Testing the Intraclass Version of Kappa Coeffcient of Agreement with Binary Scale and Sample Size Determination

The intraclass version of kappa coefficient has been commonly applied as a measure of agreement for two ratings per subject with binary outcome in reliability studies. We present an efficient statistic for testing the strength of kappa agreement using likelihood scores, and derive asymptotic power and sample size formula. Exact evaluation shows that the score test is generally conservative and more powerful than a method based on a chi‐square goodness‐of‐fit statistic (Donner and Eliasziw , 1992, Statistics in Medicine 11 , 1511–1519). In particular, when the research question is one directional, the one‐sided score test is substantially more powerful and the reduction in sample size is appreciable.     

K‐Sample Test and Sample Size Calculation for Comparing Slopes in Data with Repeated Measurements

Sample size calculations based on two‐sample comparisons of slopes in repeated measurements have been reported by many investigators. In contrast, the literature has paid relatively little attention to the design and analysis of K‐sample trials in repeated measurements studies where K is 3 or greater. Jung and Ahn (2003) derived a closed sample size formula for two‐sample comparisons of slopes by taking into account the impact of missing data. We extend their method to compare K‐sample slopes in repeated measurement studies using the generalized estimating equation (GEE) approach based on independent working correlation structure. We investigate the performance of the sample size formula since the sample size formula is based on asymptotic theory. The proposed sample size formula is illustrated using a clinical trial example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sample Size Determination for Designing a Strata-Matched Case-Control Study to Detect Multiple Risk Factors

Investigations of sample size for planning case-control studies have usually been limited to detecting a single factor. In this paper, we investigate sample size for multiple risk factors in strata-matched case-control studies. We construct an omnibus statistic for testing M different risk factors based on the jointly sufficient statistics of parameters associated with the risk factors. The statistic is non-iterative, and it reduces to the Cochran statistic when M = 1. The asymptotic power function of the test is a non-central chi-square with M degrees of freedom and the sample size required for a specific power can be obtained by the inverse relationship. We find that the equal sample allocation is optimum. A Monte Carlo experiment demonstrates that an approximate formula for calculating sample size is satisfactory in typical epidemiologic studies. An approximate sample size obtained using Bonferroni's method for multiple comparisons is much larger than that obtained using the omnibus test. Approximate sample size formulas investigated in this paper using the omnibus test, as well as the individual tests, can be useful in designing case-control studies for detecting multiple risk factors.     

Exact log-rank tests for unequal follow-up

The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.     

Variance estimation in clinical studies with interim sample size re-estimation

We consider clinical studies with a sample size re-estimation based on the unblinded variance estimation at some interim point of the study. Because the sample size is determined in such a flexible way, the usual variance estimator at the end of the trial is biased. We derive sharp bounds for this bias. These bounds have a quite simple form and can help for the decision if this bias is negligible for the actual study or if a correction should be done. An exact formula for the bias is also provided. We discuss possibilities to get rid of this bias or at least to reduce the bias substantially. For this purpose, we propose a certain additive correction of the bias. We see in an example that the significance level of the test can be controlled when this additive correction is used.     

On Exact Unconditional Test for Linear Trend in Dose‐Response Studies

The one‐degree‐of‐freedom Cochran‐Armitage (C‐A) test statistic for linear trend has been widely applied in various dose‐response studies (e.g., anti‐ulcer medications and short‐term antibiotics, animal carcinogenicity bioassays and occupational toxicant studies). This approximate statistic relies, however, on asymptotic theory that is reliable only when the sample sizes are reasonably large and well balanced across dose levels. For small, sparse, or skewed data, the asymptotic theory is suspect and exact conditional method (based on the C‐A statistic) seems to provide a dependable alternative. Unfortunately, the exact conditional method is only practical for the linear logistic model from which the sufficient statistics for the regression coefficients can be obtained explicitly. In this article, a simple and efficient recursive polynomial multiplication algorithm for exact unconditional test (based on the C‐A statistic) for detecting a linear trend in proportions is derived. The method is applicable for all choices of the model with monotone trend including logistic, probit, arcsine, extreme value and one hit. We also show that this algorithm can be easily extended to exact unconditional power calculation for studies with up to a moderately large sample size. A real example is given to illustrate the applicability of the proposed method.     

