Philip M. Dixon, Aaron M. Ellison, and Nicholas J. Gotelli. 2005. Improving the precision of estimates of the frequency of rare events. Ecology 86:1114–1123.

Appendix A. A description of the likelihood function for combining detailed observational and temporally aggregated data.

The number of captures, C, and visits V are independent of W, the total number of captured wasps in the aggregated data, because they are recorded from different plants. Also, C is conditionally independent of V. Using the distributions given in Eqs. 9 and 10 and the approximate Poisson distribution for captures in Eq. 11, the joint distribution of C, V, and W is

(A.1)

Neglecting constants, the log-likelihood function is

(A.2)

Maximum-likelihood estimates of the parameters and µ are obtained by differentiating Eq. A.2 with respect to and µ, so long as the two estimates are inside the parameter space (0   1, 0  µ). The maximum-likelihood estimates are roots of quadratic equations. Although they can be written down in closed form, the equations are not enlightening. The asymptotic variance of the estimates can be derived from the Fisher information matrix.

The data include three "observations": C (conditional on V), V, and W. Each has a Poisson distribution with a mean that depends on µ and/or  and a fixed constant (number of visits, number of plant-hours of detailed observations, or number of plant-hours of aggregate observations). The mean count, , can be written as

(A.3)

where the X variables and the offset, X_o, for each observation are:

Standard software for fitting generalized linear models or specifically for Poisson regression can be used to numerically estimate the parameters and their standard errors (e.g., Appendix B).