Appendix B. Approximate conditional variance for prevalence estimator.
Estimated prevalence is a derived parameter, expressed as the ratio of constants (raw counts) multiplied by state-specific detection probabilities with some level of uncertainty. We used the Delta Method ( Seber 1982) to derive the approximate conditional sampling variance for our prevalence estimator. Estimates of weekly detection probability were obtained from Faustino et al. (2004).
The expression for the approximate conditional sampling variance of the reduced form of estimated prevalence (Eq. 3), is
where,
= Reduced form expression for corrected estimate of prevalence in year i ( i = 2002-03), month j (j = Nov-Mar) , and week k (1,…,4),
= observed count of animals in health state s (I: infected or U: uninfected) in year i,
month j , and week k,
= the detection probability for an animal in health state s (I: infected or U: uninfected) in year i, month j, and week k,
= the variance-covariance matrix of the detection probabilities, .
Using matrix algebra, the Delta Method yields
where,
= observed count of finches in health state s (I: infected or
U : uninfected) in year i, month j , and week k,
= the estimated detection probability for a finch in health state s (I:
infected or U: uninfected) in year i, month j, and week k,
= the variance-covariance matrix of the detection probabilities, .
The approximate conditional sampling variance of the prevalence estimator is thus
LITERATURE CITED
Faustino, C. R., C. S. Jennelle, V. Connolly, A. K. Davis, E. C. Swarthout, A. A. Dhondt, and E. G. Cooch.2004. Mycoplasma gallisepticuminfection dynamics in a House Finch population: Empirical analysis of seasonal variation, encounter and transmission rate. Journal of Animal Ecology73:651–669.
Seber , G. 1982. The estimation of animal abundance and related parameters. Charles Griffin and Company, London, UK.