An efficient multivariate approach for estimating preference when individual observations are dependent

1. We discuss aspects of resource selection based on observing a given vector of resource variables for different individuals at discrete time steps. A new technique for estimating preference of habitat characteristics, applicable when there are multiple individual observations, is proposed. 2. We first show how to estimate preference on the population and individual level when only a single site- or resource component is observed. A variance component model based on normal scores in used to estimate mean preference for the population as well as the heterogeneity among individuals defined by the intra-class correlation. 3. Next, a general technique is proposed for time series of observations of a vector with several components, correcting for the effect of correlations between these. The preference of each single component is analyzed under the assumption of arbitrarily complex selection of the other components. This approach is based on the theory for conditional distributions in the multi-normal model. 4. The method is demonstrated using a data set of radio-tagged dispersing juvenile goshawks and their site characteristics, and can be used as a general tool in resource or habitat selection analysis.     

Speed Constancy or Only Slowness: What Drives the Kappa Effect

In the Kappa effect, two visual stimuli are given, and their spatial distance affects their perceived temporal interval. The classical model assumes constant speed while a competing Bayesian model assumes a slow speed prior. The two models are based on different assumptions about the statistical structure of the environment. Here we introduce a new visual experiment to distinguish between these models. When fit to the data, both the two models replicated human response, but the slowness model makes better behavioral predictions than the speed constancy model, and the estimated constant speed is close to the absolute threshold of speed. Our findings suggest that the Kappa effect appears to be due to slow speeds, and also modulated by spatial variance.     

The Inheritance of Metabolic Flux: Expressions for the within-Sibship Mean and Variance Given the Parental Genotypes

Recent developments have related quantitative trait expression to metabolic flux. The present paper investigates some implications of this for statistical aspects of polygenic inheritance. Expressions are derived for the within-sibship genetic mean and genetic variance of metabolic flux given a pair of parental, diploid, n-locus genotypes. These are exact and hold for arbitrary numbers of gene loci, arbitrary allelic values at each locus, and for arbitrary recombination fractions between adjacent gene loci. The within-sibship, genetic variance is seen to be simply a measure of parental heterozygosity plus a measure of the degree of linkage coupling within the parental genotypes. Approximations are given for the within-sibship phenotypic mean and variance of metabolic flux. These results are applied to the problem of attaining adequate statistical power in a test of association between allozymic variation and inter-individual variation in metabolic flux. Simulations indicate that statistical power can be greatly increased by augmenting the data with predictions and observations on progeny statistics in relation to parental allozyme genotypes. Adequate power may thus be attainable at small sample sizes, and when allozymic variation is scored at a only small fraction of the total set of loci whose catalytic products determine the flux.     

Resolution of genetic and cultural inheritance in twin families by path analysis: application to HDL-cholesterol.

A path model and associated statistical method for the analysis of data on twin families are introduced and applied to high density lipoprotein cholesterol (HDL-c) observations in the Swedish Twin Family Study. The proposed path model incorporates both genetic and environmental sources of familial resemblance, maternal environmental effects, intergenerational differences in heritabilities, marital resemblance due to either primary or secondary phenotypic homogamy, and twin residual environmental correlations. Application of the model to HDL-c levels resulted in parameter estimates consistent with those reported in earlier reviews and in the analysis of nuclear family and twin data. Genetic heritability was estimated as h2 = .363 +/- .243, cultural heritability as c2 = .187 +/- .082, and the proportion of phenotypic variance due to residual environmental effects as r2 = .450 +/- .207. Although the parameter estimates were comparable, the statistical tests of hypotheses were, relative to other designs, of low statistical power. It appears that environmental indices are necessary for powerful tests of hypotheses.     

