The BetaPERT 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 BetaPERT distribution, designated "classic"
and "vose"
.
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, ...)
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 BetaPERT distribution 
...

Other arguments to pass to function print and plot

The BetaPERT 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 BetaPERT 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 4parameter 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 4parameter 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’).
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
.
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.
Classic: Malcolm DG, Roseboom JH, Clark CE, Fazar W (1959) Application of a technique for research and development program evaluation. Oper Res 7(5):646669. http://dx.doi.org/10.1287/opre.7.5.646
Vose: David Vose. Risk analysis, a quantitative guide, 3rd edition. Wiley and Sons, 2000.
betaExpert
for modelling a standard Beta distribution based on expert opinion
## 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