Multivariate Variance Components Model Research Articles

Individuality in the vocalizations of infant and adult coppery titi monkeys (Plecturocebus cupreus).

As social animals, many primates use acoustic communication to maintain relationships. Vocal individuality has been documented in a diverse range of primate species and call types, many of which have presumably different functions. Auditory recognition of one's neighbors may confer a selective advantage if identifying conspecifics decreases the need to participate in costly territorial behaviors. Alternatively, vocal individuality may be nonadaptive and the result of a unique combination of genetics and environment. Pair-bonded primates, in particular, often participate in coordinated vocal duets that can be heard over long distances by neighboring conspecifics. In contrast to adult calls, infant vocalizations are short-range and used for intragroup communication. Here, we provide two separate but complementary analyses of vocal individuality in distinct call types of coppery titi monkeys (Plecturocebus cupreus) to test whether individuality occurs in call types from animals of different age classes with presumably different functions. We analyzed 600 trill vocalizations from 30 infants and 169 pulse-chirp duet vocalizations from 30 adult titi monkeys. We predicted that duet contributions would exhibit a higher degree of individuality than infant trills, given their assumed function for long-distance, intergroup communication. We estimated 7 features from infant trills and 16 features from spectrograms of adult pulse-chirps, then used discriminant function analysis with leave-one-out cross-validation to classify individuals. We correctly classified infants with 48% accuracy and adults with 83% accuracy. To further investigate variance in call features, we used a multivariate variance components model to estimate variance partitioning in features across two levels: within- and between-individuals. Between-individual variance was the most important source of variance for all features in adults, and three of four features in infants. We show that pulse-chirps of adult titi monkey duets are individually distinct, and infant trills are less individually distinct, which may be due to the different functions of the vocalizations.

Individuality in the vocalizations of adult and infant coppery titi monkeys (Plecturocebus cupreus

Many primates use acoustic communication to maintain social relationships. Specifically, territorial, pair-bonding primates participate in coordinated vocal duets. Recognition of neighbors should be selected for, as identifying conspecifics may decrease the need for costly territorial behaviors. While similar species show vocal individuality, it is unknown if vocal individuality is innate or develops over time. To understand individuality across life stages, we analyzed 169 duet vocalizations from 30 adult titi monkeys (Plecturocebus cupreus) and 600 trills from 30 infants. We estimated 16 features of adult pulse-chirp vocalizations and 9 features of infant trills from spectrograms and used discriminant function analysis with leave-one-out cross-validation to classify individuals. We correctly classified adults with 83% accuracy and infants with 60% accuracy. We used a multi-variate variance components model to estimate how variance in features was partitioned within- and between-individuals. Between-individual variance was the greatest source of variance for all features for adults, and 4/5 features for infants. Despite little sex-specificity and high overlap between duetting adults, the pulse-chirp vocalization is individually distinct. Further, the trills of infants are individually distinct, though to a lesser degree than adults. This study provides evidence for vocal individuality at multiple life stages in a territorial primate.

Association of height and pubertal timing with lipoprotein subclass profile: Exploring the role of genetic and environmental effects

Little is known about the relationship between growth and lipoprotein profile. We aimed to analyze common genetic and environmental factors in the association of height from late childhood to adulthood and pubertal timing with serum lipid and lipoprotein subclass profile. A longitudinal cohort of Finnish twin pairs (FinnTwin12) was analyzed using self-reported height at 11-12, 14, 17 years and measured stature at adult age (21-24 years). Data were available for 719 individual twins including 298 complete pairs. Serum lipids and lipoprotein subclasses were measured by proton nuclear magnetic resonance spectroscopy. Multivariate variance component models for twin data were fitted. Cholesky decomposition was used to partition the phenotypic covariation among traits into additive genetic and unique environmental correlations. In men, the strongest associations for both adult height and puberty were observed with total cholesterol, low-density lipoprotein cholesterol, intermediate-density lipoprotein cholesterol, and low-density lipoprotein particle subclasses (max. r = -0.19). In women, the magnitude of the correlations was weaker (max. r = -0.13). Few associations were detected between height during adolescence and adult lipid profile. Early onset of puberty was related to an adverse lipid profile, but delayed pubertal development in girls was associated with an unfavorable profile, as well. All associations were mediated mainly by additive genetic factors, but unique environmental effects cannot be disregarded. Early puberty and shorter adult height relate to higher concentrations of atherogenic lipids and lipoprotein particles in early adulthood. Common genetic effects behind these phenotypes substantially contribute to the observed associations.

