A Nonparametric Test for Differences in the Dispersion of Dependent Samples

A new statistic Δ to test the hypothesis of a difference in the dispersion of two dependent samples of ordinal data quality is proposed. It draws on the idea of rank assignment originally forwarded by Siegel and Tukey (1960). No exact probability levels can be given for this statistic for the time being, but it is shown that the statistic is linearly related to the so-called Hotelling-Pabst statistic D, and that one can use exact tables of the latter as a substitute in the statistical decision process with small samples. For larger samples, an approximation of Δ to the standard normal distribution is given. The problem of tied observations is not sufficiently solved yet. A conservative procedure of rank assignment is proposed as long as the exact distribution of Δ in the presence of ties is unknown.     

A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures

The problem of testing the separability of a covariance matrix against an unstructured variance‐covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first‐order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ² distribution. The tests are implemented on a real dataset from medical studies.     

A performance test for boar taint compounds in live boars

Genetically reducing boar taint using low-taint lines is considered the most sustainable and economic long-term alternative to surgical castration of male pigs. Owing to the high heritability of the main boar taint components (androstenone, skatole and indole), breeding is an excellent tool for reducing the number of tainted carcasses. To incorporate boar taint into breeding programmes, standardized performance testing is required. The objective of this study was to develop and formally present a performance test for the main boar taint compounds on live breeding candidates. First, a standardized performance test for boar taint was established. A biopsy device was developed to extract small tissue samples (200 to 300 mg) from breeding candidates. Quantification of boar taint components from these small samples using specialized chemical extraction methods proved accurate and repeatable (r = 0.938). Following establishment of the method, biopsy samples of 516 live boars (100 to 130 kg live weight) were collected in the second step. Various mixed linear models were tested for each boar taint compound; models were ranked in terms of their information content. Pedigree information of 2245 ancestors of biopsied animals was included, and genetic parameters were estimated using univariate and multivariate models. Androstenone (in μg/g liquid fat (LF): mean = 0.578, σ = 0.527), skatole (in μg/g LF: mean = 0.033, σ = 0.002) and indole (in μg/g LF: mean = 0.032, σ = 0.002) levels obtained by biopsy were plausible. Heritability estimates for androstenone calculated with univariate (0.453) and multivariate (0.452) analyses were comparable to those in the literature. Heritabilities for skatole (0.495) and indole (0.550) were higher than that for androstenone. Genetic and phenotypic correlations were similar to those published previously. Our results show that data on boar taint compounds from small adipose samples obtained by biopsy provide similar genetic parameters as that described in the literature for larger samples and are therefore a reliable performance test for boar taint in live breeding candidates.     

Resampling-based methods in single and multiple testing for equality of covariance/correlation matrices

Traditional resampling-based tests for homogeneity in covariance matrices across multiple groups resample residuals, that is, data centered by group means. These residuals do not share the same second moments when the null hypothesis is false, which makes them difficult to use in the setting of multiple testing. An alternative approach is to resample standardized residuals, data centered by group sample means and standardized by group sample covariance matrices. This approach, however, has been observed to inflate type I error when sample size is small or data are generated from heavy-tailed distributions. We propose to improve this approach by using robust estimation for the first and second moments. We discuss two statistics: the Bartlett statistic and a statistic based on eigen-decomposition of sample covariance matrices. Both statistics can be expressed in terms of standardized errors under the null hypothesis. These methods are extended to test homogeneity in correlation matrices. Using simulation studies, we demonstrate that the robust resampling approach provides comparable or superior performance, relative to traditional approaches, for single testing and reasonable performance for multiple testing. The proposed methods are applied to data collected in an HIV vaccine trial to investigate possible determinants, including vaccine status, vaccine-induced immune response level and viral genotype, of unusual correlation pattern between HIV viral load and CD4 count in newly infected patients.     

A non-iterative confidence interval estimating procedure for the intraclass kappa statistic with multinomial outcomes

We obtain the asymptotic sample variance of the intraclass kappa statistic for multinomial outcome data. A modified Wald type procedure based on this theory is then used for confidence interval construction. The results of a simulation study show that the proposed non-iterative approach performs very well in terms of confidence interval coverage and width for samples as small as 50. The procedure is illustrated with two examples from previously published medical studies.     