The variance of the index (R) of aggregation of Clark and Evans

Summary The exact formula for the variance of the Index (R) of aggregation of Clark and Evans is derived. Due to the fact that R is a dimensionless number, its variance is independent of population density, being only a function of the sample size and so can be manipulated.     

Power calculation for cross-sectional stepped wedge cluster randomized trials with variable cluster sizes

Standard sample size calculation formulas for stepped wedge cluster randomized trials (SW-CRTs) assume that cluster sizes are equal. When cluster sizes vary substantially, ignoring this variation may lead to an under-powered study. We investigate the relative efficiency of a SW-CRT with varying cluster sizes to equal cluster sizes, and derive variance estimators for the intervention effect that account for this variation under a mixed effects model—a commonly used approach for analyzing data from cluster randomized trials. When cluster sizes vary, the power of a SW-CRT depends on the order in which clusters receive the intervention, which is determined through randomization. We first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power. We then obtain an “expected” power based on a first-order approximation to the variance formula, where the expectation is taken with respect to all possible randomization sequences. Finally, we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation, instead of exact cluster sizes, are known in the design stage. We evaluate our methods through simulations and illustrate that the average power of a SW-CRT decreases as the variation in cluster sizes increases, and the impact is largest when the number of clusters is small.     

Sexual dimorphism in human cranial trait scores: Effects of population,age, and body size

Sex estimation from the skull is commonly performed by physical and forensic anthropologists using a five‐trait scoring system developed by Walker. Despite the popularity of this method, validation studies evaluating its accuracy across a variety of samples are lacking. Furthermore, it remains unclear what other intrinsic or extrinsic variables are related to the expression of these traits. In this study, cranial trait scores and postcranial measurements were collected from four diverse population groups (U.S. Whites, U.S. Blacks, medieval Nubians, and Arikara Native Americans) following Walker's protocols (total n = 499). Univariate and multivariate analyses were utilized to evaluate the accuracy of these traits in sex estimation, and to test for the effects of population, age, and body size on trait expressions. Results revealed significant effects of population on all trait scores. Sample‐specific correct sex classification rates ranged from 74% to 94%, with an overall accuracy of 85% for the pooled sample. Classification performance varied among the traits (best for glabella and mastoid scores and worst for nuchal scores). Furthermore, correlations between traits were weak or nonsignificant, suggesting that different factors may influence individual traits. Some traits displayed correlations with age and/or postcranial size that were significant but weak, and within‐population analyses did not reveal any consistent relationships between these traits across all groups. These results indicate that neither age nor body size plays a large role in trait expression, and thus does not need to be incorporated into sex estimation methods. Am J Phys Anthropol 154:259–269, 2014. © 2014 Wiley Periodicals, Inc.     

Minimum Sample Size Ensuring Validity of Classical Confidence Intervals for Means of Skewed and Platykurtic Distributions

COCHRAN (1953) and BARTCH (1957) gave formulae for the magnitude of the sample size (n) ensuring the validity of the limiting normal distribution of the sample mean x(ⁿ) obtained from a non-normal distribution with marked asymmetry and kurtosis. These formulae have been checked empirically in this paper using (a) simulated data with given asymmetry and kurtosis and (b) real data gathered from a coronary heart disease study. We find that our results are in general agreement with Bartch's formula. However, in a number of cases, the asymptotic normal distribution is attained for smaller sample size than that required by Bartch's formula.     

Is the Exact Test Better than the Asymptotic Test for Testing Marginal Homogeneity in 2 × 2 Tables?

McNemar test is commonly used to test for the marginal homogeneity in 2 × 2 contingency tables. McNemar test is an asymptotic test based either on standard normal distribution or on the chi‐square distribution. When the total sample size is small, an exact version of McNemar test is available based on the binomial probabilities. The example in the paper came from a clinical study to investigate the effect of epidermal growth factor for children who had microvillus inclusion diseases. There were only six observations available. The test results differ between the exact test and the asymptotic test. It is a common belief that with this small sample size the exact test be used. However, we claim that McNemar test performs better than the exact test even when the sample size is small. In order to investigate the performances of McNemar test and the exact test, we identify the parameters that affect the test results and then perform sensitivity analysis. In addition, through Monte Carlo simulation studies we compare the empirical sizes and powers of these tests as well as other asymptotic tests such as Wald test and the likelihood ratio test.     

