Uncertainty in Measurement: A Review of Monte Carlo Simulation Using Microsoft Excel for the Calculation of Uncertainties Through Functional Relationships,Including Uncertainties in Empirically Derived Constants

The Guide to the Expression of Uncertainty in Measurement (usually referred to as the GUM) provides the basic framework for evaluating uncertainty in measurement. The GUM however does not always provide clearly identifiable procedures suitable for medical laboratory applications, particularly when internal quality control (IQC) is used to derive most of the uncertainty estimates. The GUM modelling approach requires advanced mathematical skills for many of its procedures, but Monte Carlo simulation (MCS) can be used as an alternative for many medical laboratory applications. In particular, calculations for determining how uncertainties in the input quantities to a functional relationship propagate through to the output can be accomplished using a readily available spreadsheet such as Microsoft Excel.The MCS procedure uses algorithmically generated pseudo-random numbers which are then forced to follow a prescribed probability distribution. When IQC data provide the uncertainty estimates the normal (Gaussian) distribution is generally considered appropriate, but MCS is by no means restricted to this particular case. With input variations simulated by random numbers, the functional relationship then provides the corresponding variations in the output in a manner which also provides its probability distribution. The MCS procedure thus provides output uncertainty estimates without the need for the differential equations associated with GUM modelling.The aim of this article is to demonstrate the ease with which Microsoft Excel (or a similar spreadsheet) can be used to provide an uncertainty estimate for measurands derived through a functional relationship. In addition, we also consider the relatively common situation where an empirically derived formula includes one or more ‘constants’, each of which has an empirically derived numerical value. Such empirically derived ‘constants’ must also have associated uncertainties which propagate through the functional relationship and contribute to the combined standard uncertainty of the measurand.     

The probabilistic economic injury level: incorporating uncertainty into pest management decision-making

Despite the recognition that the economic injury level (EIL) is determined by dynamic biological and economic parameters, which can be highly variable and uncertain, there has been little effort to quantify uncertainty and to use estimates of uncertainty in the determination of EILs. In this paper, we define the probabilistic EIL (PEIL) and develop PEILs for two insect pest scenarios: alfalfa weevil larvae, Hypera postica (Gyllenhal), on early bud-stage alfalfa, and bean leaf beetle adults, Cerotoma trifurcata (Forster), on V1-stage soybean. The PEIL is an EIL that reflects its probability of occurrence. The probability of occurrence is determined by incorporating the uncertainty associated with the input variables used to calculate the EIL. We used Monte Carlo simulation, a random sampling technique in which each input variable in the model was sampled repeatedly from a range of possible values based on probability distributions. Each input variable's probability distribution was sampled such that the distribution's shape was reproduced. Then, the variability for each input was propagated into the output of the model so that the model output reflected the probability of values that could occur. This represents the first use of the Monte Carlo technique to determine EILs.     

Stochastic automaton models for interaction of excitatory and inhibitory impulse sequences in neurons

The mechanisms by which excitatory and inhibitory input impulse sequences interact in changing the spike probability in neurons are examined in the two mathematical neuron models; one is a real-time neuron model which is close to physiological reality, and the other a stochastic automaton model for the temporal pattern discrimination proposed in the previous paper (Tsukada et al., 1976), which is developed in this paper as neuron models for interaction of excitatory and inhibitory input impulse sequences. The interval distributions of the output spike train from these models tend to be multimodal and are compared with those used for experimental data, reported by Bishop et al. (1964) for geniculate neuron activity and Poisson process deleting model analyzed by Ten Hoopen et al. (1966). Special attention, moreover, should be paid to how different forms of inhibitory input are transformed into the output interval distributions through these neuron models. These results exhibit a clear correlation between inhibitory input form and output interval distribution. More detailed information on this mechanism is obtained from the computations of recurrence-time under the stationary condition to go from active state to itself for the first time, each of which is influenced by the inhibitory input forms. In addition to these facts, some resultant characteristics on interval histogram and serial correlation are discussed in relation to physiological data from the literature.     

