Negative Binomial Function Research Articles

ScLEGA: an attention-based deep clustering method with a tendency for low expression of genes on single-cell RNA-seq data.

Single-cell RNA sequencing (scRNA-seq) enables the exploration of biological heterogeneity among different cell types within tissues at a resolution. Inferring cell types within tissues is foundational for downstream research. Most existing methods for cell type inference based on scRNA-seq data primarily utilize highly variable genes (HVGs) with higher expression levels as clustering features, overlooking the contribution of HVGs with lower expression levels. To address this, we have designed a novel cell type inference method for scRNA-seq data, termed scLEGA. scLEGA employs a novel zero-inflated negative binomial (ZINB) loss function that fully considers the contribution of genes with lower expression levels and combines two distinct scRNA-seq clustering strategies through a multi-head attention mechanism. It utilizes a low-expression optimized denoising autoencoder, based on the novel ZINB model, to extract low-dimensional features and handle dropout events, and a GCN-based graph autoencoder (GAE) that leverages neighbor information to guide dimensionality reduction. The iterative fusion of denoising and topological embedding in scLEGA facilitates the acquisition of cluster-friendly cell representations in the hidden embedding, where similar cells are brought closer together. Compared to 12 state-of-the-art cell type inference methods on 15 scRNA-seq datasets, scLEGA demonstrates superior performance in clustering accuracy, scalability, and stability. Our scLEGA model codes are freely available at https://github.com/Masonze/scLEGA-main.

Concentration for multiplier empirical processes with dependent weights

<abstract><p>A novel concentration inequality for the sum of independent sub-Gaussian variables with random dependent weights is introduced in statistical settings for high-dimensional data. The random dependent weights are functions of some regularized estimators. We applied the proposed concentration inequality to obtain a high probability bound for the stochastic Lipschitz constant for negative binomial loss functions involved in Lasso-penalized negative binomial regressions. We used this bound to study oracle inequalities for Lasso estimators. Additionally, a similar concentration inequality was derived for a randomly weighted sum of independent centred exponential family variables.</p></abstract>

Loop detection using Hi-C data with HiCExplorer.

BackgroundChromatin loops are an essential factor in the structural organization of the genome; however, their detection in Hi-C interaction matrices is a challenging and compute-intensive task. The approach presented here, integrated into the HiCExplorer software, shows a chromatin loop detection algorithm that applies a strict candidate selection based on continuous negative binomial distributions and performs a Wilcoxon rank-sum test to detect enriched Hi-C interactions.ResultsHiCExplorer’s loop detection has a high detection rate and accuracy. It is the fastest available CPU implementation and utilizes all threads offered by modern multicore platforms.ConclusionsHiCExplorer’s method to detect loops by using a continuous negative binomial function combined with the donut approach from HiCCUPS leads to reliable and fast computation of loops. All the loop-calling algorithms investigated provide differing results, which intersect by n}{}sim 50% at most. The tested in situ Hi-C data contain a large amount of noise; achieving better agreement between loop calling algorithms will require cleaner Hi-C data and therefore future improvements to the experimental methods that generate the data.

Desert bighorn sheep habitat selection, group size, and mountain lion predation risk

AbstractBighorn sheep (Ovis canadensis) evolved for thousands of years in the presence of numerous predators, including mountain lions (Puma concolor). Bighorn sheep have presumably developed predator avoidance strategies; however, the effectiveness of these strategies in reducing risk of mountain lion predation is not well understood. These strategies are of increasing interest because mountain lion predation on bighorn sheep has been identified as a leading cause of mortality in some sheep populations. Therefore, we investigated how mountain lions affect both bighorn sheep habitat selection and risk of mortality in Arizona, USA. We used 2 approaches to investigate the predator‐prey relationship between mountain lions and bighorn sheep. We fit 103 bighorn sheep (81 females and 22 males) with global positioning system radio‐collars in 2 Arizona populations from 2013 to 2017, and used a negative binomial resource selection probability function to evaluate whether bighorn sheep selected for habitat features in accordance with presumed predator avoidance strategies, including terrain ruggedness, slope, topographic position, and horizontal obstruction, in 2 seasons (winter and summer). We then estimated how habitat features such as terrain ruggedness, slope, horizontal obstruction, and group size, influence the risk of mortality due to mountain lion predation using an Andersen‐Gill proportional hazards model. Generally, both sexes selected areas with lower horizontal obstruction and intermediate ruggedness and slope, but selection patterns differed between seasons and sexes. The use of more rugged areas and steeper slopes decreased the risk of mortality due to mountain lion predation, consistent with presumed predator avoidance strategies. Increased group size decreased risk of bighorn sheep mortality due to mountain lion predation but this effect became marginal at approximately 10 individuals/group. We did not identify a relationship between horizontal obstruction and bighorn sheep mortality risk. Our findings can be used in habitat and population management decisions such as the prioritization of habitat restoration sites or selection of translocation sites. In addition, we suggest that augmentation of low‐density bighorn sheep populations may reduce mountain lion predation risk by increasing group size, and that releasing large groups of bighorn sheep in population augmentation and reintroduction efforts may help to reduce mountain lion predation.

