Box 3.

Inferring trends in abundances

We consider a design in which volunteer recorders are reporting GPS locations of all detections. We further assume that some information proportional to the spent search effort exists, such as the time volunteer recorders spend looking for the alien species or the number of reports of a commonly detected species. Let us denote by $d_{\mathit{lj}}(t)$ the number of detections reported by observer $j=1,\cdots ,J$ during survey $t$ in a specific area $l=1,\cdots ,L$, for instance a specific cell of a geographic grid, and let $s_{\mathit{lj}}(t)$ be a measure proportional to the search effort spent by observer $j$ in that area. Under such a design, $n_{\mathit{lj}}$ is likely well characterized by a Poisson distribution $n_{\mathit{lj}}\sim \text{Pois}({\lambda }_{\mathit{lj}}(t)s_{\mathit{lj}})$ with unknown rates ${\lambda }_{\mathit{lj}}(t)$. Note that these rates are affected by the abundance at location $i$ as well as the detection probability of observer $j$ at that location, itself a potentially complex function of the training of the observer as well as characteristics of the location (e.g., vegetation). Assuming that the detection probabilities are constant across surveys (${\lambda }_{\mathit{lj}}\left(t_1\right)={\lambda }_{\mathit{lj}}(t_2)$), a change in the rates is reflective of a change in abundances. The interest therefore lies in inferring changes in the rates, which are independent of location or observer-specific characteristics. For a case of two surveys at $t_1,t_2$, we thus have ${\lambda }_{\mathit{lj}}(t_2)=\varphi {\lambda }_{\mathit{lj}}(t_1)$ and wish to infer $\varphi$ from the data of all observers and all locations. Following Aebischer et al. (2020), conditioning on the number of observations $n_{\mathit{lj}}=d_{\mathit{lj}}\left(1\right)+d_{\mathit{lj}}\left(2\right)$ leads to the likelihood $P\left(\mathit{d},|,\varphi ,,,\mathit{n}\right)\propto {\prod }_{l=1}^L{\prod }_{j=1}^J{p_{\mathit{lj}}\left(\varphi \right)}^{d_{\mathit{lj}}\left(t_1\right)}{(1-p_{\mathit{lj}}(\varphi ))}^{d_{\mathit{lj}}(t_2)}$ with $\mathit{d}=(d_{11},\cdots ,d_{1J},\cdots d_{\mathit{LJ}})$, $\mathit{n}=(n_{11},\cdots ,n_{1J},\cdots ,n_{\mathit{LJ}})$ and $p_{\mathit{lj}}\left(\varphi \right)=\frac{{\lambda }_{\mathit{lj}}{\left(t,1\right)s}_{\mathit{lj}}(t1)}{{\lambda }_{\mathit{lj}}{\left(t,1\right)s}_{\mathit{lj}}\left(t,1\right)+{\varphi \lambda }_{\mathit{lj}}{\left(t,1\right)s}_{\mathit{lj}}(t2)}={\left(1,+,\frac{s_{\mathit{lj}}(t2)}{s_{\mathit{lj}}(t1)},\varphi \right)}^{-1}$ Importantly, this formulation gets rid of the nuisance parameters ${\lambda }_{\mathit{ij}}(t)$. Note further that no absolute estimates of search efforts are required: since only their ratio is relevant, any quantity proportional to the search effort will do. The posterior distribution of $\varphi$ is readily inferred under Jeffrey’s prior (Aebischer et al. 2020). To illustrate this approach, we simulated data for observers that each surveyed a unique location during two consecutive surveys. As each location was surveyed by a single observer, detection probabilities and the abundances cannot be inferred individually without strong assumptions about their distribution (there are less data points than unknowns). However, a trend in abundance may still be identified. To show that, we simulated observers $j=1,\cdots ,J$ with detection probabilities $p_j\sim \text{Beta}(0.01,10)$, their search efforts as $s_j\left(t\right)\sim \text{Exp}(0.1)$ and the abundances at their location as $N_j\left(t_1\right)\sim \text{Pois}(10)$. Data simulated this way resulted in reported abundances $d_j(t_1)=0$ in $>99\%$ of all surveys, representative for community science projects targeting rare alien species. We then identified the power to detect a decreasing trend with $\varphi =0.5$, $0.9$ or $0.95$ for different numbers of observers (and corresponding locations). As shown in Fig. 6, trends are reliably identified if sufficient observers participate, with stronger declines generally easier to identify. Obviously, higher detection probabilities, higher abundances or larger search efforts would all result in higher reported abundances and render trend identification easier. In case the assumption of constant detection rates it not possible, covariates accounting for variation can be folded into $s_{\mathit{lj}}(t)$ (Link & Saur 1997).