Evaluating water quality using power priors to incorporate historical information

Abstract

SUMMARYTo assess water quality standards, measurements of water quality under the Clean Water Act are collected on a regular basis over a period of time. The data are analyzed to evaluate the percentage of samples exceeding the standard. One problem is that current data are limited by the time range and consequently the sample size is inadequate to provide necessary precision in parameter estimation. To address this issue, we present a Bayesian approach using a power prior to incorporate historical data and/or the data collected at adjacent stations. We develop a modified power prior approach and discuss its properties under the normal mean model. Several sets of water quality data are studied to illustrate the implementation of the power prior approach and its differences from alternative methods.

The Sequential Probability Ratio Test: An efficient alternative to exact binomial testing for Clean Water Act 303(d) evaluation

“…The Bayesian power prior method advocated by Duan, Ye and Smith incorporates historical and adjacent site data into a binomial-model water quality assessment via a power prior, but treats “current” data as a batch to be tested simultaneously to provide a likelihood of the parameter value of interest. 18 Similarly, the Bayesian approach encouraged by McBride and Ellis uses a simultaneous sample binomial likelihood but with beta priors. 19 The TMDL compliance method recommended by Qian and Reckhow allows for sequential updating of the likelihood, but was only developed for continuous variables (in the context of attainment of TMDL goals), 14 whereas California determines CWA compliance via a binary decision variable per sample.…”

Extrapolation of efficacy and other data to support the development of new medicines for children: A systematic review of methods

“…It has been noted that the hierarchical power prior in (1) violates the likelihood principle since it omits the normalising constant for a 0 [16, 17]. Modifying (1) to incorporate the normalising constant (2) by extending the Bayesian model for x S and x T to incorporate a parameter τ measuring the correlation between parameters of the historical and contemporary data, and stipulating that…”

“…Ibrahim and Chen [2] extend π P P in (1) to incorporate data from multiple historical studies. Versions accommodating data following generalized linear fixed and mixed effect models, proportional hazards models and cure rate models are also derived.It has been noted that the hierarchical power prior in (1) violates the likelihood principle since it omits the normalising constant for a 0 [16, 17]. Modifying (1) to incorporate the normalising constant (2) by extending the Bayesian model for x S and x T to incorporate a parameter τ measuring the correlation between parameters of the historical and contemporary data, and stipulating that…”

Modeling and validating Bayesian accrual models on clinical data and simulations using adaptive priors

“…In Bayesian analysis, the modified power prior provides an efficient way to incorporate and down weight historical data [20, 21, 22]. In our exponential waiting time model, the modified power prior can be written as $$\pi false(\theta ,Pfalse|n,Tfalse)=Cfalse(Pfalse)L{false(\theta false|n,Tfalse)}^P{\pi }_{0}false(\theta false)\pi false(Pfalse),$$ Denote the initial prior ${\pi }_{0}false(\theta false)=\frac{1}{\theta }$, which is a special inverse gamma with both shape and scale parameter equaling 0.…”

“…The power prior was evaluated and modified by several groups of researchers, which is now well recognized as the modified power prior [21, 22]. The amount of borrowing of strength of historical data is controlled by the power parameter.…”