Uncertainty Quantification in Life Cycle Assessments: Exploring Distribution Choice and Greater Data Granularity to Characterize Product Use

The life cycle environmental profile of energy‐consuming products is dominated by the products’ use stage. Variation in real‐world product use can therefore yield large differences in the results of life cycle assessment (LCA). Adequate characterization of input parameters is paramount for uncertainty quantification and has been a challenge to wider adoption of the LCA method. After emphasis in recent years on methodological development, data development has become the primary focus again. Pervasive sensing presents the opportunity to collect rich data sets and improve profiling of use‐stage parameters. Illustrating a data‐driven approach, we examine energy use in domestic cooling systems, focusing on climate change as the impact category. Specific objectives were to examine: (1) how characterization of the use stage by different probability distributions and (2) how characterizing data aggregated at successively higher granularity affects LCA modeling results and the uncertainty in output. Appliance‐level electricity data were sourced from domestic residences for 3 years. Use‐stage variables were propagated in a stochastic model and analyses simulated by Monte Carlo procedure. Although distribution choice did not necessarily significantly impact the estimated output, there were differences in the estimated uncertainty. Characterization of use‐stage power consumption in the model at successively higher data granularity reduced the output uncertainty with diminishing returns. Results therefore justify the collection of high granularity data sets representing the life cycle use stage of high‐energy products. The availability of such data through proliferation of pervasive sensing presents increasing opportunities to better characterize data and increase confidence in results of LCA.     

Chemotropism indices for polymorphonuclear leukocytes.

Trajectories of polymorphonuclear leukocytes which are responding to a chemical gradient are analyzed in order to deduce probability distributions of the angles between successive path segments. The turn angle probability distributions thus obtained are seen to be strongly dependent on the direction of locomotion prior to a turn, in that cells usually turn to maintain alignment along an axis directed towards the chemoattractant source. A mathematical model based on these observations is developed in order to understand the relationship between net chemotactic response and parameters characterizing stochastic movements of individual cells. In particular, the manner in which the chemotropism index depends on details of the turn-angle distributions is examined. When bias in the direction of turn is induced by a chemotactic field, transition from random motion to directed response occurs most abruptly if the turn-angle distribution is narrow. "Accommodation," viz., a dependence of the mean angle of turn upon prior orientation, is found to have relatively little effect on the magnitude of the response.     

New approach to the probabilistic-statistical analysis of turbulent transport processes in plasma

It is proposed to apply the statistical analysis of the increments of fluctuating particle fluxes to examine the probability characteristics of turbulent transport processes in plasma. Such an approach makes it possible to pass over to the analysis of the dynamical probability characteristics of the process under study. It is shown that, in the plasmas of the L-2M stellarator and the TAU-1 linear device, the increments of local fluctuating particle fluxes are stochastic in character and their distributions are scale mixtures of Gaussians. In particular, in TAU-1, the increments obey a Laplacian distribution (which is a scale mixture of Gaussians with an exponential mixing distribution). A mathematical model is proposed to explain such distributions. Possible physical mechanisms responsible for the random character of the increments of fluctuating particle fluxes are discussed.     

There Is No Need To Be Normal: Generalized Linear Models of Natural Variation

Both ecological field studies and attempts to extrapolate from laboratory experiments to natural populations generally encounter the high degree of natural variability and chaotic behavior that typify natural ecosystems. Regardless of this variability and non-normal distribution, most statistical models of natural systems use normal error which assumes independence between the variance and mean. However, environmental data are often random or clustered and are better described by probability distributions which have more realistic variance to mean relationships. Until recently statistical software packages modeled only with normal error and researchers had to assume approximate normality on the original or transformed scale of measurement and had to live with the consequences of often incorrectly assuming independence between the variance and mean. Recent developments in statistical software allow researchers to use generalized linear models (GLMs) and analysis can now proceed with probability distributions from the exponential family which more realistically describe natural conditions: binomial (even distribution with variance less than mean), Poisson (random distribution with variance equal mean), negative binomial (clustered distribution with variance greater than mean). GLMs fit parameters on the original scale of measurement and eliminate the need for obfuscating transformations, reduce bias for proportions with unequal sample size, and provide realistic estimates of variance which can increase power of tests. Because GLMs permit modeling according to the non-normal behavior of natural systems and obviate the need for normality assumptions, they will likely become a widely used tool for analyzing toxicity data. To demonstrate the broad-scale utility of GLMs, we present several examples where the use of GLMs improved the statistical power of field and laboratory studies to document the rapid ecological recovery of Prince William Sound following the Exxon Valdez oil spill.     

