Confidence Limits on the Ratio of the Mean Failure Times Based on Paired Exponential Observations

This paper discusses interval estimation for the ratio of the mean failure times on the basis of paired exponential observations. This paper considers five interval estimators: the confidence interval using an idea similar to Fieller's theorem (CIFT), the confidence interval using an exact parametric test (CIEP), the confidence interval using the marginal likelihood ratio test (CILR), the confidence interval assuming no matching effect (CINM), and the confidence interval using a locally most powerful test (CIMP). To evaluate and compare the performance of these five interval estimators, this paper applies Monte Carlo simulation. This paper notes that with respect to the coverage probability, use of the CIFT, CILR, or CIMP, although which are all derived based on large sample theory, can perform well even when the number of pairs n is as small as 10. As compared with use of the CILR, this paper finds that use of the CIEP with equal tail probabilities is likely to lose efficiency. However, this loss can be reduced by using the optimal tail probabilities to minimize the average length when n is small (<20). This paper further notes that use of the CIMP is preferable to the CIEP in a variety of situations considered here. In fact, the average length of the CIMP with use of the optimal tail probabilities can even be shorter than that of the CILR. When the intraclass correlation between failure times within pairs is 0 (i.e., the failure times within the same pair are independent), the CINM, which is derived for two independent samples, is certainly the best one among the five interval estimators considered here. When there is an intraclass correlation but which is small (<0.10), the CIFT is recommended for obtaining a relatively short interval estimate without sacrificing the loss of the coverage probability. When the intraclass correlation is moderate or large, either the CILR or the CIMP with the optimal tail probabilities is preferable to the others. This paper also notes that if the intraclass correlation between failure times within pairs is large, use of the CINM can be misleading, especially when the number of pairs is large.     

Weighted least-squares approach for comparing correlated kappa

In the medical sciences, studies are often designed to assess the agreement between different raters or different instruments. The kappa coefficient is a popular index of agreement for binary and categorical ratings. Here we focus on testing for the equality of two dependent kappa coefficients. We use the weighted least-squares (WLS) approach of Koch et al. (1977, Biometrics 33, 133-158) to take into account the correlation between the estimated kappa statistics. We demonstrate how the SAS PROC CATMOD can be used to test for the equality of dependent Cohen's kappa coefficients and dependent intraclass kappa coefficients with nominal categorical ratings. We also test for the equality of dependent Cohen's kappa and dependent weighted kappa with ordinal ratings. The major advantage of the WLS approach is that it allows the data analyst a way of testing dependent kappa with popular SAS software. The WLS approach can handle any number of categories. Analyses of three biomedical studies are used for illustration.     

An Estimating Equations Approach for Modelling Kappa

Agreement between raters for binary outcome data is typically assessed using the kappa coefficient. There has been considerable recent work extending logistic regression to provide summary estimates of interrater agreement adjusted for covariates predictive of the marginal probability of classification by each rater. We propose an estimating equations approach which can also be used to identify covariates predictive of kappa. Models may include an arbitrary and variable number of raters per subject and yet do not require any stringent parametric assumptions. Examples used to illustrate this procedure include an investigation of factors affecting agreement between primary and proxy respondents from a case‐control study and a study of the effects of gender and zygosity on twin concordance for smoking history.     

Optimal design of longitudinal data analysis using generalized estimating equation models

