The effective size of fluctuating populations

We consider a Wright-Fisher model whose population size is an autocorrelated stochastic process. Our interest is in the effects of autocorrelated fluctuations of the population size on the effective size. We define the inbreeding effective size and the variance effective size and show that these effective sizes are the same for this model. In the literature, it is said that the effective size is equal to the harmonic mean of population size when the size fluctuates. We will show, however, that the effective size is not the same as the harmonic mean of population size unless the fluctuations of population size are uncorrelated. The effective size is larger (resp. smaller) than the harmonic mean when the fluctuations of population size are positively (resp. negatively) autocorrelated. Further, we obtain some asymptotic expressions for effective size when the population size is very large and/or the autocorrelation of the fluctuation is very strong.     

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.     

Red-shifts and red herrings in geographical ecology

I draw attention to the need for ecologists to take spatial structure into account more seriously in hypothesis testing. If spatial autocorrelation is ignored, as it usually is, then analyses of ecological patterns in terms of environmental factors can produce very misleading results. This is demonstrated using synthetic but realistic spatial patterns with known spatial properties which are subjected to classical correlation and multiple regression analyses. Correlation between an autocorrelated response variable and each of a set of explanatory variables is strongly biased in favour of those explanatory variables that are highly autocorrelated - the expected magnitude of the correlation coefficient increases with autocorrelation even if the spatial patterns are completely independent. Similarly, multiple regression analysis finds highly autocorrelated explanatory variables "significant" much more frequently than it should. The chances of mistakenly identifying a "significant" slope across an autocorrelated pattern is very high if classical regression is used. Consequently, under these circumstances strongly autocorrelated environmental factors reported in the literature as associated with ecological patterns may not actually be significant. It is likely that these factors wrongly described as important constitute a red-shifted subset of the set of potential explanations, and that more spatially discontinuous factors (those with bluer spectra) are actually relatively more important than their present status suggests. There is much that ecologists can do to improve on this situation. I discuss various approaches to the problem of spatial autocorrelation from the literature and present a randomisation test for the association of two spatial patterns which has advantages over currently available methods.     

Testing environmental and genetic effects in the presence of spatial autocorrelation

Spatial autocorrelation is a well‐recognized concern for observational data in general, and more specifically for spatial data in ecology. Generalized linear mixed models (GLMMs) with spatially autocorrelated random effects are a potential general framework for handling these spatial correlations. However, as the result of statistical and practical issues, such GLMMs have been fitted through the undocumented use of procedures based on penalized quasi‐likelihood approximations (PQL), and under restrictive models of spatial correlation. Alternatively, they are often neglected in favor of simpler but more questionable approaches. In this work we aim to provide practical and validated means of inference under spatial GLMMs, that overcome these limitations. For this purpose, a new software is developed to fit spatial GLMMs. We use it to assess the performance of likelihood ratio tests for fixed effects under spatial autocorrelation, based on Laplace or PQL approximations of the likelihood. Expectedly, the Laplace approximation performs generally slightly better, although a variant of PQL was better in the binary case. We show that a previous implementation of PQL methods in the R language, glmmPQL, is not appropriate for such applications. Finally, we illustrate the efficiency of a bootstrap procedure for correcting the small sample bias of the tests, which applies also to non‐spatial models.     

The Determination of Sample Sizes in the Comparison of Two Multinomial Proportions from Ordered Categories

We consider sample size determination for ordered categorical data when the alternative assumption is the proportional odds model. In this paper the sample size formula proposed by Whitehead (Statistics in Medicine, 12 , 2257–2271, 1993) is compared with the methods based on exact and asymptotic linear rank tests with Wilcoxon and trend scores. We show that Whitehead's formula, which is based on a normal approximation, works well when the sample size is moderate to large but recommend the exact method with Wilcoxon scores for small sample sizes. The consequences of misspecification in models are also investigated.     

Random sampling does not exclude spatial dependence: The importance of neutral models for ecological hypothesis testing

Lájer (2007) notes that, to investigate phytosociological and ecological relationships, many authors apply traditional inferential tests to sets of relevés obtained by non-random methods. Unfortunately, this procedure does not provide reliable support for hypothesis testing because non-random sampling violates the assumptions of independence required by many parametric inferential tests. Instead, a random sampling scheme is recommended. Nonetheless, random sampling will not eliminate spatial autocorrelation. For instance, a classical law of geography holds that everything in a piece of (biotic) space is interrelated, but near objects are more related than distant ones. Because most ecological processes that shape community structure and species coexistence are spatially explicit, spatial autocorrelation is a vital part of almost all ecological data. This means that, independently from the underlying sampling design, ecological data are generally spatially autocorrelated, violating the assumption of independence that is generally required by traditional inferential tests. To overcome this drawback, randomization tests may be used. Such tests evaluate statistical significance based on empirical distributions generated from the sample and do not necessarily require data independence. However, as concerns hypothesis testing, randomization tests are not the universal remedy for ecologists, because the choice of inadequate null models can have significant effects on the ecological hypotheses tested. In this paper, I emphasize the need of developing null models for which the statistical assumptions match the underlying biological mechanisms.     

