Admissible mixing distributions for a general class of mixture survival models with known asymptotics

Statistical analysis of data on the longest living humans leaves room for speculation whether the human force of mortality is actually leveling off. Based on this uncertainty, we study a mixture failure model, introduced by Finkelstein and Esaulova (2006) that generalizes, among others, the proportional hazards and accelerated failure time models. In this paper we first, extend the Abelian theorem of these authors to mixing distributions, whose densities are functions of regular variation. In addition, taking into account the asymptotic behavior of the mixture hazard rate prescribed by this Abelian theorem, we prove three Tauberian-type theorems that describe the class of admissible mixing distributions. We illustrate our findings with examples of popular mixing distributions that are used to model unobserved heterogeneity.     

Likelihood ratio tests for a mixture of two von Mises distributions

It is proposed that the orientation of elongate objects, such as bones, may be used to identify the flow direction of ancient river deposits. If true, elongate objects could be of great value when ancient bedforms such as ripples and dunes are not visible. Two sandstone quarries were investigated wherein the paleoflow direction was determined from both bedforms and elongate dinosaur bones. A mixture of two von Mises distributions captures the observation that elongate bones transported under unidirectional flow conditions will align both parallel and perpendicular to the flow direction. Likelihood ratio tests for a mixture of two von Mises distributions are given. The power of these tests is investigated by simulation since the direction of dinosaur bones agrees with the primary bedforms if the hypothesis test comparing the dominant mean direction of the bones to the paleoflow direction fails to reject. The likelihood ratio test on the dominant mean direction has reasonable power. If the two mean directions in the mixture distribution are pi apart, a more powerful likelihood ratio test can be used. The likelihood ratio test on the hypothesis that the two mean directions are exactly pi apart is useful in determining if the assumptions of the more powerful test are satisfied.     

Protein local conformations arise from a mixture of Gaussian distributions

The classical approaches for protein structure prediction rely either on homology of the protein sequence with a template structure or on ab initio calculations for energy minimization. These methods suffer from disadvantages such as the lack of availability of homologous template structures or intractably large conformational search space, respectively. The recently proposed fragment library based approaches first predict the local structures, which can be used in conjunction with the classical approaches of protein structure prediction. The accuracy of the predictions is dependent on the quality of the fragment library. In this work, we have constructed a library of local conformation classes purely based on geometric similarity. The local conformations are represented using Geometric Invariants, properties that remain unchanged under transformations such as translation and rotation, followed by dimension reduction via principal component analysis. The local conformations are then modeled as a mixture of Gaussian probability distribution functions (PDF). Each one of the Gaussian PDF’s corresponds to a conformational class with the centroid representing the average structure of that class. We find 46 classes when we use an octapeptide as a unit of local conformation. The protein 3-D structure can now be described as a sequence of local conformational classes. Further, it was of interest to see whether the local conformations can be predicted from the amino acid sequences. To that end, we have analyzed the correlation between sequence features and the conformational classes.     

Protein local conformations arise from a mixture of Gaussian distributions

The classical approaches for protein structure prediction rely either on homology of the protein sequence with a template structure or on ab initio calculations for energy minimization. These methods suffer from disadvantages such as the lack of availability of homologous template structures or intractably large conformational search space, respectively. The recently proposed fragment library based approaches first predict the local structures,which can be used in conjunction with the classical approaches of protein structure prediction. The accuracy of the predictions is dependent on the quality of the fragment library. In this work, we have constructed a library of local conformation classes purely based on geometric similarity. The local conformations are represented using Geometric Invariants, properties that remain unchanged under transformations such as translation and rotation, followed by dimension reduction via principal component analysis. The local conformations are then modeled as a mixture of Gaussian probability distribution functions (PDF). Each one of the Gaussian PDF's corresponds to a conformational class with the centroid representing the average structure of that class. We find 46 classes when we use an octapeptide as a unit of local conformation. The protein 3-D structure can now be described as a sequence of local conformational classes. Further, it was of interest to see whether the local conformations can be predicted from the amino acid sequences. To that end,we have analyzed the correlation between sequence features and the conformational classes.     

