Unbiasedness of the Estimator of the Function of Expected Value in the Mixed Linear Model

The traditional method for estimating the linear function of fixed parameters in mixed linear model is a two-stage procedure. In the first stage of this procedure the variance components estimators are calculated and next in the second stage these estimators are taken as true values of variance components to estimating the linear function of fixed parameters according to generalized least squares method. In this paper the general mixed linear model is considered in which a matrix related to fixed parameters and or/a dispersion matrix of observation vector may be deficient in rank. It is shown that the estimators of a set of functions of fixed parameters obtained in second stage are unbiased if only the observation vector is symmetrically distributed about its expected value and the estimators of variance components from first stage are translation-invariant and are even functions of the observation vector.     

Dispersion population models discrete in time and continuous in space

We analyze a discrete-time model of populations that grow and disperse in separate phases. The growth phase is a nonlinear process that allows for the effects of local crowding. The dispersion phase is a linear process that distributes the population throughout its spatial habitat. Our study quantifies the issues of survival and extinction, the existence and stability of nontrivial steady states, and the comparison of various dispersion strategies. Our results show that all of these issues are tied to the global nature of various model parameters. The extreme strategies of staying-in place and going-everywhere-uniformly are compared numerically to diffusion strategies in various contexts. We approach the mathematical analysis of our model from a functional analysis and an operator theory point of view. We use recent results from the theory of positive operators in Banach lattices.     

Effects of muscle model parameter dispersion and multi-loop segmental interaction on the neuromuscular system performance

The effects of parameter dispersion among motor units on the neuromuscular system performance as well as interaction between muscle segments and spinal cord mechanisms are investigated. Elementary components of the system are modeled to simulate with simple models their input-output characteristics. A leaky SS-IPFM encoder with a time-dependent threshold simulates the motor-neuron encoding characteristics. An amplitude and time dependent nonlinear model represent the motor unit mechanical output to neuronal input relationship. The dispersion of parameters in the components of the whole muscle control model is investigated in the open loop mode. It is shown that the dispersion of parameters in the multi-efferent channels converging on a common tendon provides a spatial filtration generating a smoother muscle force in addition to extending the linear dynamic range compared to a similar system having identical motor units. Muscle segmental interaction is investigated in this distributed model by closing the loop through a coupling matrix, representing afferent-motorneuron interaction on the spinal cord level. A diagonal matrix represents no segmental interaction and a uniform matrix represents a uniform interaction between segments through the muscle spindles and Golgi tendon feedback elements. The close loop simulation studied shows that (a). The type of segmental interaction has little effect on the overall system performance, i.e., range of linerity and stability, which is the result of having a muscle system with a large number of motor units. (b) There are only minor differences in results between the uniform and normal parameter distributions tested. (c) A loop gain of 4 divided by 8 in the distributed model can provide linearity through the full physiological force range. (d) Type of segmental interaction has significant effects on the individual segment. A uniform matrix provides a more stable segment due to the spatial filtration resulting from the segmental interaction, while the diagonal noninteracting matrix shows instabilities on the local segmental level despite global stability. The more realistic exponentially decaying spatial interaction matrix yields both global neuromuscular and local segmental stability with the same linear dynamic range generated with the uniform or diagonal matrices.     

Modelling Heterogeneous Dispersion in Marginal Models for Longitudinal Proportional Data

Continuous proportional data is common in biomedical research, e.g., the pre‐post therapy percent change in certain physiological and molecular variables such as glomerular filtration rate, certain gene expression level, or telomere length. As shown in (Song and Tan, 2000) such data requires methods beyond the common generalised linear models. However, the original marginal simplex model of (Song and Tan, 2000) for such longitudinal continuous proportional data assumes a constant dispersion parameter. This assumption of dispersion homogeneity is imposed mainly for mathematical convenience and may be violated in some situations. For example, the dispersion may vary in terms of drug treatment cohorts or follow‐up times. This paper extends their original model so that the heterogeneity of the dispersion parameter can be assessed and accounted for in order to conduct a proper statistical inference for the model parameters. A simulation study is given to demonstrate that statistical inference can be seriously affected by mistakenly assuming a varying dispersion parameter to be constant in the application of the available GEEs method. In addition, residual analysis is developed for checking various assumptions made in the modelling process, e.g., assumptions on error distribution. The methods are illustrated with the same eye surgery data in (Song and Tan, 2000) for ease of comparison. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Factor analysis models for structuring covariance matrices of additive genetic effects: a Bayesian implementation