Protocol of a prospective and multicentre China Teratology Birth Cohort (CTBC): association of maternal drug exposure during pregnancy with adverse pregnancy outcomes

BackgroundAs reported, 27-93 % of pregnant women take at least one drug during pregnancy. However, drug exposure during pregnancy still lacks sufficient foetal safety evidence of human origin. It is urgent to fill the knowledge gap about medication safety during pregnancy for optimization of maternal disease treatment and pregnancy drug consultation.Methods and analysisThe China Teratology Birth Cohort (CTBC) was established in 2019 and is a hospital-based open-ended prospective cohort study with the aim of assessing drug safety during pregnancy. Pregnant women who set up the pregnancy health records in the first trimester or who seek drug consultation regardless of gestational age in the member hospitals are recruited. Enrolled pregnant women need to be investigated four times, namely, 6–14 and 24–28 weeks of gestational age, before discharge after hospital delivery, and 28–42 days after birth.Maternal medication exposure during pregnancy is the focus of the CTBC. For drugs, information on the type, name, and route of medication; start and end time of medication; single dose; frequency of medication; dosage form; manufacturer; and reason for medication is collected. The adverse pregnancy outcomes collected in the study include birth defects, stillbirth, spontaneous abortion, preterm birth, post-term birth, low birth weight, macrosomia, small for gestational age, large for gestational age and low Apgar score. CTBC uses an electronic questionnaire for data collection and a cloud system for data management. Biological samples are collected if informed consents are obtained. Multi-level logistic regression, mixed-effect negative binomial distribution regression and spline function regression are used to explore the effect of drugs on the occurrence of birth defects.DiscussionThe findings of the study will assist in further understanding the risk of birth defects and other adverse pregnancy outcomes associated with maternal drug exposure and developing the optimal treatment plans and drug counselling for pregnant women.Trial registrationThis study was approved by the Research Ethics Committee of the West China Second Hospital of Sichuan University and registered at the Chinese Clinical Trial Registry (http://www.chictr.org.cn/index.aspx, registration number ChiCTR1900022569).

Non-parametric modelling of temporal and spatial counts data from RNA-seq experiments.

The negative binomial distribution has been shown to be a good model for counts data from both bulk and single-cell RNA-sequencing (RNA-seq). Gaussian process (GP) regression provides a useful non-parametric approach for modelling temporal or spatial changes in gene expression. However, currently available GP regression methods that implement negative binomial likelihood models do not scale to the increasingly large datasets being produced by single-cell and spatial transcriptomics. The GPcounts package implements GP regression methods for modelling counts data using a negative binomial likelihood function. Computational efficiency is achieved through the use of variational Bayesian inference. The GP function models changes in the mean of the negative binomial likelihood through a logarithmic link function and the dispersion parameter is fitted by maximum likelihood. We validate the method on simulated time course data, showing better performance to identify changes in over-dispersed counts data than methods based on Gaussian or Poisson likelihoods. To demonstrate temporal inference, we apply GPcounts to single-cell RNA-seq datasets after pseudotime and branching inference. To demonstrate spatial inference, we apply GPcounts to data from the mouse olfactory bulb to identify spatially variable genes and compare to two published GP methods. We also provide the option of modelling additional dropout using a zero-inflated negative binomial. Our results show that GPcounts can be used to model temporal and spatial counts data in cases where simpler Gaussian and Poisson likelihoods are unrealistic. GPcounts is implemented using the GPflow library in Python and is available at https://github.com/ManchesterBioinference/GPcounts along with the data, code and notebooks required to reproduce the results presented here. The version used for this paper is archived at https://doi.org/10.5281/zenodo.5027066. Supplementary data are available at Bioinformatics online.

