Impact of Random Variables Probability Distribution on Public Health Risk Assessment from Contaminated Soil

Probabilistic methods are now being applied increasingly to public health risk assessment instead of the deterministic, conservative, point estimates. An essential part of the probabilistic methods is the selection of probability distribution functions to represent the uncertainty of the random variables considered. We study the effect of selection of different probability distribution functions on the probabilistic outcome using the first-order reliability method (FORM). An example of cancer risk resulting from dermal contact with benzo(a)pyrene (BaP)-contaminated soil is given. Cancer potency factor, soil concentration, and fraction of skin area exposed were assigned normal, lognormal, and uniform probability distribution functions, and the effect of probability of exceeding a target risk level (termed the probability of failure) and sensitivity measures were studied. We investigated the question: what happens when one assumes different distribution shapes with the same mean and standard deviation? The results indicate that the selection of a probability distribution function for the random variables had a moderate impact on the probability of failure when the target risk is at the 50th percentile level, while the impact was much larger for a 95th target risk percentile. We conclude that the probability distribution will have a large impact because in most cases the regulatory threshold risk is at the tail end of the risk distribution. The impact of the distributions on probabilistic sensitivity, however, showed a reversed trend, where the impact was slightly more appreciable for the 50th percentile than for the 95th percentile. The selection of distribution shape did not, however, alter the order of probabilistic sensitivity of the basic random variables.     

Differentiation and genetic analysis of Rhodnius prolixus and Rhodnius colombiensis by rDNA and RAPD amplification.

Domiciliated Rhodnius prolixus and sylvatic R. colombiensis were analyzed in order to confirm their genetic divergence and verify the risk that the latter represents in the domiciliation process, and to provide tools for identifying the sources of possible reinfestation by triatomines in human dwellings allowing control programs to be undertaken. Comparison of random amplified polymorphic DNA amplification patterns and cluster analysis suggests reproductive discontinuity between the two species. The calculated statistical F value of 0.24 and effective migration rate of 0.6 individuals per generation are insufficient to maintain genetic homogeneity between them and confirm the absence of present genetic flow. R. colombiensis presents higher intrapopulation variability. Polymerase chain reaction of ribosomal DNA supports these findings. The low genetic flow between the two species implies that R. colombiensis do not represent an epidemiological risk for the domiciliary transmission of Trypanosoma cruzi in the Tolima Department. The lower variability of the domiciliated R. prolixus could result in greater susceptibility to the use of pesticides in control programs.     

Agronomic and meteorological factors affecting the severity of leaf blotch caused by Mycosphaerella graminicola in commercial wheat crops in England

Factors affecting the severity of leaf blotch on the two upper leaves of wheat plants in crops at the milky ripe growth stage (GS 73–75) were investigated using survey data from 3513 randomly selected wheat crops sampled during 1985–1996. Year‐to‐year variation in disease severity was greater than spatial variability at county level, although both showed significant differences. The presence of disease above a 5% severity threshold was modelled using random effects logistic regression (Generalised Linear Mixed Model), which enabled risk variables measured at the field level to be combined with meteorological variables estimated at county level. The final model included terms for the fixed effects of disease resistance rating, date of sowing, high risk septoria periods in May and June, number of fungicide sprays and number of days with frost (≤‐2°C) in November. The percentage of crops above the threshold decreased with later sowing, increased number of November frost days and increased number of fungicide sprays. In contrast, high risk septoria periods (rain splash events) in May and June showed a positive correlation with the percentage of crops above the threshold. There were benefits from using resistant cultivars. The model showed that a range of risk variables were of broadly equivalent importance in determining the development of leaf blotch. These risk variables should be integrated in any scheme designed to support fungicide use decisions.     

The need for new methods to backcalculate soil cleanup targets in interval and probabilistic cancer risk assessments

Ordinary algebra may be used to backcalculate health‐based cleanup targets in deterministic risk assessments, but it does not work in interval or probabilistic risk assessments. Equations with interval or random variables do not follow the rules of ordinary algebra. This paper explains the need for more sophisticated methods to backcalculate soil cleanup targets when using interval or random variables.     

