Modeling the sequential sampling process of eDNA metabarcoding

We assumed that there were I focal species to be monitored and J sites sampled from an area of interest. At site j, K[j] replicates of environmental samples were collected. For each replicate, a library was prepared for DNA sequencing to obtain separate sequence reads. We denote the resulting sequence read count of species i for replicate k at site j obtained using high-throughput sequencing and subsequent bioinformatics processing as y[i, j, k] (i = 1, 2, ..., I; j = 1, 2, ..., J; k = 1, 2, ..., K[j]).

Adapted from Figure 1 in Fukaya et al. 2022 (CC BY 4.0)

The figure above shows a diagram of the minimal model that can be fitted using occumb() with default settings. The process of generating y is represented by a series of latent variables z, u, and r, and the parameters psi, theta, and phi that govern the variation of the latent variables. Although psi, theta, and phi are assumed to have species-specific values, modeling these parameters as functions of covariates allows for further variation (see the following section).

The right-pointing arrow on the left-hand side of the figure represents the ecological process of species distribution. For the DNA sequence of a species to be detected at a site, the site must be occupied by that species (i.e., the eDNA of the species must be present at the site). In this model, the site occupancy of a species is indicated by the latent variable z[i, j]. If site j is occupied by species i, z[i, j] = 1; otherwise, z[i, j] = 0. psi[i] represents the probability that species i occupies a site selected from the region of interest. Therefore, species with high psi[i] values should occur at many sites in the region, whereas those with low psi[i] values should occur at only a limited number of sites.

Next, we focus on the second right-pointing arrow from the left, which represents one of the two observation processes: the capture of the species DNA sequence. For the DNA sequence of a species to be detected, the environmental sample from an occupied site and sequencing library derived from it must contain the species DNA sequence. In this model, the inclusion of the DNA sequence of a species in a sequencing library is indicated by latent variable u[i, j, k]. If the library of replicate k at site j contains the DNA sequence of species i, u[i, j, k] = 1; otherwise, u[i, j, k] = 0. theta[i] represents the per-replicate probability that the DNA sequence of species i is captured at a site occupied by the species. The DNA sequences of species with high theta[i] values are more reliably captured at occupied sites, whereas those of species with low theta[i] values are more difficult to capture. Note that u[i, j, k] is always zero for sites not occupied by a species (i.e., z[i, j] = 0), assuming that false positives do not occur.

Finally, we examine the right-pointing arrow on the right-hand side of the figure. This model part represents another observation process, that is, the allocation of species sequence reads in high-throughput sequencing. The sequence read count vector y[1:I, j, k] is assumed to follow a multinomial distribution, with the total number of sequence reads in replicate k of site j as the number of trials. Its multinomial cell probability, pi[1:I, j, k] (not shown in the figure), is modeled as a function of the latent variables u[1:I, j, k] described above and r[1:I, j, k], which is proportional to the relative frequency of the species sequence reads. The variation in r[i, j, k] is governed by parameter phi[i], which represents the relative dominance of a species sequence. Species with higher phi[i] values tend to have more reads when the species sequence was included in the library (u[i, j, k] = 1), whereas species with lower phi[i] values tend to have fewer reads. Again, no false positives are assumed to occur at this stage; that is, pi[i, j, k] always takes zero for replicates that do not include the species DNA sequence (i.e., u[i, j, k] = 0).

In the figure, the arrows directed at psi[i], theta[i], and phi[i] indicate that the variation in these parameters is governed by a community-level multivariate normal prior distribution with a mean vector Mu and covariance matrix Sigma. The two components of Sigma are the standard deviation sigma and the correlation coefficient rho (not shown in the figure).

Covariate modeling of psi, theta, and phi

Variations in psi, theta, and phi can be modeled as functions of covariates in a manner similar to generalized linear models (GLMs). That is, the covariates are incorporated into linear predictors on the appropriate link scales for the parameters (logit for psi and theta, and log for phi). The occumb() function allows covariate modeling using the standard R formula syntax.

There are three types of related covariates: species covariates that can take on different values for each species (e.g., traits), site covariates that can take on different values for each site (e.g., habitat characteristics), and replicate covariates that can take on different values for each replicate (e.g., amount of water filtered). These covariates can be included in the data object via the spec_cov, site_cov, and repl_cov arguments of the occumbData() function and used to specify the models in the occumb() function.

The occumb() function specifies covariates for each parameter using the formula_<parameter name> and formula_<parameter name>_shared arguments. The formula_<parameter name> and formula_<parameter name>_shared arguments are used to specify species-specific effects and effects shared by all species, respectively. The following table shows examples of modeling psi using the formula_psi and formula_psi_shared arguments, where i is the species index, j is the site index, speccov1 is a continuous species covariate, and sitecov1 and sitecov2 are continuous site covariates. psi can be modeled as a function of the species and site covariates.

