One-Way Classification ANOVA Model for the Mixture of Two Multivariate Normal Populations

When a sample of size n is available from a mixture of two normal populations with different mean vectors and a common covariance matrix, SRIVASTAVA and AWAN (1982) develop one-way ANOVA analysis for testing a certain composite linear hypothesis. They show that the error and hypothesis sum of products matrices have independent noncentral Wishart densities of rank unity each. However, they do not obtain the necessary Wilks' λ for testing the desired hypothesis. The present paper obtains the density of λ. This density is doubly noncentral multivariate beta density. The derivation is based on generalized Sverdrup's lemmas, KABE (1965), (1974).     

Full conjugate analysis of normal multiple traits with missing records using a generalized inverted Wishart distribution

A Markov chain Monte Carlo (MCMC) algorithm to sample an exchangeable covariance matrix, such as the one of the error terms (R₀) in a multiple trait animal model with missing records under normal-inverted Wishart priors is presented. The algorithm (FCG) is based on a conjugate form of the inverted Wishart density that avoids sampling the missing error terms. Normal prior densities are assumed for the ''fixed'' effects and breeding values, whereas the covariance matrices are assumed to follow inverted Wishart distributions. The inverted Wishart prior for the environmental covariance matrix is a product density of all patterns of missing data. The resulting MCMC scheme eliminates the correlation between the sampled missing residuals and the sampled R₀, which in turn has the effect of decreasing the total amount of samples needed to reach convergence. The use of the FCG algorithm in a multiple trait data set with an extreme pattern of missing records produced a dramatic reduction in the size of the autocorrelations among samples for all lags from 1 to 50, and this increased the effective sample size from 2.5 to 7 times and reduced the number of samples needed to attain convergence, when compared with the ''data augmentation'' algorithm.     

Estimating correlations among demographic parameters in population models

Estimating correlations among demographic parameters is critical to understanding population dynamics and life‐history evolution, where correlations among parameters can inform our understanding of life‐history trade‐offs, result in effective applied conservation actions, and shed light on evolutionary ecology. The most common approaches rely on the multivariate normal distribution, and its conjugate inverse Wishart prior distribution. However, the inverse Wishart prior for the covariance matrix of multivariate normal distributions has a strong influence on posterior distributions. As an alternative to the inverse Wishart distribution, we individually parameterize the covariance matrix of a multivariate normal distribution to accurately estimate variances (σ²) of, and process correlations (ρ) between, demographic parameters. We evaluate this approach using simulated capture–mark–recapture data. We then use this method to examine process correlations between adult and juvenile survival of black brent geese marked on the Yukon–Kuskokwim River Delta, Alaska (1988–2014). Our parameterization consistently outperformed the conjugate inverse Wishart prior for simulated data, where the means of posterior distributions estimated using an inverse Wishart prior were substantially different from the values used to simulate the data. Brent adult and juvenile annual apparent survival rates were strongly positively correlated (ρ = 0.563, 95% CRI 0.181–0.823), suggesting that habitat conditions have significant effects on both adult and juvenile survival. We provide robust simulation tools, and our methods can readily be expanded for use in other capture–recapture or capture‐recovery frameworks. Further, our work reveals limits on the utility of these approaches when study duration or sample sizes are small.     

The S-Distribution A Tool for Approximation and Classification of Univariate,Unimodal Probability Distributions

In many statistical applications a data set needs to be evaluated but there is no solid information about which probability distribution might be most appropriate. Typical solutions to this problems are: to make assumptions that facilitate mathematical treatment; to use a family of distribution functions that contains all relevant distributions as special cases; or, to employ nonparametric methods. All three solutions have disadvantages since assumptions are usually difficult to justify, families of distributions contain too many parameters to be of practical use, and nonparametric methods make it difficult to characterize data in a succinct quantitative form. The S-distribution introduced here is a compromise between the conflicting goals of simplicity in analysis and generality in scope. It is characterized by four parameters, one of which reflects its location, the second one its spread, and the remaining two its shape; transformation to a standard form reduces the number of free parameters to two. Cumulatives and densities are computed numerically in fractions of seconds, key features like quantiles and moments are easily obtained, and results can be presented in terms of parameter values. The S-distribution rather accurately models different distribution functions, including central and noncentral distributions, and thus competes in flexibility with some distribution families. As an approximation, the S-distribution provides a graphical method for demonstrating relationships between distributions, such as the relationships between central F, χ² and χ^?2 or central and noncentral t, χ^-1, and normal.     