Environmental noise and population dynamics of the ciliated protozoa Tetrahymena thermophila in aquatic microcosms

Population theory predicts that the reddened environmental noise, especially in combination with high population growth rate, reddens population dynamics, increases population variability and strengthens environment–population correlation. We tested these predictions with axenic populations of ciliated protozoa Tetrahymena thermophila. Populations with low and high growth rate were cultured in a stable environment, and in environments with sublethal temperature fluctuations that had blue, white and red spectra (i.e. negatively autocorrelated, uncorrelated, or positively autocorrelated, respectively). Population size and biomass of individuals were determined at 3-h intervals for 18 days.
Dynamics of all populations were reddened, suggesting that internal mechanisms can redden the population spectra. However, population dynamics were reddest, variability highest, and environment–population correlation strongest in the red environment as predicted. Contrary to theoretical predictions and previous empirical findings, population growth rate (r_max being equal to 0.05 and 0.3 h⁻¹) had no effect on population dynamics.
Mean cell size and variability of cell size were affected by the presence and type of environmental noise suggesting that the physiological consequences of variability depend on colour. Environmental variability decreased mean population size and biomass and the decrease was strongest in rapidly fluctuating blue and white environments. The latter finding implies that rapid fluctuations are physiologically stressful, an effect that is not accounted for in the basic population models.     

Taxon sampling, correlated evolution, and independent contrasts

Independent contrasts are widely used to incorporate phylogenetic information into studies of continuous traits, particularly analyses of evolutionary trait correlations, but the effects of taxon sampling on these analyses have received little attention. In this paper, simulations were used to investigate the effects of taxon sampling patterns and alternative branch length assignments on the statistical performance of correlation coefficients and sign tests; "full-tree" analyses based on contrasts at all nodes and "paired-comparisons" based only on contrasts of terminal taxon pairs were also compared. The simulations showed that random samples, with respect to the traits under consideration, provide statistically robust estimates of trait correlations. However, exact significance tests are highly dependent on appropriate branch length information; equal branch lengths maintain lower Type I error than alternative topological approaches, and adjusted critical values of the independent contrast correlation coefficient are provided for use with equal branch lengths. Nonrandom samples, with respect to univariate or bivariate trait distributions, introduce discrepancies between interspecific and phylogenetically structured analyses and bias estimates of underlying evolutionary correlations. Examples of nonrandom sampling processes may include community assembly processes, convergent evolution under local adaptive pressures, selection of a nonrandom sample of species from a habitat or life-history group, or investigator bias. Correlation analyses based on species pairs comparisons, while ignoring deeper relationships, entail significant loss of statistical power and as a result provide a conservative test of trait associations. Paired comparisons in which species differ by a large amount in one trait, a method introduced in comparative plant ecology, have appropriate Type I error rates and high statistical power, but do not correctly estimate the magnitude of trait correlations. Sign tests, based on full-tree or paired-comparison approaches, are highly reliable across a wide range of sampling scenarios, in terms of Type I error rates, but have very low power. These results provide guidance for selecting species and applying comparative methods to optimize the performance of statistical tests of trait associations.     

3D analysis from micro-MRI during in situ compression on cancellous bone

A mini-compression jig was built to perform in situ tests on bovine trabecular bone monitored by micro-MRI. The MRI antenna provided an isotropic resolution of 78 μm that allows for a volume correlation method to be used. Three-dimensional displacement fields are then evaluated within the bone sample during the compression test. The performances of the correlation method are evaluated and discussed to validate the technique on trabecular bone. By considering correlation residuals and estimates of acquisition noise, the measured results are shown to be trustworthy. By analyzing average strain levels for different interrogation volumes along the loading direction, it is shown that the sample size is less than that of a representative volume element. This study shows the feasibility of the 3D-displacement and strain field analyses from micro-MRI images. Other biological tissues could be considered in future work.     