Characterization and Simulation of Uncertain Frequency Distributions: Effects of Distribution Choice,Variability, Uncertainty,and Parameter Dependence

We use bootstrap simulation to characterize uncertainty in parametric distributions, including Normal, Lognormal, Gamma, Weibull, and Beta, commonly used to represent variability in probabilistic assessments. Bootstrap simulation enables one to estimate sampling distributions for sample statistics, such as distribution parameters, even when analytical solutions are not available. Using a two-dimensional framework for both uncertainty and variability, uncertainties in cumulative distribution functions were simulated. The mathematical properties of uncertain frequency distributions were evaluated in a series of case studies during which the parameters of each type of distribution were varied for sample sizes of 5, 10, and 20. For positively skewed distributions such as Lognormal, Weibull, and Gamma, the range of uncertainty is widest at the upper tail of the distribution. For symmetric unbounded distributions, such as Normal, the uncertainties are widest at both tails of the distribution. For bounded distributions, such as Beta, the uncertainties are typically widest in the central portions of the distribution. Bootstrap simulation enables complex dependencies between sampling distributions to be captured. The effects of uncertainty, variability, and parameter dependencies were studied for several generic functional forms of models, including models in which two-dimensional random variables are added, multiplied, and divided, to show the sensitivity of model results to different assumptions regarding model input distributions, ranges of variability, and ranges of uncertainty and to show the types of errors that may be obtained from mis-specification of parameter dependence. A total of 1,098 case studies were simulated. In some cases, counter-intuitive results were obtained. For example, the point value of the 95th percentile of uncertainty for the 95th percentile of variability of the product of four Gamma or Weibull distributions decreases as the coefficient of variation of each model input increases and, therefore, may not provide a conservative estimate. Failure to properly characterize parameter uncertainties and their dependencies can lead to orders-of-magnitude mis-estimates of both variability and uncertainty. In many cases, the numerical stability of two-dimensional simulation results was found to decrease as the coefficient of variation of the inputs increases. We discuss the strengths and limitations of bootstrap simulation as a method for quantifying uncertainty due to random sampling error.     

Using a mixed effects model to estimate geographic variation in cancer rates

Commonly used methods for depicting geographic variation in cancer rates are based on rankings. They identify where the rates are high and low but do not indicate the magnitude of the rates nor their variability. Yet such measures of variability may be useful in suggesting which types of cancer warrant further analytic studies of localized risk factors. We consider a mixed effects model in which the logarithm of the mean Poisson rate is additive in fixed stratum effects (e.g., age effects) and in logarithms of random relative risk effects associated with geographic areas. These random effects are assumed to follow a gamma distribution with unit mean and variance 1/alpha, similar to Clayton and Kaldor (1987, Biometrics 43, 671-681). We present maximum likelihood and method-of-moments estimates with standard errors for inference on alpha -1/2, the relative risk standard deviation (RRSD). The moment estimates rely on only the first two moments of the Poisson and gamma distributions but have larger standard errors than the maximum likelihood estimates. We compare these estimates with other measures of variability. Several examples suggest that the RRSD estimates have advantages compared to other measures of variability.     

Non-linear compartmental systems: extensions of S. R. Bernard's urn model

One of the limitations of stochastic, linear compartmental systems is the small degree of variability in the contents of compartments. S. R. Bernard's (1981) urn model (S. R. Bernardet al., Bull. math. Biol. 43, 33–45.) which allows for bulk arrivals and departures from a one-compartment system, was suggested as a way of increasing content variability. In this paper, we show how the probability distribution of the contents of an urn model may be simply derived by studying an appropriate set of exchangeable random variables. In addition, we show how further increases in variability may be modeled by allowing the size of arrivals and departures to be random. Supported by NSF Grant No. MCS 8102215-01.     

Bayesian estimation of two-part joint models for a longitudinal semicontinuous biomarker and a terminal event with INLA: Interests for cancer clinical trial evaluation

Two-part joint models for a longitudinal semicontinuous biomarker and a terminal event have been recently introduced based on frequentist estimation. The biomarker distribution is decomposed into a probability of positive value and the expected value among positive values. Shared random effects can represent the association structure between the biomarker and the terminal event. The computational burden increases compared to standard joint models with a single regression model for the biomarker. In this context, the frequentist estimation implemented in the R package frailtypack can be challenging for complex models (i.e., a large number of parameters and dimension of the random effects). As an alternative, we propose a Bayesian estimation of two-part joint models based on the Integrated Nested Laplace Approximation (INLA) algorithm to alleviate the computational burden and fit more complex models. Our simulation studies confirm that INLA provides accurate approximation of posterior estimates and to reduced computation time and variability of estimates compared to frailtypack in the situations considered. We contrast the Bayesian and frequentist approaches in the analysis of two randomized cancer clinical trials (GERCOR and PRIME studies), where INLA has a reduced variability for the association between the biomarker and the risk of event. Moreover, the Bayesian approach was able to characterize subgroups of patients associated with different responses to treatment in the PRIME study. Our study suggests that the Bayesian approach using the INLA algorithm enables to fit complex joint models that might be of interest in a wide range of clinical applications.     

