Making sense of the directionality parameter


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I was reading this paper about pollen dispersal directionality and I am trying to make sense of the so-called directionality parameter $\delta$

On pages 4 and 5 they explain their analysis under the section `statistical procedure`. More specifically, in the first paragraph of the 5th page, they seem to describe the meaning of the parameters that are trying to estimate. One of them is the so-called `directionality parameter` $\delta$. I don't understand how to interpret this parameter $\delta$. This parameter is (I think) part of a logistic regression (although the authors do not characterize it as such) of "mating success" $y$ against variables $d$ ("distance") and $h$ ("height") and an angular variable $a = \cos(\alpha_0 - \alpha)$. ($\alpha_0$ is the "presumed prevailing direction of effective pollen dispersal," which apparently is not estimated from these data.) The corresponding parameters of the model are $\beta$, $\gamma$, and $\delta$, respectively, hence

\(\phi_j = \Pr(y_j = 1) = \frac{\exp\left(\beta d_j + \gamma h_j + \delta a_j\right)}{\sum_{k=1}^r \exp\left(\beta d_k + \gamma h_k + \delta a_k\right)}[\math]

where the index $j$ ranges over all $r$ "male(s) in the neighborhood."

**Their Results**

Using maximum likelihood, they found that the `directionality parameter`, $\hat\delta = 0.56$ (SE = $ 0.15$; bottom of page 6).


How can I make intuitive sense of the `directionality parameter` $\delta$?

For example, does a $\delta$ of 0.56 means that if we split the the pollinated area in two half disc (or two half ovales) so to maximize the number of pollen grain falling to one disk and minimize the number of pollen grain falling into the other disk, then 56% of the pollen grain falls into one disk and 44% falls into the other disk?

Thanks a lot for your help!



Burczyk, Adams, & Shimizu, *Mating patterns and pollen dispersal in a natural knobcone pine ... stand*. Heredity **77** (1996) pp 251-260.\)
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