Showing papers in "Biometrical Journal in 1979"

The Influence of External Real and White Noise on the LOTKA‐VOLTERRA Model

Abstract: The behaviour of a marginally stable predator‐prey system, the LOTKA‐VOLTERRA model is analyzed under the influence of parameter fluctuations. The cases of white and real noise are studied separately. It is shown that for white noise no stationary solution exists, but even for time tending to infinity explosion occurs only with zero probability. In the case of real noise the class of noise processes that permit a stationary solution is characterized by their spectral density. It turns out that this class consists of all stationary processes that do not contain the eigenfrequencies of the LOTKA‐VOLTERRA system. It is shown that for all other real noise processes a resonance phenomenon occurs and the solutions grow unboundedly. Copyright © 1979 WILEY‐VCH Verlag GmbH & Co. KGaA

The non-central negative binomial distribution

Abstract: The non-central negative binomial distribution is derived by mixing the POISSON distribution with a certain BESSEL function distribution, or the negative binomial distribution with the POISSON distribution. Some properties of the distribution are discussed, including a characterization. The fitting of the distribution to some data are given together with the efficiency for various procedures of estimation.

Interrupted Point Processes

Abstract: A simple random thinning procedure for point processes is considered. The model is used for the description of a pattern generated by positions of tree seedlings.

Four Applications of a Bivariate Pareto Distribution

Abstract: The following model, of “latent structure” type, is considered: in each subpopulation, X and Y are random variables drawn independently from the same exponential distribution, and the parameter of the exponential distribution varies between subpopulations with a Gamma density. Over the whole population, X and Y are then positively correlated, and jointly have a bivariate PARETO distribution. Four examples show how this distribution is useful in analysing ordered contingency tables in which the two dimensions can be regarded as alternative measures of the same thing: the injuries to the two drivers in a road accident, or the severity of a lesion present in a patient as assessed by two physicians, for instance. Two extensions are considered: (a) allowing X and Y to have Gamma distributions, with each subpopulation having the same shape parameter but different scale parameters; (b) allowing the scale parameter for Y to be correlated with the scale parameter for X, rather than being identical to it. A new bivariate distribution with three shape parameters is derived, expressed in terms of a generalised hypergeometric function.

On the KOLMOGOROV‐SMIRNOV Test for the POISSON Distribution with Unknown Mean

Abstract: The standard tables for the KOLMOGOROV-SMIRNOV test are valid only in the case of testing whether a set of observations is from a completely specified cumulative distribution, F0(X), with all parameters known. If the parameters are unknown and must be estimated from the sample, then the tables are not valid. A table is given in this paper for use with the KOLMOGOROV-SMIRNOV statistic in the case of testing whether a set of observations is from the POISSON distribution with an unknown mean that must be estimated from the sample. The table is obtained from a Monte Carlo calculation.

Prediction Models in Configural Frequency Analysis

Abstract: Configural frequency analysis models are presented for predicting a discrete criterion variable from combining 2 or more discrete predictor variables nonparametrically. Predictions are made via so-called prediction types which allow to predict a criterion class from combined predictor classes. Prediction types are implicitely defined as cells of a 2-dimensional contingeney table with rows as predictor combinations and columns as criterion classes. The CFA is extended for two criteria, and for stratified sampling.

Nonparametric Estimation of Population Density for Line Transect Sampling Using FOURIER Series

Abstract: A nonparametric, robust density estimation method is explored for the analysis of right-angle distances from a transect line to the objects sighted. The method is based on the FOURIER series expansion of a probability density function over an interval. With only mild assumptions, a general population density estimator of wide applicability is obtained.

Estimation of Attributable Risk in the Presence of Confounding

Abstract: LEVIN's attributable risk due to a risk factor is estimated in the presence of confounding in the retrospective study context. The crude attributable risk is decomposed into the attributable risk due to a risk factor and a bias factor which is shown to vanish only when the confounding factor is unassociated to either the risk factor or the Disease or to both. Asymptotic tests are provided for testing interaction, and an appropriate pooled estimate is derived when constancy of partial attributable risks prevails over the levels of the confounding factor.