Validity of the F-ratio Test for Replicated Latin Square Designs Under Relaxed Conditions

In a replicated latin square design, the F-ratio test statistic enjoys a noncentral Fisher distribution under the assumption of i.i.d. normal errors. This paper shows that the latter distribution still holds if a certain correlation structure is introduced within the sets of errors associated with each latin square. The correlation structure is characterized by the fact that, put in form of matrices, these sets of errors have exchangeable rows and exchangeable columns.     

On power and sample size calculations for likelihood ratio tests in generalized linear models

A direct extension of the approach described in Self, Mauritsen, and Ohara (1992, Biometrics 48, 31-39) for power and sample size calculations in generalized linear models is presented. The major feature of the proposed approach is that the modification accommodates both a finite and an infinite number of covariate configurations. Furthermore, for the approximation of the noncentrality of the noncentral chi-square distribution for the likelihood ratio statistic, a simplification is provided that not only reduces substantial computation but also maintains the accuracy. Simulation studies are conducted to assess the accuracy for various model configurations and covariate distributions.     

Genetic code,hamming distance and stochastic matrices

In this paper we use the Gray code representation of the genetic code C = 00, U = 10, G = 11 and A = 01 (C pairs with G, A pairs with U) to generate a sequence of genetic code-based matrices. In connection with these code-based matrices, we use the Hamming distance to generate a sequence of numerical matrices. We then further investigate the properties of the numerical matrices and show that they are doubly stochastic and symmetric. We determine the frequency distributions of the Hamming distances, building blocks of the matrices, decomposition and iterations of matrices. We present an explicit decomposition formula for the genetic code-based matrix in terms of permutation matrices, which provides a hypercube representation of the genetic code. It is also observed that there is a Hamiltonian cycle in a genetic code-based hypercube.     

Multivariate Bayesian analysis of Gaussian,right censored Gaussian,ordered categorical and binary traits using Gibbs sampling

A fully Bayesian analysis using Gibbs sampling and data augmentation in a multivariate model of Gaussian, right censored, and grouped Gaussian traits is described. The grouped Gaussian traits are either ordered categorical traits (with more than two categories) or binary traits, where the grouping is determined via thresholds on the underlying Gaussian scale, the liability scale. Allowances are made for unequal models, unknown covariance matrices and missing data. Having outlined the theory, strategies for implementation are reviewed. These include joint sampling of location parameters; efficient sampling from the fully conditional posterior distribution of augmented data, a multivariate truncated normal distribution; and sampling from the conditional inverse Wishart distribution, the fully conditional posterior distribution of the residual covariance matrix. Finally, a simulated dataset was analysed to illustrate the methodology. This paper concentrates on a model where residuals associated with liabilities of the binary traits are assumed to be independent. A Bayesian analysis using Gibbs sampling is outlined for the model where this assumption is relaxed.     

Use of Wishart Prior and Simple Extensions for Sparse Precision Matrix Estimation

A conjugate Wishart prior is used to present a simple and rapid procedure for computing the analytic posterior (mode and uncertainty) of the precision matrix elements of a Gaussian distribution. An interpretation of covariance estimates in terms of eigenvalues is presented, along with a simple decision-rule step to improve the performance of the estimation of sparse precision matrices and associated graphs. In this, elements of the estimated precision matrix that are zero or near zero can be detected and shrunk to zero. Simulated data sets are used to compare posterior estimation with decision-rule with two other Wishart-based approaches and with graphical lasso. Furthermore, an empirical Bayes procedure is used to select prior hyperparameters in high dimensional cases with extension to sparsity.     

