Definitions and relationships between parameters

Christophe Gigot


Variable definitions

Throughout the package epiphy, special attention is given to stay consistent consistent with variable and parameter names. Some of the most significant names are the followings:

Parameters of the beta-binomial distribution


The aggregation parameter, \(\theta\), can be computed from the two shape parameters of the beta-binomial distribution, \(\alpha\) and \(\beta\). Note that in epiphy, \(\theta = 1 / (\alpha + \beta)\) as in Madden, Hughes, and Bosch (2007), but this definition is not necessarily consistent with what can be found somewhere else in the literature. For example, \(\theta = \alpha + \beta\) in the package emdbook.

The intra-cluster correlation coefficient, \(\rho\) (Mak 1988), characterizes the spatial aggregation as the tendency for elements in a sampling unit to have the same disease status more frequently than expected on the basis of spatial randomness (Madden, Hughes, and Bosch 2007). In epiphy, \(\rho = \theta / (\theta + 1)\).

Base relationships

Functions of the two shape parameters of the beta-binomial distribution (\(\alpha\) and \(\beta\)):

\[p = \frac{\alpha}{\alpha + \beta}; \theta = \frac{1}{\alpha + \beta}; \rho = \frac{1}{\alpha + \beta + 1}\]

Functions of the aggregation parameter (\(\theta\)) or the intra-cluster correlation coefficient (\(\rho\)):

\[\theta = \frac{\rho}{1 - \rho}; \rho = \frac{\theta}{\theta + 1}\]

Functions of the average disease intensity (\(p\)) and the aggregation parameter (\(\theta\)):

\[\alpha = \frac{p}{\theta}; \beta = \frac{1 - p}{\theta}\]

Functions of the average disease intensity (\(p\)) and the intra-cluster correlation coefficient (\(\rho\)):

\[\alpha = \frac{p(1-\rho)}{\rho}; \beta = \frac{(1 - p)(1-\rho)}{\rho}\]

Parameters of the binary power law

There are different parametrizations of the binary form of the power law. The user should therefore be cautious when making computations and comparisons with published results. Below are reminders about definitions and relationships between these different parametrizations.

Two possible formulas for parametrization

\[ s_{obs}^2 = V_n = A_n [np(1-p)]^b = a_n [p(1-p)]^b \]

\[ s_{obs}^2 = V_p = A_p [p(1-p)/n]^b = a_p [p(1-p)]^b \]

where \(s_{obs}^2\) stands for the observed variance. The relationships between the different binary power law parameters (\(A_p\), \(a_p\), \(A_n\), \(a_n\) and \(b\)) are specified in the following relationship tables. Note that \(V_n = n^2 V_p\).

Relationship tables

Full version.

\(A_p\) \(a_p\) \(A_n\) \(a_n\)
\(A_p\) \(1\) \(A_p = a_p n^b\) \(A_p = A_n n^{2(b-1)}\) \(A_p = a_n n^{b-2}\)
\(a_p\) \(a_p = A_p n^{-b}\) \(1\) \(a_p = A_n n^{b-2}\) \(a_p = a_n n^{-2}\)
\(A_n\) \(A_n = A_p n^{2(1-b)}\) \(A_n = a_p n^{2-b}\) \(1\) \(A_n = a_n n^{-b}\)
\(a_n\) \(a_n = A_p n^{2-b}\) \(a_n = a_p n^2\) \(a_p = A_n n^b\) \(1\)

Reader-friendly version. To read it, the formula \(\text{row} = \text{col} \times \text{cell}\) must be used.

\(A_p\) \(a_p\) \(A_n\) \(a_n\)
\(A_p\) \(1\) \(n^b\) \(n^{2(b-1)}\) \(n^{b-2}\)
\(a_p\) \(n^{-b}\) \(1\) \(n^{b-2}\) \(n^{-2}\)
\(A_n\) \(n^{2(1-b)}\) \(n^{2-b}\) \(1\) \(n^{-b}\)
\(a_n\) \(n^{2-b}\) \(n^2\) \(n^b\) \(1\)

Note that the function a2a is kindly provided in epiphy to make these tricky conversions as easy as possible.

Relationship between beta-binomial and binary power law parameters

\[ \theta = \frac{a_p - f(p)/n}{f(p) - a_p} \text{, with } f(p) = [p(1-p)]^{1-b} \]


Madden, L. V., G. Hughes, and F. van den Bosch. 2007. The Study of Plant Disease Epidemics. American Phytopathological Society St Paul, MN.
Mak, T. K. 1988. “Analysing Intraclass Correlation for Dichotomous Variables.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 37 (3): 344–52.