Using Bayesian evidence synthesis to quantify uncertainty in population trends in smoking behaviour
Stephen WadePeter SarichPavla VaneckovaS. Behar HarpazPreston NgoPaul GroganSonya CressmanCoral GartnerJohn M. MurrayTony BlakelyEmily BanksMartin C. TammemägiKaren CanfellMarianne WeberMichael Caruana
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Simulation models of smoking behaviour provide vital forecasts of exposure to inform policy targets, estimates of the burden of disease, and impacts of tobacco control interventions. A key element of useful model-based forecasts is a clear picture of uncertainty due to the data used to inform the model, however, assessment of this parameter uncertainty is incomplete in almost all tobacco control models. As a remedy, we demonstrate a Bayesian approach to model calibration that quantifies parameter uncertainty. With a model calibrated to Australian data, we observed that the smoking cessation rate in Australia has increased with calendar year since the late 20th century, and in 2016 people who smoked would quit at a rate of 4.7 quit-events per 100 person-years (90% equal-tailed interval (ETI): 4.5–4.9). We found that those who quit smoking before age 30 years switched to reporting that they never smoked at a rate of approximately 2% annually (90% ETI: 1.9–2.2%). The Bayesian approach demonstrated here can be used as a blueprint to model other population behaviours that are challenging to measure directly, and to provide a clearer picture of uncertainty to decision-makers.Keywords:
Credible interval
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SUMMARY Let Y 1, . . ., Yn denote independent observations each distributed according to a distribution depending on a scalar parameter θ suppose that we are interested in constructing an interval estimate for θ. One approach is to use Bayesian inference. For a given prior density, we can construct an interval such that the posterior probability that θ lies in the interval is some specified value. In this paper, a method is given for constructing a Bayesian interval estimate such that the coverage probability of the interval is approximately equal to the posterior probability of the interval.
Credible interval
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There is always a deviation between a model prediction and the reality that the model intends to represent. The deviation is largely caused by the model uncertainty due to ignorance, assumptions, simplification, and other sources of lack of knowledge. Quantifying model uncertainty is a vital task and requires the comparison between model prediction and observation. This exercise is generally computationally intensive on the prediction side and costly on the experimentation side. In this work, a new methodology is proposed to provide an alternative implementation of model uncertainty quantification. With the new methodology, the experimental results are reported with expanded uncertainty terms around the experimental results for both model input and output. In other words, the experimental results are expressed as intervals. Then the model takes the experimental results of the input intervals and produces an interval prediction. The model uncertainty is then quantified by the difference between the model prediction and experimental observation, represented by an interval as well. By employing the standards for measurement uncertainty, the new methodology is easy to implement and could serve as a common framework for both model builders and experimenters.
Sensitivity Analysis
Uncertainty Quantification
Prediction interval
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In this article an accurate confidence interval is derived when the results of a small number of possibly biased experimental methods are combined for the determination of an unknown quantity called the consensus mean. ANOVA and a simple hierarchical Bayesian analysis of variance with locally uniform priors result in confidence intervals too wide for precision metrology. Often when deciding on experimental methods, scientists choose methods in such a way that the truth lies between the extremes of the method means. Combining this additional information with experimental data, an interval more accurate than the ANOVA interval and the simple hierarchical Bayesian interval is obtained. The estimate obtained falls within the ISO guidelines, and the mean and standard deviation used to derive the confidence interval are shown to be the posterior mean and variance of a fully Bayesian procedure.
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Typically, a measurement is regarded as being incomplete without a statement of uncertainty being provided with the result. Usually, the corresponding interval of measurement uncertainty will be an evaluated confidence interval, assuming that the classical, frequentist, approach to statistics is adopted. However, there are other types of interval that are potentially relevant, and which might wrongly be called a confidence interval. This paper describes different types of statistical interval and relates these intervals to the task of obtaining a figure of measurement uncertainty. Definitions and examples are given of probability intervals, confidence intervals, prediction intervals and tolerance intervals, all of which feature in classical statistical inference. A description is also given of credible intervals, which arise in Bayesian statistics, and of fiducial intervals. There is also a discussion of the term “coverage interval” that appears in the International Vocabulary of Metrology and in the supplements to the Guide to the Expression of Uncertainty in Measurement .
