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Bringing social standards into project evaluation under dynamic uncertainty.

Society often sets social standards that define thresholds of damage to society or the environment above which compensation must be paid to the state or other parties. In this article, we analyze the interdependence between the use of social standards and investment evaluation under dynamic uncertainty where a negative externality above a threshold established by society requires an assessment and payment of damages. Under uncertainty, the party considering implementing a project or new technology must not only assess when the project is economically efficient to implement but when to abandon a project that could potentially exceed the social standard. Using real-option theory and simple models, we demonstrate how such a social standard can be integrated into cost-benefit analysis through the use of a development option and a liability option coupled with a damage function. Uncertainty, in fact, implies that both parties interpret the social standard as a target for safety rather than an inflexible barrier that cannot be overcome. The larger is the uncertainty, in fact, the greater will be the tolerance for damages in excess of the social standard from both parties.

Journal Article↗

MODFLOW 2000 head uncertainty, a first-order second moment method.

A computationally efficient method to estimate the variance and covariance in piezometric head results computed through MODFLOW 2000 using a first-order second moment (FOSM) approach is presented. This methodology employs a first-order Taylor series expansion to combine model sensitivity with uncertainty in geologic data. MODFLOW 2000 is used to calculate both the ground water head and the sensitivity of head to changes in input data. From a limited number of samples, geologic data are extrapolated and their associated uncertainties are computed through a conditional probability calculation. Combining the spatially related sensitivity and input uncertainty produces the variance-covariance matrix, the diagonal of which is used to yield the standard deviation in MODFLOW 2000 head. The variance in piezometric head can be used for calibrating the model, estimating confidence intervals, directing exploration, and evaluating the reliability of a design. A case study illustrates the approach, where aquifer transmissivity is the spatially related uncertain geologic input data. The FOSM methodology is shown to be applicable for calculating output uncertainty for (1) spatially related input and output data, and (2) multiple input parameters (transmissivity and recharge).

Calibration↗

Reducing uncertainty in the derivation and application of health guidance values in public health practice. Dioxin as a case study.

We were requested by the U.S. Environmental Protection Agency (EPA) to clarify the relationships among the minimal risk level (MRL), action level, and environmental media evaluation guide (EMEG) for dioxin established by the Agency for Toxic Substances and Disease Registry (ATSDR). In response we developed a document entitled "Dioxin and Dioxin-Like Compounds in Soil, Part I: ATSDR Interim Policy Guideline"; and a supporting document entitled "Dioxin and Dioxin-Like Compounds in Soil, Part II: Technical Support Document". In these documents, we evaluated the key assumptions underlying the development and use of the ATSDR action level, MRL, and EMEG for dioxin. We described the chronology of events outlining these different health guidance values for dioxin and identified the areas of uncertainty surrounding these values. Four scientific assumptions were found to have had a great impact on this process; these were: (1) the specific uncertainty factors used, (2) the toxicity equivalent (TEQ) approach, (3) the fractional exposure from different pathways, and (4) the use of body burdens in the absence of exposure data. This information was subsequently used to develop a framework for reducing the uncertainties in public health risk assessment associated with exposure to other chemical contaminants in the environment. Within this framework are a number of future directions for reducing uncertainty, including physiologically based pharmacokinetic modeling (PBPK), benchmark dose modeling (BMD), functional toxicology, and the assessment of chemical mixture interactions.

Benchmarking↗

Uncertainty effects in orientation discrimination of foveally seen lines in human observers.