Applied Stochastic Geometry: A Survey (Invited Paper)

Abstract: The paper is a survey on some practical aspects of stochastic geometry in the sense of D. G. KENDALL's and G. MATHERON's & J. SERRA's schools.

Estimation of Sheet Thickness Distributions from Linear and Plane Sections

Abstract: A biological “membrane” or a “barrier” can often be modelled by a sheet, namely a portion of space comprised between two smooth, quasi-parallel faces, such that sheet thickness is in general variable but very small in relation to face extent. Sheet thickness measurements are often of interest to the morphometrist, the physiologist and the pathologist. In the present paper, some stereological techniques are developed for estimating true sheet thickness distributions, and their moments, from the apparent thickness or intercept length measurements obtained, respectively, via random plane or linear sections. An important premise for the models to work is that of isotropic random (‘non-preferential’) sheet orientation relative to the sectioning probe.

A New Nonparametric Approach to the Comparison of K Independent Samples of Response Curves II: A K Sample Generalization of the FRIEDMAN Test

Abstract: A K sample generalization of the FRIEDMAN test (1937) is introduced which can be used as a nonparametric procedure for testing the homogeneity of the profiles of K independent samples of response curves measured at T identical points of time. While a similar procedure in LEHMACHER & WALL (1978), section 3, is based on T combined tests, each of them at level a/T, here a finite and asymptotic test is presented which is based on a single test statistic. The application of the new multivariate test is illustrated by the same numerical example as in LEHMACHER & WALL (1978). The properties of this test are discussed and compared with the combined test mentioned above.

Relative Efficiency of Estimators of the Mean of a Normal Distribution when Coefficient of Variation is Known

Abstract: A biased but simple and consistent estimator of the parameter ϑ has been obtained for the normal distribution N(ϑ, aϑ2), ϑ>0 where a is a known constant. It is shown that the estimator is more efficient than the sample mean or any suitably chosen constant multiple of the sample standard deviation. It is also proved to be more efficient than the mimumum variance unbiased estimator among a typical class of unbiased estimators derived by RASUL KHAN (1968).

Some inequalities for the expectation of a product of functions of a random variable and for the multivariate distribution function at a random point

Abstract: Under certain assumptions the expectation of a product of functions of a random variable is greater (smaller) than the product of expectations. The multivariate distribution function of m independent random variables at a random point is greater than the product of the distribution functions of the m variables.

On the Accuracy of Lineal Analysis

Abstract: Formulae and tables for the error of lineal analysis method for determining volume or area fractions of spatial or planar patterns are given for three random set models.

Expected Error Rates for Logistic Regression Versus Normal Discriminant Analysis

Abstract: The expected error rates associated with using the allocation rule based on logistic regression are derived in the context of two multivariate normal populations with a common covariance matrix and compared with the corresponding error rates of the classical rule based on this normality assumption. It is shown in terms of the actual sizes of the asymptotic expected error rates that the performance of the logistic procedure does not fall far short of the normality based method, even for widely separated populations. This latter result is not obvious from previously available work on the asymptotic relative efficiency of the logistic procedure.

Development and Applications of Five New Water Quality Indices

Abstract: Two indices, P1 and I, employing ranked data and following the beta distribution, are described. These indices, which are true statistical measures of water quality, can be used with any set of parameters. Further, these indices correlate highly with biological and subjective-engineering assessments of water quality. Three indices, employing raw data and based on the chi-square distribution, are described. Two of these, B1 and B2, do not have sufficient discrimination to serve as general measures of water quality, although they have use in specific circumstances. The index C has been found to be a measure of excessive variability (at a station), and can be used to quantitatively compare the (data) variability among stations. The five indices described here are evaluated against a set of operational criteria which a mathematically sound index should meet.

On Analysis of Some Augmented Block Designs

Abstract: Augmented designs are frequently used in practice. In this paper, a simplified procedure of analysis of some types of augmented designs is discussed using the concept of efficiency-balance and partially efficiency-balance.

