gMCP: Graph based Multiple Comparison Procedures
In gMCP: Graph Based Multiple Comparison Procedures
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
Performs a graph based multiple test procedure for a given graph and
unadjusted p-values.
Usage
1
2
3
4
Arguments
graph
A graph of class graphMCP.
pvalues
A numeric vector specifying the p-values for the graph based
MCP. Note the assumptions in the details section for the parametric tests,
when a correlation is specified.
test
Should be either "Bonferroni", "Simes" or "parametric".
If not specified by default the Bonferroni-based test procedure is used if no
correlation is specified or the algorithm from Bretz et al. 2011 if a
correlation is specified. If test is set to "Simes" the weighted
Simes test will be performed for each subset of hypotheses.
correlation
Optional correlation matrix. If the weighted Simes test
is performed, it is checked whether type I error rate can be ensured and a
warning is given if this is not the case. For parametric tests the p-values
must arise from one-sided tests with multivariate normal distributed test
statistics for which the correlation is (partially) known. In that case a
weighted parametric closed test is performed (also see
generatePvals). Unknown values can be set to NA. (See details
for more information)
alpha
A numeric specifying the maximal allowed type one error rate.
approxEps
A boolean specifying whether epsilon values should be
substituted with the value given in the parameter eps.
eps
A numeric scalar specifying a value for epsilon edges.
...
Test specific arguments can be given here.
upscale
Logical. If upscale=FALSE then for each intersection
of hypotheses (i.e. each subgraph) a weighted test is performed at the
possibly reduced level alpha of sum(w)*alpha,
where sum(w) is the sum of all node weights in this subset.
If upscale=TRUE all weights are upscaled, so that sum(w)=1.
useC
Logical scalar. If TRUE neither adjusted p-values nor
intermediate graphs are returned, but the calculation is sped up by using
code written in C. THIS CODE IS NOT FOR PRODUCTIVE USE YET! If approxEps is
FALSE and the graph contains epsilon edges, a warning is thrown and
useC will be ignored.
verbose
Logical scalar. If TRUE verbose output is generated
during sequentially rejection steps.
keepWeights
Logical scalar. If FALSE the weight of a node
without outgoing edges is set to 0 if it is removed. Otherwise it keeps its
weight.
adjPValues
Logical scalar. If FALSE no adjusted p-values will
be calculated. Especially for the weighted Simes test this will result in
significantly less calculations in most cases.
Details
For the Bonferroni procedure the p-values can arise from any statistical
test, but if you improve the test by specifying a correlation matrix, the
following assumptions apply:
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 correlation 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,j') for i!=j!=j' are specified
then cor(j,j') has to be specified as well) otherwise the corresponding
intersection null hypotheses tests are not uniquely defined and an error is
returned.
For further details see the given references.
Value
An object of class gMCPResult, more specifically a list with
elements
graphslist of graphs
pvaluesp-values
rejectedlogical whether hyptheses could be rejected
adjPValuesadjusted p-values
Author(s)
Kornelius Rohmeyer rohmeyer@small-projects.de
References
Frank Bretz, Willi Maurer, Werner Brannath, Martin Posch: A
graphical approach to sequentially rejective multiple test procedures.
Statistics in Medicine 2009 vol. 28 issue 4 page 586-604.
http://www.meduniwien.ac.at/fwf_adaptive/papers/bretz_2009_22.pdf
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.
Strassburger K., Bretz F.: Compatible simultaneous lower confidence bounds
for the Holm procedure and other Bonferroni based closed tests. Statistics
in Medicine 2008; 27:4914-4927.
Hommel G., Bretz F., Maurer W.: Powerful short-cuts for multiple testing
procedures with special reference to gatekeeping strategies. Statistics in
Medicine 2007; 26:4063-4073.
Guilbaud O.: Simultaneous confidence regions corresponding to Holm's
stepdown procedure and other closed-testing procedures. Biometrical Journal
2008; 50:678-692.