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In this article an accurate confidence interval is derived when the results of a small number of possibly biased experimental methods are combined for the determination of an unknown quantity called the consensus mean. ANOVA and a simple hierarchical Bayesian analysis of variance with locally uniform priors result in confidence intervals too wide for precision metrology. Often when deciding on experimental methods, scientists choose methods in such a way that the truth lies between the extremes of the method means. Combining this additional information with experimental data, an interval more accurate than the ANOVA interval and the simple hierarchical Bayesian interval is obtained. The estimate obtained falls within the ISO guidelines, and the mean and standard deviation used to derive the confidence interval are shown to be the posterior mean and variance of a fully Bayesian procedure.
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Robust confidence intervals
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Thailand is dealing with air pollution, particularly from small particulate matter (PM), significantly impacting public health. Wind speed is pivotal in the dispersion of these particles. Due to its unpredictability, we are interested in estimating the confidence interval (CI) for mean wind speed data using a Birnbaum-Saunders (BS) distribution. We have constructed various intervals, Bootstrap confidence interval (BCI), Percentile bootstrap confidence interval (PBCI), Generalized confidence interval (GCI), Bayesian credible interval (BayCI), and The highest posterior density (HPD). Using the R statistical software, a simulation study evaluated their coverage probabilities (CP) and average lengths (AL). GCI emerged as the most effective method overall. With increased sample size and shape parameters, these intervals displayed reduced average lengths. Applying these intervals to wind speed datasets in Nong Prue subdistrict, Chonburi province, Thailand, demonstrated their effectiveness.
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interval estimation is an important mathematical statistics content,this paper Bayesian interval estimation and interval estimation of the difference between classic,introduced the normal population variance of the Bayesian confidence interval,numerical analysis shows that CI Confidence interval more accurate.
Credible interval
Interval estimation
Population variance
Tolerance interval
Prediction interval
Bayes estimator
Robust confidence intervals
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Abstract Multilevel models are designed for data that have a nested structure, for example individuals within groups, with variables measured at each level of the nested structure. The model parameters are commonly estimated with maximum likelihood methods. Recently, Bayesian estimation methods have become in use for multilevel estimation. Bayesian estimation has the advantage of working well with complex models and small samples, or with large amounts of missing data. There is also the principled advantage that Bayesian estimates have a more direct interpretation that common frequentist estimates, for example a 95% credibility interval (CI) can be interpreted as an interval that has a 95% probability of containing the real value.
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Methods are presented for construction of interval estimates on the mean of a gamma distribution when there is some prior interval information as to the location of this parameter. The methods produce posterior intervals by constructing prior distributions for the mean parameter from the prior interval information. Both Bayesian and pseudo-Bayesian approaches for the construction of the priors are considered. These concepts are illustrated by an experiment assessing the operating characteristics of a laboratory chemical analyzer.
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Reference intervals, or reference ranges, aid medical decision-making by containing a pre-specified proportion (e.g., 95%) of the measurements in a representative healthy population. We recently proposed three approaches for estimating a reference interval from a meta-analysis based on a random effects model: a frequentist approach, a Bayesian posterior predictive interval, and an empirical approach. Because the Bayesian posterior predictive interval becomes wider to incorporate estimation uncertainty, it may systematically contain greater than 95% of measurements when the number of studies is small or the between study heterogeneity is large. The frequentist and empirical approaches also captured a median of less than 95% of measurements in this setting, and 95% confidence or credible intervals for the reference interval limits were not developed. In this update, we describe how one can instead use Bayesian methods to summarize the appropriate quantiles (e.g., 2.5th and 97.5th) of the marginal distribution of individuals across studies and construct a credible interval describing the estimation uncertainty in the lower and upper limits of the reference interval. We demonstrate through simulations that this method performs well in capturing 95% of values from the marginal distribution and maintains a median coverage of near 95% of the marginal distribution even when the number of studies is small, or the between-study heterogeneity is large. We also compare the results of this method to those obtained from the three previously proposed methods in the original case study of the meta-analysis of frontal subjective postural vertical measurements.
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