A comparison of bayes and maximum likelihood estimation of the intraclass correlation coefficient

Two methods of estimating the intraclass correlation coefficient (p) for the one-way random effects model were compared in several simulation experiments using balanced and unbalanced designs. Estimates based on a Bayes approach and a maximum likelihood approach were compared on the basis of their biases (differences between estimates and true values of p) and mean square errors (mean square errors of estimates of p) in each of the simulation experiments. The Bayes approach used the median of a conditional posterior density as its estimator.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.     

Empirical Bayes estimation for the mean of the selected normal population when there are additional data

ABSTRACT

This paper is concerned with the problem of estimation for the mean of the selected population from two normal populations with unknown means and common known variance in a Bayesian framework. The empirical Bayes estimator, when there are available additional observations, is derived and its bias and risk function are computed. The expected bias and risk of the empirical Bayes estimator and the intuitive estimator are compared. It is shown that the empirical Bayes estimator is asymptotically optimal and especially dominates the intuitive estimator in terms of Bayes risk, with respect to any normal prior. Also, the Bayesian correlation between the mean of the selected population (random parameter) and some interested estimators are obtained and compared.     

Bayes least squares linear estimation of densities

Linear, least squares statistical methods in which the "parameters" are interpreted as random variables were introduced by Whittle, and further developed by Hartigan and others. They are applied here to the problem of estimating the coefficients in an orthogonal expansion of a multivariate density, given a simple random sample.     

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.     

Notes on estimation of the intraclass correlation under the AB/BA crossover trial

Under the AB/BA crossover trial, we focus our attention on estimation of the intraclass correlation in normal data. We develop both point and interval estimators in closed form for the intraclass correlation. We employ Monte Carlo simulation to study the performance of these estimators in a variety of situations. We note that the estimators developed here for the intraclass correlation remain valid even when there are possibly unexpected carry-over effects.     

Asymptotic optimality of a two-stage procedure in Bayes sequential estimation

The problem of sequential estimation of the mean with quadratic loss and fixed cost per observation is considered within the Bayesian framework. Instead of fully sequential sampling, a two-stage sampling technique is introduced to solve the problem. The proposed two-stage procedure is robust in the sense that it does not depend on the distribution of outcome variables and the prior. It is shown to be asymptotically not worse than the optimal fixed-sample-size procedures for the arbitrary distributions, and to be asymptotically Bayes for the distributions of one-parameter exponential family.     

Maximum likeihood estimation of an intraclass correlation in a bivariate normal distribution with missing observations

The maximum likeihood estimate is considered for an intraclass correlation coefficent in a bivariate normal distribution when some observations on either of the varibles are missuing. The estimate is given as the soulution of a polynomial equation of degree seven. An approximate confidence interval and a test procedure for the intraclass correlation are constricted based on an asymptotic variance stabilizing transformation of the resulting estimator. The distributional results are also considered under violation of the normality assumption. A Monte Carlo study was performed to examine the finite sample properties of the maximum likelihood estimator and to evaluate the proposed procedures for hypotheses testing and interval estimation.     

Confidence interval estimation for a linear contrast in intraclass correlation coefficients under unequal family sizes for several populations

Confidence intervals [based on F-distribution and (Z) standard normal distribution] for a linear contrast in intraclass correlation coefficients under unequal family sizes for several populations based on several independent multinormal samples have been proposed. It has been found that the confidence interval based on F-distribution consistently and reliably produced better results in terms of shorter average length of the interval than the confidence interval based on standard normal distribution for various combinations of intraclass correlation coefficient values. The coverage probability of the interval based on F-distribution is competitive with the coverage probability of the interval based on standard normal distribution. The interval based on F-distribution can be used for both small sample and large sample situations. An example with real life data has been presented.     

An objective stepwise Bayes approach to small area estimation

The term ‘small area’ or ‘small domain’ is commonly used to denote a small geographical area that has a small subpopulation of people within a large area. Small area estimation is an important area in survey sampling because of the growing demand for better statistical inference for small areas in public or private surveys. In small area estimation problems the focus is on how to borrow strength across areas in order to develop a reliable estimator and which makes use of available auxiliary information. Some traditional methods for small area problems such as empirical best linear unbiased prediction borrow strength through linear models that provide links to related areas, which may not be appropriate for some survey data. In this article, we propose a stepwise Bayes approach which borrows strength through an objective posterior distribution. This approach results in a generalized constrained Dirichlet posterior estimator when auxiliary information is available for small areas. The objective posterior distribution is based only on the assumption of exchangeability across related areas and does not make any explicit model assumptions. The form of our posterior distribution allows us to assign a weight to each member of the sample. These weights can then be used in a straight forward fashion to make inferences about the small area means. Theoretically, the stepwise Bayes character of the posterior allows one to prove the admissibility of the point estimators suggesting that inferential procedures based on this approach will tend to have good frequentist properties. Numerically, we demonstrate in simulations that the proposed stepwise Bayes approach can have substantial strengths compared to traditional methods.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Bayesian approach to estimation of intraclass correlation using reference prior

For the balanced variance component model when the intraclass correlation coefficient is of interest, Bayesian analysis is often appropriate. Berger and Bernardo’s (1992a) grouped ordering reference prior approach is used to analyze this model. The reference priors are developed and compared for the posterior inference with real and simulated data. We examine whether the reference priors satisfy the probability-matching criterion. Further, the reference prior is shown to be good in the sense of correct frequentist coverage probability of the posterior quantile.     

