A Robust Procedure for Comparing Multiple Means under Heteroscedasticity in Unbalanced Designs

Investigating differences between means of more than two groups or experimental conditions is a routine research question addressed in biology. In order to assess differences statistically, multiple comparison procedures are applied. The most prominent procedures of this type, the Dunnett and Tukey-Kramer test, control the probability of reporting at least one false positive result when the data are normally distributed and when the sample sizes and variances do not differ between groups. All three assumptions are non-realistic in biological research and any violation leads to an increased number of reported false positive results. Based on a general statistical framework for simultaneous inference and robust covariance estimators we propose a new statistical multiple comparison procedure for assessing multiple means. In contrast to the Dunnett or Tukey-Kramer tests, no assumptions regarding the distribution, sample sizes or variance homogeneity are necessary. The performance of the new procedure is assessed by means of its familywise error rate and power under different distributions. The practical merits are demonstrated by a reanalysis of fatty acid phenotypes of the bacterium Bacillus simplex from the “Evolution Canyons” I and II in Israel. The simulation results show that even under severely varying variances, the procedure controls the number of false positive findings very well. Thus, the here presented procedure works well under biologically realistic scenarios of unbalanced group sizes, non-normality and heteroscedasticity.     

A Simplified Newton Method for Computing the Factor Loadings in Maximum Likelihood Factor Analysis

A simplified Newton iteration scheme for computation of factor loadings in maximum likelihood factor analysis is described. It operates in the space of factor loadings and avoids eigenvalue problems. From this, a comparatively small computational effort results. Besides of the case of positive unique variances, also the Heywood case is considered, where some unique variances vanish. The ability of the proposed method is demonstrated in three examples, results and iteration process are compared with data given in literature.     

Intraindividual Scale Comparison in Clinical Diagnostic Methods: A Review of Elementary Methods

This text tries to summarize proposals on the statistical comparison of clinical measurement methods based on bivariate normally distributed outcome. It mainly concentrates on design adequate intraindividual comparison of variances in repeated measurements and reviews univariate graphical and numerical representations of corresponding deviations. The latter will be illustrated by the comparison of two cardiac imaging procedures in 28 probands.     

Construction and Comparison of Two Receiver Operating Characteristic Curves Derived from the Same Samples

Receiver operating characteristic (ROC) curves are used to describe the performance of diagnostic procedures. This paper proposes a simple method for the statistical comparison of two ROC curves derived from the same set of patients and the same set of healthy subjects. Generalization to studies involving more than two screening factors is straightforward. This method does not require the calculation of variances of the areas or difference of areas under the curves.     

Confidence limits for homology in protein or gene sequences. The c-myc oncogene and adenovirus E1a protein

We describe new tests, of general application, for deciding whether two proteins or DNA sequences are significantly homologous, in cases where the relationship is neither evidently true nor evidently false. Ralston and Bishop's comparison of the c-myc oncogene with the adenovirus E1a protein is discussed as an example. When the comparison matrix test is used to establish a homology between two sequences it is necessary that the number of high scores exceeds the expected mean level for random sequences by a statistically significant margin. The mean level itself is found from the double matching probability distribution. In examples where the number of high scores is larger than expected, but the highest score is not in itself exceptional, the variance of the numbers of scores expected for unrelated sequences is an important factor. We have analysed these variances by several methods. A simple binomial distribution gives only a rather inaccurate and low first estimate, but we derive a more rigorous and accurate statistical treatment, to take account of the correlations between scores in different parts of the comparison matrix. The theory is exact for random DNA or protein sequences with fluctuating compositions, selected by random draws from an infinite pool. In the more realistic situation, where sequences of fixed composition are formed by random permutations of the original sets, the deviations are smaller, and have been analysed by computer simulation. We find that although the relationship proposed by Ralston & Bishop, between the c-myc oncogene and adenovirus E1a proteins, appears to be significant in the binomial approximation, it is not supported by the full analysis. We conclude that, in general, great care is needed to establish any weak homology on the basis of comparisons that include no truly exceptional high scores, but merely have an enhanced number of scores at the upper end of the expected distribution.     