Longitudinal studies are often applied in biomedical research and clinical trials to evaluate the treatment effect. The association pattern within the subject must be considered in both sample size calculation and the analysis. One of the most important approaches to analyze such a study is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which “working correlation structure” is introduced and the association pattern within the subject depends on a vector of association parameters denoted by ρ. The explicit sample size formulas for two‐group comparison in linear and logistic regression models are obtained based on the GEE method by Liu and Liang. For cluster randomized trials (CRTs), researchers proposed the optimal sample sizes at both the cluster and individual level as a function of sampling costs and the intracluster correlation coefficient (ICC). In these approaches, the optimal sample sizes depend strongly on the ICC. However, the ICC is usually unknown for CRTs and multicenter trials. To overcome this shortcoming, Van Breukelen et al. consider a range of possible ICC values identified from literature reviews and present Maximin designs (MMDs) based on relative efficiency (RE) and efficiency under budget and cost constraints. In this paper, the optimal sample size and number of repeated measurements using GEE models with an exchangeable working correlation matrix is proposed under the considerations of fixed budget, where “optimal” refers to maximum power for a given sampling budget. The equations of sample size and number of repeated measurements for a known parameter value ρ are derived and a straightforward algorithm for unknown ρ is developed. Applications in practice are discussed. We also discuss the existence of the optimal design when an AR(1) working correlation matrix is assumed. Our proposed method can be extended under the scenarios when the true and working correlation matrix are different.     

Measuring agreement between two judges on the presence or absence of a trait.

At least a dozen indexes have been proposed for measuring agreement between two judges on a categorical scale. Using the binary (positive-negative) case as a model, this paper presents and critically evaluates some of these proposed measures. The importance of correcting for chance-expected agreement is emphasized, and identities with intraclass correlation coefficients are pointed out.     

Assessing inter-rater reliability when the raters are fixed: Two concepts and two estimates

Intraclass correlation (ICC) is an established tool to assess inter-rater reliability. In a seminal paper published in 1979, Shrout and Fleiss considered three statistical models for inter-rater reliability data with a balanced design. In their first two models, an infinite population of raters was considered, whereas in their third model, the raters in the sample were considered to be the whole population of raters. In the present paper, we show that the two distinct estimates of ICC developed for the first two models can both be applied to the third model and we discuss their different interpretations in this context.     

Estimating the generalized concordance correlation coefficient through variance components

The intraclass correlation coefficient (ICC) and the concordance correlation coefficient (CCC) are two of the most popular measures of agreement for variables measured on a continuous scale. Here, we demonstrate that ICC and CCC are the same measure of agreement estimated in two ways: by the variance components procedure and by the moment method. We propose estimating the CCC using variance components of a mixed effects model, instead of the common method of moments. With the variance components approach, the CCC can easily be extended to more than two observers, and adjusted using confounding covariates, by incorporating them in the mixed model. A simulation study is carried out to compare the variance components approach with the moment method. The importance of adjusting by confounding covariates is illustrated with a case example.     

General Derivation of Intraclass Correlation Coefficients

A general intraclass correlation coefficient is derived similarly to the general correlation coefficient given by KENDALL (1962). This coefficient II embraces as specific cases three intraclass correlations, related to Pearson's r, Kendall's tau and Spearman's rs, that were dealt with by FISHER (1921), WHITFIELD (1949), SHIRAHATA (1981) and SCHEMPER (1984). Formal relationships are presented and the qualification of the three coefficients for specific applications is discussed.     

Reliability of bulky DNA adducts measurement by the nuclease P1 32P-post-labelling technique.

The aim was to assess the reliability of bulky DNA adducts measurement by means of the 32P-post-labelling assay. The research design consisted of an intramethod reliability study. Buffy coats from 41 subjects were used to obtain two aliquots of 1-5 microg DNA for each subject; bulky DNA adducts were measured using the nuclease P1 32P-post-labelling technique. The reliability of the measurement was assessed by means of the intraclass correlation coefficient (ICC), the distribution of the differences between the two measurements and the limits of agreement. The estimated ICC was 0.977, with a 95% confidence interval between 0.921 and 0.977. The limits of agreement were +/- 0.44 (DNA adducts per 10(8) nucleotides). Only three subjects had differences lying out of such limits. Bulky DNA adduct levels measured by the 32P-post-labelling technique showed good reliability. Only one measurement is needed to use DNA adducts as a biomarker of exposure and, possibly, cancer risk. Besides, as a validation analysis, 32P-post-labelling measurements can be repeated in only 20-30% of samples.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

A Note on Interval Estimation of the Simple Difference in Data with Correlated Matched Pairs

