### Abstract:

Beta-Poisson dose-response model is a popular parametric dose response model
which is extensively applied in Microbial risk area. The advantage of Beta-Poisson
dose-response model is that it is a model which can consider the change in the risk.
The change is possibly happened because of human responses diversity, pathogen
competence diversity, or the interaction between them.
In our project, we divide it by two parts. The rst part is the Theoretical Part. We
will nd we cannot use the Method of Moments and Maximum Likelihood Estimation
directly to create the Con dence Ellipse for simultaneous estimation of Beta-Poisson
dose-response model parameters. Therefore, we need to discover a suitable approximate
distribution function for the Beta-Poisson dose-response model rst. Then, we
will infer the Maximum Likelihood Estimators for the approximation. Afterwards,
we will construct Fisher information matrix. In the nal, we are going to structure a
normal approximation that gives con dence interval for parameters of Beta-Poisson dose-response approximation. The second part is the simulation part. Since we cannot
nd that large number of real pathogen data, we are going to use R programming
to simulate 10; 000 iterations base on 8 groups of the pathogen parameters, such as
the parameters of Shigella spp., S.typhi and so on. Maximum Likelihood Estimates,
and , for parameters will be calculated after the simulation with the sample size
as n = 100; 500, and 1000. As the result of simulation, rst of all, we are going to
calculate errors from the Monte-Carlo estimations. Then, we will use Scatter Plots to
present the sets of parameters, and apply the 95% con dence ellipse from R to check
the approximate model. After that, the Histograms shows the errors of parameters
will be presented for each value of and . Finally, we will summarize the Maximum
Likelihood Estimates of and , For comparing the results of the simulation, we will
calculate the Error of Estimation, the Mean Square Error and the coverage of the
probability. The method is quite useful for the future study in Epidemiology area.

### Description:

A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science in Statistics, University of Regina. x, 89p.