The knowledge base stores for each distribution: •
Probability density or
mass functions and where available
cumulative distribution,
hazard and
survival functions. • Related quantities such as mean, median, mode and variance. •
Parameter and
support/range definitions and distribution type. •
LaTeX and
R code for mathematical functions. • Model definition and references.
Relationships ProbOnto stores in Version 2.5 over 220 relationships between univariate distributions with re-parameterizations as a special case, see figure. While this form of relationships is often neglected in literature, and the authors concentrate one a particular form for each distribution, they are crucial from the interoperability point of view. ProbOnto focuses on this aspect and features more than 15 distributions with alternative parameterizations.
Alternative parameterizations Many distributions are defined with mathematically equivalent but algebraically different formulas. This leads to issues when exchanging models between software tools. The following examples illustrate that.
Normal distribution Normal distribution can be defined in at least three ways • Normal1(μ,σ) with
mean, μ, and
standard deviation, σ P(x;\boldsymbol\mu,\boldsymbol\sigma)= \frac{1}{\sigma \sqrt{2 \pi}}\exp\Big[-\frac{(x-\mu)^2}{2\sigma^2}\Big] • Normal2(μ,υ) with mean, μ, and
variance, υ = σ^2 or P(x;\boldsymbol\mu,\boldsymbol v)= \frac{1}{\sqrt{v} \sqrt{2 \pi}}\exp\Big[-\frac{(x-\mu)^2}{2v}\Big] • Normal3(μ,τ) with mean, μ, and
precision, τ = 1/υ = 1/σ^2. P(x;\boldsymbol\mu,\boldsymbol\tau)= \sqrt{\frac{\tau}{2 \pi}} \exp\Big[-\frac{\tau}{2}(x-\mu)^2\Big]
Re-parameterization formulas The following formulas can be used to re-calculate the three different forms of the normal distribution (we use abbreviations i.e. N1 instead of Normal1 etc.) • N1(\mu,\sigma) \rightarrow N2(\mu,v): v=\sigma^2 \mbox{ and } N2(\mu,v) \rightarrow N1(\mu,\sigma): \sigma=\sqrt{v}; • N1(\mu,\sigma) \rightarrow N3(\mu,\tau): \tau=1/\sigma^2 \mbox{ and } N3(\mu,\tau) \rightarrow N1(\mu,\sigma): \sigma=1/\sqrt{\tau}; • N2(\mu,v) \rightarrow N3(\mu,\tau): \tau=1/v \mbox{ and } N3(\mu,\tau) \rightarrow N2(\mu,v): v=1/\tau.
Log-normal distribution In the case of the
log-normal distribution there are more options. This is due to the fact that it can be parameterized in terms of parameters on the natural and log scale, see figure. (supports LN1),
MCSim (LN6), Monolix (LN2 & LN3), PFIM (LN2 & LN3), Phoenix NLME (LN1, LN3 & LN6), PopED (LN7),
R (programming language) (LN1),
Simcyp Simulator (LN1), Simulx (LN1) and
winBUGS (LN5) The available forms in ProbOnto 2.0 are • LogNormal1(μ,σ) with mean, μ, and standard deviation, σ, both on the log-scale P(x;\boldsymbol\mu,\boldsymbol \tau)=\sqrt{\frac{\tau}{2 \pi}} \frac{1}{x} \exp\Big[ {-\frac{\tau}{2}(\log x-\mu)^2} \Big] • LogNormal6(m,σg) with median, m, and
geometric standard deviation, σg, both on the natural scale P(x;\boldsymbol m,\boldsymbol {\sigma_g})=\frac{1}{x \log(\sigma_g)\sqrt{2 \pi}} \exp\Big[ \frac{-[\log(x/m)]^2}{2 \log^2(\sigma_g)}\Big] • LogNormal7(μN,σN) with mean, μN, and standard deviation, σN, both on the natural scale P(x;\boldsymbol {\mu_N},\boldsymbol {\sigma_N})= \frac{1}{x \sqrt{2 \pi \log\Big(1+\sigma_N^2/\mu_N^2\Big)}} \exp\Bigg( \frac{-\Big[ \log(x) - \log\Big(\frac{\mu_N}{\sqrt{1+\sigma_N^2/\mu_N^2}}\Big)\Big]^2}{2\log\Big(1+\sigma_N^2/\mu_N^2\Big)}\Bigg) ProbOnto knowledge base stores such re-parameterization formulas to allow for a correct translation of models between tools.
Examples for re-parameterization Consider the situation when one would like to run a model using two different optimal design tools, e.g. PFIM and PopED. The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results. For the transition LN2(\mu, v) \rightarrow LN7(\mu_N, \sigma_N) following formulas hold \mu_N = \exp(\mu+v/2) \text{ and } \sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}. For the transition LN7(\mu_N, \sigma_N) \rightarrow LN2(\mu, v) following formulas hold \mu = \log\Big( \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2} \Big) \text{ and } v = \log(1+\sigma_N^2/\mu_N^2). All remaining re-parameterisation formulas can be found in the specification document on the project website. == Ontology ==