Design and analysis of experiments on random drift and inbreeding depression

While the genetic consequences of inbreeding and small population size are of fundamental importance in many areas of biology, empirical research on these phenomena has proceeded in the absence of a well-developed statistical methodology. The usual approach is to compare observed means and variances with the expectations of Wright's neutral, additive genetic model for quantitative characters. If the observations deviate from the expectations more than can be accounted for by sampling variance of the parameter estimates, the null hypothesis is routinely rejected in favor of alternatives invoking evolutionary forces such as selection or nonadditive gene action. This is a biased procedure because it treats sequential samples from the same populations as independent, and because it ignores the fact that the expectations of the neutral additive genetic model will rarely be realized when only a finite number of lines are studied. Even when genes are perfectly additive and neutral, the variation among the properties of founder populations, the random development of linkage disequilibrium within lines, and the variance in inbreeding between lines reduce the likelihood that Wright's expectations will be realized in any particular set of lines. Under most experimental designs, these sources of variation are much too large to be ignored. Formulas are presented for the variance-covariance structure of the realized within- and between-line variance under the neutral additive genetic model. These results are then used to develop statistical tests for detecting the operation of selection and/or inbreeding depression in small populations. A number of recommendations are made for the optimal design of experiments on drift and inbreeding, and a method is suggested for the correction of data for general environmental effects. In general, it appears that we can best understand the response of populations to inbreeding and finite population size by studying a very large number (>100) of self-fertilizing or full-sib mated lines in parallel with one or more stable control populations.     

Detection of rare major genes in lipid levels

Summary A statistical test of polygenic inheritance (TPI) against the alternative of a rare major gene is presented. It is designed for a random sample of quantitative observations on index cases and siblings of those index cases (probands) selected on the basis of the observed measurements of these probands. The test focuses on an increase of the variance of siblings of probands over its value under polygenic inheritance, such an increase being expected in the presence of a major gene producing a shift of the quantitative observations. Certain data on lipids are then analyzed by this test. A major gene can tentatively be confirmed for triglycerides but not for cholesterol. In addition, the values of all index cases are subjected to an analysis of a mixture of normal distributions (NOCOMP computer program), resulting in a significant second component for triglycerides but not for cholesterol. For both TPI and NOCOMP, the exponent in a power transformation is estimated by maximum likelihood simultaneously with all other parameters, so that these analysis methods are robust against a wide range of skewness in the data, which is demonstrated by manipulation of the observations and their reanalyis.     

Ranked Set Sampling: Ranking Error Models and Estimation of Visual Judgment Error Variance

Ranked set sampling where sampling is based on visual judgment of the differences between the sizes of pairs of units or on a concomitant variable is reviewed. An alternative model for judgment ranking based on ratios of sizes of pairs of units is presented. Computation of the variance of a visual ranked set sampling estimator of the mean of a distribution is enabled via maximum likelihood estimation of the visual judgment error variance. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Longitudinal data analysis for discrete and continuous outcomes

Longitudinal data sets are comprised of repeated observations of an outcome and a set of covariates for each of many subjects. One objective of statistical analysis is to describe the marginal expectation of the outcome variable as a function of the covariates while accounting for the correlation among the repeated observations for a given subject. This paper proposes a unifying approach to such analysis for a variety of discrete and continuous outcomes. A class of generalized estimating equations (GEEs) for the regression parameters is proposed. The equations are extensions of those used in quasi-likelihood (Wedderburn, 1974, Biometrika 61, 439-447) methods. The GEEs have solutions which are consistent and asymptotically Gaussian even when the time dependence is misspecified as we often expect. A consistent variance estimate is presented. We illustrate the use of the GEE approach with longitudinal data from a study of the effect of mothers' stress on children's morbidity.     

The statistical implications of autocorrelation for detection in environmental health assessment

Many environmental health and risk assessment techniques and models aim at estimating the fluctuations of selected biological endpoints through the time domain as a means of assessing changes in the environment or the probability of a particular measurement level occurring. In either case, estimates of the sample variance and mean of the sample variance are crucial to making appropriate statistical inferences. The commonly employed statistical techniques for estimating both measures presume the data were generated by a covariance stationary process. In such cases, the observations are treated as independently and identically distributed and classical statistical testing methods are applied. However, if the assumption of covariance stationarity is violated, the resulting sample variance and variance of the sample mean estimates are biased. The bias compromises statistical testing procedures by increasing the probability of detecting significance in tests of mean and variance differences. This can lead to inappropriate decisions being made about the severity of environmental damage. Accordingly, it is argued that data sets be examined for correlation in the time domain and appropriate adjustments be made to the required estimators before they are used in statistical hypothesis testing. Only then can credible and scientifically defensible decisions be made by environmental decision makers and regulators.     