A shared response model for clustered binary data in developmental toxicity studies

Existing distributions for modeling fetal response data in developmental toxicology such as the beta-binomial distribution have a tendency of inflating the probability of no malformed fetuses, and hence understating the risk of having at least one malformed fetus within a litter. As opposed to a shared probability extra-binomial model, we advocate a shared response model that allows a random number of fetuses within the same litter to share a common response. An explicit formula is given for the probability function and graphical plots suggest that it does not suffer from the problem of assigning too much probability to the event of no malformed fetuses. The EM algorithm can be used to estimate the model parameters. Results of a simulation study show that the EM estimates are nearly unbiased and the associated confidence intervals based on the usual standard error estimates have coverage close to the nominal level. Simulation results also suggest that the shared response model estimates of the marginal malformation probabilities are robust to misspecification of the distributional form, but not so for the estimates of intralitter correlation and the litter-level probability of having at least one malformed fetus. The proposed model is fitted to a set of data from the U.S. National Toxicology Program. For the same dose-response relationship, the fit based on the shared response distribution is superior to that based on the beta-binomial, and comparable to that based on the recently proposed q-power distribution (Kuk, 2004, Applied Statistics53, 369-386). An advantage of the shared response model over the q-power distribution is that it is more interpretable and can be extended more easily to the multivariate case. To illustrate this, a bivariate shared response model is fitted to fetal response data involving visceral and skeletal malformation.     

Data quality

A methodology is presented that enables incorporating expert judgment regarding the variability of input data for environmental life cycle assessment (LCA) modeling. The quality of input data in the life-cycle inventory (LCI) phase is evaluated by LCA practitioners using data quality indicators developed for this application. These indicators are incorporated into the traditional LCA inventory models that produce non-varying point estimate results (i.e., deterministic models) to develop LCA inventory models that produce results in the form of random variables that can be characterized by probability distributions (i.e., stochastic models). The outputs of these probabilistic LCA models are analyzed using classical statistical methods for better decision and policy making information. This methodology is applied to real-world beverage delivery system LCA inventory models. The inventory study results for five beverage delivery system alternatives are compared using statistical methods that account for the variance in the model output values for each alternative. Sensitivity analyses are also performed that indicate model output value variance increases as input data uncertainty increases (i.e., input data quality degrades). Concluding remarks point out the strengths of this approach as an alternative to providing the traditional qualitative assessment of LCA inventory study input data with no efficient means of examining the combined effects on the model results. Data quality assessments can now be captured quantitatively within the LCA inventory model structure. The approach produces inventory study results that are variables reflecting the uncertainty associated with the input data. These results can be analyzed using statistical methods that make efficient quantitative comparisons of inventory study alternatives possible. Recommendations for future research are also provided that include the screening of LCA inventory model inputs for significance and the application of selection and ranking techniques to the model outputs.     

Unequal cell division, growth regulation and colony size of mammalian cells: a mathematical model and analysis of experimental data.

This work describes mathematically the dynamics of expansion of cell populations from the initial division of single cells to colonies of several hundred cells. This stage of population growth is strongly influenced by stochastic (random) elements including, among others, cell death and quiescence. This results in a wide distribution of colony sizes. Experimental observations of the NIH3T3 cell line as well as for the NIH3T3 cell line transformed with the ras oncogene were obtained for this study. They include the number of cells in 4-day-old colonies initiated from single cells and measurements of sizes of sister cells after division, recorded in the 4-day-old colonies. The sister cell sizes were recorded in a way which enabled investigation of their interdependence. We developed a mathematical model which includes cell growth and unequal cell division, with three possible outcomes of each cell division: continued cell growth and division, quiescence, and cell death. The model is successful in reproducing experimental observations. It provides good fits to colony size distributions for both NIH3T3 mouse fibroblast cells and the same cells transformed with the rasEJ human cancer gene. The difference in colony size distributions could be fitted by assuming similar cell lifetimes (12-13 hr) and similar probabilities of cell death (q = 0.15), but using different probabilities of quiescence, r = 0 for the ras oncogene transformed cells and r = 0.1 for the non-transformed cells. The model also reproduces the evolution of distributions of sizes of cells in colonies, from a single founder cell of any specified size to the stable limit distribution after eight to ten cell divisions. Application of the model explains in what way both random events and deterministic control mechanisms strongly influence cell proliferation at early stages in the expansion of colonies.     