Using iterative ridge regression to explore associations between conditioned variables

We address a specific case of joint probability mapping, where the information presented is the probabilistic associations of random variables under a certain condition variable (conditioned associations). Bayesian and dependency networks graphically map the joint probabilities of random variables, though both networks may identify associations that are independent of the condition (background associations). Since the background associations have the same topological features as conditioned associations, it is difficult to discriminate between conditioned and non-conditioned associations, which results in a major increase in the search space. We introduce a modification of the dependency network method, which produces a directed graph, containing only condition-related associations. The graph nodes represent the random variables and the graph edges represent the associations that arise under the condition variable. This method is based on ridge-regression, where one can utilize a numerically robust and computationally efficient algorithm implementation. We illustrate the method's efficiency in the context of a medically relevant process, the emergence of drug-resistant variants of human immunodeficiency virus (HIV) in drug-treated, HIV-infected people. Our mapping was used to discover associations between variants that are conditioned by the initiation of a particular drug treatment regimen. We have demonstrated that our method can recover known associations of such treatment with selected resistance mutations as well as documented associations between different mutations. Moreover, our method revealed novel associations that are statistically significant and biologically plausible.     

Poor environmental tracking can make extinction risk insensitive to the colour of environmental noise

The relative importance of environmental colour for extinction risk compared with other aspects of environmental noise (mean and interannual variability) is poorly understood. Such knowledge is currently relevant, as climate change can cause the mean, variability and temporal autocorrelation of environmental variables to change. Here, we predict that the extinction risk of a shorebird population increases with the colour of a key environmental variable: winter temperature. However, the effect is weak compared with the impact of changes in the mean and interannual variability of temperature. Extinction risk was largely insensitive to noise colour, because demographic rates are poor in tracking the colour of the environment. We show that three mechanisms-which probably act in many species-can cause poor environmental tracking: (i) demographic rates that depend nonlinearly on environmental variables filter the noise colour, (ii) demographic rates typically depend on several environmental signals that do not change colour synchronously, and (iii) demographic stochasticity whitens the colour of demographic rates at low population size. We argue that the common practice of assuming perfect environmental tracking may result in overemphasizing the importance of noise colour for extinction risk. Consequently, ignoring environmental autocorrelation in population viability analysis could be less problematic than generally thought.     

Probabilistic Sensitivity Analysis of Public Health Risk Assessment from Contaminated Soil

Recently, there has been a growing trend toward using stochastic (probabilistic) methods in ecological and public health risk assessment. These methods are favored because they overcome the problem of compounded conservatism and allow the systematic consideration of uncertainty and variability typically encountered in risk assessment. This article demonstrates a new methodology for the analysis of uncertainty in risk assessment using the first-order reliability method (FORM). The reliability method is formulated such that the probability that incremental lifetime cancer risk exceeds a predefined threshold level is calculated. Furthermore, the stochastic sensitivity of this probability with respect to the random variables is provided. The emphasis is on exploring the different types of probabilistic sensitivity obtained through the reliability analysis. The method is applied to a case study given by Thompson et al. (1992) on cancer risk resulting from dermal contact with benzo(a)pyrene (BaP)-contaminated soils. The reliability results matched those of the Monte Carlo simulation method. On average, the Monte Carlo simulation method required about 35 times as many function evaluations as that of FORM to calculate the probability of exceeding the target risk level. The analysis emphasizes the significant impact that the uncertainty in cancer potency factor has on the probabilistic modeling results compared with other parameters.     