1. The effect of spatial uncertainty on line orientation thresholds was studied in normal observers. Vertical lines, 5 min of arc long, built up a matrix in which one of the line elements could be tilted to the left or right. The orientation thresholds depended strongly on the number of alternative test positions. There was a linear relation between log (threshold) and log (P), where P is the probability that a particular line element was the one being tested. 2. The uncertainty effect was shown to be time dependent. The effect was more marked for the shortest stimulus duration (1 s). However, even with a 6 s stimulus duration, allowing several re-fixations, the thresholds were significantly higher in the presence of uncertainty, compared to the situation in which the test position was fixed and known to the observer. 3. When the measurements were restricted to the centre line in a matrix, thresholds were more than twice as high when the test line could be in any of the centre 3 x 3 positions, compared to the case in which there was no uncertainty as to the test position. Foreknowledge of location of the test line within the matrix improved the threshold further, even if the whole matrix was displaced to different retinal positions. 4. It is concluded that the physiological mechanism mediating threshold improvement probably operates on a cortical processing apparatus more central than V1.

Adult↗

Incorporating model parameter uncertainty into inverse treatment planning.

Radiobiological treatment planning depends not only on the accuracy of the models describing the dose-response relation of different tumors and normal tissues but also on the accuracy of tissue specific radiobiological parameters in these models. Whereas the general formalism remains the same, different sets of model parameters lead to different solutions and thus critically determine the final plan. Here we describe an inverse planning formalism with inclusion of model parameter uncertainties. This is made possible by using a statistical analysis-based frameset developed by our group. In this formalism, the uncertainties of model parameters, such as the parameter a that describes tissue-specific effect in the equivalent uniform dose (EUD) model, are expressed by probability density function and are included in the dose optimization process. We found that the final solution strongly depends on distribution functions of the model parameters. Considering that currently available models for computing biological effects of radiation are simplistic, and the clinical data used to derive the models are sparse and of questionable quality, the proposed technique provides us with an effective tool to minimize the effect caused by the uncertainties in a statistical sense. With the incorporation of the uncertainties, the technique has potential for us to maximally utilize the available radiobiology knowledge for better IMRT treatment.

Algorithms↗

Dose uncertainty due to computed tomography (CT) slice thickness in CT-based high dose rate brachytherapy of the prostate cancer.

In computed tomography (CT)-based high dose rate (HDR) brachytherapy, the uncertainty in the localization of the longitudinal catheter-tip positions due to the discrete CT slice thickness, results in a delivered dose uncertainty. Catheter coordinates were extracted from five patients treated for prostate cancer, and three simulation scenarios were followed to mimic the longitudinal imprecision of the catheter tips, hence the dwell positions. All catheters were displaced (1) forward, (2) backward, or (3) randomly distributed within the space defined by one CT slice thickness, for thicknesses ranging from 2 to 5 mm. Average and standard deviation values of the relative dose variations are reported for the various catheter displacement scenarios. Also, the dose points were grouped according to their relative position in the prostate, inner, peripheral and outer area of prostate and base, median and apex zones, in order to estimate the spatial sensitivity of the dose errors. For scenarios (1) and (2), the dose uncertainties due to the finite slice thickness increase linearly with the slice spacing, from 3% to 8% for the slice thickness values ranging from 2 to 5 mm, respectively. The more realistic scenario (3) yields average errors ranging from 0.7% to 1.7%. The apex and the base show larger dose errors and variability of dose errors than the median of the prostate. No statistical difference was observed among different transversal sections of the prostate. A CT slice thickness of 3 mm appears to be a good compromise showing an acceptable average dose uncertainty of 1%, without unduly increasing the number of slices.

Algorithms↗

Quantifying uncertainty in geoacoustic inversion. II. Application to broadband, shallow-water data.

