Quantification of the Effects of Age on the Dose Response of Variola major in Suckling Mice

Dose-response data for Variola major (V. major), the causative agent of smallpox, were obtained from the open literature, summarized, and fitted with three dose-response models. It is known from prior outbreak experience that there is generally a difference in infectivity of the agent and its subsequent mortality depending on the age of the patient. A source of animal dose-response data were found with age delineation for the exposure group (suckling mice, intraperitoneal exposure). This delineation was used to adapt current dose-response models to include an age dependency parameter. The degree of the models' fit to the data was ascertained using maximum likelihood estimation (MLE). The effect of host age could be described quantitatively using modifications to the beta Poisson and exponential dose-response models. The modifications improvement in the accuracy of risk prediction by 72% for the beta Poisson model and 7% for the exponential model, compared to the original (unmodified) models.     

Paternal age and Down's syndrome genotypes diagnosed prenatally: No association in New York State data

Summary An investigation of a paternal age effect independent of maternal age was undertaken for 98 cases of Down's syndrome genotypes diagnosed prenatally compared to 10,329 fetuses with normal genotype diagnosed prenatally in data reported to the New York State Chromosome Registry. The mean of the difference (delta) in paternal age of cases compared to those with normal genotypes after controlling for maternal age, was slightly negative,-0.27 with a 95% confidence interval of-1.59 to +1.06. A regression analysis was also done in which the data were first fit to an equation of the type lny=(bx+c) and then to the equation ln y=(bx+dz+c) where y = rate of Down's syndrome, x = maternal age, z = paternal age, and b, d, and c are parameters. This also revealed no evidence for a paternal age effect. The value of d (the paternal age coefficient) was in fact slightly negative,-0.0058, with an asymptotic 95% confidence interval of-0.0379 to +0.0263. Lastly, multiple applications of the Mantel-Haenszel test considering various boundaries in paternal age also revealed no statistically significant evidence for a paternal age effect independent of maternal age. These results are at variance with claims of others elsewhere of a very strong paternal age effect detected in studies at prenatal diagnoses. Five different hypotheses are suggested which may account for discrepancies among studies to date in findings on paternal age effects for Down's syndrome: (i) there are temporal, geographic, or ethnic variations in paternal age effects, (ii) there is no paternal age effect and statistical fluctuation accounts for all trends to date; (iii) methologic artifacts have obscured a paternal age effect in some studies which did not find a positive outcome; (iv) methodologic artifacts are responsible for the positive results in some studies to date; (v) there is a rather weak paternal age effect independent of maternal age in most if not all populations, but because of statistical fluctuation the results are significant only in some data sets. The results of all data sets to date which we have been able to analyze by one year intervals are consistent with a mean delta of +0.04 to +0.48 and in the value of d (the paternal age coefficient) of +0.006 to +0.017, and it appears the fifth hypothesis cannot be excluded. Projections based on this assumption are presented.     

Arrhenius and Gleason revisited: new hybrid models resolve an old controversy

Aim Studies have typically employed species–area relationships (SARs) from sample areas to fit either the power relationship or the logarithmic (exponential) relationship. However, the plots from empirical data often fall between these models. This article proposes two complementary and hybrid models as solutions to the controversy regarding which model best fits sample‐area SARs. Methods The two models are and , where S_A is number of species in an area, A, where z, b, c₁ and c₂ are predetermined parameters found by calculation, and where d and n are parameters to be fitted. The number of parameters is reduced from six to two by fixing the model at either end of the scale window of the data set, a step that is justified by the condition that the error or the bias, or both, in the first and the last data points is negligible. The new hybrid models as well as the power model and the logarithmic model are fitted to 10 data sets. Results The two proposed models fit well not only to Arrhenius’ and Gleason’s data sets, but also to the other six data sets. They also provide a good fit to data sets that follow a sigmoid (or triphasic) shape in log–log space and to data sets that do not fall between the power model and the logarithmic model. The log‐transformation of the dependent variable, S, does not affect the curve fit appreciably, although it enhances the performance of the new models somewhat. Main conclusions Sample‐area SARs have previously been shown to be convex upward, convex downward (concave), sigmoid and inverted sigmoid in log–log space. The new hybrid models describe successfully data sets with all these curve shapes, and should therefore produce good fits also to what are termed triphasic SARs.     

