sensitivityIxJ (version 0.1.5)

norm_single_u_allocation_p_value: Compute the normal-approximation-based z-score and p-value for a given 2 by 2, 2 by 3, 2 by 4, 2 by 5, 3 by 2, 4 by 2, or 3 by 3 contingency table.

Description

This function performs a score test for ordinal association in contingency tables with fixed margins under a sensitivity analysis framework. It uses the test statistic \(T = \sum A_{ij} N_{ij}\) where \(A_{ij} = w_i v_j\) (treatment score X outcome score) and computes the mean and variance under the null hypothesis using the Poisson-binomial approximation for the multivariate Fisher's noncentral hypergeometric distribution.

Usage

norm_single_u_allocation_p_value(
  obs.table,
  gamma,
  delta,
  u_allocation,
  row = "treatment",
  treatment.scores,
  outcome.scores,
  shared_divisor = 1e+06
)

Value

A list containing:

T_obs

The observed test statistic

mu_T

The expected value of the test statistic under the null hypothesis

var_T

The variance of the test statistic under the null hypothesis

z_score

The standardized test statistic (T_obs - mu_T) / sqrt(var_T)

p_value

The one sided upper tailed p-value based on the normal approximation

treatment.scores

The treatment scores used

outcome.scores

The outcome scores used

Arguments

obs.table

A matrix or table object representing the contingency table. Rows represent treatments, columns represent outcomes.

gamma

A nonnegative scalar

delta

A binary vector. Must have length equal to the number of treatments. Can contain at most two unique values.

u_allocation

A numeric vector of unmeasured confounder allocations for each outcome category. Must have length equal to the number of outcomes. Each entry must be non-negative and not exceed the corresponding outcome margin.

row

Character indicating whether rows represent "treatment" or "outcome". If "outcome", the table will be transposed. Default is "treatment".

treatment.scores

A numeric vector of scores for treatments. Must be monotone (either increasing or decreasing). Higher scores typically indicate more intense treatments.

outcome.scores

A numeric vector of scores for outcomes. Must be monotone (either increasing or decreasing). Higher scores typically indicate better outcomes.

shared_divisor

A numeric value used for numerical stability in the Rcpp computations. Default is 1e6.

Details

The function implements a score test for ordinal association where the test statistic is a weighted sum of the cell counts, with weights given by the product of treatment and outcome scores. The function uses specialized Rcpp functions to compute the mean and variance-covariance structure of the free cells (those with degrees of freedom), then recovers the full mean vector and covariance matrix using the marginal constraints. The test statistic and its moments are then computed using matrix operations.

Examples

# 2x3 contingency table
obs.table <- matrix(c(10, 20, 30, 15, 25, 10), nrow = 2, byrow = TRUE)

# Sensitivity parameters and u_allocation
gamma = 0.5
delta <- c(0, 1)
u_allocation <- c(5, 10, 8)

# Ordinal scores
treatment.scores <- c(0, 1)
outcome.scores <- c(0, 1, 2)

# Perform test
result <- norm_single_u_allocation_p_value(
  obs.table = obs.table,
  gamma = gamma,
  delta = delta,
  u_allocation = u_allocation,
  treatment.scores = treatment.scores,
  outcome.scores = outcome.scores
)