This paper applies the new method of fast Gibbs sampling (FGS) to estimate the uncertainties of seabed geoacoustic parameters in a broadband, shallow-water acoustic survey, with the goal of interpreting the survey results and validating the method for experimental data. FGS applies a Bayesian approach to geoacoustic inversion based on sampling the posterior probability density to estimate marginal probability distributions and parameter covariances. This requires knowledge of the statistical distribution of the data errors, including both measurement and theory errors, which is generally not available. Invoking the simplifying assumption of independent, identically distributed Gaussian errors allows a maximum-likelihood estimate of the data variance and leads to a practical inversion algorithm. However, it is necessary to validate these assumptions, i.e., to verify that the parameter uncertainties obtained represent meaningful estimates. To this end, FGS is applied to a geoacoustic experiment carried out at a site off the west coast of Italy where previous acoustic and geophysical studies have been performed. The parameter uncertainties estimated via FGS are validated by comparison with: (i) the variability in the results of inverting multiple independent data sets collected during the experiment; (ii) the results of FGS inversion of synthetic test cases designed to simulate the experiment and data errors; and (iii) the available geophysical ground truth. Comparisons are carried out for a number of different source bandwidths, ranges, and levels of prior information, and indicate that FGS provides reliable and stable uncertainty estimates for the geoacoustic inverse problem.

Acoustics↗

Experimental validation of a nonparametric probabilistic model of nonhomogeneous uncertainties for dynamical systems.

The paper deals with an experimental validation of a nonparametric probabilistic model of nonhomogeneous uncertainties for dynamical systems. The theory used, recently introduced, allows model uncertainties and data uncertainties to be simultaneously taken into account. An experiment devoted to this validation was specifically developed. The experimental model is constituted of two simple dural rectangular plates connected together with a complex joint. In the mean mechanical model, the complex joint, which is constituted of two additional plates attached with 40 screw-bolts, is modeled by a homogeneous orthotropic continuous plate with constant thickness, as usual. Consequently, the mean model introduces a region (the joint) which has a high level of uncertainties. The objective of the paper is to present the experiment and the comparisons of the theoretical prediction with the experiments.

Journal Article↗

Signal and masker uncertainty with noise maskers of varying duration, bandwidth, and center frequency.

The detectability of sinusoidal signals added to repeatable noise bursts was measured under conditions of signal-frequency uncertainty and masker-waveform uncertainty. Either source of uncertainty raised thresholds by 2-5 dB over those measured in a fixed-signal, fixed-masker condition, while the combination of both types of uncertainty raised thresholds by 8-12 dB. The magnitude of these elevations are similar to those found in a previous study, which employed maskers composed of random sets of equal-amplitude tones (Spiegel et al., J. Acoust. Soc. AM. 70, 1015-1019 (1981)]. When masker level varied by up to 40 dB between the two intervals of a forced-choice trial, and signal thresholds were elevated by only 2.5 dB. The results support a form of profile analysis in which listeners detect signals by noticing a relative change in the masker spectrum.

Auditory Threshold↗

Perceptual learning in contrast discrimination: the effect of contrast uncertainty.

Performance in perceptual tasks improves with repetition (perceptual learning), eventually reaching a saturation level. Typically, when perceptual learning effects are studied, stimulus parameters are kept constant throughout the training and during the pre- and post-training tests. Here we investigate whether learning by repetition transfers to testing conditions in which the practiced stimuli are randomly interleaved during the post-training session. We studied practice effects with a contrast discrimination task, employing a number of training methods: (i) practice with a single, fixed pedestal (base-contrast), (ii) practice with several pedestals, and (iii) practice with several pedestals that included a spatial context. Pre- and post-training tests were carried out with the base contrast randomized across trials, under conditions of contrast uncertainty. The results showed that learning had taken place with the fixed pedestal method (i) and with the context method (iii), but only the latter survived the uncertainty test. In addition, we were able to identify a very fast learning phase in contrast discrimination that improved performance under uncertainty. We contend that learned tasks that do not pass the uncertainty test involve modification of decision strategies that require exact knowledge of the stimulus.

Contrast Sensitivity↗

Effect of uncertainty and diagnosticity on classification of multidimensional data with integral and separable displays of system status.