Phenotypic Variation in Sexual and Somatic Morphology in the Trapdoor Spider Misgolas hubbardi Wishart in Relation to its Genotypic Variation (Mygalomorphae: Idiopidae)

Spiders from 13 widely separated populations of Misgolas hubbardi Wishart are shown to exhibit distinct phenotypic and genotypic differences in the first such study on ground dwelling spiders in Australia. Electrophoretic analysis also supports the conclusion that the syntopic and morphologically constituted M. hubbardi Wishart, M. dereki Wishart and M. robertsi (Main and Mascord) are valid species. While M. hubbardi can be reliably determined from male palpal characters, the shape of the embolic apophysis and the ventum pigmentation pattern are variable and valid diagnostic characters only for particular populations and where the locality is known. the morphological differences in M. hubbardi populations are summarised and its distribution compared to the geology of the region.     

A Variation of the Method of Translation to Generate Bivariate and Multivariate Survival Distributions with Competing Risk or Reliability Applications

In the present paper, a variation of the methode of translation is discussed to generate bivariate or multivariate survival distributions starting a given bivariate or multivariate distribution which is not necessarily a life distribution. The new distribution has been called the dual of the given distribution. The duals of several bivariate and multivariate famílies of distributions are obtained, such as FRECHET'S , FARLIE -GUMBEL -MORGENSTERN'S , MARDIA'S and PLACKETT'S , among others. A necessary and sufficient condition is given for a bivariate distribution to be its own dual. Thus the present paper generates several survival distributions in addition to what are already available in the literature. These have important applications in competing risk theory or reliability of engineering systems.     

Detection of two-component mixtures of lognormal distributions in grouped, doubly truncated data: analysis of red blood cell volume distributions

We have examined the statistical requirements for the detection of mixtures of two lognormal distributions in doubly truncated data when the sample size is large. The expectation-maximization algorithm was used for parameter estimation. A bootstrap approach was used to test for a mixture of distributions using the likelihood ratio statistic. Analysis of computer simulated mixtures showed that as the ratio of the difference between the means to the minimum standard deviation increases, the power for detection also increases and the accuracy of parameter estimates improves. These procedures were used to examine the distribution of red blood cell volume in blood samples. Each distribution was doubly truncated to eliminate artifactual frequency counts and tested for best fit to a single lognormal distribution or a mixture of two lognormal distributions. A single population was found in samples obtained from 60 healthy individuals. Two subpopulations of cells were detected in 25 of 27 mixtures of blood prepared in vitro. Analyses of mixtures of blood from 40 patients treated for iron-deficiency anemia showed that subpopulations could be detected in all by 6 weeks after onset of treatment. To determine if two-component mixtures could be detected, distributions were examined from untransfused patients with refractory anemia. In two patients with inherited sideroblastic anemia a mixture of microcytic and normocytic cells was found, while in the third patient a single population of microcytic cells was identified. In two family members previously identified as carriers of inherited sideroblastic anemia, mixtures of microcytic and normocytic subpopulations were found. Twenty-five patients with acquired myelodysplastic anemia were examined. A good fit to a mixture of subpopulations containing abnormal microcytic or macrocytic cells was found in two. We have demonstrated that with large sample sizes, mixtures of distributions can be detected even when distributions appear to be unimodal. These statistical techniques provide a means to characterize and quantify alterations in erythrocyte subpopulations in anemia but could also be applied to any set of grouped, doubly truncated data to test for the presence of a mixture of two lognormal distributions.     

Fitting mixture models to grouped and truncated data via the EM algorithm

The fitting of finite mixture models via the EM algorithm is considered for data which are available only in grouped form and which may also be truncated. A practical example is presented where a mixture of two doubly truncated log-normal distributions is adopted to model the distribution of the volume of red blood cells in cows during recovery from anemia.     