Semiparametric inference for surrogate endpoints with bivariate censored data

Summary .   Considerable attention has been recently paid to the use of surrogate endpoints in clinical research. We deal with the situation where the two endpoints are both right censored. While proportional hazards analyses are typically used for this setting, their use leads to several complications. In this article, we propose the use of the accelerated failure time model for analysis of surrogate endpoints. Based on the model, we then describe estimation and inference procedures for several measures of surrogacy. A complication is that potentially both the independent and dependent variable are subject to censoring. We adapt the Theil–Sen estimator to this problem, develop the associated asymptotic results, and propose a novel resampling-based technique for calculating the variances of the proposed estimators. The finite-sample properties of the estimation methodology are assessed using simulation studies, and the proposed procedures are applied to data from an acute myelogenous leukemia clinical trial.     

A cross-cultural study of depressive symptomology

Black (N = 26), white (N = 26) and Overseas Chinese (N = 32) college students were surveyed with Zung's Self-rating Depression Scale (SDS). With the exception of two items where marginally significant differences were found, there were no quantitative differences in the results across these ethnic groups. Inter-item analyses, factor analyses and comparison of factor profiles indicated that the items clustered in different configurations for the three groups. The factors contributing the most variances differed in their item contents across the ethnic groups: a mixture of affective and somatic complaints characterizes the black group; existential and cognitive concerns characterize the white group and somatic complaints characterize the Overseas Chinese group.Discussion was focused on issues concerning cultural conception of psychological difficulties, communication styles used to communicate these difficulties and situational variables affecting performances on psychological diagnostic tests.     

Local estimation of age-dependent variance components from longitudinal twin data

In the study of longitudinal twin and family data, interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function. Various parametric models for covariance structures have been proposed but, e.g., in studies of children where growth spurts occur at various ages, it is difficult to a priori determine an appropriate parametric model for the covariance structure. In particular, there is a general lack of the visualization procedures, such as lowess, that are invaluable in the initial stages of constructing a parametric model for a mean function. Here we use kernel smoothing to modify a cross-sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data. The methods are proposed to be exploratory as an aid to parametric modeling rather than inferential, although approximate asymptotic standard errors are derived in the Appendix.     

Multiple contrast tests in the presence of heteroscedasticity

This paper proposes a general approach for handling multiple contrast tests for normally distributed data in the presence of heteroscedasticity. Three candidate procedures are described and compared by simulations. Only the procedure with both comparison-specific degrees of freedom and a correlation matrix depending on sample variances maintains the alpha-level over all situations. Other approaches may fail notably as the variances differ more. Furthermore, related approximate simultaneous confidence intervals are given. The approach will be applied to a toxicological experiment.     

Bayesian Inference for the Ratio of the Means of Two Normal Populations with Unequal Variances

The problem of making inferences about the ratio of two normal means has been addressed, both from the frequentist and Bayesian perspectives, by several authors. Most of this work is concerned with the homoscedastic case. In contrast, the situation where the variances are not equal has received little attention. Cox (1985) deals, within the frequentist framework, with a model where the variances are related to the means. His results are mainly based on Fieller's theorem whose drawbacks are well known. In this paper we present a Bayesian analysis of this model and discuss some related problems. An agronomical example is used throughout to illustrate the methods.     

Incorporating extra information in experimental design for bioassay

In a bioassay, under certain experimental circumstances, information on concentration (dose rate) and time to response for some subjects can be combined in a single analysis. An underlying logistic random variable is assumed and the resulting mixed- (continuous-quantal) response model is analyzed by likelihood methods. The estimation procedure for the mean and the variance is described, and expressions for asymptotic variances are obtained. A comparison of results from the mixed model and from the standard quantal-response model shows that there is a substantial reduction in the variance of the estimators for the mixed model. On the basis of the table of asymptotic variances, some design implications are discussed. An example from insect pheromone research is used to illustrate the main ideas.     

Statistical methods for model discrimination. Applications to gating kinetics and permeation of the acetylcholine receptor channel.