Integrative, objectlike displays have been advocated for presenting multidimensional system data. In this research two experiments assess the effect of uncertainty on the processing of integral and separable displays. In each experiment 30 subjects were trained to classify instances of system state into one of four state categories using a configural display, a bar graph display, or a digital display. In Experiment 1 the range of instances from the state categories was uniform; in Experiment 2 the distribution was biased toward those instances of highly uncertain state category membership. After training, subjects received extended practice classifying system data. In both experiments uncertainty was found to have the greatest effect on classification performance. In Experiment 1 the bar graph display was consistently superior; the configural display was superior to the digital display only under conditions of low uncertainty. In Experiment 2 the superiority of the bar graph display diminished, producing results equivalent to those of the digital display, with the configural display producing the worst performance. The effect of uncertainty on classification performance is discussed, with specific attention paid to the apparent configural and separable properties of the bar graph display.

Adolescent↗

Behavioral expressions of intolerance of uncertainty in worry. Experimental findings.

Intolerance of uncertainty has been related to increased worry in recent conceptual models of Generalized Anxiety Disorder (GAD). The present study examines the relationship between a measure of intolerance of uncertainty, the Intolerance of Uncertainty scale (IU), and behaviors associated with GAD, such as slowness of execution and amount of evidence required before decision making. A total of 49 individuals participated in the study. Three experimental tasks varying in level of ambiguity and difficulty were used. The analyses show a positive correlation between scores on the IU and number of cues required before responding in a moderately ambiguous inference task. However, scores on the IU were not significantly related to responses in an unambiguous task (regardless of the level of difficulty) or in a highly ambiguous task. The results suggest that high worriers have a lower threshold of tolerance of uncertainty for intermediate levels of ambiguity. The results are discussed in terms of theoretical models of GAD.

Adult↗

Uncertainty in Bayes.

Most standard discussions of Bayes' formula treat sensitivity, specificity, and the prior probability as fixed parameters for probability revision, but in fact these usually have associated variability. This in turn generates predictable patterns of uncertainty in the posterior probabilities. Although these have not been investigated in detail, they have important implications for the interpretation of posterior probabilities and the clinical use of Bayesian probability revision. For a test with a high likelihood ratio for a positive result, the positive predictive value (PPV) is strongly affected by uncertainties in the prior probability when the prior probability is small, but PPV is almost independent of such uncertainties at high values of the prior probability. The PPV is more affected by changes in specificity than by changes in sensitivity, and uncertainty in specificity has its maximal impact on the PPV at low prior probability values. These patterns are most pronounced for tests with high likelihood ratios of positive results. Similar results can be shown for the negative predictive value. These results imply that for suitability good tests, probability revision in certain definable ranges of prior probability may be so strongly affected by errors in the estimations of both the prior probability and the operating characteristics that the posterior probabilities may be unstable in practice. On the other hand, at other values of the prior probability, the posterior probabilities are almost constant, and formal probability revision will not have much impact. These patterns indicate limitations to the reliability and usefulness of calculated posterior probabilities, and have important implications for the clinical use of Bayes' formula.

Bayes Theorem↗

Uncertainty in decision models analyzing cost-effectiveness: the joint distribution of incremental costs and effectiveness evaluated with a nonparametric bootstrap method.

PURPOSE: To illustrate the use of a nonparametric bootstrap method in the evaluation of uncertainty in decision models analyzing cost-effectiveness. METHODS: The authors reevaluated a previously published cost-effectiveness analysis that used a Markov model comparing initial percutaneous transluminal angioplasty with bypass surgery for femoropopliteal lesions. Each probability in the model was simulated with a first-order Monte Carlo simulation to represent sampling uncertainty. Superimposed on this, a second-order Monte Carlo simulation was performed to represent parameter uncertainty, drawing the probability values from nonparametric distributions based on published data or from primary collected data as available. After simulation of a mixed (i.e., non-identical) cohort of 30,000 patients, 3,000 bootstrap samples of 1,000 patients each were drawn and the joint distribution of mean incremental costs and mean effectiveness gained was evaluated. RESULTS: Using a bootstrap sample size of 1,000 patients, 92.7% of the joint distribution of mean incremental costs and mean effectiveness gained fell in the quadrant where angioplasty dominated bypass surgery. Another 6.9% of samples demonstrated either greater effectiveness with an incremental cost-effectiveness ratio of at most $20,000/QALY gained, or cost savings with a ratio of at least $20,000 saved/QALY lost. CONCLUSION: A nonparametric bootstrap method can be used to estimate the joint distribution of mean incremental costs and mean effectiveness gained, and the results can provide an understanding of the uncertainty in a cost-effectiveness analysis based on a decision model.