Bayesian state-space modeling of metapopulation dynamics in the Glanville fritillary butterfly

The complexity of mathematical models of ecological dynamics varies greatly, and it is often difficult to judge what would be the optimal level of complexity in a particular case. Here we compare the parameter estimates, model fits, and predictive abilities of two models of metapopulation dynamics: a detailed individual-based model (IBM) and a population-based stochastic patch occupancy model (SPOM) derived from the IBM. The two models were fitted to a 17-year time series of data for the Glanville fritillary butterfly (Melitaea cinxia) inhabiting a network of 72 small meadows. The data consisted of biannual counts of larval groups (IBM) and the annual presence or absence of local populations (SPOM). The models were fitted using a Bayesian state-space approach with a hierarchical random effect structure to account for observational, demographic, and environmental stochasticities. The detection probability of larval groups (IBM) and the probability of false zeros of local populations (SPOM) in the observation models were simultaneously estimated from the time-series data and independent control data. Prior distributions for dispersal parameters were obtained from a separate analysis of mark–recapture data. Both models fitted the data about equally, but the results were more precise for the IBM than for the SPOM. The two models yielded similar estimates for a random effect parameter describing habitat quality in each patch, which were correlated with independent empirical measures of habitat quality. The modeling results showed that variation in habitat quality influenced patch occupancy more through the effects on movement behavior at patch edges than on carrying capacity, whereas the latter influenced the mean population size in occupied patches. The IBM and the SPOM explained 63% and 45%, respectively, of the observed variation in the fraction of occupied habitat area among 75 independent patch networks not used in parameter estimation. We conclude that, while carefully constructed, detailed models can have better predictive ability than simple models, this advantage comes with the cost of greatly increased data requirements and computational challenges. Our results illustrate how complex models can be helpful in facilitating the construction of effective simpler models.     

Review of Information Resources to Support Human Exposure Assessment Models

Efforts to model human exposures to chemicals are growing more sophisticated and encompass increasingly complex exposure scenarios. The scope of such analyses has increased, growing from assessments of single exposure pathways to complex evaluations of aggregate or cumulative chemical exposures occurring within a variety of settings and scenarios. In addition, quantitative modeling techniques have evolved from simple deterministic analyses using single point estimates for each necessary input parameter to more detailed probabilistic analyses that can accommodate distributions of input parameters and assessment results. As part of an overall effort to guide development of a comprehensive framework for modeling human exposures to chemicals, available information resources needed to derive input parameters for human exposure assessment models were compiled and critically reviewed. Ongoing research in the area of exposure assessment parameters was also identified. The results of these efforts are summarized and other relevant information that will be needed to apply the available data in a comprehensive exposure model is discussed. Critical data gaps in the available information are also identified. Exposure assessment modeling and associated research would benefit from the collection of additional data as well as by enhancing the accessibility of existing and evolving information resources.     

Longitudinal Data Analysis with Event Time as a Covariate

We consider the estimation of a nonparametric smooth function of some event time in a semiparametric mixed effects model from repeatedly measured data when the event time is subject to right censoring. The within-subject correlation is captured by both cross-sectional and time-dependent random effects, where the latter is modeled by a nonhomogeneous Ornstein–Uhlenbeck stochastic process. When the censoring probability depends on other variables in the model, which often happens in practice, the event time data are not missing completely at random. Hence, the complete case analysis by eliminating all the censored observations may yield biased estimates of the regression parameters including the smooth function of the event time, and is less efficient. To remedy, we derive the likelihood function for the observed data by modeling the event time distribution given other covariates. We propose a two-stage pseudo-likelihood approach for the estimation of model parameters by first plugging an estimator of the conditional event time distribution into the likelihood and then maximizing the resulting pseudo-likelihood function. Empirical evaluation shows that the proposed method yields negligible biases while significantly reduces the estimation variability. This research is motivated by the project of hormone profile estimation around age at the final menstrual period for the cohort of women in the Michigan Bone Health and Metabolism Study.     

Multi-type Galton-Watson process as a model for proliferating human tumour cell populations derived from stem cells: estimation of stem cell self-renewal probabilities in human ovarian carcinomas

A mathematical model for proliferation of tumour cell populations is developed. The cell population is assumed to be organized in a hierarchy of decreasing proliferative potential and increasing degree of differentiation. Using some elements of the theory of Multi-type Galton-Watson processes, a method is proposed for the estimation of Psr, the probability of self-renewal of tumour stem cells, from the experimental distribution of clonal unit sizes obtained in cell culture studies. Six data sets from patients with advanced adenocarcinoma of the ovary are used to demonstrate the method. Reasonable estimates are obtained, and the theoretical colony size distributions predicted by the model appear to be in good qualitative agreement with the experimental ones, and lend support to a stem cell model of tumour growth. The possible significance of Psr as a prognostic factor is briefly discussed.     

