Generate the weighting strategy based on a graphical multiple comparison procedure

A graphical multiple comparison procedure defines a closed test procedure, which tests each intersection hypothesis and reject an individual hypothesis if all intersection hypotheses involving it have been rejected. An intersection hypothesis represents the parameter space where individual null hypotheses involved are true simultaneously.

The closure based on a graph consists of all updated graphs (corresponding to intersection hypotheses) after all combinations of hypotheses are deleted. For a graphical multiple comparison procedure with \(m\) hypotheses, there are \(2^{m}-1\) updated graphs (intersection hypotheses), including the initial graph (the overall intersection hypothesis). The weighting strategy of this graph consists of hypothesis weights from all \(2^{m}-1\) updated graphs (intersection hypotheses). The algorithm to derive the weighting strategy is based on Algorithm 1 in Bretz et al. (2011).

Usage

graph_generate_weights(graph)

Arguments

graph: An initial graph as returned by graph_create().

Value

A numeric matrix of all intersection hypotheses and their hypothesis weights. For a graphical multiple comparison procedure with \(m\) hypotheses, the number of rows is \(2^{m}-1\), each of which corresponds to an intersection hypothesis. The number of columns is \(2\cdot m\). The first \(m\) columns indicate which individual hypotheses are included in a given intersection hypothesis and the second half of columns provide hypothesis weights for each individual hypothesis for a given intersection hypothesis.

Performance

Generation of intersection hypotheses is closely related to the power set of a given set of indices. As the number of hypotheses increases, the memory and time usage can grow quickly (e.g., at a rate of \(O(2^n)\)). There are also multiple ways to implement Algorithm 1 in Bretz et al. (2011). See vignette("generate-closure") for more information about generating intersection hypotheses and comparisons of different approaches to calculate weighting strategies.

References

Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., and Rohmeyer, K. (2011). Graphical approaches for multiple comparison procedures using weighted Bonferroni, Simes, or parametric tests. Biometrical Journal, 53(6), 894-913.

Examples

# A graphical multiple comparison procedure with two primary hypotheses (H1
# and H2) and two secondary hypotheses (H3 and H4)
# See Figure 1 in Bretz et al. (2011).
hypotheses <- c(0.5, 0.5, 0, 0)
transitions <- rbind(
  c(0, 0, 1, 0),
  c(0, 0, 0, 1),
  c(0, 1, 0, 0),
  c(1, 0, 0, 0)
)
g <- graph_create(hypotheses, transitions)

graph_generate_weights(g)
#>    H1 H2 H3 H4  H1  H2  H3  H4
#> 1   1  1  1  1 0.5 0.5 0.0 0.0
#> 2   1  1  1  0 0.5 0.5 0.0 0.0
#> 3   1  1  0  1 0.5 0.5 0.0 0.0
#> 4   1  1  0  0 0.5 0.5 0.0 0.0
#> 5   1  0  1  1 0.5 0.0 0.0 0.5
#> 6   1  0  1  0 1.0 0.0 0.0 0.0
#> 7   1  0  0  1 0.5 0.0 0.0 0.5
#> 8   1  0  0  0 1.0 0.0 0.0 0.0
#> 9   0  1  1  1 0.0 0.5 0.5 0.0
#> 10  0  1  1  0 0.0 0.5 0.5 0.0
#> 11  0  1  0  1 0.0 1.0 0.0 0.0
#> 12  0  1  0  0 0.0 1.0 0.0 0.0
#> 13  0  0  1  1 0.0 0.0 0.5 0.5
#> 14  0  0  1  0 0.0 0.0 1.0 0.0
#> 15  0  0  0  1 0.0 0.0 0.0 1.0