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 (kA) and mummies (kM).
For stem counts, the total number of aphids counted on all stems in a quadrant has mean nst UJ+PJ+UA+PA), where nst is the number of stems counted. The total number of mummies counted during stem counts has mean nst (M+H). The number of aphids counted in tensweeps in a quadrant has mean nsws(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 nsw 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 ntcktc (M+H), where the conversion parameter tc is the effective number of hundreds of stems surveyed in a three-minute time count and ntc 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).