Actual size of the MFET for a given \(\gamma_0\)

Finds the actual size of the MFET for a given \(\gamma_0\) by numerically maximising the local size over the nuisance parameter \(p_0 \in (0, 1)\). Supports two methods: "zoom" evaluates the local size at a grid of points and iteratively refines the maximum (matches the SAS macro default); "trust" uses a trust-region optimiser with the analytic gradient from .local_size_gradient_modified().

Usage

size_modified(
  .c,
  .odds_ratio,
  .m,
  .n,
  .df,
  .alpha,
  .precision,
  .method,
  .maze,
  .zoom_iter,
  .rejection_base = NULL
)

Arguments

.c

The \(\gamma_0\) value for which the actual size is to be computed. Supplied as a length-1 list element; typically one element of the sorted randomisation probability vector from the test frame.

.odds_ratio

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

.m

Number of trials in group 1.

.n

Number of trials in group 2.

.df

Test frame (data frame) generated by construct_test_frame(), containing critical values \(c_1\), \(c_2\) and randomisation probabilities \(\gamma_1\), \(\gamma_2\) for every possible total \(T\).

.alpha

Nominal significance level \(\alpha\). No default.

.precision

Numerical precision. No default.

.method

Numerical method: "zoom" or "trust".

.maze

Number of grid points evaluated at each zoom iteration.

.zoom_iter

Number of zoom iterations ("zoom" method only).

.rejection_base

Optional precomputed base object from .build_rejection_base(), containing the part of the rejection matrix that depends only on .df, .m, .n (not on the \(\gamma_0\) value in .c). Defaults to NULL, in which case it is built internally. Supply a prebuilt base when calling repeatedly with the same test frame across different \(\gamma_0\) candidates (as optimise_gamma0() does during its bisection search) to avoid rebuilding it on every call. If supplied, it must correspond to the same .df, .m, .n passed alongside it; this is not checked.

Value

A single numeric value: the actual size of the test, i.e. the local size maximised over the nuisance parameter, in \([0, 1]\).

See also

optimise_gamma0() which maximises this quantity to find the optimal gamma0; local_size_modified() for the local size at a fixed nuisance parameter value, of which this is the maximum; local_size_asymptotic(), local_size_probability(), local_size_randomised() for the equivalent quantity under alternative tests.

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

Other size: local_size_asymptotic(), local_size_modified(), local_size_probability(), local_size_randomised()

Examples

df <- construct_test_frame(.odds_ratio = 1, .m = 6, .n = 4,
                           .alpha = 0.05, .precision = 1e-3)
# Actual size of the test at a candidate gamma0 threshold of 0.05:
size_modified(.c = 0.05, .odds_ratio = 1, .m = 6, .n = 4, .df = df,
          .alpha = 0.05, .precision = 1e-3, .method = "zoom",
          .maze = 10, .zoom_iter = 6)
#> [1] 0.2170814