Spatial Analysis of Factors Influencing Long-Term Stress in the Grizzly Bear (Ursus arctos) Population of Alberta,Canada

Non-invasive measures for assessing long-term stress in free ranging mammals are an increasingly important approach for understanding physiological responses to landscape conditions. Using a spatially and temporally expansive dataset of hair cortisol concentrations (HCC) generated from a threatened grizzly bear (Ursus arctos) population in Alberta, Canada, we quantified how variables representing habitat conditions and anthropogenic disturbance impact long-term stress in grizzly bears. We characterized spatial variability in male and female HCC point data using kernel density estimation and quantified variable influence on spatial patterns of male and female HCC stress surfaces using random forests. Separate models were developed for regions inside and outside of parks and protected areas to account for substantial differences in anthropogenic activity and disturbance within the study area. Variance explained in the random forest models ranged from 55.34% to 74.96% for males and 58.15% to 68.46% for females. Predicted HCC levels were higher for females compared to males. Generally, high spatially continuous female HCC levels were associated with parks and protected areas while low-to-moderate levels were associated with increased anthropogenic disturbance. In contrast, male HCC levels were low in parks and protected areas and low-to-moderate in areas with increased anthropogenic disturbance. Spatial variability in gender-specific HCC levels reveal that the type and intensity of external stressors are not uniform across the landscape and that male and female grizzly bears may be exposed to, or perceive, potential stressors differently. We suggest observed spatial patterns of long-term stress may be the result of the availability and distribution of foods related to disturbance features, potential sexual segregation in available habitat selection, and may not be influenced by sources of mortality which represent acute traumas. In this wildlife system and others, conservation and management efforts can benefit by understanding spatial- and gender-based stress responses to landscape conditions.     

Sexual dimorphism in two subspecies of Ethiopian baboons (Papio hamadryas) and their hybrids

Sexual dimorphism in mammals has been related to such variables as absolute body size, ecology, and various aspects of social behaviour. Attempts to relate dimorphism to any of these variables have necessarily used small heterogeneous samples to represent the relevant species. We are concerned by the inevitable exclusion of any measure of variability in dimorphism and the consequent impossiblity of assessing the significance of observed interspecific differences. In this paper we describe aspects of sexual dimorphism in anubis,hamadryas, and hybrid baboons from Ethiopia. Samples are large enough to permit a measure of intrapopulation variability. Hamadryas baboons are more dimorphic than anubis in epigamic features, but not in postcanine dentition, nor, contrary to previous reports, in body weight or canine tooth size. Hybrid males are more hamadryas-like and hybrid females more anubis-like, as would be predicted by the proposed mechanism for the establishment of the hybrid zone, namely the capture of anubis females by hamadryas males.     

Developmental variability and stability in continuous-time host-parasitoid models

Insect host-parasitoid systems are often modeled using delay-differential equations, with a fixed development time for the juvenile host and parasitoid stages. We explore here the effects of distributed development on the stability of these systems, for a random parasitism model incorporating an invulnerable host stage, and a negative binomial model that displays generation cycles. A shifted gamma distribution was used to model the distribution of development time for both host and parasitoid stages, using the range of parameter values suggested by a literature survey. For the random parasitism model, the addition of biologically plausible levels of developmental variability could potentially double the area of stable parameter space beyond that generated by the invulnerable host stage. Only variability in host development time was stabilizing in this model. For the negative binomial model, development variability reduced the likelihood of generation cycles, and variability in host and parasitoid was equally stabilizing. One source of stability in these models may be aggregation of risk, because hosts with varying development times have different vulnerabilities. High levels of variability in development time occur in many insects and so could be a common source of stability in host-parasitoid systems.     

An ecological risk assessment paradigm using the Spatially Integrated model for Phosphorus Loading and Erosion (SIMPLE)

Ecological risk assessments provide a probabilitistic approach to analyzing and predicting ecosystem responses to stress. We are evaluating the relationship between nonpoint source (NPS) phosphorus loading and the trophic status of the aquatic ecosystem. We are using SIMPLE (the Spatially Integrated Model for Phosphorus Loading and Erosion) to identify probable phosphorus sources in a watershed, simulate the phosphorus loading to streams, and analyze the relationships between input variables and their ecological impact. The objective of this paper is to describe a risk-based paradigm using SIMPLE to characterize the probability of exceeding a critical phosphorus loading to a lotic ecosystem. We have characterized the risk of exceeding a threshold loading of 0.5 kilogram total phosphorus per hectare per year from a 2238 hectare watershed. Two-hundred-fifty random SIMPLE simulations were performed to estimate annual total phosphorus, dissolved phosphorus, and sediment-bound phosphorus loading to a lotic ecosystem from the watershed. Simulation results were analyzed statistically to determine the probabilities of exceeding the critical loadings. Based on the current land use practices in the Battle Creek watershed, the probability of exceeding the total phosphorus critical loading rate of 0.5 kg/ha/yr was approximately 11 percent, or one year in nine the total annual loading will exceed the critical loading rate. The 95 percent confidence intervals for the total phosphorus loading occurring on average once in nine years were relatively close (0.45 to 0.60 kg/ha/yr), assuming the only variability from year to year was due to natural variability in weather.     

