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Environmental Monitoring and Assessment – Normal Response Models
Published in Song S. Qian, Mark R. DuFour, Ibrahim Alameddine, Bayesian Applications in Environmental and Ecological Studies with R and Stan, 2023
Song S. Qian, Mark R. DuFour, Ibrahim Alameddine
In equation (4.12), θ1 and θ4 are the lower (when ) and upper (when ) bounds of the curve, θ3 is the x value when the expected value of y is , and is the shape parameter. This model can be fit using the R function nls with the self-starter function in Qian [2016, Chapter 6]. We note that the four-parameter logistic function defined in Qian [2016, equation (6.4)] () uses α4 as the lower bound and α1 as the upper bound of the curve. There is no advantage of using the Bayesian method if fitting the model is the only goal. The advantage of the Bayesian estimation method is in the estimation of unknown concentrations. That is, the unknown concentration of x0 associated with the observed response of y0. When using the fitted coefficients (i.e., ), the inverse solution of x0 does not exist when the observed y0 is outside of the range defined by the estimated θ4 and θ1. Furthermore, the estimator of x0 based on the inverse function of equation (4.12)
Network Meta-Analysis
Published in Ding-Geng (Din) Chen, Karl E. Peace, Applied Meta-Analysis with R and Stata, 2021
Bayesian inference involves a process of fitting a probability model to a set of observed data and summarizes the results for the unobserved parameters or unobserved data given the observed data (Gelman 2014). The essential characteristic of Bayesian methods is that the use of probability for quantifying uncertainty in inferences based on statistical data analysis and the process of Bayesian data analysis can be divided into three steps as following: (1) setting up a joint probability model for all observable data and unobservable parameters in a problem; (2) calculating and interpreting the appropriate posterior distribution, which is known as the conditional probability distribution of the unobserved parameters of interest, given the observed data; (3) evaluating the model fitting and the implications of the resulting posterior distribution (Gelman 2014).
Bayesian Cure Model
Published in Yingwei Peng, Binbing Yu, Cure Models, 2021
Recently, the Bayesian methods are increasingly used in clinical trials. For example, informative priors that incorporate historical data or expert opinions are necessary because of limited data. Bayesian design that use historical data may be used for clinical trials for pediatric populations or rare indications with cure rate (Psioda and Ibrahim, 2018). It is of interest to explore how to use power prior to utilize historical data to reduce the number of patients in the control group (Chen et al., 2002).
Optimised adaptive procedures and analysis methods for conducting speech-in-noise tests
Published in International Journal of Audiology, 2023
Nicholas Herbert, Matthias Keller, Peter Derleth, Volker Kühnel, Olaf Strelcyk
There do exist other stimulus placement procedures and analysis methods which were not investigated in this study. It was simply not feasible to test all methods in existence, so the most recent and relevant examples were chosen. We chose specifically to focus on Bayesian methods to compare against the current clinical standard. Furthermore, we chose not to include the maximum likelihood analysis method reported by Brand and Kollmeier (2002), as such procedures can be sensitive to search parameterisation (Prins 2019). We considered that this could be problematic when defining the maximum likelihood search for a wide range of data (e.g. in clinical settings). Thus, we argue that the use of the Beta method was more appropriate. One consideration in this regard is the availability and ease of administration for clinicians. Whilst most of the methods and procedures investigated here have toolboxes available for their implementation, a more user-friendly graphical user interface would greatly assist clinicians using these methodologies. This is particularly true given the relative complexity of some methods compared to the clinical standard.
Improved maximum likelihood estimation of the shape-scale family based on the generalized progressive hybrid censoring scheme
Published in Journal of Applied Statistics, 2022
In statistical inference, it is known that the maximum likelihood estimation method is biased when sample sizes are small or when the data are heavily censored and ineffective as the Bayesian method. These biases can mislead subsequent inferences and in some distributions, contain nonlinear equations that require numerical techniques. Therefore, the challenge in this paper is to improve the maximum likelihood estimation method, using the Runge–Kutta technique. The simulations and real data set results indicated that the improved method was more efficient than Bayesian method, even using informative and kernel priors. Thus, the statistical significance of this method is its efficiency compared to most estimation methods, and it is reliable and easy to apply, especially for researchers in social sciences and psychology. To illustrate this, we applied the proposed method to a general lifetime distribution which contains some of the lifetime distributions most commonly used in reliability and survival analyzes such as Weibull and Weibull extension models.
Variational Bayesian inference for association over phylogenetic trees for microorganisms
Published in Journal of Applied Statistics, 2022
Xiaojuan Hao, Kent M. Eskridge, Dong Wang
The kernel adjusted association test was applied to the same simulation setting as in Simulation I. For simulated data sets, 67% resulted in p-values less than 0.05, and 23% resulted in p-values less than 0.01. Though it is not straightforward to compare results from frequentist and Bayesian frameworks, it does suggest that this kernel adjusted association test could potentially be used to identify linear associations. As the kernel-based test requires the maximum likelihood estimator for the rate parameter λ for constructing the kernel matrix as well as using permutations, it takes similar amount of computation time as the variational Bayesian method. A major advantage of the Bayesian method is the ability to provide detailed information on the mode of action through the posterior distribution, which is not available from the kernel adjusted association test.