Methods are described for discrimination of models of the gating kinetics and permeation of single ionic channels. Both maximum likelihood and regression procedures are discussed. In simple situations, where models are nested, standard hypothesis tests can be used. More commonly, however, non-nested models are of interest, and several procedures are described for model discrimination in these cases, including Monte Carlo methods, which allow the comparison of models at significance levels of choice. As an illustration, the methods are applied to single-channel data from acetylcholine receptor channels.     

Common Slope Tests for Bivariate Errors‐in‐Variables Models

Likelihood ratio tests are derived for bivariate normal structural relationships in the presence of group structure. These tests may also be applied to less restrictive models where only errors are assumed to be normally distributed. Tests for a common slope amongst those from several datasets are derived for three different cases – when the assumed ratio of error variances is the same across datasets and either known or unknown, and when the standardised major axis model is used. Estimation of the slope in the case where the ratio of error variances is unknown could be considered as a maximum likelihood grouping method. The derivations are accompanied by some small sample simulations, and the tests are applied to data arising from work on seed allometry.     

Improving data interpretation of fragmentary data-sets on invertebrate dispersal with permutation tests

Biological data often tend to have heterogeneous, discontinuous non-normal distributions. Statistical non-parametric tests, like the Mann–Whitney U-test or the extension for more than two samples, the Kruskal–Wallis test, are often used in these cases, although they assume certain preconditions which are often ignored. We developed a permutation test procedure that uses the ratio of the interquartile distances and the median differences of the original non-classified data to assess the properties of the real distribution more appropriately than the classical methods. We used this test on a heterogeneous, skewed biological data set on invertebrate dispersal and showed how different the reactions of the Kruskal–Wallis test and the permutation approach are. We then evaluated the new testing procedure with reproducible data that were generated from the normal distribution. Here, we tested the influence of four different experimental trials on the new testing procedure in comparison to the Kruskal–Wallis test. These trials showed the impact of data that were varying in terms of (a) negative correlation between variances and means of the samples, (b) changing variances that were not correlated with the means of the samples, (c) constant variances and means, but different sample sizes and in trials (d) we evaluated the testing power of the new procedure. Due to the different test statistics, the permutation test reacted more sensibly to the data presented in trials (a) and c) and non-uniformly in trial (b). In the evaluation of the testing power, no significant differences between the Kruskal–Wallis test and the new permutation testing procedure could be detected. We consider this test to be an alternative for working on heterogeneous data where the preconditions of the classical non-parametric tests are not met.     

Small-sample inference for the comparison of means of log-normal distributions

We propose a likelihood-based test for comparing the means of two or more log-normal distributions, with possibly unequal variances. A modification to the likelihood ratio test is needed when sample sizes are small. The performance of the proposed procedures is compared with the F-ratio test using Monte Carlo simulations.     

A Simple Method for Estimating Average Number of Nucleotide Substitutions within and between Populations from Restriction Data

A simple method is proposed for estimating the average number of nucleotide substitutions per site within and between populations for the case where a large number of individuals are examined for many restriction enzymes. This method gives essentially the same results as those obtained by Nei and Li's method but saves a large amount of computer time. The variances of the quantities estimated can be obtained by the jackknife method, and these variances are very similar to those obtained by Nei and Jin's more sophisticated method. A similar method can also be applied to DNA sequence data.     