The stability–complexity relationship at age 40: a random matrix perspective

Since the work of Robert May in 1972, the local asymptotic stability of large ecological systems has been a focus of theoretical ecology. Here we review May's work in the light of random matrix theory, the field of mathematics devoted to the study of large matrices whose coefficients are randomly sampled from distributions with given characteristics. We show how May's celebrated “stability criterion” can be derived using random matrix theory, and how extensions of the so-called circular law for the limiting distribution of the eigenvalues of large random matrix can further our understanding of ecological systems. Our goal is to present the more technical material in an accessible way, and to provide pointers to the primary mathematical literature on this subject. We conclude by enumerating a number of challenges, whose solution is going to greatly improve our ability to predict the stability of large ecological networks.     

Spherical Deconvolution of Multichannel Diffusion MRI Data with Non-Gaussian Noise Models and Spatial Regularization

Spherical deconvolution (SD) methods are widely used to estimate the intra-voxel white-matter fiber orientations from diffusion MRI data. However, while some of these methods assume a zero-mean Gaussian distribution for the underlying noise, its real distribution is known to be non-Gaussian and to depend on many factors such as the number of coils and the methodology used to combine multichannel MRI signals. Indeed, the two prevailing methods for multichannel signal combination lead to noise patterns better described by Rician and noncentral Chi distributions. Here we develop a Robust and Unbiased Model-BAsed Spherical Deconvolution (RUMBA-SD) technique, intended to deal with realistic MRI noise, based on a Richardson-Lucy (RL) algorithm adapted to Rician and noncentral Chi likelihood models. To quantify the benefits of using proper noise models, RUMBA-SD was compared with dRL-SD, a well-established method based on the RL algorithm for Gaussian noise. Another aim of the study was to quantify the impact of including a total variation (TV) spatial regularization term in the estimation framework. To do this, we developed TV spatially-regularized versions of both RUMBA-SD and dRL-SD algorithms. The evaluation was performed by comparing various quality metrics on 132 three-dimensional synthetic phantoms involving different inter-fiber angles and volume fractions, which were contaminated with noise mimicking patterns generated by data processing in multichannel scanners. The results demonstrate that the inclusion of proper likelihood models leads to an increased ability to resolve fiber crossings with smaller inter-fiber angles and to better detect non-dominant fibers. The inclusion of TV regularization dramatically improved the resolution power of both techniques. The above findings were also verified in human brain data.     

Simulation of hyper-inverse Wishart distributions in graphical models

We introduce and exemplify an efficient method for direct samplingfrom hyper-inverse Wishart distributions. The method reliesvery naturally on the use of standard junction-tree representationof graphs, and couples these with matrix results for inverseWishart distributions. We describe the theory and resultingcomputational algorithms for both decomposable and nondecomposablegraphical models. An example drawn from financial time seriesdemonstrates application in a context where inferences on astructured covariance model are required. We discuss and investigatequestions of scalability of the simulation methods to higher-dimensionaldistributions. The paper concludes with general comments aboutthe approach, including its use in connection with existingMarkov chain Monte Carlo methods that deal with uncertaintyabout the graphical model structure.     

Sex Differences in Task Distribution and Task Exposures among Danish House Painters: An Observational Study Combining Questionnaire Data with Biomechanical Measurements

Objectives

Sex differences in occupational biomechanical exposures may be part of the explanation why musculoskeletal complaints and disorders tend to be more common among women than among men. We aimed to determine possible sex differences in task distribution and task-specific postures and movements of the upper extremities among Danish house painters, and to establish sex-specific task exposure matrices.

