# betaPERT: Calculate the parameters of a Beta-PERT distribution In prevalence: Tools for Prevalence Assessment Studies

## Description

The Beta-PERT methodology allows to parametrize a generalized Beta distribution based on expert opinion regarding a pessimistic estimate (minimum value), a most likely estimate (mode), and an optimistic estimate (maximum value). The `betaPERT` function incorporates two methods of calculating the parameters of a Beta-PERT distribution, designated `"classic"` and `"vose"`.

## Usage

 ```1 2 3 4 5 6``` ```betaPERT(a, m, b, k = 4, method = c("classic", "vose")) ## S3 method for class 'betaPERT' print(x, conf.level = .95, ...) ## S3 method for class 'betaPERT' plot(x, y, ...) ```

## Arguments

 `a` Pessimistic estimate (Minimum value) `m` Most likely estimate (Mode) `b` Optimistic estimate (Maximum value) `k` Scale parameter `method` `"classic"` or `"vose"`; see details below `x` Object of class `betaExpert` `y` Currently ignored `conf.level` Confidence level used in printing quantiles of resulting Beta-PERT distribution `...` Other arguments to pass to function `print` and `plot`

## Details

The Beta-PERT methodology was developed in the context of Program Evaluation and Review Technique (PERT). Based on a pessimistic estimate (minimum value), a most likely estimate (mode), and an optimistic estimate (maximum value), typically derived through expert elicitation, the parameters of a Beta distribution can be calculated. The Beta-PERT distribution is used in stochastic modeling and risk assessment studies to reflect uncertainty regarding specific parameters.

Different methods exist in literature for defining the parameters of a Beta distribution based on PERT. The two most common methods are included in the `BetaPERT` function:

Classic:

The standard formulas for mean, standard deviation, α and β, are as follows:

mean = (a + k*m + b) / (k + 2)

sd = (b - a) / (k + 2)

α = { (mean - a) / (b - a) } * { (mean - a) * (b - mean) / sd^{2} - 1 }

β = α * (b - mean) / (mean - a)

The resulting distribution is a 4-parameter Beta distribution: Beta(α, β, a, b).

Vose:

Vose (2000) describes a different formula for α:

(mean - a) * (2 * m - a - b) / { (m - mean) * (b - a) }

Mean and β are calculated using the standard formulas; as for the classical PERT, the resulting distribution is a 4-parameter Beta distribution: Beta(α, β, a, b).

Note: If m = mean, α is calculated as 1 + k/2, in accordance with the mc2d package (see 'Note').

## Value

A list of class `"betaPERT"`:

 `alpha ` Parameter α (shape1) of the Beta distribution `beta ` Parameter β (shape2) of the Beta distribution `a ` Pessimistic estimate (Minimum value) `m ` Most likely estimate (Mode) `b ` Optimistic estimate (Maximum value) `method ` Applied method

Available generic functions for class `"betaPERT"` are `print` and `plot`.

## Note

The mc2d package provides the probability density function, cumulative distribution function, quantile function and random number generation function for the PERT distribution, parametrized by the `"vose"` method.

## Author(s)

Brecht Devleesschauwer <[email protected]>

## References

Classic:

Malcolm DG, Roseboom JH, Clark CE, Fazar W (1959) Application of a technique for research and development program evaluation. Oper Res 7(5):646-669.

Vose:

David Vose. Risk analysis, a quantitative guide, 2nd edition. Wiley and Sons, 2000.
PERT distribution in ModelRisk (Vose software)

`betaExpert`, for modelling a standard Beta distribution based on expert opinion

## Examples

 ```1 2 3 4 5 6 7 8``` ```## The value of a parameter of interest is believed to lie between 0 and 50 ## The most likely value is believed to be 10 # Classical PERT betaPERT(a = 0, m = 10, b = 50, method = "classic") # Vose parametrization betaPERT(a = 0, m = 10, b = 50, method = "vose") ```

### Example output

```Loading required package: rjags