Angioplasty, Balloon↗

Breast screening: adverse psychological consequences one month after placing women on early recall because of a diagnostic uncertainty. A multicentre study.

BACKGROUND: It was the original intention of the UK National Health Service Breast Screening Programme (NHSBSP) to place women who were not diagnosed with cancer on three yearly routine recall (RR). In 1994-5 approximately 16,500 women, aged 50 to 64, were placed on early recall (ER) at a shorter time interval, of which about 98% will have a normal result. This large number exceeds the expectations of the NHSBSP. OBJECTIVE: To establish the adverse psychological consequences (PCs) for women one month after placement on ER because of a diagnostic uncertainty, and if detected, to suggest practical solutions to reduce them. METHODS: Thirteen breast screening centres throughout the UK participated in the study. From March to October 1995 all women who were placed on ER because of a diagnostic uncertainty were identified and compared with groups of women placed on RR (after mammography, assessment, fine needle aspiration, and a benign biopsy). These women were invited to complete a postal questionnaire one month after they were placed on ER or RR. One reminder was sent. RESULTS: Overall 75% of women completed the questionnaire. The adverse PCs of placing women on ER because of a diagnostic uncertainty were higher (63%; n = 81 of 130) than those of women placed on RR after mammography (29%; n = 38 of 130) (P < 0.00001) or assessment (50%; n = 64 of 128) (P < 0.05), but lower than the adverse PCs of women who underwent a benign biopsy (87%; n = 26 of 30) (P < 0.05). Factors that were significantly associated with subsequent adverse PCs were identified. CONCLUSIONS: The adverse PCs of being placed on ER because of a diagnostic uncertainty were significantly higher than those of women who turned out to have a false-positive mammographic result after assessment. Possible practical solutions are discussed.

Breast Neoplasms↗

Application of knowledge-driven spatial modelling approaches and uncertainty management to a study of Rift Valley fever in Africa.

BACKGROUND: There are few studies that have investigated uncertainties surrounding the scientific community's knowledge of the geographical distribution of major animal diseases. This is particularly relevant to Rift Valley fever (RVF), a zoonotic disease causing destructive outbreaks in livestock and man, as the geographical range of the disease is widening to involve previously unaffected regions. In the current study we investigate the application of methods developed in the decision sciences: multiple criteria decision making using weighted linear combination and ordered weighted averages, and Dempster-Shafer theory, implemented within the geographical information system IDRISI, to obtain a greater understanding of uncertainty related to the geographical distribution of RVF. The focus is on presenting alternate methods where extensive field data are not available and traditional, model-based approaches to disease mapping are impossible to conduct. RESULTS: Using a compensatory multiple criteria decision making model based on weighted linear combination, most of sub-Saharan Africa was suitable for endemic circulation of RVF. In contrast, areas where rivers and lakes traversed semi-arid regions, such as those bordering the Sahara, were highly suitable for RVF epidemics and wet, tropical areas of central Africa had low suitability. Using a moderately non-compensatory model based on ordered weighted averages, the areas considered suitable for endemic and epidemic RVF were more restricted. Varying the relative weights of the different factors in the models did not affect suitability estimates to a large degree, but variations in model structure had a large impact on our suitability estimates. Our Dempster-Shafer analysis supported the belief that a range of semi-arid areas were suitable for RVF epidemics and the plausibility that many other areas of the continent were suitable. Areas where high levels of uncertainty were highlighted included the Ethiopian Highlands, southwest Kenya and parts of West Africa. CONCLUSION: We have demonstrated the potential of methods developed in the decision sciences to improve our understanding of uncertainties surrounding the geographical distribution of animal diseases, particularly where information is sparse, and encourage wider application of the decision science methodology in the field of animal health.

