prevalence (version 0.4.1)

betaPERT: 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 betaPERT
print(x, conf.level = .95, …)
# S3 method for 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 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

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.

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: $$mean = \frac{a + k*m + b}{k + 2}$$ $$sd = \frac{b - a}{k + 2}$$ $$\alpha = \frac{mean - a}{b - a} * \left\{ (mean - a) * \frac{b - mean}{sd^{2}} - 1 \right\} $$ $$\beta = \alpha * \frac{b - mean}{mean - a}$$ The resulting distribution is a 4-parameter Beta distribution: Beta(\(\alpha\), \(\beta\), a, b).

Vose:

Vose (2000) describes a different formula for \(\alpha\): $$(mean - a) * \frac{2 * m - a - b}{(m - mean) * (b - a)}$$ Mean and \(\beta\) 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').

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)

See Also

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

Examples

# NOT RUN {
## 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")
# }