Kevin Gross, Anthony R. Ives, and Erik V. Nordheim. 2005. Estimating fluctuating vital rates from time-series data: a case study of aphid biocontrol. Ecology 86:740–752.

Appendix C. Observation model.

We use negative binomial models for measurements of aphid and mummy abundance. These can be motivated heuristically by assuming that stem counts in each quadrant are iid Poisson random variates conditional on a per-quadrant mean, and that the per-quadrant mean is a Gamma random variate with mean equal to the field-level mean (from the process model) and shape parameter k. Different shape parameters are used for aphids (k_A) and mummies (k_M). 

For stem counts, the total number of aphids counted on all stems in a quadrant has mean n_st UJ+PJ+UA+PA), where n_st is the number of stems counted. The total number of mummies counted during stem counts has mean n_st (M+H). The number of aphids counted in tensweeps in a quadrant has mean n_sws(UA+PA + 0.75(UJ+PJ)), where the conversion factor _s is the effective number of hundreds of stems swept in a single tensweep, and n_sw is the number of tensweeps in a given quadrant. Using data from a previous field season, we estimated that 25% of juveniles are not counted in a ten-sweep.  The number of mummies observed during time counts in a quadrant has mean n_tck_tc (M+H), where the conversion parameter _tc is the effective number of hundreds of stems surveyed in a three-minute time count and n_tc is the number of three-minute time counts.

For the dissection data, we model the number of parasitized adult aphids in each quadrant as a beta-binomial with mean PA/(UA+PA) and overdispersion parameter _W. For the hyperparasitism data, not all mummies emerged, so we fit two parameters to model the fraction of (unhyperparasitized) wasps and hyperparasitoids that emerge –– _W and _H, respectively. We model the total of emerged parasitoids, emerged hyperparasitoids, and unemerged mummies as a Dirichlet-trinomial (the extension of a beta-binomial) with overdispersion parameter _H. The expected fraction of emerged parasitoids is _WM /(M+H), and the expected fraction of emerged hyperparasitoids is _HH /(M+H).