Dose-response time modelling for highly pathogenic avian influenza A (H5N1) virus infection

Aims: To develop time‐dependent dose–response models for highly pathogenic avian influenza A (HPAI) of the H5N1 subtype virus. Methods and Results: A total of four candidate time‐dependent dose–response models were fitted to four survival data sets for animals (mice or ferrets) exposed to graded doses of HPAI H5N1 virus using the maximum‐likelihood estimation. A beta‐Poisson dose–response model with the N₅₀ parameter modified by an exponential‐inverse‐power time dependency or an exponential dose–response model with the k parameter modified by an exponential‐inverse time dependency provided a statistically adequate fit to the observed survival data. Conclusions: We have successfully developed the time‐dependent dose–response models to describe the mortality of animals exposed to an HPAI H5N1 virus. The developed model describes the mortality over time and represents observed experimental responses accurately. Significance and Impact of the Study: This is the first study describing time‐dependent dose–response models for HPAI H5N1 virus. The developed models will be a useful tool for estimating the mortality of HPAI H5N1 virus, which may depend on time postexposure, for the preparation of a future influenza pandemic caused by this lethal virus.     

Muscle power, locomotor performance and flexibility in aging mentally-retarded adults with and without Down's syndrome

Longer life expectancy is resulting in increasing numbers of elderly adults with mental retardation (MR). The objective of the study was to compare lower limb isokinetic muscle power, locomotor performance and flexibility of aged adult mentally-retarded individuals with and without Down's syndrome (DS). Nine subjects with MR and DS (mean age 61), and sixteen subjects with MR and without DS (mean age 63), performed leg power testing on a Biodex dynamometer. Parameters measured were dynamic torque, dynamic torque % body weight, and average power % body weight. Functional performance tests including "Timed Get-Up and Go" and flexibility were also analyzed and compared. Results indicate that in knee extension and flexion isokinetic power the MR group without DS showed significantly higher scores than the MR group with DS. The functional performance of elderly adults with MR and DS was significantly impaired compared with MR adults without DS, although no differences were observed between the two groups in the flexibility tests. It was concluded that muscle leg power, and gross motor performance of elderly mentally-retarded individuals without Down's syndrome is better than in those with Down's syndrome.     

A simple function for maternal-age-specific rates of Down syndrome in the 20-to-49-year age range and its biological implications.

The familial increase in the rate of Down syndrome with maternal age can be represented by a simple equation, consisting of the sum of a constant term plus an exponential term that is a first-order function of masternal age: y = a + exp (b + cx), where y is the rate in live births, x is maternal age, and a, b, and c are constants. Unlikely analyses in which two separate equations were derived from different segments of the 20 to 49 maternal age range, this single, simple equation can be applied to the entire range. An unlike previous complex equations that were derived by regression analysis for the entire age range, the component terms can be readily understood as contributions by different etiologic categories. This model fits the data recently available by 1-year intervals about as well as the approach that used separate equations, but it has fewer parameters and requires no ad hoc division of the age range. However, it does not postulate a sharp transition in biological processes around maternal age 30, but, rather, a process continuously accumulating at a constant exponential rate (analogous to that produced by an infectious mechanism), superimposed upon a constant background rate.     

The natural variability of vital rates and associated statistics

The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.     

Reexamination of paternal age effect in Down's syndrome

Summary The recent discovery that the extra chromosome in about 30% of cases of 47, trisomy 21 is of paternal origin has revived interest in the possibility of paternal age as a risk factor for a Down syndrome birth, independent of maternal age. Parental age distribution for 611 Down's syndrome 47,+21 cases was studied. The mean paternal age was 0.16 year greater than in the entire population of live births after controlling for maternal age. There was no evidence for a significant paternal age effect at the 0.05 level. For 242 of these Down's syndrome cases, control subjects were selected by rigidly matching in a systematic manner. Paternal age was the variable studied, with maternal age and time and place of birth controlled. There was no statistically significant association between paternal age and Down's syndrome. After adjustment for maternal age, these two studies were not consistent with an increase of paternal age in Down's syndrome.     

The incidence of Down's syndrome and progress towards its reduction

Down's syndrome is the most common autosomal aberration and single cause of mental retardation in man. There is a close relation between advanced maternal age and Down's syndrome. The limitation of family size has made a considerable impact on the incidence of Down's syndrome. In Denmark in the 1950s, 50% of Down's syndrome cases were born to mothers over the age of 35. The percentage went down to 25% in the 1970s and was reduced by prenatal diagnosis to 8% in the 1980s. For the period 1980-85 we followed the birth prevalence closely for different maternal age groups. The birth prevalence was lowered for the age group over 35, but there was a steady rise for the age groups below 35. Early diagnosis, high rate of survival of light-for-date babies and babies with congenital heart defect, and, possibly, exogenous factors working on gametogenesis might be an explanation. To achieve a reduction in incidence, maternal alpha-fetoprotein (AFP)-serum screening for low values may be a possibility. So far, avoidance, but not primary prevention, of Down's syndrome is available.     

