the prevalence package
tools for prevalence assessment studies.

## Calculate the parameters of a Beta-PERT distribution

### 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

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 x Object of class betaPERT 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, $$\alpha$$ and $$\beta$$, are as follows

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

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

$\alpha = { (mean - a) / (b - a) } * { (mean - a) * (b - mean) / sd^{2} - 1 }$

$\beta = \alpha * (b - mean) / (mean - a)$

The resulting distribution is a 4-parameter Beta distribution: $$Beta(\alpha, \beta, a, b)$$.

Vose

Vose (2000) describes a different formula for :

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

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

Note: If $$m = mean$$, $$\alpha$$ is calculated as $$1 + k/2$$, in accordance with the mc2d package (see ‘Note’).

### Value

A list of class "betaPERT":

 alpha Parameter $$\alpha$$ (shape1) of the Beta distribution beta Parameter $$\beta$$ (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.

### References

betaExpert for modelling a standard Beta distribution based on expert opinion

### Examples

## 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")
#>    method alpha  beta a  b mean   median mode      var     2.5%    97.5%
#> 1 classic 1.968 4.592 0 50   15 13.93669   10 69.44444 2.247894 33.37723
## .. Plot method
plot(betaPERT(a = 0, m = 10, b = 50, method = "classic"))

## .. Vose parametrization
betaPERT(a = 0, m = 10, b = 50, method = "vose")
#>   method alpha beta a  b mean   median mode var     2.5%    97.5%
#> 1   vose   1.8  4.2 0 50   15 13.83361   10  75 1.960636 34.15672