Multivariate linear models are increasingly important in quantitative genetics. In high dimensional specifications, factor analysis (FA) may provide an avenue for structuring (co)variance matrices, thus reducing the number of parameters needed for describing (co)dispersion. We describe how FA can be used to model genetic effects in the context of a multivariate linear mixed model. An orthogonal common factor structure is used to model genetic effects under Gaussian assumption, so that the marginal likelihood is multivariate normal with a structured genetic (co)variance matrix. Under standard prior assumptions, all fully conditional distributions have closed form, and samples from the joint posterior distribution can be obtained via Gibbs sampling. The model and the algorithm developed for its Bayesian implementation were used to describe five repeated records of milk yield in dairy cattle, and a one common FA model was compared with a standard multiple trait model. The Bayesian Information Criterion favored the FA model.     

Use of Generalised Linear Mixed Models for the Analysis of Clustered Survival Data

Data from a litter matched tumorigenesis experiment are analysed using a generalised linear mixed model (GLMM) approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same litter. Maximum likelihood (ML) and residual maximum likelihood (REML) estimates of risk variable parameters, variance component parameters and the prediction of random effects are given. Estimation of treatment effect parameter (carcinogen effect) has good agreement with previous analyses obtained in the literature though the dependence structure within a litter is modelled in different ways. The variance component estimation provides the estimated dispersion of the random effects. The prediction of random effects, is useful, for instance, in identifying high risk litters and individuals. The present analysis illustrates its wider application to detecting increased risk of occurrence of disease in particular families of a study population.     

Simple equations to estimate light interception by isolated trees from canopy structure features: assessment with three-dimensional digitized apple trees

* Simple models of light interception are useful to identify the key structural parameters involved in light capture. We developed such models for isolated trees and tested them with virtual experiments. Light interception was decomposed into the projection of the crown envelope and the crown porosity. The latter was related to tree structure parameters. * Virtual experiments were conducted with three-dimensional (3-D) digitized apple trees grown in Lebanon and Switzerland, with different cultivars and training. The digitized trees allowed actual values of canopy structure (total leaf area, crown volume, foliage inclination angle, variance of leaf area density) and light interception properties (projected leaf area, silhouette to total area ratio, porosity, dispersion parameters) to be computed, and relationships between structure and interception variables to be derived. * The projected envelope area was related to crown volume with a power function of exponent 2/3. Crown porosity was a negative exponential function of mean optical density, that is, the ratio between total leaf area and the projected envelope area. The leaf dispersion parameter was a negative linear function of the relative variance of leaf area density in the crown volume. * The resulting models were expressed as two single equations. After calibration, model outputs were very close to values computed from the 3-D digitized databases.     

On Plasmon Polariton Propagation Along Metallic Nano-Chain

The collective wave type plasmon polariton self–modes in the metallic (Au, Ag) nano-chain were determined and analyzed with respect to the nano-sphere size and chain separation parameters. At some regions for parameters, the undamped modes were identified when the interaction had been assumed as the near-field-zone dipole coupling. These modes were found on the rim of stability of the linear theory, which indicates artifact of the model of near-field coupling. Inclusion of the medium- and far-field zone contributions to dipole interaction removes, however, instability and allows for fully analytical demonstration of quenching of irradiation losses of plasmon polaritons in the chain to the level of only ohmic attenuation. The plasmon polariton dispersion and the group velocity of plasmon polariton wave packets were examined with respect to nano-sphere and chain parameters and mode polarization. Previous numerical results related to long-range plasmon polariton propagation in the chain are transparently interpreted within the analytical approach.     

