Markov branching processes with varying growth rates

A branching processZ(t) which behaves as Markov branching processesZ ₁(t) andZ ₂(t) during the free and dead times of a counter process is considered. Expression forE[Z(t)] is given.     

Pairwise Comparisons Using Trimmed Means or M-Estimators when Working with Dependent Groups

For J dependent groups, let θ_j, j = 1, …, J, be some measure of location associated with the jth group. A common goal is computing confidence intervals for the pairwise differences, θ_j — θ_k, j < k, such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy-tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey-McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one-step M-estimator is also considered but found to be less satisfactory.     

The Matched Pairs Problem with Exponential Variates

Two classes of tests for the hypothesis of bivariate symmetry are studied. For paired exponential survival times (t_1j, t_2j), the classes of tests are those based on t_1j-t_2j and those based on log t_1j–log t_2j. For each class the sign, signed ranks, t and likelihood ratio tests are compared via Pitman's criterion of asymptotic relative efficiency (ARE). For tests based on t_1j — t_2j, it is found that: (1) the efficacy of the paired t depends on the coefficient of variation (CV) of the pair means, (2) the signed rank test has the same ARE to the sign test as for the usual location problem. For tests based on log t_1j — log t_2j, the ARE comparisons reduce to the well-known results for the one-sample location problem for samples from a logistic density. Hence, the signed rank test is asymptotically efficient. Furthermore, analyses based on log t_1j — log t_2j are not complicated by the underlying pairing mechanism.     

Using a Normal Approximation to Test for the Binomial Distribution

The commonly used method to test for the binomial distribution is the x²-test. In this paper, we introduce an alternative method to test for the binomial distribution. Suppose N is the number of sample groups with n individuals each, x_ij is the jth sample in ith group, a Bernoulli variable with parameter and VVI=s²/[m(1 - m)/n]. Then it is well know that the asymptotic distribution of the statistic (N - 1) VVI is x²(N - 1) under the hypothesis p₁ = p₂ = … = p_N. Here we find that VVI has an asymptotic normal distribution N(1, 2(1 - 1/n)/(N - 1)). Unlike the x²-statistic, the variance of the normal test statistic is a function of n. This method is convenient in detecting spatial patterns and dispersion in the study of diseased organisms (e.g., plants) in field samples.     

Kinetics of biopolymerization on nucleic acid templates

The kinetics of biopolymerization on nucleic acid templates is discussed. The model introduced allows for the simultaneous synthesis of several chains, of a given type, on a common template, e.g., the polyribosome situation. Each growth center [growing chain end plus enzyme(s)] moves one template site at a time, but blocks L adjacent sites. Solutions are found for the probability n_j(t) that a template has a growing center that occupies the sites j — L + 1,…, j at time t. Two special sets of solutions are considered, the uniform-density solutions, for which n_j(t) = n, and the more general steady-state solutions, for which dn_j(t)/dt = 0. In the uniform-density case, there is an upper bound to the range of rates of polymerization that can occur. Corresponding to this maximum rate, there is one uniform solution. For a polymerization rate less than this maximum, there are two uniform solutions that give the same rate. In the steady-state case, only L = 1 is discussed. For a steady-state polymerization rate less than the maximum uniform-density rate, the steady-state solutions consist of either one or two regions of nearly uniform density, with the density value(s) assumed in the uniform region(s) being either or both of the uniform-density solutions corresponding to that polymerization rate. For a steady-state polymerization rate equal to or slightly larger than the maximum uniform-density rate, the steady-state solutions are nearly uniform to the single uniform-density solution for the maximum rate. The boundary conditions (rate of initiation and rate, of release of completed chains from the template) govern the choice among the possible solutions, i.e., determine the region(s) of uniformity and the value(s) assumed in the uniform region(s).     

Properties and Application of the Location-Test V (n,k)

The present paper is concerned with the properties of a test statistic V(n, k) to test location differences in the one-sample case with known hypothetical distribution G(x). The test is similar to the WILCOXON two-sample statistic after replacement of the second sample by quantiles of the hypothetical distribution. A comparison with the exact distribution of V(n, k) shows that an approximation by means of the normal distribution provides good results even for small sample sizes. The V-test is unbiased against one-tailed alternatives and it is consistent with a restriction which is hardly relevant in practical applications. With regard to the application we are interested especially in the power and robustness against extreme observations for small sample size n. It is shown that in a normal distribution with known standard deviation V(n, k) is more powerful than STUDENT's t for small n and more robust in the sense considered here. The test statistic is based on grouping of the observations into classes of equal expected frequency. A generalization to arbitrary classes provides an essential extension of applicability such as to discrete distributions and to situations where only relative frequencies of G(x) in fixed classes are known.     

