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Z-variabilty in POD of rare taxa

This repo contains all code and analyses used to examine depth variability in the probability of detecting cetaceans in eDNA samples. A living version of the manuscript in progress can be found here. Herein, we analyse marine mammal detections from water samples collected throughout the California Current and at depths of 0-500m with the goal of answering the following three questions:

  1. Does the probability of a detecting cetaceans in eDNA samples vary with sample depth?
    • H0: Probability of detection does not vary with depth.
    • H1: Probability of detection varies across depth agnostic to species or functional group.
    • H2: Probability of detection varies across depth according to species or functional group.
  2. Does the probability of a detecting cetaceans in eDNA samples change with the number of technical replicates?
    • NOTE should we wrap dilution into this question as well?
    • H0: Detection does not vary with # technical replicates.
    • H1: Detection varies with # technical replicated agnostic to species/fuctional group or depth.
    • H2: Detection varies with # technical replicates according to species/functional group, depth, or a combination of the two.
  3. Does depth distribution of detections vary across xy spatial distribution?
    • H0: Depth distribution of detections does not vary across xy spatial distribution.
    • H1: Depth distribution of detection does vary across xy spatial distribution agnostic to oceanography (e.g. upwelling).
    • H2: Depth distribution of detection does vary across xy spatial distribution according to oceanography (e.g. upwelling).

The Plan

All raw data are in "Data", and intermediate data products are in "ProcessedData." Dive data come from here and here.

Analysis 1: POD by depth, collapse data across X and Y

  • Model 1: POD ~ z
  • Model 2: POD ~ z with intercept by species
  • Model 3: POD ~ z with smooth by species
  • Model 4: POD ~ z with smooth by family
  • Model 5: POD ~ z with smooth by prey category (invert, fish, squid)
  • Run model diagnostics, select best model, interpret results within ecological context, develop recommendations for eDNA monitoring of marine mammals.

Analysis 2: POD by primer, freeze/thaw, tech rep. and vary with depth?

  • Using GAM splines from Analysis 1, retest varying number of replicates with GAMs
  • Compare to alternative Bayesian occupancy model: Brice's replication model without assuming species' presence and adding depth as a covariate (see "Taking Brice's approach" below).
  • Run model diagnostics, select best model, interpret results within ecological context, develop recommendations for eDNA monitoring of marine mammals.

Analysis 3: depth variability across xy or oceanography

  • Model 1: POD ~ xy + xy x species + z + z x species + xy x species x z
  • Run model diagnostics, report significant covariates, interpret results within ecological context, develop recommendations for eDNA monitoring of marine mammals.

Taking Brice's approach

We don't have observations of the presence or absence of marine mammals at each site, but we can still treat site occupancy as a latent variable (per species).

We have: multiple sites, each of which contain biological samples at multiple depths (not replicates), then multiple primers x tech reps. Each has an associated volume and dilution.

  • need to add a dilution coefficient, or does this happen at the tech rep level?

Hierarchical Model Structure

  1. Site-Level Occurrence The presence of a marine mammal species at site $s$ is modeled as a Bernoulli random variable:

    $Z_s \sim \text{Bernoulli}(\psi)$

    where $Z_s$ is the site-level occurrence indicator, and $\psi$ is the overall occurrence probability, drawn from a Beta prior:

    $\psi \sim \text{Beta}(1,1)$

    --> note that in our case, particularly since we are working at the species level, psi might come from some field over X, Y

  2. **Capture within a depth x primer x tech rep reaction ** The logit-linear model for capture probability at depth $d$ is:

    $\text{logit}(p_{\text{capture},d}) = \beta_0 + \beta_{\text{vol}} \cdot X_{\text{vol},d} + \beta_{\text{depth}} \cdot X_{\text{depth},d} + \gamma_{s} + \delta_{\text{primer}[b]}$

    Where:

    • $p_{\text{capture},d}$ is the capture probability for depth $d$
    • $\beta_0$ is the intercept (site-level capture probability hyperparameter)
    • $\beta_{\text{vol}}$ is the volume coefficient
    • $\beta_{\text{depth}}$ is the depth coefficient
    • $X_{\text{vol},d}$ is the centered water volume
    • $X_{\text{depth},d}$ is the centered sampling depth
    • $\gamma_{s}$ is the site-specific random effect # EKJ note is this what we want?
    • $\delta_{\text{primer}[b]}$ is the primer-specific fixed effect, with method effects constrained such that: $\delta_{\text{MFU}} = 0$ and $\delta_{\text{MV1}} \sim \text{Normal}(0, 1.7)$ # EKJ note may need to change these $\delta_{\text{DLP}} \sim \text{Normal}(0, 1.7)$

    The depth capture for a given site, volume, primer is then modeled as:

    $Y_{\text{capture},d} \sim \text{Bernoulli}(Z_{s} \cdot p_{\text{capture},d})$

    Need to rewrite this bc it's only one Bernoulli trial in our case

    Conditional on capture in that volume x primer reaction, technical replicates are modeled as:

    $Y_{\text{detect},i} \sim \text{Bernoulli}(p_{\text{detect}} \cdot Y_{\text{capture},b[i]})$

    where $p_{\text{detect}}$ is the detection probability, drawn from a Beta prior:

    $p_{\text{detect}} \sim \text{Beta}(1,1)$