What Information am I looking for?
Think about a tree that is sending pollen all over the place. Because of wind, most pollen grain will go toward one direction. Imagine, we split the 2D area around the tree where pollen grains fall into two half disks of equal size. We chose the disks so that the number of pollen grains falling into one half-disk is minimized and the quantity of pollen falling in the other half-disk is maximized.
The information I need is what proportion of pollen grain falls into each disk? Is it $\frac{0.5}{0.5}$ (in which case the wind would have no effect) or is it something like $\frac{0.8}{0.2}$?
Where to get the information from?
I was reading this paper about pollen dispersal directionality and was trying to extract the info I need.
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 part of a logistic regression I think (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
$$\phi_j = \Pr(y_j = 1) = \frac{\exp\left(\beta d_j + \gamma h_j + \delta a_j\right)}{\sum_{k=1}^r \exp\left(\beta d_k + \gamma h_k + \delta a_k\right)}$$
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).
Question
How can I relate their finding to the information I need (even if it is just a rough approximation)?
I think I have to understand the so-called directionality paramter
$\delta$ Can you help me to interpret the meaning of this parameter $\delta$?
Reference:
Burczyk, Adams, & Shimizu, Mating patterns and pollen dispersal in a natural knobcone pine ... stand. Heredity 77 (1996) pp 251-260.
Not receiving any answer on Biology.SE, I asked the same question here on Mathematics.SE but sadly, the question doesn't receive any attention!