Does conventional energy pricing induce innovation in renewable energy? New evidence from a nonlinear approach

AbstractThis article theoretically and empirically studies the association between research and development investments in renewable energy (RE) and oil prices at the country level. We find a positive and robust association between patent counts for RE and price of crude oil using a panel of 46 countries for 1991–2013 in a fixed‐effects hurdle Negative Binomial control function framework. Results suggest that net conventional energy importing countries are more likely to invest in RE when oil prices increase. Countries with more electricity production, less emissions, and greater energy utilization generate more RE patents per year.JEL CLASSIFICATIONQ13; Q42; Q55

Mapping oak shoot browning in SW Spain using online imagery as virtual prospecting tool

Oak shoot browning is spatially widespread throughout the region of Extremadura (SW Spain) following an aggregative distribution pattern, with a much higher prevalence in cork than in holm oak. Shoot browning is considered a potential contributing or inciting factor to oak decline in SW Iberia. Two causal agents are typically involved, Coraebus florentinus Herbst and a fungal complex in the genus Diplodia Fries (teleomorph Botryosphaeria De Cesati & De Notaris). Our goals were to map oak shoot browning incidence in Extremadura and to explore geographical/environmental factors potentially involved using a novel image-based prospecting method. We used online imagery provided by Google Maps/Google Street View. A virtual sampling protocol permitted to inspect, characterise and georeference forest stands and damaged oaks throughout > 5000 km of navigation. Geostatistical interpolation analyses (Indicator Kriging algorithm) and validation processes were conducted to generate probability maps of shoot browning incidence. We prospected 6478 forest stands in which 3410 oaks with shoot browning symptoms were identified and georeferenced. Damage had a clustered distribution pattern both at the regional scale (IK-based probability maps) and the stand scale (fitting to negative binomial function). Factors such as oak species, stand type (pure vs. mixed), treetop orientation, tree aspect, altitude, ground slope and some interactions among them significantly affected shoot browning incidence. We provide the first integrating overview on the distribution and prevalence of oak shoot browning at a regional scale in SW Spain and we also define the effects of some geographical/ecological factors involved.

Regional disparity in outcomes among patients hospitalized for Takotsubo cardiomyopathy in the United States✰

IntroductionTakotsubo cardiomyopathy (TTCM), an entity first described in Japan over two decades ago following myocardial stunning cases without evidence of coronary stenosis, has emerged as a unique entity with global recognition. We sought to investigate the extent and magnitude of regional variations in its outcomes. MethodWe used the National Inpatient Sample (NIS) of the Agency for Healthcare Research and Quality (2010–2014). Risk-adjusted rates of outcomes across the US geographical regions were calculated by fitting a Poisson regression model with a robust error variance under generalized estimating equations. Discrete numeric variables with over-dispersed count distributions -length of stay and continuous variables with a right skewed spread- cost of hospitalization were modeled using a generalized linear regression with a negative binomial function and gamma function respectively. ResultWe found significant regional variations in-patient mortality. While there was significantly higher risk of in-hospital death in the West (5.28 [4.34–6.44]) vs 4.40 [3.57–5.43] vs 4.10 [3.38–498] vs 4.78 [3.96–5.77]), there was a different pattern of variation in the length of days with longer hospital stay in the Northeast. Likewise, the risk-adjusted rate of non-routine home discharges was highest for Northeast. The West had the highest cost of hospitalization (West: $40,217 vs. South: $28,465) ConclusionSignificant geographic variation exists in the cost of hospitalization and in-hospital mortality of TTCM across the US. Understanding this variation requires a detailed understanding of the processes of care and identification of effective strategies to eliminate these disparities.

Lack of interest in physical activity - individual and environmental attributes in adults across Europe: The SPOTLIGHT project

A considerable proportion of European adults report little or no interest in physical activity. Identifying individual-level and environmental-level characteristics of these individuals can help designing effective interventions and policies to promote physical activity. This cross-sectional study additionally explored associations between level of interest and physical activity, after controlling for other individual and environmental variables. Measures of objective and perceived features of the physical environment of residence, self-reported physical activity and other lifestyle behaviors, barriers towards physical activity, general health, and demographics were obtained from 5205 European adults participating in the 2014 online SPOTLIGHT survey. t-Tests, chi-square tests, and generalized estimating equations with negative binomial log-link function were conducted. Adults not interested in physical activity reported a higher BMI and a lower self-rated health, were less educated, and to a smaller extent female and less frequently employed. They were more prone to have less healthy eating habits, and to perceive more barriers towards physical activity. Only minor differences were observed in environmental attributes: the non-interested were slightly more likely to live in neighborhoods objectively characterized as less aesthetic and containing more destinations, and perceived as less functional, safe, and aesthetic. Even after controlling for other individual and environmental factors, interest in physical activity remained a significant correlate of physical activity, supporting the importance of this association. This study is among the first to describe characteristics of individuals with reduced interest in physical activity, suggesting that (lack of) interest is a robust correlate of physical activity in several personal and environmental conditions.

