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Appendix B. Statistical adjustment of spatial coverage bias.

Here we address several issues related to modeling the bias induced by variable route length in the analysis of the MHB data.

First, we address an alternative motivation of the parameterization that was used in the model. This parameterization of route length can be motivated more formally as follows. Let N(s) be the quadrat population size and let M(s) ≤ N(s) be the population size exposed to sampling by a route of prescribed length. A natural expression of the relationship between the quadrat population size and the population exposed to sampling is to suppose that M(s) ~ Binomial(N(s),φ(s)) where φ(s) depends on route length. For example, if we choose

Then, as before, for β₁ < 0, as sample route length increases φ(s) → 1, and the exposed population size increases to the actual quadrat population size N(s). Now, with N(s) ~ Poisson(exp(β₀)), the marginal distribution of M(s) (i.e., unconditional on N(s)) is Poisson with mean exp(β₀)φ(s), precisely equal to

(B.1)

Secondly, we have chosen a particular model that has a single parameter. In the case of the Swiss survey, we feel that more complex models describing incomplete quadrat coverage would be difficult to evaluate formally because the variation in route length is insufficient. For example, we might prefer to have a parameterization of incomplete coverage that has a non-zero intercept as route length tends to zero, such as

In this case, Length = 0 corresponds to the exposed quadrat population for a “point count” (i.e., a route length of 0) which does have positive sample area. However, this model is weakly identified owing to the range of observed route lengths being concentrated away from zero. Given that sample route length cannot be standardized in the MHB survey, we believe that some effort directed toward understanding the relationship between expected count, sampled area, and route length would be useful. One possibility is to carry out designed studies with more variation in observed route length, including some routes with near 0 length.