Negative Binomial-Gamma Family for Overdispersed Count/Effort Ratios

Two-process model where the numerator follows a negative binomial distribution and the denominator (effort/exposure) follows a Gamma distribution. This fills the gap between Poisson-Gamma (equidispersed counts) and NegBin-NegBin (integer denominator): use it when counts are overdispersed and effort is continuous.

Use cases:

Overdispersed catch per unit effort (CPUE)
Overdispersed observations per continuous exposure
Any ratio of overdispersed counts over continuous positive effort

Usage

ratiod_negbin_gamma(link_num = "log", link_denom = "log")

Arguments

link_num: Link function for the count rate: "log".
link_denom: Link function for the effort mean: "log" (default), "identity" or "inverse".

Value

A ratiod_family object

Examples

# Create family object
fam <- ratiod_negbin_gamma()
print(fam)
#> tulpaRatio family: negbin_gamma 
#> Negative binomial numerator, Gamma denominator (overdispersed CPUE-type) 
#> 
#> Numerator:  neg_binomial_2(log)
#> Denominator: gamma (log) 

# Simulate overdispersed CPUE data
set.seed(123)
n <- 60
df <- data.frame(
  catch = rnbinom(n, size = 5, mu = 8),
  effort_hours = rgamma(n, shape = 4, rate = 1),
  depth = rnorm(n),
  vessel = factor(rep(1:6, each = n/6))
)

# \donttest{
# Fit model
fit <- tratio(
  catch | effort_hours ~ depth + (1 | vessel),
  data = df,
  family = ratiod_negbin_gamma(),
  control = list(iter = 400, chains = 1, seed = 1)
)
summary(fit)
#> ratio model summary
#> ===================
#> 
#> Inference: Exact (Tier 1) via hmc
#> Family: negbin_gamma 
#> 
#> Fixed effects (numerator):
#>        parameter  mean    sd   2.5% 97.5%  rhat ess_bulk
#>  num:(Intercept) 2.025 0.078  1.843 2.140 1.022       32
#>        num:depth 0.070 0.069 -0.071 0.189 1.026       57
#> 
#> Fixed effects (denominator):
#>          parameter  mean    sd   2.5% 97.5%  rhat ess_bulk
#>  denom:(Intercept) 1.319 0.100  1.041 1.468 1.040       23
#>        denom:depth 0.029 0.077 -0.122 0.186 1.034      102
#> 
#> Variance components:
#>  parameter  mean    sd  2.5%  97.5%  rhat ess_bulk
#>  phi_denom 3.794 0.663 2.760  5.283 0.997      103
#>    phi_num 9.841 4.212 4.272 19.860 1.006      128
#>   sigma_re 0.087 0.074 0.006  0.302 1.002       46
#> 
#> Diagnostics:
#>   Divergences: 0 
#>   Avg. acceptance: 0.86 
#>   Chains: 1 
#>   Iterations: 400 
#>   Warmup: 200 
#>   R-hat > 1.01 for 4 parameter(s), largest 1.040 (denom:(Intercept)).
#>   Bulk ESS < 100 for 4 parameter(s), smallest 23 (denom:(Intercept)).
#>   Run longer chains or reparameterise before using these estimates.
# }