A Bayesian approach to analysis of covariance in balanced randomized block experiments

Analysis of covariance in designed experiments has a long history dating back to the middle of the twentieth century. Given the popularity of Bayesian approaches to statistical modelling and inference, it is somewhat surprising that there is so little literature on the application of Bayesian methods in this context. This paper proposes methods based on a recent formulation of the problem in terms of a multivariate variance components model which allows for a conjugate Bayesian analysis of balanced randomized block experiments with concomitant information. The analysis is complicated by a linear constraint involving two covariance matrices. Two solutions are proposed and implemented using Markov chain Monte Carlo methods.

Genetic and environmental influences on growth from late childhood to adulthood: A longitudinal study of two Finnish twin cohorts

Human growth is a complex process that remains insufficiently understood. We aimed to analyze genetic and environmental influences on growth from late childhood to early adulthood. Two cohorts of monozygotic and dizygotic (same sex and opposite sex) Finnish twin pairs were studied longitudinally using self-reported height at 11-12, 14, and 17 years and adult age (FinnTwin12) and at 16, 17, and 18 years and adult age (FinnTwin16). Univariate and multivariate variance component models for twin data were used. From childhood to adulthood, genetic differences explained 72-81% of the variation of height in boys and 65-86% in girls. Environmental factors common to co-twins explained 5-23% of the variation of height, with the residual variation explained by environmental factors unique to each twin individual. Common environmental factors affecting height were highly correlated between the analyzed ages (0.72-0.99 and 0.91-1.00 for boys and girls, respectively). Genetic (0.58-0.99 and 0.70-0.99, respectively) and unique environmental factors (0.32-0.78 and 0.54-0.82, respectively) affecting height at different ages were more weakly, but still substantially, correlated. The genetic contribution to height is strong during adolescence. The high genetic correlations detected across the ages encourage further efforts to identify genes affecting growth. Common and unique environmental factors affecting height during adolescence are also important, and further studies are necessary to identify their nature and test whether they interact with genetic factors.

Reconsidering the asymptotic null distribution of likelihood ratio tests for genetic linkage in multivariate variance components models under complete pleiotropy

Accurate knowledge of the null distribution of hypothesis tests is important for valid application of the tests. In previous papers and software, the asymptotic null distribution of likelihood ratio tests for detecting genetic linkage in multivariate variance components models has been stated to be a mixture of chi-square distributions with binomial mixing probabilities. For variance components models under the complete pleiotropy assumption, we show by simulation and by theoretical arguments based on the geometry of the parameter space that all aspects of the previously stated asymptotic null distribution are incorrect-both the binomial mixing probabilities and the chi-square components. Correcting the null distribution gives more conservative critical values than previously stated, yielding P values that can easily be 10 times larger. The true mixing probabilities give the highest probability to the case where all variance parameters are estimated positive, and the mixing components show severe departures from chi-square distributions. Thus, the asymptotic null distribution has complex features that raise challenges for the assessment of significance of multivariate linkage findings. We propose a method to generate an asymptotic null distribution that is much faster than other empirical methods such as permutation, enabling us to obtain P values with higher precision more efficiently.

A Multivariate Variance Components Model for Analysis of Covariance in Designed Experiments

Traditional methods for covariate adjustment of treatment means in designed experiments are inherently conditional on the observed covariate values. In order to develop a coherent general methodology for analysis of covariance, we propose a multivariate variance components model for the joint distribution of the response and covariates. It is shown that, if the design is orthogonal with respect to (random) blocking factors, then appropriate adjustments to treatment means can be made using the univariate variance components model obtained by conditioning on the observed covariate values. However, it is revealed that some widely used models are incorrectly specified, leading to biased estimates and incorrect standard errors. The approach clarifies some issues that have been the source of ongoing confusion in the statistics literature.

Genetic regulation of growth from birth to 18 years of age: The Swedish young male twins study

Growth is a complex process, and only little is known on the genetic regulation of it. We analyzed the effect of genetic and environmental factors on growth in a longitudinal Swedish cohort of 231 monozygotic and 144 dizygotic twin pairs born 1973-1979 with length or height measured annually from birth to age 18. The data were analyzed by two different multivariate variance component models for twin data using the Mx statistical package. At birth and 1 year of age, a substantial part of the variation in length was because of common environment (50 and 57%, respectively) and the effect of genetic factors was minor. After 2 years of age, 91-97% of the variation of height could be explained by genetic differences whereas the rest was because of environmental variation not shared by twins. The genetic correlation between heights at ages 2 and 18 was 0.73 (95% confidence intervals 0.68-0.77) showing that 53% of the genes affecting height at these ages are the same or closely linked; with increasing age the correlation with genetic effects at age 18 become subsequently stronger. Especially in mid-childhood, growth was largely regulated by the same genetic factors. During puberty new genetic factors started to affect height, but also genetic variation affecting height at previous ages remained. These results suggest that genetic regulation of growth is rather uniform, which is encouraging for further efforts to identify genes affecting growth.

