13.5 Distribution formulas

This section provides the formulas TreeAge Pro uses to generate distributions.

Hint: To find distributions quickly, use the search option in your browser and enter the distribution name.

13.5.1  Normal distribution

Formula

Domain
Parameters

Mean:

Standard deviation:

 

13.5.2  Uniform distribution continuous – (real numbers)

Formula

Domain
Parameters

LB lower bound
UB upper bound

 

13.5.3  Uniform distribution discrete (integer numbers)

Formula

or x is not an integer.



Domain
Parameters

LB and UB integers.

LB lower bound
UB upper bound

 

13.5.4  Triangular distribution

Formula


Domain Min ≤ x ≤ Max
Parameters Min, Mode (likeliest), Max

 

13.5.5  Fractile distributions (10/50/90 et al)

Formula

  • The distribution generates samples from specified list of Low Value (LV), Median Value (MV) and High Value (HV).

  • HV and LV are drawn with probability of 0.30 or 0.25; MV is returned with probability of 0.40 or 0.50 respectively depending on the selected option Gaussian or Swanson’s mean within the distribution dialog.
Domain LV, MV, HV
Parameters LV, MV, HV

 

13.5.6  Beta distribution discrete (integer numbers)

Formula
Domain
Parameters
Details

 

13.5.7  Beta distribution continuous (real numbers)

Formula
Domain
Parameters
Details
Parameters a and b can be parameterized from a mean μ and standard deviation σ:

 

More continuous distributions

13.5.8  Dirichlet distribution (multivariate, normalized beta)

Formula
where are sampled from Gamma distributions:

The parameters are specified in the Alphas list of the Dirichlet Distribution dialog using
Domain where
Parameters
Details See section Sampling probabilities from a multivariate Dirichlet distribution .

 

13.5.9  Chi distribution

Formula
Domain
Parameters

 

13.5.10  Chi-Squared distribution

Formula
Domain
Parameters

 

13.5.11  Erlang distribution

Formula
Domain
Parameters integer,

Please note that the Erlang probability density function can also be represented using a related formula (as shown on Wikipedia):

To convert the above scale parameter to TreeAge Pro’s scale parameter please use the following formula:

13.5.12  Exponential distribution

Formula
Domain
Parameters

Survival Function information:

TreeAge Pro survival function form:

Corresponding parameters from other software packages (STATA and SAS):

R SURVREG function output. (Note: negative sign before intercept!)

R FLEXSURVREG function output.

 

13.5.13  Gamma distribution

Formula
Domain
Parameters
Details

The parameters a and can be parameterized from a mean and standard deviation :

 

13.5.14  Generalized Gamma distribution

Formula
Domain
Parameters

Survival Function information:

TreeAge Pro survival function form:

where

Corresponding parameters from other software packages:

  • STATA implementation of GenGamma is not compatible

  • SAS does not support GenGamma

  • R SURVREG does not support GenGamma

  • R FLEXSURVREG function supports GenGamma.orig and GenGamma.

GenGamma Original has more straightforward correspondence to TreeAge Pro

GenGamma has a more complex parameterization. For positive Q parameter use the following formulas:

 

13.5.15  Gompertz distribution

Formula
Domain
Parameters

Alternative

 

TreeAge Pro also implements
optional alternative Gompertz
distribution parameterization
(as shown in Wikipedia):
Parameters where

Survival Function information:

TreeAge Pro survival function form:

Corresponding parameters from other software packages:

 

13.5.16  Hyper-exponential distribution

Formula
Domain
Parameters

 

13.5.17  Laplace distribution

Formula
Domain
Parameters

 

13.5.18  Logistic distribution

Formula
Domain
Parameters

 

13.5.19  Log-Logistic distribution

Formula
Domain
Parameters

Survival Function information:

TreeAge Pro survival function form:

Corresponding parameters from other software packages (Stata and SAS):

R SURVREG function output:

R FLEXSURVREG function output:

 

13.5.20  Lognormal distribution

Formula

Domain
Parameters

Parameters

The parameters μ and σ are the mean and standard deviation respectively, from the distribution of the ln(x), and can be approximated:

Survival Function information:

TreeAge Pro survival function form:

where is the cumulative distribution function of the standard normal distribution (i.e., N(0,1)).

Corresponding parameters from other software packages:

R SURVREG function output:

R FLEXSURVREG function output:

R FLEXSURVREG (newer version 2023?) function output:

The R statistical packages keep refining the parameterizations of distributions, please examine the associated documentation for their survival functions to determine which parameterization is appropriate for your model.

 

13.5.21  Maxwell distribution

Formula

Domain
Parameters

 

13.5.22  PERT distribution continuous

Formula

This is the same formula as Beta continuous distribution, but with z rescaled by the PERT parameters.
Domain

for

Parameters

Min, Mode (Likeliest), Max, Shape
Details

The underlying Beta parameters a and b are functions of the PERT parameters:
Pert Sigma

 

13.5.23  Rayleigh distribution

Formula
Domain
Parameters

 

13.5.24  Weibull distribution

Formula
Domain
Parameters
Alternative representation of PDF

Please note that the Weibull probability density function can also be represented using a related formula (as shown on Wikipedia):

To convert the above scale parameter to TreeAge Pro’s scale parameter please use the following formula:

The shape parameter k is identical in both formulas.
Note For values of k smaller than 0.05 the resulting samples are likely to fall outside of numerical precision of x<~5.0∙10^(-324) or x>~1.8∙10^308, these values will result in calculation errors, where expressions could not be evaluated.

Survival Function information:

TreeAge Pro survival function form:

Corresponding parameters from other software packages (Stata, SAS):

R SURVREG function output (Use TreeAge Pro alternate parameterization).

R FLEXSURVREG function output (Use TreeAge Pro alternate parameterization)

 

More discrete distributions

13.5.25  Binomial distribution

Formula
Domain
Parameters

 

13.5.26  Poisson distribution

Formula
Domain
Parameters

 

Other distributions

13.5.27  TableProb distribution

The TableProb distribution takes as its input either a probability table or cumulative probability table. Typical use case of this distribution is to sample time to death directly from a mortality table.

The first entry in the probability table and cumulative probability table has to be equal to 0 and the last entry has to be equal to 1. The index for these tables represents time in cycles (typically year). Make sure that your model time horizon is shorter than the last time entry in the table. For model time horizon of 80 years make sure that the probability table or the cumulative probability table has entries beyond 80 years, at least there is an entry 1 at index 81.

Sampling from cumulative probability table is straightforward. A random number between 0 and 1 is sampled. The random number is than used for reverse look- up of the indices (cycle time) that contain the random number. Finally linear

interpolation is performed to return the fraction cycle time corresponding to the random number.

Sampling from probability table involves an internal step of converting the probability table to cumulative probability table. The following calculations are performed to convert probability table to cumulative probability table:

Additional option for conversion of probability tables to cumulative probability tables is interpolation. If the probability table has definitions for each integer value from 0, 1, ..., N-1, N you do not need to use interpolation. However, if the probability table is “sparse” e.g. 0, 5, 10, 15, ..., N then interpolation option will enable creation of cumulative probability table that will approximate results obtained from a Markov model using the “sparse” probability table.