A Modified Bayesian Power Prior Approach with Applications in Water Quality Evaluation

This research is motivated by an issue frequently encountered in environmental water quality evaluation. Many times, the sample size of water monitoring data is too small to have adequate power. Here, we present a Bayesian power prior approach by incorporating the current data and historical data and/or the data collected at neighboring stations to make stronger statistical inferences on the parameters of interest.

The elicitation of power prior distributions is based on the availability of historical data, and is realized by raising the likelihood function of the historical data to a fractional power. The power prior Bayesian analysis has been proven to be a useful class of informative priors in Bayesian inference. In this dissertation, we propose a modified approach to constructing the joint power prior distribution for the parameter of interest and the power parameter. The power parameter, in this modified approach, quantifies the heterogeneity between current and historical data automatically, and hence controls the influence of historical data on the current study in a sensible way. In addition, the modified power prior needs little to ensure its propriety. The properties of the modified power prior and its posterior distribution are examined for the Bernoulli and normal populations. The modified and the original power prior approaches are compared empirically in terms of the mean squared error (MSE) of parameter estimates as well as the behavior of the power parameter. Furthermore, the extension of the modified power prior to multiple historical data sets is discussed, followed by its comparison with the random effects model.

Several sets of water quality data are studied in this dissertation to illustrate the implementation of the modified power prior approach with normal and Bernoulli models. Since the power prior method uses information from sources other than current data, it has advantages in terms of power and estimation precision for decisions with small sample sizes, relative to methods that ignore prior information.