Comparing and predicting between several methods of measurement

In studies designed to compare different methods of measurement where more than two methods are compared or replicate measurements by each method are available, standard statistical approaches such as computation of limits of agreement are not directly applicable. A model is presented for comparing several methods of measurement in the situation where replicate measurements by each method are available. Measurements are viewed as classified by method, subject and replicate. Models assuming exchangeable as well as non-exchangeable replicates are considered. A fitting algorithm is presented that allows the estimation of linear relationships between methods as well as relevant variance components. The algorithm only uses methods already implemented in most statistical software.     

Masking unmasked in the proportional hazards model

Influence measures based on the pairwise deletion approach and the differentiation approach are developed for unmasking observations masked by other observations in the proportional hazards model. These influential observations might have substantial impact on statistical inference and might provide important information for model adequacy. One numerical example based on real data is presented and discussed.     

Use and abuse of variance models in regression

In (nonlinear) regression with heteroscedastic errors, introduction of a variance model can be useful to obtain good estimators of the regression parameter. For example, the variance model can be used to obtain the optimal weights in weighted least squares. Methodology of this kind is often used in the analysis of assay data in clinical chemistry, pharmacokinetics, and toxicology. In a series of papers in the pharmacological literature, Sheiner and Beal and others advocate the extended least squares (ELS) methodology that combines regression and variance model into a single objective function based on normal-theory maximum likelihood. The inadequacy of this method is folklore in the (mathematical) statistical literature. In this article it is pointed out that this methodology may lead to inconsistent estimators in practically relevant situations. A review is given of other methods that may be preferable to ELS.     

Model Building and Analysis for Block Designs with Nested Rows and Columns

The paper deals with two main problems: it gives a unified approach to model building for observations obtained in a block design with nested rows and columns and it provides the analysis of variance for such data. Also, some statistical properties of the design are examined.     

The sampling variance of the correlation coefficients estimated from two-fold nested and offspring-parent regression analyses

Summary A formula is presented for the large-sample variance of phenotypic, genetic and environmental correlation coefficients, estimated from two-fold nested genetic analyses of balanced or unbalanced offspring data. Where parents of these offspring are scored, offspring on parent regression estimates of genetic parameters may be obtained for each parental level of the nested model. The large-sample variance of the offspring-parent genetic correlation coefficients, computed from either level or by using mid-parent data, is given. The first formula is a correction and generalisation of an expression given by Grossman (1970), while the second formula is an extension of a relationship derived by Reeve (1955).     

Visual homing using an associative memory

Homing is the process by which an autonomous system guides itself to a particular location on the basis of sensory input. In this paper, a method of visual homing using an associative memory based on a simple pattern classifier is described. Homing is accomplished without the use of an explicit world model by utilizing direct associations between learned visual patterns and system motor commands. The method is analyzed in terms of a pattern space and conditions obtained that allow the system performance to be predicted on the basis of statistical measurements on the environment. Results of experiments utilizing the method to guide a robot-mounted camera in a three-dimensional environment are presented.     

The Statistical Analysis for Incomplete Diallel Cross Series. Fixed Model Analysis of Variance

This paper presents modern statistical methods used for quantitative genetics and plant breeding. It contains biometrioal analysis principles of fixed model for an identical, incomplete diallel cross series. The theory of vector space is used for estimation of parameters assumed in the model and for performance of the analysis of variance. As a result, the estimation of all groups of parameters treated in the model, is independent. Theoretical results are illustrated by numerical examples from winter wheat breeding.     

Comment on analyzing flow cytometric data for comparison of mean values of the coefficient of variation of the G1 peak.