The MAD legacy: How meaningful is mean age-at-death in skeletal samples

This study examines the representativeness of palaeodemographic reconstructions from human skeletal remains. Mean age-at-death (MAD) is the primary statistic used in interpretations of changing patterns of health and well being from palaeodemographic analyses. A series of sampling experiments were conducted on three documented 19th century samples representing the total cemetery population from which skeletal samples could be drawn. Comparisons of the age-at-death distributions of simulated skeletal samples to the parent population were made to assess the relative magnitude of deviation associated with different types of bias (age, sex, temporal). From the examples presented, variability in age-at-death distribution is high in samples of less than 100, suggesting that for samples of less than 100analyzable individuals, it is probable that the mortality profiles constructed are not an accurate reflection of the cemetery. It is proposed that whateverprocess mean age-at-death reflects for past populations (fertility or mortality), is irrelevant if the sample on which the statistic is calculated is not representative of the population. Given that most cemetery samples will be subject, differentially, to biases at a variety of levels, comparative studies based on palaeodemographic data cannot be considered reliablewithout careful control for those biases. It is suggested that representativeness is the primary theoretical obstacle for researches to overcome, and that it is necessary to shift our focus to rigorously exploring those factors that bias our samples. Without some direct quantification of the representativeness of a sample, palaeodemographic estimators such as mean age-at-death are meaningless and any subsequent interpretations regarding the past, dubious at best.     

Use of Ranks in Comparisons of Variability in the Non-Normal Populations

A method of analysis for comparing the variability of two samples drawn from two populations has been developed. The method is also suitable for the nonnumeric form of data. A test based on ordered observations for testing the null hypothesis of equality of two variances has been given. The test statistic is a function of the sum of ranks assigned to smaller size sample. Ranking procedure has been modified to depict the variability in the data by the sum of ranks. The null distribution of the test-statistic has been worked out for small samples and it turns out to be chi-square distribution for large samples. The analytical procedure has been explained by a numerical example on the productivity and production of rice and wheat in India from 1950–51 to 1983–84.     

Illustration of SSMD, z score, SSMD*, z* score, and t statistic for hit selection in RNAi high-throughput screens

Hit selection is the ultimate goal in many high-throughput screens. Various analytic methods are available for this purpose. Some commonly used ones are z score, z* score, strictly standardized mean difference (SSMD), SSMD*, and t statistic. It is critical to know how to use them correctly because the misusage of them can readily produce misleading results. Here, the author presents basic concepts, elaborates their commonality and difference, describes some common misusage that people should avoid, and uses simulated simple examples to illustrate how to use them correctly.     

Correlation between the different therapeutic properties of Chinese medicinal herbs and delayed luminescence

In the practice and principle of Chinese medicine, herbal materials are classified according to their therapeutic properties. ‘Cold’ and ‘heat’ are the most important classes of Chinese medicinal herbs according to the theory of traditional Chinese medicine (TCM). In this work, delayed luminescence (DL) was measured for different samples of Chinese medicinal herbs using a sensitive photon multiplier detection system. A comparison of DL parameters, including mean intensity and statistic entropy, was undertaken to discriminate between the ‘cold’ and ‘heat’ properties of Chinese medicinal herbs. The results suggest that there are significant differences in mean intensity and statistic entropy and using this method combined with statistical analysis may provide novel parameters for the characterization of Chinese medicinal herbs in relation to their energetic properties. Copyright © 2015 John Wiley & Sons, Ltd.     

Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

In geo-statistics, the Durbin-Watson test is frequently employed to detect the presence of residual serial correlation from least squares regression analyses. However, the Durbin-Watson statistic is only suitable for ordered time or spatial series. If the variables comprise cross-sectional data coming from spatial random sampling, the test will be ineffectual because the value of Durbin-Watson’s statistic depends on the sequence of data points. This paper develops two new statistics for testing serial correlation of residuals from least squares regression based on spatial samples. By analogy with the new form of Moran’s index, an autocorrelation coefficient is defined with a standardized residual vector and a normalized spatial weight matrix. Then by analogy with the Durbin-Watson statistic, two types of new serial correlation indices are constructed. As a case study, the two newly presented statistics are applied to a spatial sample of 29 China’s regions. These results show that the new spatial autocorrelation models can be used to test the serial correlation of residuals from regression analysis. In practice, the new statistics can make up for the deficiencies of the Durbin-Watson test.     