When we employ cluster sampling to collect data with matched pairs, the assumption of independence between all matched pairs is not likely true. This paper notes that applying interval estimators, that do not account for the intraclass correlation between matched pairs, to estimate the simple difference between two proportions of response can be quite misleading, especially when both the number of matched pairs per cluster and the intraclass correlation between matched pairs within clusters are large. This paper develops two asymptotic interval estimators of the simple difference, that accommodate the data of cluster sampling with correlated matched pairs. This paper further applies Monte Carlo simulation to compare the finite sample performance of these estimators and demonstrates that the interval estimator, derived from a quadratic equation proposed here, can actually perform quite well in a variety of situations.     

Bayesian inference for kappa from single and multiple studies

Cohen's kappa coefficient is a widely popular measure for chance-corrected nominal scale agreement between two raters. This article describes Bayesian analysis for kappa that can be routinely implemented using Markov chain Monte Carlo (MCMC) methodology. We consider the case of m > or = 2 independent samples of measured agreement, where in each sample a given subject is rated by two rating protocols on a binary scale. A major focus here is on testing the homogeneity of the kappa coefficient across the different samples. The existing frequentist tests for this case assume exchangeability of rating protocols, whereas our proposed Bayesian test does not make any such assumption. Extensive simulation is carried out to compare the performances of the Bayesian and the frequentist tests. The developed methodology is illustrated using data from a clinical trial in ophthalmology.     

Bias of maximum likelihood estimator of intraclass correlation

Summary A bias correction was derived for the maximum likelihood estimator (MLE) of the intraclass correlation. The bias consisted of two parts: a correction from MLE to the analysis of variance estimator (ANOVA) and the bias of ANOVA. The total possible bias was always negative and depended upon both the degree of correlation and the design size and balance. The first part of the bias was an exact algebraic expression from MLE to ANOVA, and the corrected estimator by this part was ANOVA. It was also shown that the first correction term was equivalent to Fisher's reciprocal bias correction on hisZ scores. The total possible bias of MLE was large for small and moderate samples. Relative biases were larger for small parametric values and vice versa. To ensure a relative bias less than 10% assuming an intraclass correlation of 0.025, which is not unusual in most of the animal genetic studies, the total number of observations (N) should be not less than 500. From a design point of view, minimum bias occurred atn = 2, the minimum family size possible, underN fixed.     

A Coefficient of Agreement Adjusted for Bias in Paired Ordered Categorical Data

The quality of ordered categorical recordings is determined from repeated measurements on the same subject in order to assess the level of agreement between raters, scales or occasions. The presented rating-invariant method for ordered categorical data provides means of analysing the quality of single-item rating scales, irrespective of the number of possible response values and the marginal distributions. Marginal heterogeneity implies systematic disagreement, so-called bias. An augmented ranking approach is the basis for the separation of inter-rater disagreement into systematic and random components. Correlation between pairs of augmented rank values provides a measure of agreement to the best common ordering of paired classifications, given inter-rater bias. The essential differences in interpretation and applicability of the proposed coefficient of agreement and the Spearman rank-order correlation for ordered categorical data are discussed.     

Inference Procedures for Assessing Interobserver Agreement among Multiple Raters

We propose a new procedure for constructing inferences about a measure of interobserver agreement in studies involving a binary outcome and multiple raters. The proposed procedure, based on a chi-square goodness-of-fit test as applied to the correlated binomial model (Bahadur, 1961, in Studies in Item Analysis and Prediction, 158-176), is an extension of the goodness-of-fit procedure developed by Donner and Eliasziw (1992, Statistics in Medicine 11, 1511-1519) for the case of two raters. The new procedure is shown to provide confidence-interval coverage levels that are close to nominal over a wide range of parameter combinations. The procedure also provides a sample-size formula that may be used to determine the required number of subjects and raters for such studies.     

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.     

