method is illustrated using a data set on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Robustness of weighted L p–depth and L p–median

Summary: L ^p –norm weighted depth functions are introduced and the local and global robustness of these weighted L ^p –depth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a notion of finite sample breakdown point is introduced. The weighted L ^p –depth functions turn out to have the same low breakdown point as some other popular depth functions. Their influence functions are also unbounded. On the other hand, the weighted L ^p –depth induced medians are globally robust with the highest possible breakdown point for any reasonable estimator. The weighted L ^p –medians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted L ^p depth and medians are easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted L ^p depth and medians, respectively.*The author thanks the referees for their very insightful and constructive comments and suggestions which led to corrections and substantial improvements. Supported in part by NSF Grants DMS-0071976 and DMS-0134628.