A new meta-ANOVA approach for synthesizing information under signal-heterogeneity setting with application to nuclear magnetic resonance spectroscopic data

A new synthesizing statistical methodology is proposed to resolve issues of signal-heterogeneity in data sets collected through high-resolution ¹H nuclear magnetic resonance (NMR) spectroscopy. This signal-heterogeneity is typically caused by subjective operations for processing spectral profiles and measuring peak areas, non-homogeneous biological phases of experimental subjects, and variations of systems in multi-center. All these causes are likely to simultaneously impact signals of metabolic changes and their precision in a nonlinear fashion. As a combined effect, signal-heterogeneity chiefly manifests through non-homomorphic patterns of standardized treatment mean deviations spanning all experiments, and makes most remedial statistical models with linearity structure invalid. By avoiding a huge and very complex model, we develop a simple meta-ANOVA approach to synthesize many one-way-layout ANOVA analyses from individual experiments. A scale-invariant F-ratio statistic is taken as the summarizing sufficient statistic of a non-centrality parameter that supposedly captures the information about metabolic change from each experiment. Then a joint-likelihood function of a common non-centrality is constructed as the basis for maximum likelihood estimation and Chi-square likelihood ratio testing for statistical inference. We apply the meta-ANOVA to detect metabolic changes of three metabolites identified through pattern recognition on NMR spectral profiles obtained from muscle and liver tissues. We also detect effect differences among different treatments via meta-ANOVA multiple comparison.     

Empirical comparison of distance equations using discrete traits

The use of the Grewal-Smith statistic in measuring biological distance among skeletal population samples has been questioned since it was first applied to human populations. Recently, in an attempt to stabilize the variance of the Grewal-Smith statistic for use with non-metric analysis, Sjøvold ('73) and Green and Suchey ('76) have introduced corrections and alternative transformations which may enhance the meaning of biological distance among population samples. Their recommendations improve the statistics for specific variable ranges; i.e., small sample size and low trait frequencies. Thirteen equations representing Grewal-Smith, Freeman-Tukey, Anscombe, and Bartlett transformations and/or corrections, were compared using rank order correlation statistics on actual biological distances generated by real population data as presented in existing literature. Results from testing these actual distance models show little variation between equations based on the populational data sets used. Based on these findings, the distance model resulting from the Grewal-Smith statistic is not inferior to the more sophisticated models, although the latter may be superior by allowing specific improvements for small sample size and/or low trait frequencies.     

Influence functions and robust Bayes and empirical Bayes small area estimation

We introduce new robust small area estimation procedures basedon area-level models. We first find influence functions correspondingto each individual area-level observation by measuring the divergencebetween the posterior density functions of regression coefficientswith and without that observation. Next, based on these influencefunctions, properly standardized, we propose some new robustBayes and empirical Bayes small area estimators. The mean squarederrors and estimated mean squared errors of these estimatorsare also found. A small simulation study compares the performanceof the robust and the regular empirical Bayes estimators. Whenthe model variance is larger than the sample variance, the proposedrobust empirical Bayes estimators are superior.     

Mutation and sexual selection: a test using barn swallows from Chernobyl

Abstract Secondary sexual characters have been hypothesized to be particularly susceptible to the deleterious effects of mutation because the expression of such characters is usually influenced by many more metabolic pathways than are ordinary morphological characters. We tested this hypothesis using the elevated mutation rates in the barn swallow ( Hirundo rustica ) of the Chernobyl region of Ukraine as a model system. A great deal is known about the relative importance of different characters for male mating success in this species. The importance of phenotypic characters for male mating success was quantified based on a long-term study of a Danish breeding population, by expressing phenotypic differences between mated and unmated males as the difference between log-transformed mean values. For field samples from Ukraine we likewise expressed the difference in male phenotype between individuals living in a relatively uncontaminated area and individuals from the Chernobyl region as the difference between log-transformed mean values. The standardized difference in male phenotype between the two regions in Ukraine for the 41 different characters was strongly positively correlated with the standardized difference in male phenotype between mated and unmated males from Denmark. The standardized difference in male phenotype between the two regions in Ukraine was significantly positively associated with sexual size dimorphism. However, the standardized difference in male phenotype between mated and unmated males was a much better predictor of standardized difference in male phenotype between the two regions in Ukraine than was the standardized difference in sexual size dimorphism, expressed as the difference between log-transformed mean values for males and females. These findings are consistent with the hypothesis that traits most important for sexual selection are particularly susceptible to the effects of deleterious mutations.     

