Weighted kappa as a function of unweighted kappas

The kappa coefficient is a widely used measure for assessing agreement on a nominal scale. Weighted kappa is an extension of Cohen's kappa that is commonly used for measuring agreement on an ordinal scale. In this article, it is shown that weighted kappa can be computed as a function of unweighted kappas. The latter coefficients are kappa coefficients that correspond to smaller contingency tables that are obtained by merging categories.     

Symmetric kappa as a function of unweighted kappas

It is shown that a symmetric kappa corresponding to a c × c table with c ? 3 categories can be written as a function of the unweighted kappa corresponding to the same table and the c(c ? 1)/2 distinct unweighted kappas associated with the (c ? 1) × (c ? 1) tables that are obtained by combining two categories. The result is a new MGB-type result.     

Evaluation of a weighted multiple comparison procedure

We consider the problem of accounting for multiplicity for two correlated endpoints in the comparison of two treatments using weighted hypothesis tests. Various weighted testing procedures are reviewed, and a more powerful method (a variant of the weighted Simes test) is evaluated for the general bivariate normal case and for a particular clinical trial example. Results from these evaluations are summarized and indicate that the weighted methods perform in a manner similar to unweighted methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimally weighted,fixed sequence and gatekeeper multiple testing procedures

We consider a class of closed multiple test procedures indexed by a fixed weight vector. The class includes the Holm weighted step-down procedure, the closed method using the weighted Fisher combination test, and the closed method using the weighted version of Simes’ test. We show how to choose weights to maximize average power, where “average power” is itself weighted by importance assigned to the various hypotheses.Numerical computations suggest that the optimal weights for the multiple test procedures tend to certain asymptotic configurations. These configurations offer numerical justification for intuitive multiple comparisons methods, such as downweighting variables found insignificant in preliminary studies, giving primary variables more emphasis, gatekeeping test strategies, pre-determined multiple testing sequences, and pre-determined sequences of families of tests. We establish that such methods fall within the envelope of weighted closed testing procedures, thus providing a unified view of fixed sequences, fixed sequences of families, and gatekeepers within the closed testing paradigm. We also establish that the limiting cases control the familywise error rate (or FWE), using well-known results about closed tests, along with the dominated convergence theorem.     

A simple method for estimating a regression model for κ between a pair of raters

Agreement studies commonly occur in medical research, for example, in the review of X-rays by radiologists, blood tests by a panel of pathologists and the evaluation of psychopathology by a panel of raters. In these studies, often two observers rate the same subject for some characteristic with a discrete number of levels. The κ-coefficient is a popular measure of agreement between the two raters. The κ-coefficient may depend on covariates, i.e. characteristics of the raters and/or the subjects being rated. Our research was motivated by two agreement problems. The first is a study of agreement between a pastor and a co-ordinator of Christian education on whether they feel that the congregation puts enough emphasis on encouraging members to work for social justice (yes versus no). We wish to model the κ-coefficient as a function of covariates such as political orientation (liberal versus conservative) of the pastor and co-ordinator. The second example is a spousal education study, in which we wish to model the κ-coefficient as a function of covariates such as the highest degree of the father of the wife and the father of the husband. We propose a simple method to estimate the regression model for the κ-coefficient, which consists of two logistic (or multinomial logistic) regressions and one linear regression for binary data. The estimates can be easily obtained in any generalized linear model software program.     

Statistical inference of agreement coefficient between two raters with binary outcomes

Abstract

Scott’s pi and Cohen’s kappa are widely used for assessing the degree of agreement between two raters with binary outcomes. However, many authors have pointed out its paradoxical behavior, that comes from the dependence on the prevalence of a trait under study. To overcome the limitation, Gwet [Computing inter-rater reliability and its variance in the presence of high agreement. British Journal of Mathematical and Statistical Psychology 61(1):29–48] proposed an alternative and more stable agreement coefficient referred to as the AC₁. In this article, we discuss a likelihood-based inference of the AC₁ in the case of two raters with binary outcomes. Construction of confidence intervals is mainly discussed. In addition, hypothesis testing, and sample size estimation are also presented.     

Probability inequalities for weighted distributions

In this paper, properties of weighted distributions for general weight functions are investigated. We establish the subadditivity [superadditivity] property of weighted distributions for log-concave [log-convex] weight functions in the sense of the usual stochastic order. The main result generalizes Lemma 2.3 in Brown (2006). Several interesting moment inequalities are presented.     