BACKGROUND: The coefficient of variation (CV) is often used to characterize and summarize the flow cytometry analysis of nuclear DNA of the Go/G1 peak in a cell population within an individual organism. CV values are frequently used in subsequent statistical analysis to compare experimental groups of individuals. METHODS: We explain why the conventional analysis of variance, linear comparisons and regressions that employ the F and t-tests are not appropriate for analyzing CV data sets. The weighted least squares procedure which utilizes the chi-square test is presented as an adequate method. We further explain why this type of data needs to be analyzed by this procedure. RESULTS: To illustrate the application of the weighted least squares procedure, we analyzed a real data set that had been previously analyzed by conventional methods. We found that a non-significant result (p = 1) using the latter was significant when re-analyzed with the weighted least squares procedure (p = 0.032). CONCLUSIONS: Significant differences between treatments established by the weighted least squares often go unidentified by the conventional analysis. Use of the weighted least squares procedure is recommended for analyzing CV data sets.     

Spatial models for spatial statistics: some unification

A general statistical framework is proposed for comparing linear models of spatial process and pattern. A spatial linear model for nested analysis of variance can be based on either fixed effects or random effects. Greig-Smith (1952) originally used a fixed effects model, but there are also examples of random effects models in the soil science literature. Assuming intrinsic stationarity for a linear model, the expectations of a spatial nested ANOVA and two term local variance (TTLV, Hill 1973) are functions of the variogram, and several examples are given. Paired quadrat variance (PQV, Ludwig & Goodall 1978) is a variogram estimator which can be used to approximate TTLV, and we provide an example from ecological data. Both nested ANOVA and TTLV can be seen as weighted lag-1 variogram estimators that are functions of support, rather than distance. We show that there are two unbiased estimators for the variogram under aggregation, and computer simulation shows that the estimator with smaller variance depends on the process autocorrelation.     

Maximum Likelihood based comparison of the specific growth rates for P. aeruginosa and four mutator strains

The specific growth rate for P. aeruginosa and four mutator strains mutT, mutY, mutM and mutY-mutM is estimated by a suggested Maximum Likelihood, ML, method which takes the autocorrelation of the observation into account. For each bacteria strain, six wells of optical density, OD, measurements are used for parameter estimation. The data is log-transformed such that a linear model can be applied. The transformation changes the variance structure, and hence an OD-dependent variance is implemented in the model. The autocorrelation in the data is demonstrated, and a correlation model with an exponentially decaying function of the time between observations is suggested. A model with a full covariance structure containing OD-dependent variance and an autocorrelation structure is compared to a model with variance only and with no variance or correlation implemented. It is shown that the model that best describes data is a model taking into account the full covariance structure. An inference study is made in order to determine whether the growth rate of the five bacteria strains is the same. After applying a likelihood-ratio test to models with a full covariance structure, it is concluded that the specific growth rate is the same for all bacteria strains. This study highlights the importance of carrying out an explorative examination of residuals in order to make a correct parametrization of a model including the covariance structure. The ML method is shown to be a strong tool as it enables estimation of covariance parameters along with the other model parameters and it makes way for strong statistical tools for inference studies.     

Patterns of temporal variation in Lake Titicaca. A high altitude tropical lake. I. Background,physical and chemical processes,and primary production

A statistical analysis is presented of patterns of variation in some physical, chemical, and biological variables for a 6 year series of data from the tropical, high altitude Lake Titicaca (Peru-Bolivia). ANOVA techniques and autocorrelation analyses were used to partition the variance in Titicaca, and in some comparison tropical and temperate series, into components with repeatable annual cycles and components attributable to other kinds of patterns.In Titicaca, insolation and stratification are highly seasonal in pattern of variation, although the amount of variance relative to means is small compared to temperate lakes. However, the seasonal pattern of physical variation is only weakly imposed on chemical and biological processes, to judge from analyses of silicate, oxygen, and primary production series. Comparable temperate series of primary production and chlorophyll a are much more seasonal.