Distributions of the use frequencies of amino acids in the hypervariable regions of immunoglobulins

Two types of distributions for the frequencies of occurrence of amino acids in each position of hypervariable regions CDR-1 and CDR-2 were obtained for 2,000 immunoglobulins. The results show that some positions fit an inverse power-law distribution, while others fit an exponential-type distribution. As a result of comparison with structural data in the literature it is proposed that sites in which the frequency distribution fits the inverse power law are critical to maintaining canonical shapes of the recognition regions or are involved in modulating these canonical conformations, while those sites where the distribution fits the exponential law are those which should be exclusively involved in the recognition mechanism. Correspondence to: F. Lara-Ochoa     

Modeling Human Mortality from All Diseases in the Five Most Populated Countries of the European Union

Age affects mortality from diseases differently than it affects mortality from external causes, such as accidents. Exclusion of the latter leads to the “all-diseases” category. The age trajectories of mortality from all diseases are studied in the five most populated countries of the EU, and the shape of these 156 age trajectories is investigated in detail. The arithmetic mean of ages where mortality reaches a minimal value is 8.47 years with a 95% confidence interval of [8.08, 8.85] years. Two simple deterministic models fit the age trajectories on the two sides of the mortality minimum. The inverse relationship is valid in all cases prior to this mortality minimum and death rates exactly decreased to three thousandths of its original size during the first 3000 days. After the mortality minimum, the standard Gompertz model fits the data in 63 cases, and the Gompertz model extended by a small quadratic element fits the remaining 93 cases. This analysis indicates that the exponential increase begins before the age of 15 years and that it is overshadowed by non-biological causes. Therefore, the existence of a mechanism switching that would explain the exponential increase in mortality after the age of 35 years is unlikely.     

Process‐based functions for seed retention on animals: a test of improved descriptions of dispersal using multiple data sets

Studies of external seed transport on animals usually assume that the probability of detachment is constant, so that seed retention should show a simple exponential relationship with time. This assumption has not been tested explicitly, and may lead to inaccurate representation of long distance seed dispersal by animals. We test the assumption by comparing the fit to empirical data of simple, two‐parameter functions. Fifty‐two data sets were obtained from five published studies, describing seed retention of 32 plant species on sheep, cattle, deer, goats and mice. Model selection suggested a simple exponential function was adequate for data sets in which seed retention was followed for short periods ( <48 h). The data gathered over longer periods (49–219 days) were best described by the power exponential function, a form of the stretched exponential which allows a changing dropping rate. In these cases the power exponential showed that seed dropping rate decreased with time, suggesting that seeds vary in attachment, with some seeds becoming deeply buried or wound up in the animal's coat. Comparison of fitted parameters across all the data sets also confirmed that seeds with adhesive structures have lower dropping rates than those without. We conclude that the seed dropping rate often changes with time during external transport on animals and that the power exponential is an effective function to describe this change. We advise that, to analyse seed dropping rates adequately, retention should be measured over reasonable time periods – until most seeds are dropped – and both the simple and power exponential functions should be fitted to the resulting data. To increase its utility, we provide functions describing the seed dropping rate and the dispersal kernel resulting from the power exponential relationship.     

The gambin model provides a superior fit to species abundance distributions with a single free parameter: evidence,implementation and interpretation

The species abundance distribution (SAD) has been a central focus of community ecology for over fifty years, and is currently the subject of widespread renewed interest. The gambin model has recently been proposed as a model that provides a superior fit to commonly preferred SAD models. It has also been argued that the model's single parameter (α) presents a potentially informative ecological diversity metric, because it summarises the shape of the SAD in a single number. Despite this potential, few empirical tests of the model have been undertaken, perhaps because the necessary methods and software for fitting the model have not existed. Here, we derive a maximum likelihood method to fit the model, and use it to undertake a comprehensive comparative analysis of the fit of the gambin model. The functions and computational code to fit the model are incorporated in a newly developed free‐to‐download R package (gambin). We test the gambin model using a variety of datasets and compare the fit of the gambin model to fits obtained using the Poisson lognormal, logseries and zero‐sum multinomial distributions. We found that gambin almost universally provided a better fit to the data and that the fit was consistent for a variety of sample grain sizes. We demonstrate how α can be used to differentiate intelligibly between community structures of Azorean arthropods sampled in different land use types. We conclude that gambin presents a flexible model capable of fitting a wide variety of observed SAD data, while providing a useful index of SAD form in its single fitted parameter. As such, gambin has wide potential applicability in the study of SADs, and ecology more generally.     