Morphometric testing of structural hypotheses of the supraorbital region in modern humans

Our understanding of the functional morphology of the primate supraorbital region is based largely on previous morphometric and in vivo mechanical tests of hypotheses in non-human anthropoids. Prior tests of two structural hypotheses explaining morphological variation in the supraorbital region, the craniofacial size hypothesis and the spatial hypothesis, did not fully consider modern humans. We extend these previous findings to include modern humans by conducting morphometric tests of these two hypotheses in a sample of adult Melanesian crania. Morphometric correlates of structural predictions for the craniofacial size and spatial hypotheses were developed and compared to measurements of the supraorbital region via bivariate product-moment correlations. Measurements of the supraorbital region are significantly correlated with a craniofacial size estimate across individuals from this Melanesian sample. This result supports the prediction of the craniofacial size hypothesis that the magnitude of the supraorbital region is proportional to craniofacial size. The predicted link between the degree of neural-orbital disjunction and the magnitude of the supraorbital region, explicated in the spatial hypothesis, receives mixed support in the correlation analysis. These two results agree with previous research indicating that support for the craniofacial size and spatial hypotheses can be found across and within anthropoid primate species, including modern humans. Correlational support for both the craniofacial size and spatial hypotheses suggests multiple factors influence variation in the modern human supraorbital region. Thus, a single hypothesis cannot fully account for modern human variation in this region. The low bivariate correlation coefficients in this study further question whether existing hypotheses can adequately explain morphological variation in the supraorbital region in a primate population sample. Novel functional, structural, behavioral and developmental ideas must be explored if we are to better understand morphological variation in the modern human supraorbital region.     

Chisquare Tests From Clusters of Observations

A simple model for categorical observations that are correlated in clusters is developed. The implied correlation coefficient is used to define an effective number of observations, and a natural estimation procedure for the correlation coefficient is proposed. By grouping the clusters of observations, and by relating them to models of the probabilities for the implied categories, common chisquare tests are formulated and motivated asymptotically. As an example of applications we look at forage registrations for radio-collared moose.     

Homogeneity score test for the intraclass version of the kappa statistics and sample-size determination in multiple or stratified studies

When the intraclass correlation coefficient or the equivalent version of the kappa agreement coefficient have been estimated from several independent studies or from a stratified study, we have the problem of comparing the kappa statistics and combining the information regarding the kappa statistics in a common kappa when the assumption of homogeneity of kappa coefficients holds. In this article, using the likelihood score theory extended to nuisance parameters (Tarone, 1988, Communications in Statistics-Theory and Methods 17(5), 1549-1556) we present an efficient homogeneity test for comparing several independent kappa statistics and, also, give a modified homogeneity score method using a noniterative and consistent estimator as an alternative. We provide the sample size using the modified homogeneity score method and compare it with that using the goodness-of-fit method (GOF) (Donner, Eliasziw, and Klar, 1996, Biometrics 52, 176-183). A simulation study for small and moderate sample sizes showed that the actual level of the homogeneity score test using the maximum likelihood estimators (MLEs) of parameters is satisfactorily close to the nominal and it is smaller than those of the modified homogeneity score and the goodness-of-fit tests. We investigated statistical properties of several noniterative estimators of a common kappa. The estimator (Donner et al., 1996) is essentially efficient and can be used as an alternative to the iterative MLE. An efficient interval estimation of a common kappa using the likelihood score method is presented.     

Effects of age and distance on the composition of mixed deciduous forest fragments in an agricultural landscape

Abstract. Forest patches in central Belgium were inventoried twice for the presence or absence of forest plant species to study the effects of age and distance on species composition. All forests in the study area were subdivided based on their land use history. To avoid effects of autocorrelated environmental characteristics on species composition, habitat homogeneity was indirectly investigated using a TWINSPAN classification of the vegetation data. Two major habitats (alluvial and non‐alluvial forests) were distinguished and analysed separately. Patterns of species composition were investigated at the landscape level using Mantel tests. Species composition similarity measures were calculated between all pairs of fragments based on the floristic data. A highly significant correlation was found between species composition similarity and inter‐patch distance. Difference in age, which we used as a measure for habitat quality, was less important in explaining species composition patterns. The effects of spatial configuration became significant when difference in age was accounted for, and a partial correlation was calculated between inter‐patch distance and species composition similarity. Different results were found for alluvial and non‐alluvial forest types. Alluvial forests were more influenced by the spatial configuration than the non‐alluvial. For the non‐alluvial forest type effects measured with the difference in age between forest fragments obscured the effects of inter‐patch distance. Based on our findings we suggest that species composition is not only internally generated, but external processes such as differential colonization caused by varying degrees of isolation may be of overriding importance.     