Nonlinear pattern selection in a mechanical model for morphogenesis

We present a numerical study of the nonlinear mechanical model for morphogenesis proposed by Oster et al. (1983) with the aim of establishing the pattern forming capability of the model. We present a technique for mode selection based on linear analysis and show that, in many cases, it is a reliable predictor for nonlinear mode selection. In order to determine the set of model parameters that can generate a particular pattern we develop a technique based on nonlinear least square fitting to a dispersion relation. As an application we present a scenario for sequential pattern formation of dermal aggregations in chick embryos which leads to the hexagonal array of cell aggregations observed in feather germ formation in vivo.     

Determination of solubility parameters for poly(3-hydroxyalkanoates).

The three dimensional solubility parameters defined by Hansen are based on dispersion forces between structural units, interaction between polar groups and hydrogen bonding. For polar polymers such as poly(3-hydroxyalkanoates), P(3HA), this approach was used to obtain the three coordinates of a solubility parameter in terms of: a dispersion part, a polar part and a hydrogen bonding part. Thirty-eight different solvents for poly(3-hydroxybutyrate), PHB, which are mentioned in the literature are examined by this method and the theoretical predictions are compared with the experimental reports. Another overall comparison between PHA polymers provides their Hansen and Hildebrand parameters for side chain lengths up to C13. In this series a linear progression in calculated solubility parameters with side chain length was found. An Appendix provides information and data on calculation of the solubility parameters. While the solubility information is limited and only covers homopolymers, it should help to highlight some of the contradictions regarding PHB solubility. This semi-empirical approach is only valid for amorphous polymers hence crystallinity effects, which are important with PHB, as well as molecular weight effects still require analysis.     

Theory for modelling means and covariances in a two-breed population with dominance inheritance

This paper presents theory and methods to compute genotypic means and covariances in a two breed population under dominance inheritance, assuming multiple unlinked loci. It is shown that the genotypic mean is a linear function of five location parameters and that the genotypic covariance between relatives is a linear function of 25 dispersion parameters. Recursive procedures are given to compute the necessary identity coefficients. In the absence of inbreeding, the number of parameters for the mean is reduced from five to three and the number for the covariance is reduced from 25 to 12. In a two-breed population, for traits exhibiting dominance, the theory presented here can be used to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood.Supported in part by the Illinois Agricultural Experiment Station, Hatch Projects 35-0345 (R.L.F.) and 35-0367 (M.G.). A computer program implementing the methods described here is available upon request to R.L.F.     

A nonproportional hazards Weibull accelerated failure time regression model

We present a study of risk factors measured in mean before age 50 and subsequent incidence of heart disease over 32 years of follow-up. The data are from the Framingham Heart Study. The standard accelerated failure time model assumes the logarithm of time until an event has a constant dispersion parameter and a location parameter that is a linear function of covariates. Parameters are estimated by maximum likelihood. We reject a standard Weibull model for these data in favor of a model with the dispersion parameter depending on the location parameter. This model suggests that the cumulative hazard ratio for two individuals shrinks towards unity over the follow-up period. Thus, not only the standard Weibull, but also the semiparametric proportional hazards (Cox) model is inadequate for this data. The model improvement appears particularly valuable when estimating the difference in predicted outcome probabilities for two individuals.     

Interaction of a high-current electron beam with hybrid waveguide and plasma modes in a dielectric Cherenkov maser with a plasma layer

The characteristics of a high-current electron beam-driven microwave amplifier—a dielectric Cherenkov maser—are investigated in the framework of linear theory for the case of a plasma layer present at the surface of the maser slow-wave structure. The dispersion relation for axisymmetric perturbations is obtained for the conventional configuration (a circular dielectric-lined waveguide and a thin annular beam propagating within the vacuum region inside the annular plasma) in the model of a fully magnetized plasma and beam. The results of numerically solving the dispersion relation for different beam and plasma parameters are presented, and an analysis based on these results is given with regard to the features of the beam interaction with the hybrid waves of the system (both hybrid waveguide and hybrid plasma modes). For the hybrid waveguide mode, the dependences of the spatial growth rate on the frequency demonstrate an improvement in the gain at moderate plasma densities, along with narrowing the amplification band and shifting it toward higher frequencies. For the hybrid plasma mode, the interaction with a mildly relativistic (200–250 keV) beam, when the wave phase velocity is close to the speed of light in the dielectric medium, is most interesting and, therefore, has been studied in detail. It is shown that, depending on the beam and plasma parameters, different regimes of the hybrid plasma mode coupling to the hybrid waveguide mode or a usual, higher order plasma mode take place; in particular, a flat gain vs. frequency dependence is possible over a very broad band. The parameters at which the ?3-dB bandwidth calculated for the 30-dB peak gain exceeds an octave are found.     