Africa South of the Sahara↗

Geostatistical analysis of disease data: visualization and propagation of spatial uncertainty in cancer mortality risk using Poisson kriging and p-field simulation.

BACKGROUND: Smoothing methods have been developed to improve the reliability of risk cancer estimates from sparsely populated geographical entities. Filtering local details of the spatial variation of the risk leads however to the detection of larger clusters of low or high cancer risk while most spatial outliers are filtered out. Static maps of risk estimates and the associated prediction variance also fail to depict the uncertainty attached to the spatial distribution of risk values and does not allow its propagation through local cluster analysis. This paper presents a geostatistical methodology to generate multiple realizations of the spatial distribution of risk values. These maps are then fed into spatial operators, such as in local cluster analysis, allowing one to assess how risk spatial uncertainty translates into uncertainty about the location of spatial clusters and outliers. This novel approach is applied to age-adjusted breast and pancreatic cancer mortality rates recorded for white females in 295 US counties of the Northeast (1970-1994). A public-domain executable with example datasets is provided. RESULTS: Geostatistical simulation generates risk maps that are more variable than the smooth risk map estimated by Poisson kriging and reproduce better the spatial pattern captured by the risk semivariogram model. Local cluster analysis of the set of simulated risk maps leads to a clear visualization of the lower reliability of the classification obtained for pancreatic cancer versus breast cancer: only a few counties in the large cluster of low risk detected in West Virginia and Southern Pennsylvania are significant over 90% of all simulations. On the other hand, the cluster of high breast cancer mortality in Niagara county, detected after application of Poisson kriging, appears on 60% of simulated risk maps. Sensitivity analysis shows that 500 realizations are needed to achieve a stable classification for pancreatic cancer, while convergence is reached for less than 300 realizations for breast cancer. CONCLUSION: The approach presented in this paper enables researchers to generate a set of simulated risk maps that are more realistic than a single map of smoothed mortality rates and allow the propagation of cancer risk uncertainty through local cluster analysis. Coupled with visualization and querying capabilities of geographical information systems, animated display of realizations can highlight areas that depart consistently from the general behavior observed across the region, guiding further investigation and control activities.

Cluster Analysis↗

Chronic beryllium disease and cancer risk estimates with uncertainty for beryllium released to the air from the Rocky Flats Plant.

Beryllium was released into the air from routine operations and three accidental fires at the Rocky Flats Plant (RFP) in Colorado from 1958 to 1989. We evaluated environmental monitoring data and developed estimates of airborne concentrations and their uncertainties and calculated lifetime cancer risks and risks of chronic beryllium disease to hypothetical receptors. This article discusses exposure-response relationships for lung cancer and chronic beryllium disease. We assigned a distribution to cancer slope factor values based on the relative risk estimates from an occupational epidemiologic study used by the U.S. Environmental Protection Agency (EPA) to determine the slope factors. We used the regional atmospheric transport code for Hanford emission tracking atmospheric transport model for exposure calculations because it is particularly well suited for long-term annual-average dispersion estimates and it incorporates spatially varying meteorologic and environmental parameters. We accounted for model prediction uncertainty by using several multiplicative stochastic correction factors that accounted for uncertainty in the dispersion estimate, the meteorology, deposition, and plume depletion. We used Monte Carlo techniques to propagate model prediction uncertainty through to the final risk calculations. We developed nine exposure scenarios of hypothetical but typical residents of the RFP area to consider the lifestyle, time spent outdoors, location, age, and sex of people who may have been exposed. We determined geometric mean incremental lifetime cancer incidence risk estimates for beryllium inhalation for each scenario. The risk estimates were < 10(-6). Predicted air concentrations were well below the current reference concentration derived by the EPA for beryllium sensitization.

Adult↗