Respiratory and heart rate patterns in infants destined to be victims of sudden infant death syndrome: average rates and their variability measured over 24 hours.

From a prospective study in which 24 hour recordings of the electrocardiogram and respiratory activity (abdominal wall movement) were made on a population of full term infants, 22 recordings were obtained from 16 infants who later were victims of the sudden infant death syndrome. The average heart rate, average heart rate variability, average breath to breath interval, and average breath to breath interval variability over the whole of each recording for the 22 recordings were compared with those from a control group of 324 infants selected at random from the rest of the population. No significance was found in the number of recordings from those infants who suffered the sudden infant death syndrome which lay outside the 5th-95th percentile range of the control group for the four variables studied. In a group comparison no difference was found between the sudden infant death syndrome group and the controls either in terms of the respiratory variables studied or in terms of the average heart rate variability. The results did, however, suggest that there may be a group difference in terms of the average instantaneous heart rate.     

Geographical information system-based forest fire risk assessment integrating national forest inventory data and analysis of its spatiotemporal variability

Improvements in forest fire risk estimation and mapping fire risk zones are vital to reduce the negative impacts of fire and to facilitate planning for the protection of forested areas. This is especially important for places with little previous data on fire history. This paper presents an improved conceptual scheme for the assessment and mapping of fire risk using a Forest Resource Inventory Database, based on four aspects of topographical, human activity, climate, and forest characteristics factors. We selected 12 variables based on our defined conceptual scheme to generate a synthetic forest fire risk index (FRI) to quantify potential forest fire risk and map risk zones in the Wuyishan Scenery District (WSD), a world heritage site that located in the northwest of Fujian province, People's Republic of China. Spatial statistics were used to examine the spatio-temporal variation of FRI. The results showed the main fire risk zones in the WSD were in the low or moderate categories (accounting for 76.7% of the total area of the WSD in 1997 and 79.2% in 2009). The spatial heterogeneity of FRI showed anisotropic variability characteristics which changed over time. From 1997 to 2009, there was an increasing influence from both autocorrelation factors and random factors. Moreover, these factors played almost equally important roles in forest fire processes in the WSD. The fire risk map was applied to assess the vulnerability of cultural heritage resources in the WSD. Most were located in low- or moderate-risk areas, and therefore would be at low risk from potential fire damage.     

Metabolic Variability in Micro-Populations

Biological cells in a population are variable in practically every property. Much is known about how variability of single cells is reflected in the statistical properties of infinitely large populations; however, many biologically relevant situations entail finite times and intermediate-sized populations. The statistical properties of an ensemble of finite populations then come into focus, raising questions concerning inter-population variability and dependence on initial conditions. Recent technologies of microfluidic and microdroplet-based population growth realize these situations and make them immediately relevant for experiments and biotechnological application. We here study the statistical properties, arising from metabolic variability of single cells, in an ensemble of micro-populations grown to saturation in a finite environment such as a micro-droplet. We develop a discrete stochastic model for this growth process, describing the possible histories as a random walk in a phenotypic space with an absorbing boundary. Using a mapping to Polya’s Urn, a classic problem of probability theory, we find that distributions approach a limiting inoculum-dependent form after a large number of divisions. Thus, population size and structure are random variables whose mean, variance and in general their distribution can reflect initial conditions after many generations of growth. Implications of our results to experiments and to biotechnology are discussed.     

Which metrics derived from airborne laser scanning are essential to measure the vertical profile of ecosystems?

In a recent perspective (Diversity and Distributions, 29, 39–50), ‘10 variables’ were proposed to measure vegetation structure from airborne laser scanning (ALS) for assessing species distributions and habitat suitability. We worry about this list because the variables predominantly represent variation in vegetation height, the vertical variability of vegetation biomass is insufficiently captured, and variables of vegetation cover are ill-defined or not ecosystem agnostic. We urge for a better defined, more comprehensive and more balanced list, and for assessing which information from ALS point clouds is truly essential to measure the major dimensions of 3D vegetation structure within and across ecosystems and animal habitats. We think that the currently proposed ‘list of 10 ALS metrics’ is premature and that researchers and stakeholders should be cautious in adopting this list.