A general model for sample size determination for collecting germplasm

The paper develops a general model for determining the minimum sample size for collecting germplasm for genetic conservation with an overall objective of retaining at least one copy of each allele with preassigned probability. It considers sampling from a large heterogeneous 2k-ploid population under a broad range of mating systems leading to a general formula applicable to a fairly large number of populations. It is found that the sample size decreases as ploidy levels increase, but increases with the increase in inbreeding. Under exclusive selfing the sample size is the same, irrespective of the ploidy level, when other parameters are held constant. Minimum sample sizes obtained for diploids by this general formula agree with those already reported by earlier workers. The model confirms the conservative characteristics of genetic variability of polysomic inheritance under chromosomal segregation.     

Sample Size Reassessment in Adaptive Clinical Trials Using a Bias Corrected Estimate

Point estimation in group sequential and adaptive trials is an important issue in analysing a clinical trial. Most literature in this area is only concerned with estimation after completion of a trial. Since adaptive designs allow reassessment of sample size during the trial, reliable point estimation of the true effect when continuing the trial is additionally needed. We present a bias adjusted estimator which allows a more exact sample size determination based on the conditional power principle than the naive sample mean does.     

Exact coalescent for the Wright-Fisher model

The Kingman coalescent, which has become the foundation for a wide range of theoretical as well as empirical studies, was derived as an approximation of the Wright-Fisher (WF) model. The approximation heavily relies on the assumption that population size is large and sample size is much smaller than the population size. Whether the sample size is too large compared to the population size is rarely questioned in practice when applying statistical methods based on the Kingman coalescent. Since WF model is the most widely used population genetics model for reproduction, it is desirable to develop a coalescent framework for the WF model, which can be used whenever there are concerns about the accuracy of the Kingman coalescent as an approximation. This paper described the exact coalescent theory for the WF model and develops a simulation algorithm, which is then used, together with an analytical approach, to study the properties of the exact coalescent as well as its differences to the Kingman coalescent. We show that the Kingman coalescent differs from the exact coalescent by: (1) shorter waiting time between successive coalescent events; (2) different probability of observing a topological relationship among sequences in a sample; and (3) slightly smaller tree length in the genealogy of a large sample. On the other hand, there is little difference in the age of the most recent common ancestor (MRCA) of the sample. The exact coalescent makes up the longer waiting time between successive coalescent events by having multiple coalescence at the same time. The most significant difference among various summary statistics of a coalescent examined is the sum of lengths of external branches, which can be more than 10% larger for exact coalescent than that for the Kingman coalescent. As a whole, the Kingman coalescent is a remarkably accurate approximation to the exact coalescent for sample and population sizes falling considerably outside the region that was originally anticipated.     

Polygenic risk for psychiatric disorders correlates with executive function in typical development

Executive functions are a diverse and critical suite of cognitive abilities that are often disrupted in individuals with psychiatric disorders. Despite their moderate to high heritability, little is known about the molecular genetic factors that contribute to variability in executive functions and how these factors may be related to those that predispose to psychiatric disorders. We examined the relationship between polygenic risk scores built from large genome‐wide association studies of psychiatric disorders and executive functioning in typically developing children. In our discovery sample (N = 417), consistent with previous reports on general cognitive abilities, polygenic risk for autism spectrum disorder was associated with better performance on the Dimensional Change Card Sort test from the NIH Cognition Toolbox, with the largest effect in the youngest children. Polygenic risk for major depressive disorder was associated with poorer performance on the Flanker test in the same sample. This second association replicated for performance on the Penn Conditional Exclusion Test in an independent cohort (N = 3681). Our results suggest that the molecular genetic factors contributing to variability in executive function during typical development are at least partially overlapping with those associated with psychiatric disorders, although larger studies and further replication are needed.     