Experimental investigation of alternative transmission functions: Quantitative evidence for the importance of nonlinear transmission dynamics in host-parasite systems.

Understanding pathogen transmission is crucial for predicting and managing disease. Nonetheless, experimental comparisons of alternative functional forms of transmission remain rare, and those experiments that are conducted are often not designed to test the full range of possible forms.To differentiate among 10 candidate transmission functions, we used a novel experimental design in which we independently varied four factors—duration of exposure, numbers of parasites, numbers of hosts and parasite density—in laboratory infection experiments.We used interactions between amphibian hosts and trematode parasites as a model system and all candidate models incorporated parasite depletion. An additional manipulation involving anaesthesia addressed the effects of host behaviour on transmission form.Across all experiments, nonlinear transmission forms involving either a power law or a negative binomial function were the best‐fitting models and consistently outperformed the linear density‐dependent and density‐independent functions. By testing previously published data for two other host–macroparasite systems, we also found support for the same nonlinear transmission forms.Although manipulations of parasite density are common in transmission studies, the comprehensive set of variables tested in our experiments revealed that variation in density alone was least likely to differentiate among competing transmission functions. Across host–pathogen systems, nonlinear functions may often more accurately represent transmission dynamics and thus provide more realistic predictions for infection.

The Effects of Socioeconomic and Environmental Factors on the Incidence of Dengue Fever in the Pearl River Delta, China, 2013

BackgroundAn outbreak of dengue fever (DF) occurred in Guangdong Province, China in 2013 with the highest number of cases observed within the preceding ten years. DF cases were clustered in the Pearl River Delta economic zone (PRD) in Guangdong Province, which accounted for 99.6% of all cases in Guangdong province in 2013. The main vector in PRD was Aedes albopictus. We investigated the socioeconomic and environmental factors at the township level and explored how the independent variables jointly affect the DF epidemic in the PRD.Methodology/Principal FindingsSix factors associated with the incidence of DF were identified in this project, representing the urbanization, poverty, accessibility and vegetation, and were considered to be core contributors to the occurrence of DF from the perspective of the social economy and the environment. Analyses were performed with Generalized Additive Models (GAM) to fit parametric and non-parametric functions to the relationships between the response and predictors. We used a spline-smooth technique and plotted the predicted against the observed co-variable value. The distribution of DF cases was over-dispersed and fit the negative binomial function better. The effects of all six socioeconomic and environmental variables were found to be significant at the 0.001 level and the model explained 45.1% of the deviance by DF incidence. There was a higher risk of DF infection among people living at the prefectural boundary or in the urban areas than among those living in other areas in the PRD. The relative risk of living at the prefectural boundary was higher than that of living in the urban areas. The associations between the DF cases and population density, GDP per capita, road density, and NDVI were nonlinear. In general, higher “road density” or lower “GDP per capita” were considered to be consistent risk factors. Moreover, higher or lower values of “population density” and “NDVI” could result in an increase in DF cases.ConclusionIn this study, we presented an effect analysis of socioeconomic and environmental factors on DF occurrence at the smallest administrative unit (township level) for the first time in China. GAM was used to effectively detect the nonlinear impact of the predictors on the outcome. The results showed that the relative importance of different risk factors may vary across the PRD. This work improves our understanding of the differences and effects of socioeconomic and environmental factors on DF and supports effectively targeted prevention and control measures.

Interval-valued time series models: Estimation based on order statistics exploring the Agriculture Marketing Service data

The current regression models for interval-valued data ignore the extreme nature of the lower and upper bounds of intervals. A new estimation approach is proposed; it considers the bounds of the interval as realizations of the max/min order statistics coming from a sample of nt random draws from the conditional density of an underlying stochastic process {Yt}. This approach is important for data sets for which the relevant information is only available in interval format, e.g., low/high prices. The interest is on the characterization of the latent process as well as in the modeling of the bounds themselves. A dynamic model is estimated for the conditional mean and conditional variance of the latent process, which is assumed to be normally distributed, and for the conditional intensity of the discrete process {nt}, which follows a negative binomial density function. Under these assumptions, together with the densities of order statistics, maximum likelihood estimates of the parameters of the model are obtained. They are needed to estimate the expected value of the bounds of the interval. This approach is implemented with the time series of livestock prices, of which only low/high prices are recorded making the price process itself a latent process. It is found that the proposed model provides an excellent fit of the intervals of low/high returns with an average coverage rate of 83%. In addition, a comparison with current models for interval-valued data is offered.

