Question on trend estimation in hierarchical abundance models

[Sarah Jean Kiuama Frey]

Hi all,

I’m working on sooty grouse abundance trend estimation using a binomial N-mixture model in JAGS, following the approach in Kéry et al. 2009. My model includes site random effects on abundance and observer random effects on detection, and I’ve been testing different ways to calculate the trend line (based on growth rate r) to plot alongside yearly mean abundance (Nmn[k]) and raw abundance counts.

Issues I'm facing:

• Yearly abundance estimates (Nmn[k]) seem reasonable and align well with results from other frameworks (ubms, unmarked).
• However, my modeled trend (lambda_trend[k]) appears too low, aligning more closely with raw abundance means than with Nmn[k].
• Correcting for detection probability (mean_p[k]) inflates lambda_trend unrealistically (values >30).
• A post hoc linear model approach produces a trend that aligns with Nmn[k] and reflects the estimated growth rate (r), but I’m unsure if this is an appropriate way to derive the trend line.

Likelihood model in JAGS:

# Likelihood

 for (k in 1:nyear) {

  for (i in 1:nsite) {

    log(lambda[i, k]) <- loglam0 + r * (k - nyear/2) + eta.site[i]

    N[i, k] ~ dpois(lambda[i, k]) 

    for (t in 1:nrep) {

      logit(p[i, t, k]) <- logitp0 + bpts * PTS[i, t, k] + teta.p[i, t, k]

      C[i, t, k] ~ dbin(p[i, t, k], N[i, k])  # Observation process

    }

  }

} 

# Derived parameters: Mean and total abundance over time

 for (k in 1:nyear) {

   Nmn[k] <- mean(N[1:nsite, k])  # Mean abundance per year

   Npop[k] <- sum(N[1:nsite, k])  # Total abundance per year

}

Trend line derivation in JAGS:

for (k in 1:nyear) {

  # Growth rate model

  log(lambda_trend[k]) <- loglam0 + r * (k - nyear/2)

 

  # Including sit random effects

  log(lambda_trend_site[k]) <- mean(log(lambda[1:nsite, k]))

 

  # Compute year-specific detection probability

  mean_p[k] <- mean(p[,,k])

 

  # Correcting for detection probability

  lambda_trend_corrected[k] <- lambda_trend[k] / mean_p[k]

  lambda_trend_site_corrected[k] <- lambda_trend_site[k] / mean_p[k]

}

 Questions:

1. What is the best way to structure lambda_trend in a JAGS model to ensure it reflects the estimated growth rate (r) while still aligning with Nmn[k]?
2. Is applying a post hoc linear regression a valid approach, or should the correction for detection probability (mean_p[k]) be handled differently within the model?
3. Would it be more appropriate to derive lambda_trend using Nmn[1] as a baseline? If so, how would this best be implemented in JAGS?

Any insights or suggestions would be greatly appreciated. Thanks in advance for your help!

Best,
Sarah