Alternative Bayes factors for model selection

Several alternative Bayes factors have been recently proposed in order to solve the problem of the extreme sensitivity of the Bayes factor to the priors of models under comparison. Specifically, the impossibility of using the Bayes factor with standard noninformative priors for model comparison has led to the introduction of new automatic criteria, such as the posterior Bayes factor (Aitkin 1991), the intrinsic Bayes factors (Berger and Pericchi 1996b) and the fractional Bayes factor (O'Hagan 1995). We derive some interesting properties of the fractional Bayes factor that provide justifications for its use additional to the ones given by O'Hagan. We further argue that the use of the fractional Bayes factor, originally introduced to cope with improper priors, is also useful in a robust analysis. Finally, using usual classes of priors, we compare several alternative Bayes factors for the problem of testing the point null hypothesis in the univariate normal model.     

Empirical Bayes sequential estimation of a finite population mean

In this article, we consider the Bayes and empirical Bayes problem of the current population mean of a finite population when the sample data is available from other similar (m-1) finite populations. We investigate a general class of linear estimators and obtain the optimal linear Bayes estimator of the finite population mean under a squared error loss function that considered the cost of sampling. The optimal linear Bayes estimator and the sample size are obtained as a function of the parameters of the prior distribution. The corresponding empirical Bayes estimates are obtained by replacing the unknown hyperparameters with their respective consistent estimates. A Monte Carlo study is conducted to evaluate the performance of the proposed empirical Bayes procedure.     

Adaptive wavelet empirical Bayes estimation of a location or a scale parameter

Assume that in independent two-dimensional random vectors (X₁,θ₁),…,(X_n,θ_n), each θ_i is distributed according to some unknown prior density function g. Also, given θ_i=θ, X_i has the conditional density function q(x−θ), x,θ(−∞,∞) (a location parameter case), or θ⁻¹q(x/θ), x,θ(0,∞) (a scale parameter case). In each pair the first component is observable, but the second is not. After the (n+1)th pair (X_n+1,θ_n+1) is obtained, the objective is to construct an empirical Bayes (EB) estimator of θ. In this paper we derive the EB estimators of θ based on a wavelet approximation with Meyer-type wavelets. We show that these estimators provide adaptation not only in the case when g belongs to the Sobolev space H with an unknown , but also when g is supersmooth.     

Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.     

Bayesian estimation of log odds ratios over two-way contingency tables with intraclass correlated cells

In this article, a Bayesian approach is proposed for the estimation of log odds ratios and intraclass correlations over a two-way contingency table, including intraclass correlated cells. Required likelihood functions of log odds ratios are obtained, and determination of prior structures is discussed. Hypothesis testing for log odds ratios and intraclass correlations by using the posterior simulations is outlined. Because the proposed approach includes no asymptotic theory, it is useful for the estimation and hypothesis testing of log odds ratios in the presence of certain intraclass correlation patterns. A family health status and limitations data set is analyzed by using the proposed approach in order to figure out the impact of intraclass correlations on the estimates and hypothesis tests of log odds ratios. Although intraclass correlations are small in the data set, we obtain that even small intraclass correlations can significantly affect the estimates and test results, and our approach is useful for the estimation and testing of log odds ratios in the presence of intraclass correlations.     

Robust hierarchical Bayes estimation of exchangeable means

Estimation of the mean of a multivariate normal distribution is considered. The components of the mean vector θ are assumed to be exchangeable; this is modelled in a hierarchical fashion with independent Cauchy distributions as the first-stage prior. The resulting generalized Bayes estimator is calculated and shown to be robust with respect to the presence of outlying means. Alternative estimators that have similar behaviour but are cheaper to compute are also derived.     

On the estimation of the intraclass correlation

The problem of estimating the intraclass correlation when the sampling design is unbalanced is discussed. The method of moments is used to derive an approximation to the distribution of the estimate of the intraclass correlation obtained by the variance components approach. The maximum likelihood estimator is also presented, along with a simple procedure due to Richard (1961), for numerically maximizing a likelihood function of several parameters. Finally, the issue of optimal study design is considered for both balanced and unbalanced situations. For given power we determine the number of sets required to detect different values of the intraclass correlation.