compute rejection bounds for z-scores of each elementary hypotheses within each intersection hypotheses

1 2 | ```
generateBounds(g, w, cr, al = 0.05, hint = generateWeights(g, w),
upscale = FALSE)
``` |

`g` |
graph defined as a matrix, each element defines how much of the local alpha reserved for the hypothesis corresponding to its row index is passed on to the hypothesis corresponding to its column index |

`w` |
vector of weights, defines how much of the overall alpha is initially reserved for each elementary hypothesis |

`cr` |
correlation matrix if p-values arise from one-sided tests with multivariate normal distributed test statistics for which the correlation is partially known. Unknown values can be set to NA. (See details for more information) |

`al` |
overall alpha level at which the family error is controlled |

`hint` |
if intersection hypotheses weights have already been computed
(output of |

`upscale` |
if |

It is assumed that under the global null hypothesis
*(Φ^{-1}(1-p_1),...,Φ^{-1}(1-p_m))* follow a multivariate normal
distribution with correlation matrix `cr`

where *Φ^{-1}* denotes
the inverse of the standard normal distribution function.

For example, this is the case if *p_1,..., p_m* are the raw p-values
from one-sided z-tests for each of the elementary hypotheses where the
correlation between z-test statistics is generated by an overlap in the
observations (e.g. comparison with a common control, group-sequential
analyses etc.). An application of the transformation *Φ^{-1}(1-p_i)*
to raw p-values from a two-sided test will not in general lead to a
multivariate normal distribution. Partial knowledge of the correlation
matrix is supported. The correlation matrix has to be passed as a numeric
matrix with elements of the form: *correlation[i,i] = 1* for diagonal
elements, *correlation[i,j] = ρ_{ij}*, where *ρ_{ij}* is the
known value of the correlation between *Φ^{-1}(1-p_i)* and
*Φ^{-1}(1-p_j)* or `NA`

if the corresponding correlation is
unknown. For example correlation[1,2]=0 indicates that the first and second
test statistic are uncorrelated, whereas correlation[2,3] = NA means that
the true correlation between statistics two and three is unknown and may
take values between -1 and 1. The correlation has to be specified for
complete blocks (ie.: if cor(i,j), and cor(i,k) for i!=j!=k are specified
then cor(j,k) has to be specified as well) otherwise the corresponding
intersection null hypotheses tests are not uniquely defined and an error is
returned.

The parametric tests in (Bretz et al. (2011)) are defined such that the
tests of intersection null hypotheses always exhaust the full alpha level
even if the sum of weights is strictly smaller than one. This has the
consequence that certain test procedures that do not test each intersection
null hypothesis at the full level alpha may not be implemented (e.g., a
single step Dunnett test). If `upscale`

is set to `FALSE`

(default) the parametric tests are performed at a reduced level alpha of
sum(w) * alpha and p-values adjusted accordingly such that test procedures
with non-exhaustive weighting strategies may be implemented. If set to
`TRUE`

the tests are performed as defined in Equation (3) of (Bretz et
al. (2011)).

Returns a matrix of rejection bounds. Each row corresponds to an intersection hypothesis. The intersection corresponding to each line is given by conversion of the line number into binary (eg. 13 is binary 1101 and corresponds to (H1,H2,H4))

Florian Klinglmueller

Bretz F, Maurer W, Brannath W, Posch M; (2008) - A graphical approach to sequentially rejective multiple testing procedures. - Stat Med - 28/4, 586-604

Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, Willi Maurer, Kornelius Rohmeyer (2011): Graphical approaches for multiple comparison procedures using weighted Bonferroni, Simes or parametric tests. Biometrical Journal 53 (6), pages 894-913, Wiley. http://onlinelibrary.wiley.com/doi/10.1002/bimj.201000239/full

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
## Define some graph as matrix
g <- matrix(c(0,0,1,0,
0,0,0,1,
0,1,0,0,
1,0,0,0), nrow = 4,byrow=TRUE)
## Choose weights
w <- c(.5,.5,0,0)
## Some correlation (upper and lower first diagonal 1/2)
c <- diag(4)
c[1:2,3:4] <- NA
c[3:4,1:2] <- NA
c[1,2] <- 1/2
c[2,1] <- 1/2
c[3,4] <- 1/2
c[4,3] <- 1/2
## Boundaries for correlated test statistics at alpha level .05:
generateBounds(g,w,c,.05)
``` |

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