Association of Seasonal Climate Variability and Age-Specific Mortality in Northern Sweden before the Onset of Industrialization

Background and aims: Little is known about health impacts of climate in pre-industrial societies. We used historical data to investigate the association of temperature and precipitation with total and age-specific mortality in Skellefteå, northern Sweden, between 1749 and 1859. Methods: We retrieved digitized aggregated population data of the Skellefteå parish, and monthly temperature and precipitation measures. A generalized linear model was established for year to year variability in deaths by annual and seasonal average temperature and cumulative precipitation using a negative binomial function, accounting for long-term trends in population size. The final full model included temperature and precipitation of all four seasons simultaneously. Relative risks (RR) with 95% confidence intervals (CI) were calculated for total, sex- and age-specific mortality. Results: In the full model, only autumn precipitation proved statistically significant (RR 1.02; CI 1.00–1.03, per 1cm increase of autumn precipitation), while winter temperature (RR 0.98; CI 0.95–1.00, per 1 °C increase in temperature) and spring precipitation (RR 0.98; CI 0.97–1.00 per 1 cm increase in precipitation) approached significance. Similar effects were observed for men and women. The impact of climate variability on mortality was strongest in children aged 3–9, and partly also in older children. Infants, on the other hand, appeared to be less affected by unfavourable climate conditions. Conclusions: In this pre-industrial rural region in northern Sweden, higher levels of rain during the autumn increased the annual number of deaths. Harvest quality might be one critical factor in the causal pathway, affecting nutritional status and susceptibility to infectious diseases. Autumn rain probably also contributed to the spread of air-borne diseases in crowded living conditions. Children beyond infancy appeared most vulnerable to climate impacts.

Modelling multiple fishing gear efficiencies and abundance for aggregated populations using fishery or survey data

Abstract Fish and wildlife often exhibit an aggregated distribution pattern, whereas local abundance changes constantly due to movement. Estimating population density or size and survey detectability (i.e. gear efficiency in a fishery) for such elusive species is technically challenging. We extend abundance and detectability (N-mixture) methods to deal with this difficult situation, particularly for application to fish populations where gear efficiency is almost never equal to one. The method involves a mixture of statistical models (negative binomial, Poisson, and binomial functions) at two spatial scales: between-cell and within-cell. The innovation in this approach is to use more than one fishing gear with different efficiencies to simultaneously catch (sample) the same population in each cell at the same time-step. We carried out computer simulations on a range of scenarios and estimated the relevant parameters using a Bayesian technique. We then applied the method to a demersal fish species, tiger flathead, to demonstrate its utility. Simulation results indicated that the models can disentangle the confounding parameters in gear efficiency and abundance, and the accuracy generally increases as sample size increases. A joint negative binomial–Poisson model using multiple gears gives the best fit to tiger flathead catch data, while a single gear yields unrealistic results. This cross-sampling method can evaluate gear efficiency cost effectively using existing fishery catch data or survey data. More importantly, it provides a means for estimating gear efficiency for gear types (e.g. gillnets, traps, hook and line, etc.) that are extremely difficult to study using field experiments.

PHS18 - Economic burden of otitis media among children in the United States: Results from a national database

