Estimate parameters for multiple Beta probability distributions or summarise distributions for specified parameters

Introduction

Calculate the alpha and beta parameters for Beta probability distributions, based on either specified values for the mode and 5th or 95th percentile of the distribution, or on count data, or summarise Beta probability distributions for given alpha or beta parameters. See the User Guide or Suess et al. (2002) for more details on parameter estimation based on mode and percentiles.

What is a Beta distribution?

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. Alternatively alpha and beta parameters can be estimated from the mode and a given percentile, if suitable data is not available.

If there is no prior information on which to base a prior distribution, alpha = beta = 1 should be used. This results in a uniform (uninformed) distribution, in which all values between 0 and 1 have equal probability of occurrence.

Inputs

Parameters can be estimated for as many distributions as you wish, simply by providing the appropriate data. To estimate parameters:

• Select the type of data being entered;
• paste the data into the data submission area; and
• Click on the submit button.

The program expects two columns of data, either mode and percentile, counts, or alpha and beta parameters. Distribution parameters will be calculated and distribution summaries presented for each pair of values provided:

1. For mode and percentile data: Paste at least two columns labeled "mode" and "pc". Column order is not important, but column names must be included. Values must be expressed as proportions (between 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. Column order is not important, but column names must be included.
2. For count data: Paste at least two columns labeled "n" (sample size or number of trials) and "x" (number of successes). Values must be integers: n must be positive integers while x must be non-negative integers less than or equal to the corresponding n value. Column order is not important, but column names must be included.
3. To summarise distributions: Paste at least two columns labeled "alpha" (first or alpha parameter) and "beta" (second or beta parameter). Both alpha and beta values must be positive numbers. Column order is not important, but column names must be included.

Outputs

Outputs from this program are the alpha and beta parameters for each distribution, which can then be used as inputs for other analyses. Numeric summaries and density plots for each distribution are also provided.