Estimating marginal proportions and intraclass correlations with clustered binary data

A logistic regression with random effects model is commonly applied to analyze clustered binary data, and every cluster is assumed to have a different proportion of success. However, it could be of interest to obtain the proportion of success over clusters (i.e. the marginal proportion of success). Furthermore, the degree of correlation among data of the same cluster (intraclass correlation) is also a relevant concept to assess, but when using logistic regression with random effects it is not possible to get an analytical expression of the estimators for marginal proportion and intraclass correlation. In our paper, we assess and compare approaches using different kinds of approximations: based on the logistic‐normal mixed effects model (LN), linear mixed model (LMM), and generalized estimating equations (GEE). The comparisons are completed by using two real data examples and a simulation study. The results show the performance of the approaches strongly depends on the magnitude of the marginal proportion, the intraclass correlation, and the sample size. In general, the reliability of the approaches get worsen with low marginal proportion and large intraclass correlation. LMM and GEE approaches arises as reliable approaches when the sample size is large.     

Confidence Limits for the Intraclass Correlation in Compound-Poisson Sampling

When the underlying responses are discrete, the interval estimation of the intraclass correlation derived from the normality assumption is not strictly valid for use. This paper focuses the interval estimation on the intraclass correlation under the negative binomial distribution, that has been commonly applied in epidemiological or consumer purchasing behaviour studies. This paper develops two simple asymptotic interval estimation procedures in closed forms for the intraclass correlation. To evaluate the performance of these procedures, a Monte Carlo simulation is carried out for a variety of situations. An example about consumer purchasing behaviors is also included to illustrate the use of the two proposed interval estimation procedures.     

Reliability of bulky DNA adducts measurement by the nuclease P1 32P-post-labelling technique

The aim was to assess the reliability of bulky DNA adducts measurement by means of the ³²P-post-labelling assay. The research design consisted of an intramethod reliability study. Buffy coats from 41 subjects were used to obtain two aliquots of 1–5 μg DNA for each subject; bulky DNA adducts were measured using the nuclease P1 ³²P-post-labelling technique. The reliability of the measurement was assessed by means of the intraclass correlation coefficient (ICC), the distribution of the differences between the two measurements and the limits of agreement. The estimated ICC was 0.977, with a 95% confidence interval between 0.921 and 0.977. The limits of agreement were ±0.44 (DNA adducts per 10⁸ nucleotides). Only three subjects had differences lying out of such limits. Bulky DNA adduct levels measured by the ³²P-post-labelling technique showed good reliability. Only one measurement is needed to use DNA adducts as a biomarker of exposure and, possibly, cancer risk. Besides, as a validation analysis, ³²P-post-labelling measurements can be repeated in only 20–30% of samples.     

"Noise" as a unifying concept in the theory of schizophrenia

The psychological effects of pharmacologically active substances can only be adequately described in terms of the psychological processes upon which they act. Psychopharmacology therefore depends on the development of psychological “models”, i.e. formally stated and testable hypotheses concerning the psychological processes required for the performance of any given task. Information theory is not concerned with the physical nature of events but only with those features which confer specificity upon them. It therefore provides a suitable theoretical language in which to describe interactions between biochemical, neurophysiological and psychological events. (i) The first section of this paper defines the information-theory concept of noise. Starting from first principles with the “noisy channel” and progressing to the “noisy system”, some of its psychophysiological implications are explored. (ii) An “order-memory” task is described which was used for three years to study the thinking of acutely disturbed young adult psychiatric patients, including many with acute schizophrenia. On the basis of a simple model, it is possible to calculate the value of a “critical” noise-level which marks the dividing line between two qualitatively different modes of functioning. (iii) A number of results are either reported or summarized, which show the functional significance of a supra-critical noise-level and its connection with the acute schizophrenic state. (iv) A “control-theory” approach to schizophrenia is outlined, which shows how specific and non-specific hereditary factors could be accommodated in the hypothesis but mainly emphasizes the concept of a mutual struggle for control between parent and child, in which the “loser” overloads the regulatory capacity of the other by “going noisy”. In this way the “loser” escapes from control but is more or less disabled by his own cognitive noise.