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The non-conservative, size-\(\alpha\) modified Fisher exact test

Source: R/modified_fisher.R
modified_fisher_exact_test.Rd

Computes the non-conservative, size-\(\alpha\) modified Fisher exact test for comparing two proportions \(u/m\) and \(v/n\), testing \(H_0\): OR = odds_ratio. Returns an htest object containing test-based two-sided p-values and \((1 - \alpha)\) confidence intervals for the odds ratio, which agree by construction. The test maximises the actual size over the nuisance parameter \(p_0\) subject to it remaining no greater than \(\alpha\), making it less conservative than the standard Fisher exact test while strictly controlling the Type I error rate.

Usage

modified_fisher_exact_test(
  u,
  m,
  v,
  n,
  odds_ratio,
  alpha = 0.05,
  precision = 0.001,
  message = FALSE,
  maze = 10,
  method = "zoom",
  zoom_iter = 6,
  conf_int = TRUE,
  pvalue = TRUE,
  local_size_data = FALSE,
  power = TRUE,
  superiority = FALSE,
  power_at_pi1 = 0.5,
  power_at_pi2 = 0.75
)

Arguments

u

Number of successes observed in group 1 (out of m trials). No default.

m

Number of trials in group 1. No default.

v

Number of successes observed in group 2 (out of n trials). No default.

n

Number of trials in group 2. No default.

odds_ratio

The null hypothesis odds ratio \(\theta_0\). No default.

alpha

Nominal significance level \(\alpha\). Defaults to 0.05.

precision

Numerical precision for p-values, confidence limits, and size calculations. Defaults to 1e-03.

message

Logical. If TRUE, prints progress messages during execution. Defaults to FALSE.

maze

Number of grid points evaluated at each zoom iteration. Defaults to 10.

method

Numerical method for maximising the local size over the nuisance parameter. Either "zoom" (default) or "trust" (trust-region method from the trust package).

zoom_iter

Number of zoom iterations. Defaults to 6.

conf_int

Logical. If FALSE, skips the \((1 - \alpha)\) two-sided confidence interval. Defaults to TRUE.

pvalue

Logical. If FALSE, skips the two-sided test-based p-value. Defaults to TRUE.

local_size_data

Logical. If TRUE, attaches a local.size.data data frame to the results for plotting the size of the test as a function of the nuisance parameter. Defaults to FALSE.

power

Logical. If FALSE, skips the power calculation. Defaults to TRUE.

superiority

Logical. If TRUE, power is computed only over tables where the observed rate in group 2 exceeds that in group 1. Defaults to FALSE.

power_at_pi1

Success probability in group 1 at which to evaluate power. Defaults to 0.5.

power_at_pi2

Success probability in group 2 at which to evaluate power. Defaults to 0.75.

Value

An object of class htest: a list with components p.value (the two-sided test-based p-value, if pvalue = TRUE), estimate (the sample odds ratio), conf.int (the two-sided \((1 - \alpha)\) confidence interval for the odds ratio, if conf_int = TRUE), null.value (the null odds ratio), alternative, method, and data.name. It additionally carries support.data (the test frame from construct_test_frame()), gamma0 (the optimal threshold), power (the power at power_at_pi1 and power_at_pi2, if power = TRUE), local.size.data (a data frame for plotting the size as a function of the nuisance parameter, if local_size_data = TRUE), and fn_args (the arguments used).

See also

construct_test_frame() for the underlying test frame of critical values and randomisation probabilities; optimise_gamma0() for the gamma0 optimisation; size_modified() for the size of the test maximised over the nuisance parameter; local_size_modified() for the local size at a fixed nuisance parameter value; power_modified() for the power of the test.

Other modified: construct_test_frame(), local_size_modified(), optimise_gamma0(), power_modified(), size_modified()

Examples

# Example 1 of van der Meulen, Raymond & van der Meulen (2021):
# 5/12 successes versus 7/11, testing H0: OR = 1 at alpha = 0.05.
modified_fisher_exact_test(u = 5, m = 12, v = 7, n = 11, odds_ratio = 1)
#> 
#> 	Non-Conservative Size-α Modified Fisher's Exact Test
#> 
#> data:  u = 5, v = 7, m = 12, n = 11
#> p-value = 0.3208
#> alternative hypothesis: true odds ratio is not equal to 1
#> 95 percent confidence interval:
#>  0.072034 2.209549
#> sample estimates:
#> odds ratio 
#>  0.4081633 
#> 

# \donttest{
# The size of the test as a function of the nuisance parameter can be
# returned for diagnostic plotting (slower, as the size is evaluated on a
# grid of 101 nuisance-parameter values):
res <- modified_fisher_exact_test(u = 5, m = 12, v = 7, n = 11,
                                  odds_ratio = 1, local_size_data = TRUE)
head(res$local.size.data)
#>    pi1         size method
#> 1 0.00 0.0000000000   zoom
#> 2 0.01 0.0006921852   zoom
#> 3 0.02 0.0092790921   zoom
#> 4 0.03 0.0393074943   zoom
#> 5 0.04 0.1038314178   zoom
#> 6 0.05 0.2116501340   zoom
# }