Methods

To obtain task distributions, we sent out a questionnaire to all members of the Painters'' Union in Denmark (N = 9364), of whom 53% responded. Respondents reported their task distributions in a typical week. To obtain task exposures, postures and movements were measured in 25 male and 25 female house painters for one whole working day per person. We used goniometers on the wrists, and inclinometers on the forehead and the upper arms. Participants filled in a logbook allowing task-specific exposures to be identified. Percentiles and % time with non-neutral postures were used to characterise postures. Velocity, range of motion, repetitiveness, and variation were used as measures of movement. Cochran-Mantel-Haenszel statistics and unpaired double-sided t-tests with post-hoc Bonferroni correction were used to evaluate sex differences.

Results

Statistically significant (p<0.05) sex differences were revealed in task proportions, but the proportions differed by less than 4%. For task exposures, no statistically significant sex differences were found.

Conclusions

Only minor sex differences were found in task distribution and task exposures regarding postures and movements among Danish house painters. Sex-specific task exposure matrices were established.     

Predicting fundamental and realized distributions based on thermal niche: A case study of a freshwater turtle

Species distribution models (SDM) have been broadly used in ecology to address theoretical and practical problems. Currently, there are two main approaches to generate SDMs: (i) correlative, which is based on species occurrences and environmental predictor layers and (ii) process-based models, which are constructed based on species' functional traits and physiological tolerances. The distributions estimated by each approach are based on different components of species niche. Predictions of correlative models approach species realized niches, while predictions of process-based are more akin to species fundamental niche. Here, we integrated the predictions of fundamental and realized distributions of the freshwater turtle Trachemys dorbigni. Fundamental distribution was estimated using data of T. dorbigni's egg incubation temperature, and realized distribution was estimated using species occurrence records. Both types of distributions were estimated using the same regression approaches (logistic regression and support vector machines), both considering macroclimatic and microclimatic temperatures. The realized distribution of T. dorbigni was generally nested in its fundamental distribution reinforcing theoretical assumptions that the species' realized niche is a subset of its fundamental niche. Both modelling algorithms produced similar results but microtemperature generated better results than macrotemperature for the incubation model. Finally, our results reinforce the conclusion that species realized distributions are constrained by other factors other than just thermal tolerances.     

On statistical methods for comparison of intrasample morphometric variability: Zalavár revisited

In studies of morphology, methods for comparing amounts of variability are often important. Three different ways of utilizing determinants of covariance matrices for testing for surplus variability in a hypothesis sample compared to a reference sample are presented: an F-test based on standardized generalized variances, a parametric bootstrap based on draws on Wishart matrices, and a nonparametric bootstrap. The F-test based on standardized generalized variances and the Wishart-based bootstrap are applicable when multivariate normality can be assumed. These methods can be applied with only summary data available. However, the nonparametric bootstrap can be applied with multivariate nonnormally distributed data as well as multivariate normally distributed data, and small sample sizes. Therefore, this method is preferable when raw data are available. Three craniometric samples are used to present the methods. A Hungarian Zalavár sample and an Austrian Berg sample are compared to a Norwegian Oslo sample, the latter employed as reference sample. In agreement with a previous study, it is shown that the Zalavár sample does not represent surplus variability, whereas the Berg sample does represent such a surplus variability.     

Evidence of Commingling in Human Eating Behavior

This investigation tested whether distributions of certain aspects of eating behavior were consistent with the notion of a “mixture model;” that is, two or more distinct Commingled component distributions, consistent with the possibility of major gene action. Undergraduates (n=901) completed self-report trait measures of hunger, disinhibition, and dietary restraint. Variables were residualized for gender and age and transformed to remove skewness. Residualized transformed distributions were tested for departure from unimodality with Hartigan's (14) dip statistic. The distributions of all three aspects of eating behavior were significantly non-unimodal. Next, component multivariate normal distributions were estimated via maximum likelihood. Likelihood ratio tests were employed to compare nested models. A mixture of four distributions with unequal variance-covariance matrices tit significantly better than any more parsimonious model. In sum, these data strongly suggest that the distributions of several measures of eating behavior are composed of four component distributions. This finding is consistent with the possibility of major gene effects for eating behavior.