See Also
Examples
1
2
3
4
5
6
7
8
9
10
11
g <- BonferroniHolm(5)
gMCP(g, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7))
# Simple Bonferroni with empty graph:
g2 <- matrix2graph(matrix(0, nrow=5, ncol=5))
gMCP(g2, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7))
# With 'upscale=TRUE' equal to BonferroniHolm:
gMCP(g2, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7), upscale=TRUE)
# Entangled graphs:
g3 <- Entangled2Maurer2012()
gMCP(g3, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7), correlation=diag(5))
Example output
OpenJDK 64-Bit Server VM warning: Can't detect primordial thread stack location - find_vma failed
gMCP-Result
Initial graph:
A graphMCP graph
H1 (weight=0.2)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
Edges:
H1 -( 0.25 )-> H2
H1 -( 0.25 )-> H3
H1 -( 0.25 )-> H4
H1 -( 0.25 )-> H5
H2 -( 0.25 )-> H1
H2 -( 0.25 )-> H3
H2 -( 0.25 )-> H4
H2 -( 0.25 )-> H5
H3 -( 0.25 )-> H1
H3 -( 0.25 )-> H2
H3 -( 0.25 )-> H4
H3 -( 0.25 )-> H5
H4 -( 0.25 )-> H1
H4 -( 0.25 )-> H2
H4 -( 0.25 )-> H3
H4 -( 0.25 )-> H5
H5 -( 0.25 )-> H1
H5 -( 0.25 )-> H2
H5 -( 0.25 )-> H3
H5 -( 0.25 )-> H4
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.05 0.08 0.12 0.12 0.70
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE FALSE FALSE FALSE FALSE
Final graph after 1 steps:
A graphMCP graph
H1 (rejected, weight=0)
H2 (weight=0.25)
H3 (weight=0.25)
H4 (weight=0.25)
H5 (weight=0.25)
Edges:
H2 -( 0.333333333333333 )-> H3
H2 -( 0.333333333333333 )-> H4
H2 -( 0.333333333333333 )-> H5
H3 -( 0.333333333333333 )-> H2
H3 -( 0.333333333333333 )-> H4
H3 -( 0.333333333333333 )-> H5
H4 -( 0.333333333333333 )-> H2
H4 -( 0.333333333333333 )-> H3
H4 -( 0.333333333333333 )-> H5
H5 -( 0.333333333333333 )-> H2
H5 -( 0.333333333333333 )-> H3
H5 -( 0.333333333333333 )-> H4
gMCP-Result
Initial graph:
A graphMCP graph
H1 (weight=0.2)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
No edges.
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.05 0.10 0.20 0.20 1.00
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE FALSE FALSE FALSE FALSE
Final graph after 1 steps:
A graphMCP graph
Sum of weight: 0.8
H1 (rejected, weight=0)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
No edges.
gMCP-Result
Initial graph:
A graphMCP graph
H1 (weight=0.2)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
No edges.
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.05 0.08 0.12 0.12 0.70
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE FALSE FALSE FALSE FALSE
Final graph after 1 steps:
A graphMCP graph
H1 (rejected, weight=0)
H2 (weight=0.25)
H3 (weight=0.25)
H4 (weight=0.25)
H5 (weight=0.25)
No edges.
gMCP-Result
Initial graph:
An object of class "entangledMCP"
Slot "subgraphs":
[[1]]
A graphMCP graph
H1 (weight=1)
H2 (weight=0)
H3 (weight=0)
H4 (weight=0)
H5 (weight=0)
Edges:
H1 -( 1 )-> H3
H2 -( 1 )-> H5
H3 -( 1 )-> H4
H4 -( 1 )-> H2
[[2]]
A graphMCP graph
H1 (weight=0)
H2 (weight=1)
H3 (weight=0)
H4 (weight=0)
H5 (weight=0)
Edges:
H1 -( 1 )-> H4
H2 -( 1 )-> H3
H3 -( 1 )-> H5
H5 -( 1 )-> H1
Slot "weights":
[1] 0.5 0.5
Slot "graphAttr":
list()
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.0199 0.0396 0.0400 0.0784 0.7000
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE TRUE TRUE FALSE FALSE
gMCP documentation built on May 2, 2019, 6:07 p.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
Performs a graph based multiple test procedure for a given graph and unadjusted p-values.
Usage
1 2 3 4 |
Arguments
graph |
A graph of class |
pvalues |
A numeric vector specifying the p-values for the graph based MCP. Note the assumptions in the details section for the parametric tests, when a correlation is specified. |
test |
Should be either |
correlation |
Optional correlation matrix. If the weighted Simes test
is performed, it is checked whether type I error rate can be ensured and a
warning is given if this is not the case. For parametric tests the p-values
must arise from one-sided tests with multivariate normal distributed test
statistics for which the correlation is (partially) known. In that case a
weighted parametric closed test is performed (also see
|
alpha |
A numeric specifying the maximal allowed type one error rate. |
approxEps |
A boolean specifying whether epsilon values should be
substituted with the value given in the parameter |
eps |
A numeric scalar specifying a value for epsilon edges. |
... |
Test specific arguments can be given here. |
upscale |
Logical. If |
useC |
Logical scalar. If |
verbose |
Logical scalar. If |
keepWeights |
Logical scalar. If |
adjPValues |
Logical scalar. If |
Details
For the Bonferroni procedure the p-values can arise from any statistical test, but if you improve the test by specifying a correlation matrix, the following assumptions apply:
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 correlation 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,j') for i!=j!=j' are specified
then cor(j,j') has to be specified as well) otherwise the corresponding
intersection null hypotheses tests are not uniquely defined and an error is
returned.
For further details see the given references.
Value
An object of class gMCPResult, more specifically a list with
elements
graphslist of graphs
pvaluesp-values
rejectedlogical whether hyptheses could be rejected
adjPValuesadjusted p-values
Author(s)
Kornelius Rohmeyer rohmeyer@small-projects.de
References
Frank Bretz, Willi Maurer, Werner Brannath, Martin Posch: A graphical approach to sequentially rejective multiple test procedures. Statistics in Medicine 2009 vol. 28 issue 4 page 586-604. http://www.meduniwien.ac.at/fwf_adaptive/papers/bretz_2009_22.pdf
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.