Statistical Analysis of Joint Effects of Major Genes and Polygenes in Quantitative Genetics

Abstract: In this paper we analyze a quantitative genetic character which is controlled by both major genes and polygenes. Assuming that there are no epistatic effects, no linkage and no genetic-environmental interactions, we follow TAN and CHANG (1972) to derive the probability distributions for segregating populations. The numbers of major genes and polygenes, and the additive and dominance effects of major genes and polygenes are then estimated by using the procedures developed in TAN and CHANG (1972) and the POWELL-FLETCHER search procedure for maximum values. In this paper, we consider the case involving data from P1, P2, F2, B1 (Backcross to P1) and B2 (Backcross to P2) as this type of experiment is common in practical applications. The analyses are applied to a simulated model generated by using binomial, multinomial and normal variables and to the data of an experiment on kernel weight of sorghum plant provided to the authors by Professor GEORGE H. L. LIANG of Kansas State University. The analysis of these data indicate clearly that the method derived in this paper is useful and desirable.

Regarding the Optimality of the Formula \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{n_0 }}{n} = \sqrt p $\end{document} for the Ratio of Sample Sizes of a Control and of p Treated Groups

Abstract: FINNEY (1955) recommended without argument the ratio where p is the number of treated groups to be compared with a control and n and n0 are the sizes of the treated samples and of the control. The question is does this allocation result in an optimum, anywise. RASCH, HERRRENDorfer und BOCK (1972) asserted that this ratio with t-tests is followed by maximal expected values of the t-statistics. Here is shown that this allocation with t-tests provides minimal comparisonwise type II error rates. Moreover, with the t-test and the DUNNETT test the mean number of type II errors becomes a minimum. Nearly optimal is this allocation with the t-test and the DUNNETT test regarding the experimentwise type II error rates. Not optimal is this allocation regarding the experimentwise type I error rates.

A General Computing Algorithm for Factor Analysis

Abstract: A general algorithm for the factor analysis is developed, which contains the alpha factor analysis, the maximum likelihood method and the minres method as special cases. This algorithm, which is a multidimensional NEWTON algorithm, uses the partial derivatives of the reproduced communalities with respect to the inserted ones. The partial derivatives are got by means of the perturbation theory.

On the Efficiency Factor for a Contrast of Treatment Parameters

Abstract: In this paper a formula for computing the efficiency factor of a block design for any estimable contrast of treatment parameters is given. The formula is expressible in terms of the efficiency factors of some basic contrasts.

A Modified MANN‐WHITNEY Test with Improved Asymptotic Relative Efficiency

Abstract: A slight modification of the MANN-WHITNEY test is introduced here. When the underlying distribution is normal the ARE of the modified test with respect to t-test is 99.22% (MANN-WHITNEY test: 95.45%). The lower bound of the ARE of the modified test is 92.50% for all distributions with a continuous density function and a finite variance (MANN-WHITNEY test: 86.4%).

Quantitative Description and Mathematical Modelling of Biological Systems by Means of Nonlinear Approximations – Demonstrated at the Example of the Body Length Growth of Man

Abstract: After general remarks upon the biomathematical tasks of quantitative description or mathematical modelling of biological systems and phenomena on different levels of organization the growth process of body length of human beings serves as illustration of the possibilities of such mathematical treatments. A short survey is given about the attempts of using differential equations for reflecting basic principles of growth processes. Further on analytical functions (as the solutions of growth differential equations) preferably may be used for an objective quantitative reproducing of measured courses of growth variables. The values of parameters contained in such expressions will be calculated by nonlinear fitting procedures. In nearly all cases we got a very high numerical precision in approximating parts of the body length growth process of man which extends from conception until the age of 18. Our attempts of a phenomenologic-mathematical modelling of this growth process mean a subdividing of the whole process in single growth spurts with a symmetric-sigmoidal shape of their courses, and describing the single spurt by a hyperbolic tangent function. An advantage of this kind of modelling is the fact that each of the terms is valid for the whole time interval of growth. Within the process of forming the model heuristical hints are taken into consideration arising from biological background and which would supply the possibility of interpretation and verification of the analyzed growth spurts.

