# 11 - Estimation of alpha and beta parameters for prior Beta distributions and summarisation of Beta distributions for specified alpha and beta parameters

This utility calculates the alpha and beta parameters for prior Beta distributions, based on the values specified for the mode and 5th or 95th percentile of the distribution. See Suess et al. (2002) for more details. An accompanying utility allows the calculation of summary values for Beta distributions for specified alpha and beta parameters.

Beta distributions are a type of probability distribution that is commonly used to describe uncertainty about the true value of a proportion, such as sensitivity, specificity or prevalence. They are appropriate distributions to express uncertainty about the prior values for prevalence, sensitivity or specificity in the Gibbs sampler ( Joseph et al., 1995; Vose, 2000). When used for this purpose, the Beta distribution can be defined by the two parameters, alpha and beta (written as Beta(alpha, beta)), with alpha = x + 1 and beta = n - x + 1, where x is the number of positive events out of n trials. As n increases, the degree of uncertainty (the width of the distribution) about the estimated proportion (x/n) decreases. The mean value of the distribution can be calculated as alpha/(alpha+beta) and the mode is (alpha-1)/(alpha+beta-2).

Required inputs for the parameterisation utility are:

• mode and 5/95th percentile for at least one distribution; and
• mode and 5/95th percentile for each distribution required, up to a maximum of six distributions.

Required inputs for the summarisation utility are:

• alpha and beta parameters for at least one distribution; and
• alpha and beta parameters for each distribution required, up to a maximum of six distributions.

If any input cell is blank or contains an invalid value results will not be calculated for that or any subsequent distributions.

For parameterisation of Beta distributions, all values entered must be >0 and <1. It is suggested that, where the mode is less than 0.5, you enter the 95th percentile, and where the mode is greater than 0.5 enter the 5th percentile. For example, if the most likely value for the prior estimate of prevalence is 0.1, and you are 95% confident that the true prevalence is less than 0.2, enter a mode of 0.1 and a 95th percentile value of 0.2. Conversely, if the most likely value for test sensitivity is 0.95 and you are 95% confident that the true value is >0.75, use the values 0.95 and 0.75. For summarisation of Beta distributions, all parameter values must be positive real (decimal or integer) numbers.

Output from the parameterisation utility is a table of input values (mode and percentile) and corresponding alpha and beta values for each distribution requested. Output from the summarisation utility is a table of alpha and beta values and corresponding percentile values, mean, mode and standard deviation for each distribution requested. Density curves for each distribution are also generated and can be viewed or downloaded by clicking on the icon.

Contents
1 Introduction
2 Overview
3 Bayesian vs Frequentist methods
4 Fixed pool size and perfect tests
5 Fixed pool size and known Se & Sp
6 Fixed pool size and uncertain Se & Sp
7 Variable pool size and perfect tests
8 Pooled prevalence using a Gibbs sampler
9 True prevalence using one test
10 Estimated true prevalence using two tests with a Gibbs sampler
11 Estimation of parameters for prior Beta distributions
12 Sample size for fixed pool size and perfect test
13 Sample size for fixed pool size and known test sensitivity and specificity
14 Sample size for fixed pool size and uncertain test sensitivity and specificity
15 Simulate sampling for fixed pool size
16 Simulate sampling for variable pool sizes
17 Important Assumptions
18 Pooled prevalence estimates are biased!