Penalized inverse probability weighted estimators for weighted rank regression with missing covariates

Abstract

In this article, we study the variable selection and estimation for linear regression models with missing covariates. The proposed estimation method is almost as efficient as the popular least-squares-based estimation method for normal random errors and empirically shown to be much more efficient and robust with respect to heavy tailed errors or outliers in the responses and covariates. To achieve sparsity, a variable selection procedure based on SCAD is proposed to conduct estimation and variable selection simultaneously. The procedure is shown to possess the oracle property. To deal with the covariates missing, we consider the inverse probability weighted estimators for the linear model when the selection probability is known or unknown. It is shown that the estimator by using estimated selection probability has a smaller asymptotic variance than that with true selection probability, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for penalized rank estimator with the covariates missing in the linear model. Some numerical examples are provided to demonstrate the performance of the estimators.     

Design of experiments for locally weighted regression

We consider the design of experiments when estimation is to be performed using locally weighted regression methods. We adopt criteria that consider both estimation error (variance) and error resulting from model misspecification (bias). Working with continuous designs, we use the ideas developed in convex design theory to analyze properties of the corresponding optimal designs. Numerical procedures for constructing optimal designs are developed and applied to a variety of design scenarios in one and two dimensions. Among the interesting properties of the constructed designs are the following: (1) Design points tend to be more spread throughout the design space than in the classical case. (2) The optimal designs appear to be less model and criterion dependent than their classical counterparts.(3) While the optimal designs are relatively insensitive to the specification of the design space boundaries, the allocation of supporting points is strongly governed by the points of interest and the selected weight function, if the latter is concentrated in areas significantly smaller than the design region. Some singular and unstable situations occur in the case of saturated designs. The corresponding phenomenon is discussed using a univariate linear regression example.     

Efficient weighted bootstraps for the mean

The weighted bootstrap due to Mason and Newton (1992. Ann. Statist. 20, 1611–1624.) is applied to Studentized statistics in view of deriving efficient confidence intervals for the mean. First, we give conditions on the moments of the weights to ensure that the weighted bootstrap approximation leads to uniformly correct two-sided confidence intervals up to the rate O(n^−3/2). Then, we discuss the practical choice of the random weights in order to construct one-sided confidence intervals accurate up to O(n^−3/2) and two-sided confidence intervals up to higher orders. Simulations are given to illustrate the practical efficiency of our approach.     

An algorithm for weighted bilinear regression

Bilinear models in which the expectation of a two-way array is the sum of products of parameters are widely used in spectroscopy. In this paper we present an algorithm called combined-vector successive overrelaxation (COV-SOR) for bilinear models, and compare it with methods like alternating least squares, singular value decomposition, and the Marquardt procedure. Comparisons are done for missing data also.     

Relations for reliability measures of weighted distributions

Relibility measures of weighted distribution of alifeistribution have been derived Sufficientconditions on the weight function have been obtained for the weighted distribution of an IFR distribution to be IFR. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context.     

The weighted jackknife for ratio estimation

Using the idea of impirical influence function, Hinkley (1977), the weighted jackknife technique is extended to ratio estimation. A weighted jackknife variance estimator for the ratio estimator is developed. Using the prediction theory approach, the properties of the weighted jackknifed variance estimator are examined. The implications of the failures of regression model on the behaviour of the weighted jackknifed variance estimator, for ratio estimation, are also studied.     

A test statistic for weighted runs

A new test statistic based on runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic χ² test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute p-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.     

Peakedness for weighted sums of symmetric random variables

In this paper, we provide sufficient conditions on weighted coefficients under which the peakedness comparison between weighted sums of independent random variables can be carried out. These results extend and enrich the existing peakedness results in the literature including those presented by Proschan (1965) and Ma (1998).     

A weighted spatial median for clustered data

A weighted spatial median is proposed for the multivariate one-sample location problem with clustered data. Its limiting distribution is derived under mild conditions (no moment assumptions) and it is shown to be multivariate normal. Asymptotic as well as finite sample efficiencies and breakdown properties are considered, and the theoretical results are supplied with illustrative examples. It turns out that there is a potential for meaningful gains in estimation efficiency: the weighted spatial median has superior efficiency to the unweighted spatial median particularly when the cluster sizes are widely disparate and in the presence of strong intracluster correlation. The unweighted spatial median for clustered data was considered earlier by Nevalainen et al. (Can J Statist, in press, 2007). The proposed weighted estimators provide companion estimates to the weighted affine invariant sign test proposed recently by Larocque et al. (Biometrika, in press, 2007). An affine equivariant weighted spatial median is discussed in parallel.     

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.     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.