Interval estimation for a difference between intraclass kappa statistics

Model-based inference procedures for the kappa statistic have developed rapidly over the last decade. However, no method has yet been developed for constructing a confidence interval about a difference between independent kappa statistics that is valid in samples of small to moderate size. In this article, we propose and evaluate two such methods based on an idea proposed by Newcombe (1998, Statistics in Medicine, 17, 873-890) for constructing a confidence interval for a difference between independent proportions. The methods are shown to provide very satisfactory results in sample sizes as small as 25 subjects per group. Sample size requirements that achieve a prespecified expected width for a confidence interval about a difference of kappa statistic are also presented.     

Use of the Gibbs Sampler to Obtain Conditional Tests, with Applications

A random sample is drawn from a distribution which admits aminimal sufficient statistic for the parameters. The Gibbs sampleris proposed to generate samples, called conditionally sufficientor co-sufficient samples, from the conditional distributionof the sample given its value of the sufficient statistic. Theprocedure is illustrated for the gamma distribution. Co-sufficientsamples may be used to give exact tests of fit; for the gammadistribution these are compared for size and power with approximatetests based on the parametric bootstrap.     

Rank Analysis of Block Designs Having Different Cell Frequencies

A distribution–free test is considered for testing the treatment effects in block designs with different cell frequencies. A test statistic which is a function of treatment ranks has been proposed which is distributed as chi-square for large samples. The null distribution of the test statistic has been obtained. The entire procedure has been explained by a numerical example.     

Der standardisierte mittelwert als wachstumsunabhängiger parameter morphologischer merkmale

Quantitative morphological data usually depend not only on shape but also on size. Therefore, a new parameter is proposed for comparison of different samples, the standardized mean, which is derived from the reduced major axis eliminating the influence of size. It is explained, how to calculate confidence interval and tests to compare two or more standardized means (Student’s t-test resp. analysis of variance).     

A new u-statistic with superior design sensitivity in matched observational studies

Summary In an observational or nonrandomized study of treatment effects, a sensitivity analysis indicates the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a naïve analysis that presumes adjustments for observed covariates suffice to remove all bias. The power of sensitivity analysis is the probability that it will reject a false hypothesis about treatment effects allowing for a departure from random assignment of a specified magnitude; in particular, if this specified magnitude is “no departure” then this is the same as the power of a randomization test in a randomized experiment. A new family of u‐statistics is proposed that includes Wilcoxon's signed rank statistic but also includes other statistics with substantially higher power when a sensitivity analysis is performed in an observational study. Wilcoxon's statistic has high power to detect small effects in large randomized experiments—that is, it often has good Pitman efficiency—but small effects are invariably sensitive to small unobserved biases. Members of this family of u‐statistics that emphasize medium to large effects can have substantially higher power in a sensitivity analysis. For example, in one situation with 250 pair differences that are Normal with expectation 1/2 and variance 1, the power of a sensitivity analysis that uses Wilcoxon's statistic is 0.08 while the power of another member of the family of u‐statistics is 0.66. The topic is examined by performing a sensitivity analysis in three observational studies, using an asymptotic measure called the design sensitivity, and by simulating power in finite samples. The three examples are drawn from epidemiology, clinical medicine, and genetic toxicology.     

A generalized nonparametric test for lattice-ordered means

Treatment means in factorial experiments are lattice ordered when there is an increase in mean response as the level of any factor is increased while holding the other factors fixed. Such means occur naturally in many experiments. A nonparametric test for lattice-ordered means involving a Kendall-type statistic will be summarized for k-factor factorial experiments. Specifically, the form of the test statistic and variance under the null hypothesis will be presented. In addition, a normalized version of the test statistic will be discussed and applied to relevant data.