# generateTest

### Description

generates a test function for the multiple comparison procedure with correlated test statistics defined by a graph

### Usage

1 | ```
generateTest(g, w, cr, al, 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) |

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

`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 upscale 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. If set to
`TRUE`

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

### Value

Returns a function that will take a vector of z-scores to which the test will be applied. This function in turn will return a boolean vector with elements false if the particular elementary hypothesis can not be rejected and true otherwise.

### 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 | ```
## 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
## Test function for further use:
myTest <- generateTest(g,w,c,.05)
myTest(c(3,2,1,2))
``` |

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