Molecular-Dynamics Simulations for the Density Autocorrelation Function in a Supercooled Fluid Phase

Abstract

We have re-calculated the self part of the density autocorrelation function F_s(k, t) (incoherent scattering function) for the binary soft-sphere fluid with a much longer molecular-dynamics (MD) simulation than our previous MD calculations, and with a larger system size (N = 4000) to a longer time window as well as to study a system-size dependence, if it exists. The full density autocorrelation function F(k, t) was also computed. It is found that all F(k, t)'s that we have computed in this work can be fitted over a wide range of time steps (at least over three figures of the decay) by a Williams-Watts stretched exponential function F_s(k, t) = A exp [— (t/t ₀)^β], where A, β and t ₀ are adjustable parameters. Other significant dynamical behaviours were also presented in mean square displacements and non-Gaussian parameters for highly supercooled fluids with N = 4000. The present results are compatible to our previous computations with N = 500, but a significant size dependence is suggested.     

Almost-one Parameter Hypothesis of Individuality in Height Growth

Thus far an individual height growth curve h_ij(t) of the i-th person in the j-th period, t being his (or her) age, has been studied as a function of t associated with its velocity curve. In this note we introduce a natural scale X(t) in place of t, which linearizes this personal curve and facilitates its analysis, in the sense that this equation of growth contains apparently two personal parameters for one period but one of them plays an essential role. The effectiveness of this approach will be seen in four figures.     

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

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.     

Malthusian parameters,reproductive values and change under selection in self fertilizing age-structured populations

A population, reproducing wholly by selfing, is assumed to be observed at times . Individuals between x–1 and x units of age at time t are said to be in age class x at that time. The rate of increase in the long run of individuals of type A_iAj is denoted by m_ij+1=m_ji+1. For each genotype there is also a set of reproductive values, corresponding to all age classes and genotypes of individuals having descendants of that genotype. Then, if the number of individuals of each sort of ancestor is multiplied by its reproductive value and the products are summed, the result is the total value, which is V_ij(t) for genotype A_iAj. Then V_ij(t+1)–V_ij(t) is equal to m_ijV_ij(t), where m_ij is the Malthusian parameter for A_iAj. Furthermore, if the mean and variance at time t of the m_ijs, weighted by their corresponding reproductive values, are respectively (t) and _m²(t), then m¯(t+1)–m¯(t)=_m²(t)/(1+m¯(t)).     

Improved power of familywise error rate procedures for discrete data under dependency

In many applications where it is necessary to test multiple hypotheses simultaneously, the data encountered are discrete. In such cases, it is important for multiplicity adjustment to take into account the discreteness of the distributions of the p‐values, to assure that the procedure is not overly conservative. In this paper, we review some known multiple testing procedures for discrete data that control the familywise error rate, the probability of making any false rejection. Taking advantage of the fact that the exact permutation or exact pairwise permutation distributions of the p‐values can often be determined when the sample size is small, we investigate procedures that incorporate the dependence structure through the exact permutation distribution and propose two new procedures that incorporate the exact pairwise permutation distributions. A step‐up procedure is also proposed that accounts for the discreteness of the data. The performance of the proposed procedures is investigated through simulation studies and two applications. The results show that by incorporating both discreteness and dependency of p‐value distributions, gains in power can be achieved.     

A mathematical model of biological evolution

In order to understand generally how the biological evolution rate depends on relevant parameters such as mutation rate, intensity of selection pressure and its persistence time, the following mathematical model is proposed: dN _n (t)/dt=(m _n (t-)N _n (t)+N _n-1(t) (n=0,1,2,3...), where N _n (t) and m _n (t) are respectively the number and Malthusian parameter of replicons with step number n in a population at time t and is the mutation rate, assumed to be a positive constant. The step number of each replicon is defined as either equal to or larger by one than that of its parent, the latter case occurring when and only when mutation has taken place. The average evolution rate defined by is rigorously obtained for the case (i) m _n (t)=m _n is independent of t (constant fitness model), where m _n is essentially periodic with respect to n, and for the case (ii) (periodic fitness model), together with the long time average m of the average Malthusian parameter . The biological meaning of the results is discussed, comparing them with the features of actual molecular evolution and with some results of computer simulation of the model for finite populations.An early version of this study was read at the International Symposium on Mathematical Topics in Biological held in kyoto, Japan, on September 11–12, 1978, and was published in its Procedings.     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

ALGORITHM FOR APPLICATION OF FOURIER ANALYSIS FOR BIORHYTHMIC BASELINES OF PHARMACODYNAMIC INDIRECT RESPONSE MODELS

The change of an indirect pharmacological response R(t) can be described by a periodic time-dependent production rate k_in (t) and a first-order loss constant k_out. If k_in(t) follows some biological rhythm (e.g., circadian), then the response R(t) also displays a periodic behavior. A new approach for describing the input function in indirect response models with biorhythmic baselines of physiologic substances is introduced. The present approach uses the baseline (placebo) response R_b(t) to recover the equation for k_in(t). Fourier analysis provides an approximate equation for R_b(t) that consists of terms (usually two or three) of the Fourier series (harmonics) that contribute most to the overall sum. The model differential equation is solved backward for k_in(t), yielding the equation involving R_b(t). A computer program was developed to perform the square L²-norm approximation technique. Fourier analysis was also performed based on nonlinear regression. Cortisol suppression after inhalation of fluticasone propionate (FP) was modeled based on the inhibition of the secretion rate k_in(t) using ADAPT II. The pharmacodynamic parameters k_out and IC₅₀ were estimated from the model equation with k_in(t) derived by the new approach. The proposed method of describing the input function needs no assumption about the behavior of k_in(t), is as efficient as methods used previously, and is more flexible in describing the baseline data than the nonlinear regression method. (Chronobiology International, 17(1), 77–93, 2000)     

