ProbOnto was initially designed to facilitate the encoding of nonlinear-mixed effect models and their annotation in Pharmacometrics Markup Language (PharmML)[3][4] developed by DDMoRe,[5][6] an Innovative Medicines Initiative project. However, ProbOnto, due to its generic structure can be applied in other platforms and modeling tools for encoding and annotation of diverse models applicable to discrete (e.g. count, categorical and time-to-event) and continuous data.
Overview of supported distributions in ProbOnto, version 2.5, and relationships between univariate probability distributions.
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.[7] The following examples illustrate that.
The following formulas can be used to re-calculate the three different forms of the normal distribution (we use abbreviations i.e. instead of etc.)
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.
Overview of parameterizations of the log-normal distributions.Support of different parameterizations of the log-normal distributions in various tools and there connections, see text for examples. Tools visualised are Matlab (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[8]
LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale
LogNormal3(m,σ) with median, m, on the natural scale and standard deviation, σ, on the log-scale[8]
LogNormal7(μN,σN) with mean, μN, and standard deviation, σN, both on the natural scale[14]
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[15] and PopED.[16] 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 following formulas hold
.
For the transition following formulas hold
.
All remaining re-parameterisation formulas can be found in the specification document on the project website.[2]
The knowledge base is built from a simple ontological model. At its core, a probability distribution is an instance of the class thereof, a specialization of the class of mathematical objects. A distribution relates to a number of other individuals, which are instances of various categories in the ontology. For example, these are parameters and related functions associated with a given probability distribution. This strategy allows for the rich representation of attributes and relationships between domain objects. The ontology can be seen as a conceptual schema in the domain of mathematics and has been implemented as a PowerLoom knowledge base.[17] An OWL version is generated programmatically using the Jena API.[18]
Output for ProbOnto are provided as supplementary materials and published on or linked from the probonto.org website. The OWL version of ProbOnto is available via Ontology Lookup Service (OLS)[19] to facilitate simple searching and visualization of the content. In addition the OLS API provides methods to programmatically access ProbOnto and to integrate it into applications. ProbOnto is also registered on the BioSharing portal.[20]
A PharmML interface is provided in form of a generic XML schema for the definition of the distributions and their parameters. Defining functions, such as probability density function (PDF), probability mass function (PMF), hazard function (HF) and survival function (SF), can be accessed via methods provided in the PharmML schema.
Use example
This example shows how the zero-inflated Poisson distribution is encoded by using its codename and declaring that of its parameters (‘rate’ and ‘probabilityOfZero’). Model parameters Lambda and P0 are assigned to the parameter code names.
To specify any given distribution unambiguously using ProbOnto, it is sufficient to declare its code name and the code names of its parameters.
More examples and a detailed specification can be found on the project website.[2]
Swat MJ. et al. (2015). Pharmacometrics Markup Language (PharmML): Opening New Perspectives for Model Exchange in Drug Development. CPT Pharmacometrics Syst Pharmacol, 4(6):316-9.
Retout S, Duffull S, Mentré F (2001) Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs. Comp Meth Pro Biomed 65:141–151
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![{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {v}})={\frac {1}{{\sqrt {v}}{\sqrt {2\pi }}}}\exp {\Big [}-{\frac {(x-\mu )^{2}}{2v}}{\Big ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6294655c5a8137251de9c61996d58ade53312cc6)
![{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\tau }})={\sqrt {\frac {\tau }{2\pi }}}\exp {\Big [}-{\frac {\tau }{2}}(x-\mu )^{2}{\Big ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/451cd82e363bad450815f08cadfaafadc829559b)
![{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp {\Big [}{\frac {-(\log x-\mu )^{2}}{2\sigma ^{2}}}{\Big ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f2203c6532a8f2fb656f5d094facaa68c1d71da6)
![{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {v}})={\frac {1}{x{\sqrt {v}}{\sqrt {2\pi }}}}\exp {\Big [}{\frac {-(\log x-\mu )^{2}}{2v}}{\Big ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/70707ef688d2544dd7177b97b28c009935010536)
![{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp {\Big [}{\frac {-[\log(x/m)]^{2}}{2\sigma ^{2}}}{\Big ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d066a6571c7ae5e27583711fc86d687d9deaef35)
![{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {cv}})={\frac {1}{x{\sqrt {\log(cv^{2}+1)}}{\sqrt {2\pi }}}}\exp {\Big [}{\frac {-[\log(x/m)]^{2}}{2\log(cv^{2}+1)}}{\Big ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e62d3a100a18af22f94f3d47307e0b170994cd2c)
![{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\tau }})={\sqrt {\frac {\tau }{2\pi }}}{\frac {1}{x}}\exp {\Big [}{-{\frac {\tau }{2}}(\log x-\mu )^{2}}{\Big ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/754af28c18e6d700d524fca45427c1df19ffeaf5)
![{\displaystyle 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 ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d6f421a0dce7d8e99c7a9318b73317c5cd1ea362)
![{\displaystyle 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 )}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/96cda4673155da96615dcb0b7390b0d36d23fb25)
