poisson_test_b: Poisson tests

In bayesics: Bayesian Analyses for One- and Two-Sample Inference and Regression Methods

Poisson tests

Description

Make inference on one or two populations using Poisson distributed count data

Usage

poisson_test_b(
  x,
  offset,
  r,
  ROPE,
  prior = "jeffreys",
  prior_shape_rate,
  CI_level = 0.95,
  plot = TRUE,
  seed = 1,
  mc_error = 0.002
)

Arguments

x	Number of events. A vector of length one or two.
offset	Time, area, etc. measured in the Poisson process. NOTE: Do not take the log!
r	optional. If provided and inference is being made for a single population, poisson_test_b will return the posterior probability that the population rate is less than this value.
ROPE	ROPE for rate ratio if inference is being made for two populations. Provide either a single value or a vector of length two. If the former, the ROPE will be taken as (1/ROPE,ROPE). If the latter, these will be the bounds of the ROPE.
prior	Either "jeffreys" (Gamma(1/2,0)) or "flat" (Gamma(0.001,0.001)). This is ignored if prior_shape_rate is provided.
prior_shape_rate	Vector of length two, giving the shape and rate parameters for the gamma distribution that will act as the prior on the population rates.
CI_level	The posterior probability to be contained in the credible intervals.
plot	logical. Should a plot be shown?
seed	Always set your seed! (Unused for a single population rate)
mc_error	The number of posterior draws will ensure that with 99% probability the bounds of the credible intervals of \lambda_1/\lambda_2 will be within \pm mc_error. (Ignored for a single population rate.)

Details

The likelihood is

y \sim Poi(\lambda t),

where \lambda is the rate, and t is the time or area observed and is given by the argument offset.

The prior is given by

\lambda \sim \Gamma(a,b),

where a and b are given by the argument prior_shape_rate. If prior_shape_rate is missing and prior = "jeffreys", then a Jeffrey's prior will be used, i.e., \Gamma(0.5,0) (improper), while if prior = "flat", \Gamma(0.001,0.001) will be used.

Value

(returned invisible) A list with the following:

• x, offset: data and offset(s)

• posterior_mean, posterior_mean_pop1, posterior_mean_pop2: posterior means of the Poisson rates

• CI, CI_pop1, CI_pop2: Credible interval bounds for the rates

• CI_lambda1_over_lambda2: Credible interval bounds for the rate ratio (rate of population 1 over the rate of population 2)

• Pr_less_than_r: (1 sample analysis only) If r was supplied, the posterior probability that the rate is less than r.

• Pr_rate_ratio_lt_one: (2 sample analysis only) Posterior probability that the rate ratio is less than 1

• Pr_rateratio_in_ROPE: (2 sample analysis only) Posterior probability that the rate ratio is in the ROPE (based on Pr_rate_ratio_lt_one)

• rate_plot: Posterior and prior plots for the rates

• posterior_parameters: Posterior parameters for rates for the gamma posterior distribution

Examples

# One sample
poisson_test_b(x = 12)
## You can compute the posterior probability that the rate is less than r
poisson_test_b(x = 12,
               r = 8)

# Two samples
poisson_test_b(x = c(12,20))

# Offsets can be included:
poisson_test_b(x = c(12,20),
               offset = c(10,9))

# Different priors can be used
poisson_test_b(x = c(12,20),
               prior = "flat")
poisson_test_b(x = c(12,20),
               prior_shape_rate = c(20,1.5))

bayesics documentation built on March 11, 2026, 5:07 p.m.