bayesics (version 2.1.0)

chisq_test_b: Test of independence for 2-way contingency tables

Description

Test of independence for 2-way contingency tables

Usage

independence_b(
  x,
  sampling_design = "multinomial",
  ROPE,
  prior = "jeffreys",
  prior_shapes,
  CI_level = 0.95,
  seed = 1,
  mc_error = 0.002
)

Value

(returned invisible) A list with the following elements:

• posterior_shapes: posterior Dirichlet shape parameters

• posterior_mean: posterior mean

• lower_bound: lower credible interval bounds

• upper_bound: upper credible interval bounds

• individual_ROPE: Probability that joint probabilities are in the ROPE around independent probabilities

• overall_ROPE: Overall probability of falling in the ROPE (i.e., all probabilities are near the product of the marginal probabilities)

• prob_pij_less_than_p_i_times_p_j: (If multinomial sampling design) Probabilities that each joint probability is less than the product of the marginal probabilities

• prob_p_j_given_i_less_than_p_j: (If fixed rows or columns sampling design) Probabilities that each conditional probability is less than the marginal probabilities

• prob_direction: Probability of direction for the joint or conditional (depending on sampling scheme) probabilities (based on prob_pij_less_than_p_i_times_p_j or prob_p_j_given_i_less_than_p_j)

• BF_for_dependence_vs_independence: Bayes factor testing dependence vs. independence (higher values favor dependence, lower values favor independence)

• BF_evidence: Kass and Raftery's interpretation of the level of evidence of the Bayes factor

Arguments

x

Either a table or a matrix of counts

sampling_design

Either "multinomial", "fixed rows", or "fixed columns"

ROPE

vector of positive values giving ROPE boundaries for each regression.

prior

Either "jeffreys" (Dirichlet(1/2)) or "uniform" (Dirichlet(1)). This is ignored if prior_shapes is provided.

prior_shapes

Either a single positive scalar, in which case a symmetric Dirichlet is used, or else a matrix matching the dimensions of x or a vector of length prod(dim(x)).

CI_level

The posterior probability to be contained in the credible interval.

seed

Always set your seed!

mc_error

This is the error in probability from the posterior CDF evaluated at the ROPE bounds. Note that if it is estimated that these probabilities are between 0.11 and 0.89, the more relaxed value of 0.01 is used.

Details

For a 2-way contingency table with R rows and C columns, evaluate the probability that

• the joint probabilities \(p_{ij}\) are all within the ROPE of \(p_{i\cdot}\times p_{\cdot j}\) for sampling_design = "multinomial"

• the probabilities \(p_{j|i}\) are all within the ROPE of \(p_{\cdot j}\) if sampling_design = "fixed rows" or "fixed columns"

References

Gunel, Erdogan & Dickey, James (1974). Bayes factors for independence in contingency tables, Biometrika, 61(3), Pages 545–557, https://doi.org/10.1093/biomet/61.3.545

Kass, R. E., & Raftery, A. E. (1995). Bayes Factors. Journal of the American Statistical Association, 90(430), 773–795.

Examples

# \donttest{
# Generate data
set.seed(2025)
N = 500
nR = 5
nC = 3
dep_probs = 
  extraDistr::rdirichlet(1,rep(2,nR*nC)) |> 
  matrix(nR,nC)

# Multinomial sampling
## Test independence
independence_b(round(N * dep_probs))

## Use other priors
independence_b(round(N * dep_probs),
               prior = "uniform")
independence_b(round(N * dep_probs),
               prior_shapes = 2)
independence_b(round(N * dep_probs),
               prior_shapes = matrix(1:(nR*nC),nR,nC))

# Fixed marginals
independence_b(round(N * dep_probs),
               sampling_design = "rows")
independence_b(round(N * dep_probs),
               sampling_design = "cols")
# }