Estimation of fine root mortality and growth from simple measurements: a method based on system dynamics

This study is a further development of a dynamic compartment-flow analysis, intended as an analytical tool for the empirical estimation of fine root growth, mortality and decomposition in forest soil. General properties of the dynamic system are utilised to interpret relatively simple measurements of standing biomass, necromass, and decomposition, in order to derive estimates of the process rates. The method is based on the finding that the ratio of fine root necromass to biomass is related to the specific rates of decomposition, mortality, and net growth. If the decomposition rate is measured and the net growth trend is determined from live root measurements, mortality and gross growth can be estimated using these relationships, provided certain regularity requirements are met. These requirements are explicated, such that the estimates can be easily assessed for reliability. To illustrate the use of the method, it was applied to the estimation of specific mortality rates in seven Scots pine stands of different ages and site types. A reanalysis of a previous sequential coring study yielded consistent results. The advantage of this method is that, unlike the standard analysis of sequential cores, it accounts for the possibility of simultaneous growth, mortality and decomposition. It is therefore applicable to situations with no apparent fluctuations or trend in the biomass and necromass levels. No minimum sampling interval is required; hence the method also allows for more extensive or prolonged studies. Received: 1 June 1999 / Accepted: 20 January 2000     

A simple implementation of a normal mixture approach to differential gene expression in multiclass microarrays

MOTIVATION: An important problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. We provide a straightforward and easily implemented method for estimating the posterior probability that an individual gene is null. The problem can be expressed in a two-component mixture framework, using an empirical Bayes approach. Current methods of implementing this approach either have some limitations due to the minimal assumptions made or with more specific assumptions are computationally intensive. RESULTS: By converting to a z-score the value of the test statistic used to test the significance of each gene, we propose a simple two-component normal mixture that models adequately the distribution of this score. The usefulness of our approach is demonstrated on three real datasets.     

Kinetics of an enzyme-catalyzed reaction measured by electrospray ionization mass spectrometry using a simple rapid mixing attachment

Mass spectrometry offers a potential means of measuring virtually all enzyme-catalyzed reactions by simultaneously measuring the concentrations of substrates, products, and intermediates where there are differences in mass between them. To perform these measurements the reaction mixture must be aged for different times and then ionized. Electrospray ionization mass spectrometry provides the most direct means of measuring these reactions. Here we describe a simple reaction mixing and ageing attachment for an electrospray ionization mass spectrometer, built from commercially available components. We have employed this device to measure the kinetics of a model reaction, namely the hydrolysis of N2-(carbobenzyloxy)-L-lysine-p-nitrophenyl ester-catalyzed by trypsin. In this way we were able to measure the kinetics of substrate depletion, product formation, and changes in both free enzyme and acyl-enzyme intermediate concentration in the approach to steady state. With this device we were able to measure reaction times down to about 640 ms.     

The proportion of genetic deviates in the tails of a normal population

Summary If genetic and environmental effects upon a quantitative phenotype X=G+E are normally and independently distributed then the probability distribution of genetic value G among individuals of fixed phenotypic value X is likewise a normal distribution. The mean of this a posteriori distribution of genetic values is +h ² (X− ) and the variance is σ _g ² (1−h ²), where is the a priori mean of X, h ² is the heritability ratio, and σ _g ² is genetic variance. For any fixed values of h ² and σ _g ² the a posteriori probability that the genetic value G associated with a given phenotype X exceeds the population mean by any specified amount can therefore be read directly from the tables of the standard normal distribution. The expected proportion of these superior genetic deviates among individuals whose phenotypic value exceeds some specified constant may also be calculated (by numerical analysis) and is presented here in graphical form. If phenotypic selection is practiced by choosing the best out of N phenotypes then N should be large enough to assure high probability of obtaining a superior genetic deviate. The operating characteristics of this type of selection are displayed in tabular form, again based upon numerical integration.

