The /stan folder in this folder contains Bayesian model specifications written in the Stan probabalistic programming language. Each file corresponds to a variation of a model (originally developed in Keller et al., 2022) that uses environmental DNA (eDNA) data and “traditional” survey data to jointly estimate parameters. These model variations are accessed based on the type of input data and/or user-defined input parameters, including distributional assumptions.

Probability distributions were chosen for the model specifications using the model developed in Keller et al. 2022. These original models use:

1. a binomial distribution to represent the probability of a qPCR detection (1) or non-detection (0) (Lahoz-Monfort et al., 2016)
2. a poisson or negative binomial distribution to represent how traditional survey count data are generated from an expected catch rate (Lindén and Mäntyniemi, 2011).
3. a normal distribution as the prior on the probability of a false positive eDNA detection. This prior is informative and specified by the user as an input argument when running joint_model() (Lahoz-Monfort et al., 2016).
4. a gamma distribution as the prior for the overdispersion parameter for traditional survey count data, if a negative binomial distribution is used. This prior can be informative and can be specified by the user as an input argument when running joint_model().

Other variations on this original model specification include:

1. a gamma distribution to represent how continuous traditional survey data are generated from an expected catch rate.
2. a gear scaling coefficient if multiple traditional survey gear types are used and have different catchabilities
3. a regression to include site-level covariates that scale the sensitivity of eDNA sampling relative to traditional surveys. A “shrinkage” prior is used for these covariates as a form of Bayesian penalization (van Erp et al., 2019).

This folder also contains ‘traditional models’, which can be used to model the traditional survey data in isolation. These models can be used as a comparison with the joint model that adds eDNA survey data to determine if and how the addition of eDNA data affects inference.

The four files in the /stan folder represent four model variations:

1. joint_continuous.stan: joint model with continuous traditional survey data
2. joint_count.stan: joint model with discrete count traditional survey data
3. traditional_continuous.stan: traditional model with continuous survey data
4. traditional_count.stan: traditional model with discrete count survey data

The /stan/functions folder contains helper function for the above files.

References

Keller, A.G., Grason, E.W., McDonald, P.S., Ramon-Laca, A., Kelly, R.P. (2022). Tracking an invasion front with environmental DNA. Ecological Applications. 32(4): e2561. https://doi.org/10.1002/eap.2561

Lahoz-Monfort, J., Guillera-Arroita, G., Tingley, R. (2016). Statistical approaches to account for false-positive errors in environmental DNA samples. Molecular Ecology Resources. 16(3): 673-685. https://doi.org/10.1111/1755-0998.12486

Lindén, A., Mäntyniemi, S. (2011). Using the negative binomial distribution to model overdispersion in ecological count data. Ecology. 92(7): 1414-1421. https://doi.org/10.1890/10-1831.1

van Erp, S., Oberski, D.L., Mulder, J. (2019). Shrinkage priors for Bayesian penalized regression. Journal of Mathematical Psychology. 89: 31-50. https://doi.org/10.1016/j.jmp.2018.12.004