Strassburger K., Bretz F.: Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni based closed tests. Statistics in Medicine 2008; 27:4914-4927.
Hommel G., Bretz F., Maurer W.: Powerful short-cuts for multiple testing procedures with special reference to gatekeeping strategies. Statistics in Medicine 2007; 26:4063-4073.
Guilbaud O.: Simultaneous confidence regions corresponding to Holm's stepdown procedure and other closed-testing procedures. Biometrical Journal 2008; 50:678-692.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 | g <- BonferroniHolm(5)
gMCP(g, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7))
# Simple Bonferroni with empty graph:
g2 <- matrix2graph(matrix(0, nrow=5, ncol=5))
gMCP(g2, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7))
# With 'upscale=TRUE' equal to BonferroniHolm:
gMCP(g2, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7), upscale=TRUE)
# Entangled graphs:
g3 <- Entangled2Maurer2012()
gMCP(g3, pvalues=c(0.01, 0.02, 0.04, 0.04, 0.7), correlation=diag(5))
|
Example output
OpenJDK 64-Bit Server VM warning: Can't detect primordial thread stack location - find_vma failed
gMCP-Result
Initial graph:
A graphMCP graph
H1 (weight=0.2)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
Edges:
H1 -( 0.25 )-> H2
H1 -( 0.25 )-> H3
H1 -( 0.25 )-> H4
H1 -( 0.25 )-> H5
H2 -( 0.25 )-> H1
H2 -( 0.25 )-> H3
H2 -( 0.25 )-> H4
H2 -( 0.25 )-> H5
H3 -( 0.25 )-> H1
H3 -( 0.25 )-> H2
H3 -( 0.25 )-> H4
H3 -( 0.25 )-> H5
H4 -( 0.25 )-> H1
H4 -( 0.25 )-> H2
H4 -( 0.25 )-> H3
H4 -( 0.25 )-> H5
H5 -( 0.25 )-> H1
H5 -( 0.25 )-> H2
H5 -( 0.25 )-> H3
H5 -( 0.25 )-> H4
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.05 0.08 0.12 0.12 0.70
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE FALSE FALSE FALSE FALSE
Final graph after 1 steps:
A graphMCP graph
H1 (rejected, weight=0)
H2 (weight=0.25)
H3 (weight=0.25)
H4 (weight=0.25)
H5 (weight=0.25)
Edges:
H2 -( 0.333333333333333 )-> H3
H2 -( 0.333333333333333 )-> H4
H2 -( 0.333333333333333 )-> H5
H3 -( 0.333333333333333 )-> H2
H3 -( 0.333333333333333 )-> H4
H3 -( 0.333333333333333 )-> H5
H4 -( 0.333333333333333 )-> H2
H4 -( 0.333333333333333 )-> H3
H4 -( 0.333333333333333 )-> H5
H5 -( 0.333333333333333 )-> H2
H5 -( 0.333333333333333 )-> H3
H5 -( 0.333333333333333 )-> H4
gMCP-Result
Initial graph:
A graphMCP graph
H1 (weight=0.2)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
No edges.
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.05 0.10 0.20 0.20 1.00
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE FALSE FALSE FALSE FALSE
Final graph after 1 steps:
A graphMCP graph
Sum of weight: 0.8
H1 (rejected, weight=0)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
No edges.
gMCP-Result
Initial graph:
A graphMCP graph
H1 (weight=0.2)
H2 (weight=0.2)
H3 (weight=0.2)
H4 (weight=0.2)
H5 (weight=0.2)
No edges.
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.05 0.08 0.12 0.12 0.70
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE FALSE FALSE FALSE FALSE
Final graph after 1 steps:
A graphMCP graph
H1 (rejected, weight=0)
H2 (weight=0.25)
H3 (weight=0.25)
H4 (weight=0.25)
H5 (weight=0.25)
No edges.
gMCP-Result
Initial graph:
An object of class "entangledMCP"
Slot "subgraphs":
[[1]]
A graphMCP graph
H1 (weight=1)
H2 (weight=0)
H3 (weight=0)
H4 (weight=0)
H5 (weight=0)
Edges:
H1 -( 1 )-> H3
H2 -( 1 )-> H5
H3 -( 1 )-> H4
H4 -( 1 )-> H2
[[2]]
A graphMCP graph
H1 (weight=0)
H2 (weight=1)
H3 (weight=0)
H4 (weight=0)
H5 (weight=0)
Edges:
H1 -( 1 )-> H4
H2 -( 1 )-> H3
H3 -( 1 )-> H5
H5 -( 1 )-> H1
Slot "weights":
[1] 0.5 0.5
Slot "graphAttr":
list()
P-values:
H1 H2 H3 H4 H5
0.01 0.02 0.04 0.04 0.70
Adjusted p-values:
H1 H2 H3 H4 H5
0.0199 0.0396 0.0400 0.0784 0.7000
Alpha: 0.05
Hypothesis rejected:
H1 H2 H3 H4 H5
TRUE TRUE TRUE FALSE FALSE
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