Beta-binomial family for overdispersed proportions

Two-process model where the numerator follows a beta-binomial distribution, allowing for overdispersion in binomial data. The denominator is the number of trials, which can be fixed or modelled.

Use cases:

• Overdispersed proportions (e.g., infection rates across sites)

• Hierarchical binomial with extra-binomial variation

• Proportions with correlation within clusters

Usage

ratiod_beta_binomial(link = "logit", denominator_known = TRUE)

Arguments

link

Link function for probability (default: "logit")

denominator_known

Logical; if TRUE, denominator is treated as fixed and known. If FALSE, denominator is also modelled.

Value

A ratiod_family object

Details

The beta-binomial distribution arises when the success probability itself follows a beta distribution. This creates overdispersion (variance > binomial variance) and is useful when:

• There is unmodelled heterogeneity in success probability

• Observations within clusters are correlated

• The binomial assumption of independent trials is violated

The overdispersion parameter \(\rho\) (intra-class correlation) ranges from 0 (binomial) to 1 (maximum overdispersion).

Examples

# Create family object
fam <- ratiod_beta_binomial()
print(fam)
#> tulpaRatio family: beta_binomial_fixed 
#> Beta-binomial numerator with fixed denominator (overdispersed proportions) 
#> 
#> Numerator:  beta_binomial(logit)
#> Denominator: fixed (fixed) 

# Simulate overdispersed proportion data
set.seed(123)
n <- 40
df <- data.frame(
  infected = rbinom(n, size = 20, prob = rbeta(n, 2, 5)),
  tested = rep(20L, n),
  treatment = factor(rep(c("control", "treated"), each = n/2)),
  site = factor(rep(1:8, each = n/8))
)

if (FALSE) { # \dontrun{
# Fit model (slow, not run on CRAN)
fit <- tratio(
  infected | tested ~ treatment + (1 | site),
  data = df,
  family = ratiod_beta_binomial(),
  control = list(iter = 200, warmup = 100, chains = 1)
)
} # }