Measures of precision for dissimilarity‐based multivariate analysis of ecological communities

Ecological studies require key decisions regarding the appropriate size and number of sampling units. No methods currently exist to measure precision for multivariate assemblage data when dissimilarity‐based analyses are intended to follow. Here, we propose a pseudo multivariate dissimilarity‐based standard error (MultSE) as a useful quantity for assessing sample‐size adequacy in studies of ecological communities. Based on sums of squared dissimilarities, MultSE measures variability in the position of the centroid in the space of a chosen dissimilarity measure under repeated sampling for a given sample size. We describe a novel double resampling method to quantify uncertainty in MultSE values with increasing sample size. For more complex designs, values of MultSE can be calculated from the pseudo residual mean square of a permanova model, with the double resampling done within appropriate cells in the design. R code functions for implementing these techniques, along with ecological examples, are provided.     

Estimating Treatment Effects of Longitudinal Designs using Regression Models on Propensity Scores

Summary We derive regression estimators that can compare longitudinal treatments using only the longitudinal propensity scores as regressors. These estimators, which assume knowledge of the variables used in the treatment assignment, are important for reducing the large dimension of covariates for two reasons. First, if the regression models on the longitudinal propensity scores are correct, then our estimators share advantages of correctly specified model‐based estimators, a benefit not shared by estimators based on weights alone. Second, if the models are incorrect, the misspecification can be more easily limited through model checking than with models based on the full covariates. Thus, our estimators can also be better when used in place of the regression on the full covariates. We use our methods to compare longitudinal treatments for type II diabetes mellitus.     

Evaluation of Noether's Method of Sample Size Determination for the Wilcoxon-Mann-Whitney Test

NOETHER (1987) proposed a method of sample size determination for the Wilcoxon-Mann-Whitney test. To obtain a sample size formula, he restricted himself to alternatives that differ only slightly from the null hypothesis, so that the unknown variance o² of the Mann-Whitney statistic can be approximated by the known variance under the null hypothesis which depends only on n. This fact is frequently forgotten in statistical practice. In this paper, we compare Noether's large sample solution against an alternative approach based on upper bounds of σ² which is valid for any alternatives. This comparison shows that Noether's approximation is sufficiently reliable with small and large deviations from the null hypothesis.     

Predicting loneliness with polygenic scores of social,psychological and psychiatric traits

Loneliness is a heritable trait that accompanies multiple disorders. The association between loneliness and mental health indices may partly be due to inherited biological factors. We constructed polygenic scores for 27 traits related to behavior, cognition and mental health and tested their prediction for self‐reported loneliness in a population‐based sample of 8798 Dutch individuals. Polygenic scores for major depressive disorder (MDD), schizophrenia and bipolar disorder were significantly associated with loneliness. Of the Big Five personality dimensions, polygenic scores for neuroticism and conscientiousness also significantly predicted loneliness, as did the polygenic scores for subjective well‐being, tiredness and self‐rated health. When including all polygenic scores simultaneously into one model, only 2 major depression polygenic scores remained as significant predictors of loneliness. When controlling only for these 2 MDD polygenic scores, only neuroticism and schizophrenia remain significant. The total variation explained by all polygenic scores collectively was 1.7%. The association between the propensity to feel lonely and the susceptibility to psychiatric disorders thus pointed to a shared genetic etiology. The predictive power of polygenic scores will increase as the power of the genome‐wide association studies on which they are based increases and may lead to clinically useful polygenic scores that can inform on the genetic predisposition to loneliness and mental health.     

Converting Amino Acid Alignment Scores into Measures of Evolutionary Time: A Simulation Study of Various Relationships

Amino acid substitution tables are essential for the proper alignment of protein sequences, and alignment scores based on them can be transformed into distance measures by various means. In the simplest case, the negative log of the score is used. This Poisson relationship assumes that all sites are equally likely to change, however. A more accurate relationship would correct for different rates of change at each residue position. Recently, Grishin (J. Mol. Evol. 41:675–679, 1995) published a set of simple equations that correct for various circumstances, including different rates of change at different sites. We have used these equations in conjunction with similarity scores that take into account constraints on amino acid interchange. Simulation studies show a linear relationship between these calculated distances and the numbers of allowed mutations based on the observed variation of rate at all sites in various proteins. Received: 25 January 1996 / Accepted: 1 October 1996