Evolution of phenotypic fluctuation under host-parasite interactions

Robustness and plasticity are essential features that allow biological systems to cope with complex and variable environments. In a constant environment, robustness, i.e., insensitivity of phenotypes, is expected to increase, whereas plasticity, i.e., the changeability of phenotypes, tends to diminish. Under a variable environment, existence of plasticity will be relevant. The robustness and plasticity, on the other hand, are related to phenotypic variances. As phenotypic variances decrease with the increase in robustness to perturbations, they are expected to decrease through the evolution. However, in nature, phenotypic fluctuation is preserved to a certain degree. One possible cause for this is environmental variation, where one of the most important “environmental” factors will be inter-species interactions. As a first step toward investigating phenotypic fluctuation in response to an inter-species interaction, we present the study of a simple two-species system that comprises hosts and parasites. Hosts are expected to evolve to achieve a phenotype that optimizes fitness. Then, the robustness of the corresponding phenotype will be increased by reducing phenotypic fluctuations. Conversely, plasticity tends to evolve to avoid certain phenotypes that are attacked by parasites. By using a dynamic model of gene expression for the host, we investigate the evolution of the genotype-phenotype map and of phenotypic variances. If the host–parasite interaction is weak, the fittest phenotype of the host evolves to reduce phenotypic variances. In contrast, if there exists a sufficient degree of interaction, the phenotypic variances of hosts increase to escape parasite attacks. For the latter case, we found two strategies: if the noise in the stochastic gene expression is below a certain threshold, the phenotypic variance increases via genetic diversification, whereas above this threshold, it is increased mediated by noise-induced phenotypic fluctuation. We examine how the increase in the phenotypic variances caused by parasite interactions influences the growth rate of a single host, and observed a trade-off between the two. Our results help elucidate the roles played by noise and genetic mutations in the evolution of phenotypic fluctuation and robustness in response to host–parasite interactions.     

Tests for differentiation in gene expression using a data-driven order or weights for hypotheses

In the analysis of gene expression by microarrays there are usually few subjects, but high-dimensional data. By means of techniques, such as the theory of spherical tests or with suitable permutation tests, it is possible to sort the endpoints or to give weights to them according to specific criteria determined by the data while controlling the multiple type I error rate. The procedures developed so far are based on a sequential analysis of weighted p-values (corresponding to the endpoints), including the most extreme situation of weighting leading to a complete order of p-values. When the data for the endpoints have approximately equal variances, these procedures show good power properties. In this paper, we consider an alternative procedure, which is based on completely sorting the endpoints, but smoothed in the sense that some perturbations in the sequence of the p-values are allowed. The procedure is relatively easy to perform, but has high power under the same restrictions as for the weight-based procedures.     

Variance of Neutral Genetic Variances within and between Populations for a Quantitative Character

The variances of genetic variances within and between finite populations were systematically studied using a general multiple allele model with mutation in terms of identity by descent measures. We partitioned the genetic variances into components corresponding to genetic variances and covariances within and between loci. We also analyzed the sampling variance. Both transient and equilibrium results were derived exactly and the results can be used in diverse applications. For the genetic variance within populations, sigma 2 omega, the coefficient of variation can be very well approximated as [formula: see text] for a normal distribution of allelic effects, ignoring recurrent mutation in the absence of linkage, where m is the number of loci, N is the effective population size, theta 1(0) is the initial identity by descent measure of two genes within populations and t is the generation number. The first term is due to genic variance, the second due to linkage disequilibrium, and third due to sampling. In the short term, the variation is predominantly due to linkage disequilibrium and sampling; but in the long term it can be largely due to genic variance. At equilibrium with mutation [formula: see text] where u is the mutation rate. The genetic variance between populations is a parameter. Variance arises only among sample estimates due to finite sampling of populations and individuals. The coefficient of variation for sample gentic variance between populations, sigma 2b, can be generally approximated as [formula: see text] when the number of loci is large where S is the number of sampling populations.     

Simultaneous Selection of Extreme Populations: A Subset Selection Approach

The problem of selecting a “best” (largest mean, or smallest mean) population from a collection of k independent populations was formulated and solved by Bechhofer (1954). Gupta (1965) solved another important problem, that of selecting a subset of populations containing the “best” population from the original collection of populations. Since then many variations of the problem have been considered. Tong (1969) and Lewis (1980) have investigated the problem of selecting extreme populations (populations with a largest, and populations with a smallest, mean) with respect to one and two standard populations, respectively. In this paper we study the selection of extreme populations in absence of any standard population. We formulate subset-selection procedures when variances are known and equal, and also in the most general case when they are unknown and unequal. Nonexistence of a single-stage procedure is noted for this latter case (even if variances are equal). A two-stage procedure and some of its associated properties are discussed. Tables needed for application are provided, as is a worked example.