Goodness-of-Fit Tests and Model Diagnostics for Negative Binomial Regression of RNA Sequencing Data

This work is about assessing model adequacy for negative binomial (NB) regression, particularly (1) assessing the adequacy of the NB assumption, and (2) assessing the appropriateness of models for NB dispersion parameters. Tools for the first are appropriate for NB regression generally; those for the second are primarily intended for RNA sequencing (RNA-Seq) data analysis. The typically small number of biological samples and large number of genes in RNA-Seq analysis motivate us to address the trade-offs between robustness and statistical power using NB regression models. One widely-used power-saving strategy, for example, is to assume some commonalities of NB dispersion parameters across genes via simple models relating them to mean expression rates, and many such models have been proposed. As RNA-Seq analysis is becoming ever more popular, it is appropriate to make more thorough investigations into power and robustness of the resulting methods, and into practical tools for model assessment. In this article, we propose simulation-based statistical tests and diagnostic graphics to address model adequacy. We provide simulated and real data examples to illustrate that our proposed methods are effective for detecting the misspecification of the NB mean-variance relationship as well as judging the adequacy of fit of several NB dispersion models.     

Modeling the effect of collagen fibril alignment on ligament mechanical behavior

Ligament mechanical behavior is primarily regulated by fibrous networks of type I collagen. Although these fibrous networks are typically highly aligned, healthy and injured ligament can also exhibit disorganized collagen architecture. The objective of this study was to determine whether variations in the collagen fibril network between neighboring ligaments can predict observed differences in mechanical behavior. Ligament specimens from two regions of bovine fetlock joints, which either exhibited highly aligned or disorganized collagen fibril networks, were mechanically tested in uniaxial tension. Confocal microscopy and FiberFit software were used to quantify the collagen fibril dispersion and mean fibril orientation in the mechanically tested specimens. These two structural parameters served as inputs into an established hyperelastic constitutive model that accounts for a continuous distribution of planar fibril orientations. The ability of the model to predict differences in the mechanical behavior between neighboring ligaments was tested by (1) curve fitting the model parameters to the stress response of the ligament with highly aligned fibrils and then (2) using this model to predict the stress response of the ligament with disorganized fibrils by only changing the parameter values for fibril dispersion and mean fibril orientation. This study found that when using parameter values for fibril dispersion and mean fibril orientation based on confocal imaging data, the model strongly predicted the average stress response of ligaments with disorganized fibrils (\(R^{2}=0.97\)); however, the model only successfully predicted the individual stress response of ligaments with disorganized fibrils in half the specimens tested. Model predictions became worse when parameters for fibril dispersion and mean fibril orientation were not based on confocal imaging data. These findings emphasize the importance of collagen fibril alignment in ligament mechanics and help advance a mechanistic understanding of fibrillar networks in healthy and injured ligament.     

A numerical model of birch pollen emission and dispersion in the atmosphere. Model evaluation and sensitivity analysis

An evaluation of performance of the System for Integrated modeLling of Atmospheric coMposition (SILAM) in application to birch pollen dispersion is presented. The system is described in a companion paper whereas the current study evaluates the model sensitivity to details of the pollen emission module parameterisation and to the meteorological input data. The most important parameters are highlighted. The reference year considered for the analysis is 2006. It is shown that the model is capable of predicting about two-thirds of allergenic alerts, with the odds ratio exceeding 12 for the best setup. Several other statistics corroborate with these estimations. Low-pollen concentration days are also predicted correctly in more than two-thirds of cases. The model experiences certain difficulties only with intermediate pollen concentrations. It is demonstrated that the most important input parameter is the near-surface temperature, the bias of which can easily jeopardise the results. The model sensitivity to random fluctuations of temperature is much lower. Other parameters important at various stages of pollen development, release, and dispersion are precipitation and ambient humidity, as well as wind direction.     