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Zusammenfassung Wenn genetische und umweltbedingte Effekte auf einen quantitativen Ph?notyp X=G+E von einander unabh?ngig und normal verteilt sind, dann entspricht die Wahrscheinlichkeitsverteilung des genetischen Werts G unter Individuen mit fixiertem ph?notypischem Wert X gleichfalls einer Normalverteilung. Das Mittel dieser a posteriori-Verteilung der genetischen Werte ist +h ² (X− ) und die Varianz ist σ _g ² (1−h ²); wobei das a priori-Mittel von X, h ² der Heritabilit?tskoeffizient und σ _g ² die genetische Varianz sind. Für jeden fixierten Wert von h ² und σ _g ² kann daher die a posteriori-Wahrscheinlichkeit, da? der genetische Wert G das Populationsmittel in Verbindung mit einem gegebenen Ph?notyp X um einen bestimmten Wert übersteigt, direkt aus den Tabellen einer standardisierten Normalverteilung abgelesen werden. Der erwartete Anteil dieser überlegenen, genetisch bedingten Abweichung unter Individuen, deren ph?notypischer Wert einen vorgegebenen Konstantwert übersteigt, kann ebenfalls numerisch errechnet werden. Er wird im vorliegenden Fall graphisch dargestellt. Wenn eine ph?notypische Selektion zur Auswahl der besten Ph?notypen aus N Individuen erfolgt, sollte N gro? genug sein, um mit hoher Wahrscheinlichkeit eine überlegene, genetisch bedingte Abweichung zu erhalten. Die wirksamen Charakteristiken dieses Typs der Selektion werden in tabellierter Form wiedergegeben, die gleichfalls auf numerischer Integration beruht.

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Dedicated to Dr. George F. Sprague on the occasion of his 65^th birthday.

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Cooperative investigations of the Colorado Agricultural Experiment Station, the Crops Research Division, Agricultural Research Service, U.S. Department of Agriculture, and the Beet Sugar Development Foundation. Approved by the Colorado Agricultural Experiment Station for publication as Scientific Series Article No. 880. Paper No. BU-78, Biometrics Unit, and Paper No. 529, Plant Breeding Department, Cornell University.

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Geneticist, Crops Research Division, Agricultural Research Service, U.S. Department of Agriculture; now deceased.

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Formerly Research Assistant, Colorado State University; now Assistant Professor of Biological Statistics, Cornell University.     

Quantitative measurements of mixing intensity in shake-flasks and stirred tank reactors. Use of the Mixmeter, a mixing process analyzer

A new mixing probe has been developed which measures the motions of the fluid during mixing as pressure fluctuations and converts the measurements into a mixing signal (MS). The MS is the root mean square (RMS) pressure fluctuation in the 1-64Hz range as determined by a sensitive pressure sensor and a digital signal processor specifically designed for the purpose. The MS is a measure of the actual mixing flow of the fluid rather than a measurement of the input motions or energies into the reactor system (e.g. RPM, torque or power). In other studies, the MS has been measured as a function of mixing speed in numerous sized reactors from 10 to 1000l, and provides consistent and reproducible measurements. The MS increases monotonically as a function of mixing speed, with a change of slope corresponding to the transition from laminar to turbulent mixing regimes. Maps of MS as a function of location in the reactor are useful in understanding stirred tank reactor design and performance. Quantitative measurements of mixing are especially useful during process development as a tool to increase the success of scale-up during the transition from process development to manufacturing. Measurements at a fixed location in a given reactor are useful in understanding changes in mixing that occur during the course of a given process, and are useful in manufacturing situations where validated documentation of lot-to-lot consistency of mixing is required (e.g. pharmaceutical manufacturing). In addition, the probe has been used to measure mixing in vessels with vibrational mixers with similar results. The probe has been successfully used in feedback loops to control either mixing speed or vibrational mixing amplitude in order to maintain constant mixing of the fluid during processing. With this system it is possible to maintain constant mixing over a wide range of fluid volumes in a given reactor, and, for instance, to compensate for changes in viscosity throughout the course of the process. Adaptations of this system for the measurement of mixing in shake-flasks is described in this paper.