Spore Dispersal and Plant Disease Gradients; a Comparison between two Empirical Models

Power law and exponential models were fitted to 325 sets of observations which described decreases with distance in deposition of air-borne or splash-borne spores, or pollen, or in amounts of plant disease caused by fungi, bacteria or viruses. There, was generally little difference between the models in the goodness of fit to these data, although deposition gradients for spores borne in splash droplets were fitted better by exponential models and gradients for fungi with air-borne spores less than 10 μm in diameter were fitted better by power law models. The exponential model has the property that the observed variable decreases by half as the distance from the source increases by a constant increment (the half-distance); this provides a measureof the gradient that is more easy to visualize than the exponent in power law model. The half-distances of gradients for air-borne pathogens were greater than those for splash-borne or soil-borne pathogens. The exponential model is easier to incorporate into models of disease development than the power law model because the boundary condition at the source (the estimated number of spores or amount of disease at the source) is finite rather than infinite. However, both these empirical models have limitations and should not be extrapolated to distances outside the observed range.     

Shapes and functions of species–area curves (II): a review of new models and parameterizations

This paper has extended and updated my earlier list and analysis of candidate models used in theoretical modelling and empirical examination of species–area relationships (SARs). I have also reviewed trivariate models that can be applied to include a second independent variable (in addition to area) and discussed extensively the justifications for fitting curves to SARs and the choice of model. There is also a summary of the characteristics of several new candidate models, especially extended power models, logarithmic models and parameterizations of the negative-exponential family and the logistic family. I have, moreover, examined the characteristics and shapes of trivariate linear, logarithmic and power models, including combination variables and interaction terms. The choice of models according to best fit may conflict with problems of non-normality or heteroscedasticity. The need to compare parameter estimates between data sets should also affect model choice. With few data points and large scatter, models with few parameters are often preferable. With narrow-scale windows, even inflexible models such as the power model and the logarithmic model may produce good fits, whereas with wider-scale windows where inflexible models do not fit well, more flexible models such as the second persistence (P2) model and the cumulative Weibull distribution may be preferable. When extrapolations and expected shapes are important, one should consider models with expected shapes, e.g. the power model for sample area curves and the P2 model for isolate curves. The choice of trivariate models poses special challenges, which one can more effectively evaluate by inspecting graphical plots.     

The use of a correlated binomial model for the analysis of certain toxicological experiments

In certain toxicological experiments with laboratory animals, the outcome of interest is the occurrence of dead or malformed fetuses in a litter. Previous investigations have shown that the simple one-parameter binomial and Poisson models generally provide poor fits to this type of binary data. In this paper, a type of correlated binomial model is proposed for use in this situation. First, the model is described in detail and is compared to a beta-binomial model proposed by Williams (1975). These two-parameter models are then contrasted for goodness of fit to some real-life data. Finally, numerical examples are given in which likelihood ratio tests based on these models are employed to assess the significance of treatment-control differences.     

Paternal age and Down syndrome.

The frequency of Down syndrome (DS) in infants of older fathers has been examined in two sets of data. The effect of maternal age was controlled by single years of age. Lack of tight control has been an important weakness of other studies on this subject. Data obtained in metropolitan Atlanta by an intensive case-ascertainment program showed no overall excess of DS infants born to older fathers. Nor was there evidence of such an effect in recent birth certificate data made available by the National Center for Health Statistics. The Atlanta data suggest an increased number of DS infants born to older fathers who had children by women less than or equal to 34 years. However, there was a small deficiency of DS infants born to older fathers by women greater than or equal to 35 years. The possibility of a paternal-age effect remains open, but the available data suggest that, if it exists, it is quite small.     

Reexamination of paternal age effect in Down's syndrome

Summary Paternal age distribution for 1279 cases of Down's syndrome born in 1952–1968 was compared with the corresponding distribution for the general population, corrected for the maternal age as well as for the year of birth of the patients. Although there was no difference in the mean paternal age, the two distributions differed significantly, largely due to the excess of fathers aged 55 years and over and to the deficit of those aged 40–44 years in the patients born to mothers aged 30 years and over. The overall pattern of the relative incidence of Down's syndrome with advancing paternal age, with maternal age controlled, seems consistent with the hypothesis proposed by Stene et al. (1977). It increased from 0.8 for fathers aged 20–24 years slowly up to 1.2 for those aged 45–49 years, though with an intermediate drop to 0.8 at the age of 40–44 years, and then sharply to 2.4 for those aged 55 years and over. This rising pattern of the relative incidence with paternal age was essentially the same for the patients born in 1952–1960 and for those born in 1961–1968, although the slope was less steep in the latter than in the former group.This paper is dedicated to Professor Heinrich Schade in honor of his 70th birthday