# generatePvals

### Description

compute adjusted p-values either for the closed test defined by the graph or for each elementary hypotheses within each intersection hypotheses

### Usage

1 2 | ```
generatePvals(g, w, cr, p, adjusted = TRUE, hint = generateWeights(g, w),
upscale = FALSE)
``` |

### Arguments

`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) |

`p` |
vector of observed unadjusted p-values, that belong to
test-statistics with a joint multivariate normal null distribution with
(partially) known correlation matrix |

`adjusted` |
logical, if TRUE (default) adjusted p-values for the closed test are returned, else a matrix of p-values adjusted only for each intersection hypothesis is returned |

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

`upscale` |
if |

### Details

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: *cr[i,i] = 1* for diagonal elements,
*cr[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 cr[1,2]=0
indicates that the first and second test statistic are uncorrelated, whereas
cr[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)).

### Value

If adjusted is set to true returns a vector of adjusted p-values. Any elementary null hypothesis is rejected if its corresponding adjusted p-value is below the predetermined alpha level. For adjusted set to false a matrix with p-values adjusted only within each intersection hypotheses is returned. 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)). If any adjusted p-value within a given line falls below alpha, then the corresponding intersection hypotheses can be rejected.

### Author(s)

Florian Klinglmueller

### References

Bretz F, Maurer W, Brannath W, Posch M; (2008) - A graphical approach to sequentially rejective multiple testing procedures. - Stat Med - 28/4, 586-604 Bretz F, Posch M, Glimm E, Klinglmueller F, Maurer W, Rohmeyer K; (2011) - Graphical approaches for multiple endpoint problems using weighted Bonferroni, Simes or parametric tests - to appear

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
## 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
## p-values as Section 3 of Bretz et al. (2011),
p <- c(0.0121,0.0337,0.0084,0.0160)
## Boundaries for correlated test statistics at alpha level .05:
generatePvals(g,w,c,p)
g <- Entangled2Maurer2012()
generatePvals(g=g, cr=diag(5), p=rep(0.1,5))
``` |