Estimating the correlation of pairwise relatedness along chromosomes

The 'spatial' pattern of the correlation of pairwise relatedness among loci within a chromosome is an important aspect for an insight into genomic evolution in natural populations. In this article, a statistical genetic method is presented for estimating the correlation of pairwise relatedness among linked loci. The probabilities of identity-in-state (IIS) are related to the probabilities of identity-by-descent (IBS) for the two- and three-loci cases. By decomposing the joint probabilities of two- or three-loci IBD, the probability of pairwise relatedness at a single locus and its correlation among linked loci can be simultaneously estimated. To provide effective statistical methods for estimation, weighted least square (LS) and maximum likelihood (ML) methods are evaluated through extensive Monte Carlo simulations. Results show that the ML method gives a better performance than the weighted LS method with haploid genotypic data. However, there are no significant differences between the two methods when two- or three-loci diploid genotypic data are employed. Compared with the optimal size for haploid genotypic data, a smaller optimal sample size is predicted with diploid genotypic data.     

A general approach to sample size determination for prevalence surveys that use dual test protocols

We develop a Bayesian simulation based approach for determining the sample size required for estimating a binomial probability and the difference between two binomial probabilities where we allow for dependence between two fallible diagnostic procedures. Examples include estimating the prevalence of disease in a single population based on results from two imperfect diagnostic tests applied to sampled individuals, or surveys designed to compare the prevalences of two populations using diagnostic outcomes that are subject to misclassification. We propose a two stage procedure in which the tests are initially assumed to be independent conditional on true disease status (i.e. conditionally independent). An interval based sample size determination scheme is performed under this assumption and data are collected and used to test the conditional independence assumption. If the data reveal the diagnostic tests to be conditionally dependent, structure is added to the model to account for dependence and the sample size routine is repeated in order to properly satisfy the criterion under the correct model. We also examine the impact on required sample size when adding an extra heterogeneous population to a study.     

A comparison of confidence interval methods for the intraclass correlation coefficient

Different methods of obtaining confidence intervals for the intraclass correlation coefficient rho in the unbalanced one-way random-effects model are investigated, focusing on applications to family studies. Methods based on simple modifications of formulas for the case of equal group sizes are found to provide adequate coverage at small to moderate values of rho. A method based on the large-sample standard error of the sample intraclass correlation, as derived by Smith (1956, Annals of Human Genetics 21, 363-373), is shown to provide consistently good coverage at all values of rho. A method proposed by Thomas and Hultquist (1978, Annals of Statistics 6, 582-587) also provides consistently good coverage, but generates mean interval widths substantially greater than those generated by Smith's method at values of rho likely to arise in practice.     

An attempt to define 1qh+, 9qh+, and 16qh+

Summary The lengths of the secondary constrictions of chromosomes 1, 9, and 16 vary with the degree of contraction of the chromosomes but these constrictions contract to a lesser degree than the euchromatic portions of the chromosomes. The regression coefficient for the regression of the length of the secondary constriction on the length of the euchromatic part of the chromosomes is shown to be larger for large constrictions. It is furthermore shown that there is a linear correlation between the regression coefficient and the size of the secondary constriction in question. This linear correlation makes it possible to correct the lengths of the secondary constrictions to the lengths expected when contraction is average. The correction method is used in a sample of 30 couples, and on the basis of this sample, the normal limits for the lengths of the secondary constrictions in chromosomes 1, 9, and 16 are defined.     

Importance of Reproducibility in Responsiveness Issues

In cohort studies, longitudinal changes in individuals are often studied. The sample size requirements are then directly related to the responsiveness of the measuring instrument/scale, which is defined as the ability to detect a clinically meaningful change over time. We demonstrate that whatever the index used to assess responsiveness (effect size, standardized response mean, Guyatt's index), it is enhanced by an increase in the intraclass correlation coefficient (ICC), which assesses the reproducibility of the measuring instrument/scale. Therefore, an improvement in reproducibility leads to an increase in responsiveness and thus an increase in power. In the special case of one observation per subject and time, we also prove that an increase in the ICC reduces the coefficient of variation of the estimators of the standardized response mean and of Guyatt's index. This result has also been observed for the effect size on an example. Such a result implies that the better the reproducibility, the lower the sample size requirements for confident statistical inference of responsiveness results.     

Effective population size of a population with stochastically varying size

For a Wright–Fisher model with mutation whose population size fluctuates stochastically from generation to generation, a heterozygosity effective population size is defined by means of the equilibrium average heterozygosity of the population. It is shown that this effective population size is equal to the harmonic mean of population size if and only if the stochastic changes of population size are uncorrelated. The effective population size is larger (resp. smaller) than the harmonic mean when the stochastic changes of population size are positively (resp. negatively) autocorrelated. These results and those obtained so far for other stochastic models with fluctuating population size suggest that the property that effective population sizes are always larger than the harmonic mean under the fluctuation of population size holds only for continuous time models such as diffusion and coalescent models, whereas effective population sizes can be equal to or smaller than the harmonic mean for discrete time models.