Check whether Gamma can be reliably fitted from median + IQR summary statistics

The Gamma distribution (dist_type = 2) has a single shape parameter \(\phi\) that controls the coefficient of variation CV = 1/\(\sqrt{\phi}\). When only median + IQR (summary type 2) data are available, the likelihood gradient with respect to \(\phi\) comes exclusively from central order statistics at p = 0.25, 0.50, 0.75. These central quantiles are most sensitive to location and spread, but carry weak information about shape compared with tail quantiles (type-1: min/max) or full frequency data (types 4/5).

As \(\phi\) grows large the gamma distribution approaches a normal distribution: the three central quantiles become nearly symmetric around the mean, and the likelihood surface flattens in the \(\phi\) direction. MCMC consequently exhibits very small step sizes, high autocorrelation, and inference that is dominated by the prior on log_phi.

Note that this problem does not apply to type-1 (median + range) or type-3 (mean + SD) data. Type-1 uses extreme order statistics (min, max) which lie in the tails where gamma shape sensitivity is largest. Type-3 directly identifies \(\phi\) via \(\phi = (\text{mean}/\text{SD})^2\), giving a sharp, well-defined gradient regardless of how concentrated the distribution is.

The heuristic uses the moment estimator $$\hat{\phi} = \left(\frac{1.35 \times \text{median}}{\text{IQR}}\right)^2$$ (equivalent to CV\(^{-2}\), the method-of-moments gamma shape estimate) computed from each type-2 dataset. If the median implied shape across type-2 datasets exceeds max_implied_shape, the function returns FALSE.

Usage

gamma_type2_reliable(
  datasets,
  max_implied_shape = 20,
  min_n = 50,
  verbose = TRUE
)

Arguments

datasets

A named list of datasets as accepted by prepare_stan_data_from_datasets().

max_implied_shape

Numeric scalar (default 20). Maximum tolerated median implied shape across type-2 datasets. Corresponds to CV \(\approx\) 0.22 and IQR/median \(\approx\) 0.30. Reduce to be stricter; increase to be more permissive.

min_n

Integer scalar (default 50). Minimum acceptable sample size for a type-2 dataset. If more than half of the type-2 datasets fall below this threshold, an advisory message is printed but FALSE is not returned — use this as a soft warning only.

verbose

Logical (default TRUE). Print a one-line verdict with the reason a check failed.

Value

TRUE if type-2 data appear adequate for gamma fitting, FALSE if the implied shape is too large for reliable inference. Returns TRUE silently when no type-2 datasets are present (the heuristic is not relevant in that case).

Examples

if (FALSE) { # \dontrun{
# Concentrated distribution — high implied shape, likely slow
ds_concentrated <- list(
  d1 = list(median = 10, Q1 = 9.2, Q3 = 10.8, n = 50),
  d2 = list(median = 12, Q1 = 11.1, Q3 = 12.9, n = 60)
)
gamma_type2_reliable(ds_concentrated)   # expected: FALSE

# Dispersed distribution — low implied shape, reliable
ds_dispersed <- list(
  d1 = list(median = 10, Q1 = 7, Q3 = 14, n = 80),
  d2 = list(median = 8,  Q1 = 5, Q3 = 12, n = 100)
)
gamma_type2_reliable(ds_dispersed)      # expected: TRUE
} # }