Biomass equations for birch in Finland

Biomass equations were compiled for the above- and below-ground tree components of birch (Betula pendula Roth and Betula pubescens Ehrh.). The equations were based on 127 sample trees in 24 birch stands located on mineral soil sites. The study material consisted of 20 temporary plots and ten plots from four thinning experiments with different thinning intensities (unthinned, moderately and heavily thinned plots). The equations were estimated for the following individual tree components: stem wood, stem bark, living and dead branches, foliage, stump, and roots. In the data analysis, a multivariate procedure was applied in order to take into account the statistical dependency among the equations. Three multivariate variance component models were constructed for the above-ground biomass components, and one for the below-ground biomass components. The multivariate model (1) was mainly based on tree diameter and height, and in the multivariate models (2) and (3) additional commonly measured tree variables were used. The equations provided logical biomass predictions for a number of tree components, and were comparable with other functions used in Finland and Sweden. The applied statistical method generated equations that gave more reliable biomass estimates than the equations presented earlier. Furthermore, the structure of the multivariate models enables more flexible application of the equations, especially for research purposes.

Genetic Regulation of Growth in Height and Weight from 3 to 12 Years of Age: A Longitudinal Study of Dutch Twin Children

Human growth is a complex and poorly understood process. We studied the effect of genetic and environmental factors on height and body mass index (BMI, kg/m(2)) based on maternal reports at 3, 4, 5, 7, 10 and 12 years of age in a large longitudinal cohort of Dutch twins (7755 complete twin pairs at age 3). Several multivariate variance component models for twins were fitted to the data using the Mx statistical package. The first-born twin was taller until age 10 and heavier until age 12 than the second-born co-twin. Heritability estimates were high for height (a(2) = .58-.91) and BMI (a(2) = .31-.82), but common and unshared environmental factors were also important. The phenotypic correlations across the ages for height and BMI were mainly explained by correlated additive genetic factors (r(a) = .77-.96 for height and .43-.92 for BMI), but common (r(c) = .40-.84 and .09-.78, respectively) and specific environmental correlations (r(e) = .50-.81 and .42-.80, respectively) were also significant. Additive genetic factors decreased with increasing age difference for both height and BMI. However, the full Cholesky model, which does not make any assumptions regarding the underlying genetic structure, had the best fit. High genetic correlations across the ages, especially for height, may help further molecular genetic studies of human growth. Environmental factors affecting height and BMI during growth period are also important, and further studies are needed to identify these factors and test whether they interact with genetic factors.

Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions

For Wishart density functions, there remains a long-time question unsolved. That is whether there exists the closed-form MLEs of mean matrices over the partially Löwner ordering sets. In this note, we provide an affirmative answer by demonstrating a unified procedure on exactly how the closed-form MLEs are obtained for the simple ordering case. Under the Kullback–Leibler loss function, a property of obtained MLEs is further studied. Some applications of the obtained closed-form MLEs, including the comparison between our ML estimates and Calvin and Dykstra's [Maximum likelihood estimation of a set of covariance matrices under Löwner order restrictions with applications to balanced multivariate variance components models, Ann. Statist. 19 (1991) 850–869.] which obtained by iterative algorithm, are also made.

Statistical analysis of a class of factor time series models

For a class of factor time series models, which is called a multivariate time series variance component (MTV) models, we consider the problem of testing whether an observed time series belongs to this class. We propose the test statistic, and derive its symptotic null distribution. Asymptotic optimality of the proposed test is discussed in view of the local asymptotic normality. Also, numerical evaluation of the local power illuminates some interesting features of the test.

Restricted expected multivariate least squares

A new approach of estimating parameters in multivariate models is introduced. A fitting function will be used. The idea is to estimate parameters so that the fitting function equals or will be close to its expected value. The function will be decomposed into two parts. From one part, which will be independent of the mean parameters, the dispersion matrix is estimated. This estimator is inserted in the second part which then yields the estimators of the mean parameters. The Growth Curve model, extended Growth Curve model and a multivariate variance components model will illustrate the approach.

Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

We consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley–Hotelling and Bartlett–Nanda–Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed. Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels. For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable. As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results.

Empirical best linear unbiased and empirical Bayes prediction in multivariate small area estimation

Abstract Small area estimation plays a prominent role in survey sampling due to a growing demand for reliable small area estimates from both public and private sectors. Popularity of model-based inference is increasing in survey sampling, particularly, in small area estimation. The estimates of the small area parameters can profitably ‘borrow strength’ from data on related multiple characteristics and/or auxiliary variables from other neighboring areas through appropriate models. Fay (1987, Small Area Statistics, Wiley, New York, pp. 91–102) proposed multivariate regression for small area estimation of multiple characteristics. The success of this modeling rests essentially on the strength of correlation of these dependent variables. To estimate small area mean vectors of multiple characteristics, multivariate modeling has been proposed in the literature via a multivariate variance components model. We use this approach to empirical best linear unbiased and empirical Bayes prediction of small area mean vectors. We use data from Battese et al. (1988, J. Amer. Statist. Assoc. 83, 28 –36) to conduct a simulation which shows that the multivariate approach may achieve substantial improvement over the usual univariate approach.

Improved Estimation of Covariante Matrices in Balanced Hierarchical Multivariate Variance Components Models

The problem of simultaneous estimation of covariance matrices in balanced hierarchical multivariate variance components models is considered. A new class of estimators is proposed which dominates the usual sensible estimators with respect to total variability (sum of squared error losses). These estimators shrink towards a multiple of an identity matrix, the multiple being the geometric mean of the characteristic roots of the component Wishart matrices. Numerical illustrations are considered to exhibit the improvement in risk under a simple model.

REML Estimation of Covariance Matrices with Restricted Parameter Spaces

Abstract Restricted parameter spaces for covariance matrices, such as ∑ = σ2 I or ∑ = αI + βJ, are often used to simplify estimation. In addition, fixed upper and/or lower bounds may be needed to ensure that estimates satisfy a priori hypotheses. With multivariate variance components models, several covariance matrices need to be simultaneously estimated and, even with a reduced parameter space, estimation can be difficult. In earlier work we have discussed estimation for a widely-used class of models where the variance components matrices need only be nonnegative definite. In this article we extend these results to handle a wide class of restricted parameter spaces. We state the conditions required for a parameterization to be a member of the class, discuss the implementation of the results for several different members of the class, and discuss estimation with both balanced and unbalanced data. We give several examples to demonstrate the results.

REML Estimation of Covariance Matrices with Restricted Parameter Spaces

Abstract Restricted parameter spaces for covariance matrices, such as ∑ = σ2 I or ∑ = αI + βJ, are often used to simplify estimation. In addition, fixed upper and/or lower bounds may be needed to ensure that estimates satisfy a priori hypotheses. With multivariate variance components models, several covariance matrices need to be simultaneously estimated and, even with a reduced parameter space, estimation can be difficult. In earlier work we have discussed estimation for a widely-used class of models where the variance components matrices need only be nonnegative definite. In this article we extend these results to handle a wide class of restricted parameter spaces. We state the conditions required for a parameterization to be a member of the class, discuss the implementation of the results for several different members of the class, and discuss estimation with both balanced and unbalanced data. We give several examples to demonstrate the results.

An algorithm for restricted maximum likelihood estimation in balanced multivariate variance components models

Restricted maximum likelihood (REML) estimation of variance components in multivariate linear models is a computationally intensive task. Calvin and Dykstra (1991a) have developed a general maximum likelihood estimation algorithm for a product of Wishart densities when the parameter matrices of the Wishart distributions follow a partial order of the Lowner type (Lowner, 1934). The algorithm is iterative in nature and does not require the computation of derivatives. It is guaranteed to converge to the correct solution and, for an important class of models, can be applied to the problem of REML estimation of variance component. In this paper, we define the class of models for which the algorithm is applicable. This class includes all two-factor models and all nested models as special cases. We also relate the general algorithm to the problem of REML estimation for balanced multivariate variance components models. An imple-mentation is supplied in the Appendix and computational aspects of the algorithm are a...

Least Squares Estimation of Covariance Matrices in Balanced Multivariate Variance Components Models

Least Squares Estimation of Covariance Matrices in Balanced Multivariate Variance Components Models