Modeling of a vapor-phase fungi bioreactor for the abatement of hexane: fluid dynamics and kinetic aspects

During some previous works, a packed-bed lab-scale biofilter (177 . 10(-6) m3), inoculated with a selected strain of Aspergillus niger had been tested for the abatement of hexane vapors, showing a maximum elimination capacity of 200 g hexane/m3 reactor/h. A steady-state mathematical model taking into account axial dispersion effect was applied to describe the process and predict experimental results, but many model parameters could not be calculated from experimental data. The aim of the present work was to carry out further investigations to accurately determine the dispersion coefficient and the kinetics parameters to verify the effective validity of the model. Analysis of residential time distribution revealed the presence of a certain degree of axial dispersion (dispersion coefficient D of 1.22 . 10(-4) m2/s). Experimental data from kinetic trials carried out in reduced height reactors, together with data from full-scale runs, were elaborated to estimate the kinetic saturation constant (K(s)), the coefficient yield (Y), the maximum growth rate (mu(max)) and maximum substrate degradation rate (r(max)). All these parameters were introduced into the model, which was then solved by simulation software finding a good correlation between experimental and theoretical results.     

Ion exchange chromatography of proteins-prediction of elution curves and operating conditions. I. Theoretical considerations

A mathematical model is proposed for the elution of proteins on ion exchange columns by a linear gradient increase and stepwise increase of ionic strength in order to predict relationships between the elution characteristics (the peak position, the peak width, etc.) and the operating conditions (the flow rate, the slope of gradient, etc). This model is in principle based on the continuous-flow plate theory, in which the protein concentration and ionic strength dependent distibution coefficient between proteins and ion exchangers and zone sperading effects are taken into consideration. The advantage of this model is its simplicity since it requires only two parameters: The distribution coefficient and the number of plates. Since the distribution coefficient of proteins depends on both the protein concentration and ionic strength of the elution buffer, the number of plates should vary with time. However, it is extremely difficult to take into consideration the time-dependent number of plates. Therefore, we assume that the number of plates is constant and related to that number derived from a mass balance model which includes longitudinal dispersion and gel phase diffusion. On the basis of these assumptions, a method for determining the number of plates by the moment method is presented. Although the dependencies of the peak position and peak width on the slope of linear gradient are predictable by numerical calculations of the present model, simpler methods for prediction of these dependencies are desirable. A graphical method is proposed for prediction of the peak position. For prediction of the peak width, an asymptotic solution is derived from a quasi-steady-state model.     

Breakthrough performance of plasmid DNA on ion-exchange membrane columns

Breakthrough performance of plasmid DNA adsorption on ion-exchange membrane columns was theoretically and experimentally investigated using batch and fixed-bed systems. System dispersion curves showed the absence of flow non-idealities in the experimental arrangement. Breakthrough curves (BTC) were significantly affected by inlet flow rate and solute concentration. In the theoretical analysis, a model was integrated by the serial coupling of the membrane transport model and the system dispersion model. A transport model that considers finite kinetic rate and column dispersed flow was used in the study. A simplex optimization routine, coupled to the solution of the partial differential model equations, was employed to estimate the maximum adsorption capacity constant, the equilibrium desorption constant, and the forward interaction rate constant, which are the parameters of the membrane transport model. The analysis shows that as inlet concentration or flow rate increases, the deviation of the model from the experimental behavior decreases. The BTCs displacement as inlet concentration increases was explained in terms of a greater degree of column saturation reached and more efficient operation accomplished. The degree of column saturation was not influenced by inlet flow rate. It was necessary to consider in the column model the slight variation in the BTC produced by the axial dispersion, in order to accomplish the experimental curve dispersion. Consequently, the design criteria that for Pe > 40 the column axial dispersion can be neglected should be taken with precaution.