Previous studies have documented the impact of otitis media among children, but recent estimates of cost of otitis media (OM) among children in the United States (US) are not available. The objective of this study was to estimate the incremental direct medical expenditures of treating OM among children in the US. Retrospective analysis was conducted using the 2010 Medical Expenditure Panel Survey (MEPS) data. Among all children (age<18 years; n=8,710), those with OM were identified using International Classification of Diseases (ICD)-9 diagnosis codes 381 or 382. Incremental total expenditures associated with OM were estimated using a regression model adjusting for age, gender, race, ethnicity, education, geographic region, insurance status and number of medications used (proxy for comorbidity). Given the skewed distribution of expenditure variables, multiple model specifications including ordinary least squares regression, generalized linear model (GLM) with Poisson, gamma and negative binomial variance functions were evaluated. The prevalence of OM among children in the US was estimated at 11.32%, i.e., 8.47 million persons (95% CI: 10.30% to 12.33%). A majority of children with OM were male (51.7%), white (84.2%), and insured (97.3%) with mean age of 4.7±0.20 years and education of 0.9±0.11 years. In unadjusted analysis among patients with OM, physician office visits comprised the largest proportion of total expenditures at $769.06 (SE: 54.99), followed by prescription medications at $606.56 (SE: $41.72), and inpatient visits at $500.73 (SE: $148.42). After controlling for covariates, children with OM had 48% higher total expenditures than those without OM (Estimate: 1.48; p=0.001). The annual adjusted mean incremental total expenditure associated with OM was $543.10 (SE: $192.0; p=0.005) per person. Given the prevalence of OM in children and its associated incremental expenditures, the annual direct medical expenditure for treating OM in children is estimated at approximately $4.6 billion in 2012 US dollars.

Przhevalskiana silenus myiasis among slaughter goats in northern Jordan

During the period December 1998–May 2000, 900 local goats slaughtered at the Irbid Abattoir (northern Jordan) were examined for the larval instars of Przhevalskiana silenus. Of 900 goats, 10% (95% CI: 9,13) were infested with P. silenus larvae. Only the second and third larval instars were seen. A multiple-regression analysis (with the error variance described by the negative-binomial function) suggested that infestation depended on the month of sampling, and that infestation with live larvae was associated with a poorer carcass. The percentage of infested goats and the mean monthly total number of larvae per goat peaked in samples taken in the autumn and winter. Larval numbers were highly aggregated: most animals had no larvae but the maximum was 69. Analysis of the pattern of aggregation suggested that the best model fit was one in which the larvae counts per goat varied with the monthly prevalence.

Leaching losses of N under grazed irrigated and non-irrigated pastures

High spatial and temporal variability of nitrogen application via livestock urine on grazed pasture make meaningful measurement of leaching of nitrate very difficult. The present study was aimed at estimating the mean annual nitrogen loadings and concentrations leached to an underlying aquifer beneath unfertilized irrigated and non-irrigated paddocks of a dairy farm at OB Flat 10 km south of the city of Mount Gambier (37°50′S, 140°40′E), South Australia. Urine was collected in the milking shed and applied to 1 m deep monolith lysimeters in three separate seasons in 1992. Some lysimeters received two applications. Drainage and N fluxes and concentrations were recorded 1 year before urine application and for 3 years following urine applications. A negative binomial function was used to specify the appropriate proportions of pasture subject to 0, 1 or 2 loadings in a year, and the lysimeter drainage measurements were then scaled across seasons and paddock area to estimate mean annual leaching under the whole of each paddock. It was found that the timing of urine application had a profound effect on the leaching of nitrogen. Results indicated that the nitrate flux leaching from the root zone in the irrigated paddock was 26–33 kg N/ha/year at 13–17 mg N/litre and in the non-irrigated paddock, 10–13 kg N/ha/year at 9–11 mg N/litre. These estimates are not inconsistent with nitrate concentrations measured in groundwater but cannot be independently verified. The current transition from non-irrigated to irrigated pastures is likely to double nitrogen loads to the aquifer and cause nitrate concentrations in groundwater to exceed drinking water guidelines, unless alternative pasture management strategies are developed.

Parasite aggregations in host populations using a reformulated negative binomial model.

The negative binomial distribution model is reformulated and used to demarcate a host population at a specific level of infection by defining an attribute spanning a range of parasite aggregations. The upper limit of the range specifies the boundary for the classification of the host population and provides a technique to determine the cumulative probability at any level of parasite infection to a high degree of accuracy. This approach also leads to the evaluation of the k parameter, i.e. an inverse measure of dispersion of parasite aggregation, for each fraction of the host population with a discrete level of infection. The basic mathematical premise of the negative binomial function is unaltered in developing this reformulation which was applied to data on the distribution of the trichostrongylid nematode Heligmosomoides polygyrus in populations of the field mouse, Apodemus sylvaticus.

A generalized model for the lifetime of microelectronic components, applied to storage conditions

To improve the quality of microcircuit lifetime prediction in storage and non-operating conditions, a new model, based on physical principles and generally applicable to microcircuits, is proposed. This model is derived from the fundamentals of manufacturing yield statistics and uses compound Poisson statistics with a time-dependent negative binomial shaping function. Comparison of the model to statistical distributions created from available data shows that the model is applicable to many situations that may occur in actual populations.