On washout in nonlinear compartmental systems

Let x(t) be a solution of a compartmental system. If, for some compartment j, x_j(t)→0 as t→∞, then we say that the compartment j washes out. We show that a compartment washes out if it always reaches (along a fixed path) either the environment or another compartment for which there is no return path. Additional criteria, particularly regarding exponential convergence, are also presented. Examples are drawn from tracer kinetics, enzyme reactions, and epidemic models.     

Cumulative Damage Survival Models for the Randomized Complete Block Design

A continuous time discrete state cumulative damage process {X(t), t ≥ 0} is considered, based on a non‐homogeneous Poisson hit‐count process and discrete distribution of damage per hit, which can be negative binomial, Neyman type A, Polya‐Aeppli or Lagrangian Poisson. Intensity functions considered for the Poisson process comprise a flexible three‐parameter family. The survival function is S(t) = P(X(t) ≤ L) where L is fixed. Individual variation is accounted for within the construction for the initial damage distribution {P(X(0) = x) | x = 0, 1, …,}. This distribution has an essential cut‐off before x = L and the distribution of L – X(0) may be considered a tolerance distribution. A multivariate extension appropriate for the randomized complete block design is developed by constructing dependence in the initial damage distributions. Our multivariate model is applied (via maximum likelihood) to litter‐matched tumorigenesis data for rats. The litter effect accounts for 5.9 percent of the variance of the individual effect. Cumulative damage hazard functions are compared to nonparametric hazard functions and to hazard functions obtained from the PVF‐Weibull frailty model. The cumulative damage model has greater dimensionality for interpretation compared to other models, owing principally to the intensity function part of the model.     

Structure and magnetic properties of a tetranuclear Cu4O4 open-cubane in [Cu(L)]4·4H2O with L = (E)-N′-(2-oxy-3-methoxybenzylidene)benzohydrazide

The in-situ formed hydrazone Schiff base ligand (E)-N′-(2-oxy-3-methoxybenzylidene)benzohydrazide (L²⁻) reacts with copper(II) acetate to a tetranuclear open cubane [Cu(L)]₄ complex which crystallizes as two symmetry-independent (Z′ = 2) S₄-symmetrical molecules in different twofold special positions with a homodromic water tetramer. The two independent (A and B) open- or pseudo-cubanes with Cu₄O₄ cores of 4 + 2 class (Ruiz classification) each have three different magnetic exchange pathways leading to an overall antiferromagnetic coupling with J_1B = J_2B = −17.2 cm⁻¹, J_1A = −36.7 cm⁻¹, J_2A = −159 cm⁻¹, J_3A = J_3B = 33.5 cm⁻¹, g = 2.40 and ρ = 0.0687. The magnetic properties have been analysed using the H = −Σ_i,jJ_ij(S_iS_j) spin Hamiltonian.     

On a Characterization of the Exponential Distribution by Weak Homoscedasticity of Record Values

A sequence {X_n, n≤1} of independent and identically distributed random values with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j ≤ max (X₁, X₂, …, X_j?1). We define L(n)=min.(j!j>L(n?1.), X_j<X_L(n?1)), with L(0) = 1. Let Z_n=X_L(n)? X_L(n?1), n ≤ 1. We will show that the conditional variance of Z_n given X_L(n?1)=x does not depend on × if and only if F(x) is exponential.     

A model for continuous production of thermally induced recombinant proteins

Summary General conditions for continuous expression of heterologous genes fromP _L promoter in two fermenters connected in series have been established. The induction time of the bacterial cells is calculated as a function of the retention time in the inducing reactor. Using this model, it is possible to adapt fermentation parameters to the particular behaviour of any specific recombinant clone.Nomenclature F(t) flow at timet [ml/min] - M _T (t) culture induced, at timeT of fermentation, during a period up tot [ml] - N _T (t) culture induced, at timeT of fermentation, during a period fromt tot+dt [ml] - p(t) product yield in a discontinuous culture [units/ml] - P(t) product yield at the outlet of the fermenter [units/ml] - v(t) volume of culture entered into the inducing reactor up to timet [ml] - V volume of the inducing reactor [ml] Greek letters retention time in the inducing reactor [min] - (t) average induction time at timet [min]