Percentage Points of Multivariate t Distribution with Zero Correlations and their Application

Abstract: The k-variate t distribution plays an important role in the application of Multiple Comparison Procedures, Ranking and Selection Procedures, and Estimation of Ranked Parameters. This distribution was considered by several authors in the past and is now widely used. In this paper, tables of some percentage points of the multivariate t distribution are tabulated for given probability level γ = 0.80, 0.90, 0.95, and 0.99; population size k = 2(1)20; and degrees of freedom v=2(1) 30(5)60 with common correlation coefficient ϱ = 0. The accuracy of the percentage points is specifically determined (and is up to the fourth decimal place). An application of these tables is considered for interval estimation of the largest mean of k(≧2) normal populations.

Signed‐Rank Tests in the Presence of Ties (with extended tables)

Abstract: We begin with a review of the areas of application of the signed-rank tests (SRTs) and we conclude that the results are exact only if no ties of non-null differences exist. In order to apply the SRTs according to WILCOXON and according to PRATT also in the presence of ties, by assigning midranks, we derive their null distributions. As special cases the null distributions for the problem without ties are obtained. In order to save the practising statistician the time-consuming calculations of the distribution functions, we compute tables of critical values (for reasons of volume they will be published as part of the reprints only). For N0 = 0 (1) 5 null differences and M = = 1(1) 10 non-null differences the critical values of all distributions with all possible tie vectors are calculated. Instructions are provided and an example serves to illustrate the use of the table. The extension of the tables are obtained by means of counting formulas given in the text. Approximations are provided in order to make the application of tests possible for larger samples as well. It is shown that the approximation of the null distribution in the presence of ties by the null distributions under the assumption of no ties in some cases overstates and sometimes understates the exact rejection probability. For N0 = 0 (1) 10 and M = 1 (1) 10 all distributions with all possible tie vectors for the SRTs with WILCOXON and PRATT ranking are examined with respect to the lattice type of the test statistic. The result is given in table 6. It is evident that the portion of PRATT-distributions with lattice character decreases as the number of null differences increases. Continuity corrections are obtained for the asymptotic normal distribution which take into account the lattice character of the distribution of the test statistic.

Mathematical Models for the Spread and Control of Foot-and-Mouth Disease During the 1973 Epidemic in Austria

Abstract: Two models are studied to describe the spread of foot-and-mouth disease during the 1973 epidemic in Austria. The models are modifications of a three-class-model (due to KERMACK and MCKENDRICK) taking into account the special character of the disease as well as control methods (stamping-out-method, vaccination programs).

Group Divisible Second Order Rotatable Designs

Abstract: The Group Divisible Rotatable (GDR) designs are the designs in which the factors get divided into groups such that for the factors within group, the designs are rotatable. In the present paper we have obtained a series of Group Divisible Second Order Rotatable designs, by decomposing the v-dimensional space corresponding to v-factors under consideration into three mutually orthogonal spaces. We have given the least squares estimates of the parameters, the analysis and construction of such designs.

The Heritability of Difference Scores when Environments are Correlated

Abstract: The statistical analysis of difference scores (contrasts) is a fundamental problem in all learning, feeding, and training experiments and tests, and in longitudinal studies of growth and development. Outgoing from the analogy between the mathematical models of classical psychological test theory and quantitative genetics, as well as between the parameters reliability and heritability of these models, the present paper derives the formulas of the heritability of difference scores in general cases where it is not assumed that environmental deviations on distinct tests and measurements are uncorrelated. Contrary to the assertion, made by FELDMAN and LEWONTIN, heritabilities in the broad sense can be used as ideal weighting factors in long-range personnel index selection. Longitudinal studies of twins and the cotwin method are powerful experimental designs to estimate heritabilities of differences.