In occumb(), species-specific effects on psi are denoted by gamma, and the shared effects on psi are denoted by gamma_shared. The first row of the table specifies a default intercept-only model where logit(psi[i]) is determined only by the intercept term gamma[i, 1]. As in this most straightforward case, occumb() always estimates the species-specific intercept gamma[i, 1]. In the second case, the species-specific effect gamma[i, 2] of the site covariate sitecov1 are incorporated. Note that the site subscript j is added to psi on the left-hand side of the equation because the value of psi now varies from site to site depending on the value of sitecov1[j]. In the third and fourth cases, another site covariate, sitecov2, is specified in addition to sitecov1. In the fourth case, interaction is specified using the * operator.

In the fifth case, the formula_psi_shared argument specifies the shared effect of the species covariate speccov1. Note that the effect gamma_shared[1] of speccov1[i] in the linear predictor does not have subscript i. Because species-specific effects cannot be estimated for species covariates, occumb() accepts species covariates and their interactions only in its formula_<parameter name>_shared argument. Introducing species covariates does not change the dimension of psi (note that it has only the subscript i), but may help reveal variations in site occupancy probability associated with species characteristics.

In the sixth case, the site covariate sitecov1 is specified in the formula_psi_shared argument. Note that, in contrast to the second case, sitecov1[j] has a shared effect gamma_shared[1]. Because species are often expected to respond differently to site characteristics, site covariates are likely to be introduced using the formula_psi argument. Nevertheless, the formula_psi_shared argument can be used when consistent covariate effects across species are expected or when the data support doing so.

A similar approach can be applied to theta and phi, which can be modeled as functions of species, site, and replicate covariates. The following table shows examples of theta modeling, where i is the species index, j is the site index, k is the replicate index, speccov1 is a continuous species covariate, sitecov1 is a continuous site covariate, and replcov1 is a continuous replicate covariate.

In occumb(), species-specific effects on theta are denoted as beta and shared effects on theta are denoted as beta_shared. The first row of the above table specifies an intercept-only model. As in the case of psi, occumb() always estimates the species-specific intercept beta[i, 1]. The second and third cases can be contrasted with the second case of the psi example with a single covariate specified in the formula_psi argument, and the remaining cases with the fifth and sixth cases of the psi example with a single covariate specified in the formula_psi_shared argument. Because the replicate covariate replcov1 has both site index j and replicate index k, specifying it adds these two indices to theta.

The same rule applies to the phi modeling. The following is an example of a more complex case involving interactions between different types of covariates.

In occumb(), species-specific effects on phi are denoted as alpha and shared effects on phi are denoted as alpha_shared. Similar to the other two parameters, occumb() always estimates the species-specific intercept alpha[i, 1].

The following table summarizes the covariate types accepted by each formula argument.

Prior distributions

A hierarchical prior distribution is specified for the species-specific effects alpha, beta, and gamma. Specifically, the vector of these effects is assumed to follow a multivariate normal distribution, and a prior distribution is specified for the elements of its mean vector Mu and covariance matrix Sigma. The element values of Mu and Sigma are estimated from the data; as these hyperparameters summarize the variation in species-specific effects at the community level, their estimates may be of interest in assessing e.g., whether covariates have a consistent effect on a wide range of species.

For each element of Mu, a normal prior distribution with a mean of 0 and precision (i.e., the inverse of the variance) prior_prec is specified. The prior_prec value is determined by the prior_prec argument of the occumb() function, which by default is set to a small value of 1e-4 to specify vague priors.

Sigma is decomposed into the elements of standard deviation sigma and correlation coefficient rho, each of which is specified by a different vague prior. Specifically, a uniform prior distribution with a lower limit of 0 and an upper limit of prior_ulim is specified for sigma, and a uniform prior with a lower limit of −1 and an upper limit of 1 is set for rho. The value of prior_ulim is determined by the prior_ulim argument of the occumb() function and is set to 1e4 by default.

For each of the shared effects alpha_shared, beta_shared, and gamma_shared, a normal prior distribution with mean 0 and precision prior_prec is specified.

After all, which are the relevant parameters?

The latent variables and parameters of the model to be estimated and saved using the occumb() function are as follows. Note that occumb() will not save u and r, but their function pi. The posterior samples of these latent variables and parameters can be accessed using get_post_samples() or get_post_summary() functions.

z

Site occupancy status of species.

pi

Multinomial probabilities of species sequence read counts.

phi

Sequence relative dominance of species.

theta

Sequence capture probabilities of species.

psi

Site occupancy probabilities of species.

alpha

Species-specific effects on sequence relative dominance (phi).

beta

Species-specific effects on sequence capture probabilities (theta).

gamma

Species-specific effects on site occupancy probabilities (psi).

alpha_shared

Effects on sequence relative dominance (phi) common across species.

beta_shared

Effects on sequence capture probabilities (theta) that are common across species.

gamma_shared

Effects on site occupancy probabilities (psi) that are common across species.

Mu

Community-level averages of species-specific effects (alpha, beta, gamma).

sigma

Standard deviations of species-specific effects (alpha, beta, gamma).

rho

Correlation coefficients of the species-specific effects (alpha, beta, gamma).