A stochastic collocation method for uncertainty quantification and propagation in cardiovascular simulations

Simulations of blood flow in both healthy and diseased vascular models can be used to compute a range of hemodynamic parameters including velocities, time varying wall shear stress, pressure drops, and energy losses. The confidence in the data output from cardiovascular simulations depends directly on our level of certainty in simulation input parameters. In this work, we develop a general set of tools to evaluate the sensitivity of output parameters to input uncertainties in cardiovascular simulations. Uncertainties can arise from boundary conditions, geometrical parameters, or clinical data. These uncertainties result in a range of possible outputs which are quantified using probability density functions (PDFs). The objective is to systemically model the input uncertainties and quantify the confidence in the output of hemodynamic simulations. Input uncertainties are quantified and mapped to the stochastic space using the stochastic collocation technique. We develop an adaptive collocation algorithm for Gauss-Lobatto-Chebyshev grid points that significantly reduces computational cost. This analysis is performed on two idealized problems--an abdominal aortic aneurysm and a carotid artery bifurcation, and one patient specific problem--a Fontan procedure for congenital heart defects. In each case, relevant hemodynamic features are extracted and their uncertainty is quantified. Uncertainty quantification of the hemodynamic simulations is done using (a) stochastic space representations, (b) PDFs, and (c) the confidence intervals for a specified level of confidence in each problem.     

Modeling species distributions by breaking the assumption of self-similarity

Species distributions are commonly measured as the number of sites, or geographic grid cells occupied. These data may then be used to model species distributions and to examine patterns in both intraspecific and interspecific distributions. Harte et al. (1999) used a model based on a bisection rule and assuming self-similarity in species distributions to do so. However, this approach has also been criticized for several reasons. Here we show that the self-similarity in species distributions breaks down according to a power relationship with spatial scales, and we therefore adopt a power-scaling assumption for modeling species occupancy distributions. The outcomes of models based on these two assumptions (self-similar and power-scaling) have not previously been compared. Based on Harte's bisection method and an occupancy probability transition model under these two assumptions (self-similar and power-scaling), we compared the scaling pattern of occupancy (also known as the area-of-occupancy) and the spatial distribution of species. The two assumptions of species distribution lead to a relatively similar interspecific occupancy frequency distribution pattern, although the spatial distribution of individual species and the scaling pattern of occupancy differ significantly. The bimodality in occupancy frequency distributions that is common in species communities, is confirmed to a result for certain mathematical and statistical properties of the probability distribution of occupancy. The results thus demonstrate that the use of the bisection method in combination with a power-scaling assumption is more appropriate for modeling species distributions than the use of a self-similarity assumption, particularly at fine scales.     

Newer Classification and Regression Tree Techniques: Bagging and Random Forests for Ecological Prediction

The task of modeling the distribution of a large number of tree species under future climate scenarios presents unique challenges. First, the model must be robust enough to handle climate data outside the current range without producing unacceptable instability in the output. In addition, the technique should have automatic search mechanisms built in to select the most appropriate values for input model parameters for each species so that minimal effort is required when these parameters are fine-tuned for individual tree species. We evaluated four statistical models—Regression Tree Analysis (RTA), Bagging Trees (BT), Random Forests (RF), and Multivariate Adaptive Regression Splines (MARS)—for predictive vegetation mapping under current and future climate scenarios according to the Canadian Climate Centre global circulation model. To test, we applied these techniques to four tree species common in the eastern United States: loblolly pine (Pinus taeda), sugar maple (Acer saccharum), American beech (Fagus grandifolia), and white oak (Quercus alba). When the four techniques were assessed with Kappa and fuzzy Kappa statistics, RF and BT were superior in reproducing current importance value (a measure of basal area in addition to abundance) distributions for the four tree species, as derived from approximately 100,000 USDA Forest Service’s Forest Inventory and Analysis plots. Future estimates of suitable habitat after climate change were visually more reasonable with BT and RF, with slightly better performance by RF as assessed by Kappa statistics, correlation estimates, and spatial distribution of importance values. Although RTA did not perform as well as BT and RF, it provided interpretive models for species whose distributions were captured well by our current set of predictors. MARS was adequate for predicting current distributions but unacceptable for future climate. We consider RTA, BT, and RF modeling approaches, especially when used together to take advantage of their individual strengths, to be robust for predictive mapping and recommend their inclusion in the ecological toolbox.