structured Holm {cherry}

Testing of hypotheses, forming a DAG, by using a variant of Holm's method.

Description

Tests all hypotheses in a given DAG while controlling the FWER, using a user-specified local test.

Usage

 structuredHolm (DAGstructure, test, alpha_max = 0.05, isadjusted = FALSE, 
optimization = "none", pvalues = NULL, verbose = FALSE)

Arguments

DAGstructure

DAGstructure object, as returned by the function construct.

test

A function that performs the local test. The function should have a set as input and return a p-value.

alpha_max

The significance level of the test procedure.

isadjusted

If set to TRUE, adjusted p-values will be calculated. Otherwise, the p-values of all rejected hypotheses will equal alpha_max.

optimization

Can be, in ascending order of accuracy and computational costs: "none", "LP" (linear programming) or "ILP" (integer linear programming).

pvalues

Optional (in case of stored p-values): a vector in which the raw p-values of the exact sets as found in the DAGstructure argument are stored (in the same order). If the test function is provided, this argument is not necessary.

verbose

If set to TRUE, while running the method, a counter will indicate how many hypotheses are already rejected.

Details

The function structuredHolm tests all possible hypotheses within a given DAG structure, while controlling the familywise error rate.

Value

The function structuredHolm returns an object of class DAG.

Author(s)

Rosa Meijer: [e-mail]

References

Meijer and Goeman (2015) Briefings in Bioinformatics, submitted.

See Also

DAG, DAGstructure, construct, DAGpick.

Examples

  
  #Generate data, where the response Y is associated with two (out of 4) covariates
  set.seed(1)
  n=100
  p=4
  X <- matrix(rnorm(n*p),n,p)
  beta <- c(0,0.5,0.5,0)
  Y <- X %*% beta + rnorm(n)

  # Let us assume we have the following sets that we want to test: 
  sets <- list(c(1,2,3,4), c(1,2), c(2,3,4), c(2,3), 1, 2, 3, 4)
  names(sets) <- c(1234, 12, 234, 23, 1, 2, 3, 4)

  # Start by making the corresponding graph structure
  struct <- construct(sets)
  
  # Check whether the DAG has toway logical relations: 
  istwoway(struct)
  
  # Define the local test to be used in the closed testing procedure. 
  # This test expects a set as input. 
  mytest <- function(set)
  { 
    X <- X[,set,drop=FALSE]
    lm.out <- lm(Y ~ X)
    x <- summary(lm.out)
    return(pf(x$fstatistic[1],x$fstatistic[2],x$fstatistic[3],lower.tail=FALSE))  
  }

  # Perform the structuredHolm procedure.
  DAG <- structuredHolm(struct, mytest, isadjusted=TRUE)
  summary(DAG)
  
  # What are the smallest sets that are found to be significant? 
  implications(DAG)
  
  # What is the adjusted p-value of the null-hypothesis corresponding to the fourth set, 
  # which is set c(2,3)? 
  # To look up the pvalue, the function uses the index or name of the set 
  # in the list of sets stored in the DAGstructure. 
  # (Note that, if there were duplicate sets in the original list, this index can be different from 
  # the one in the original list given to \code{construct})
  pvalue(DAG,4)
  pvalue(DAG, "23") #as above, but while using names
  
  # How many of the elementary hypotheses (the last 4 sets) have to be false 
  # with probability 1-alpha? 
  # Sets (don't have to be elementary hypotheses in general) must be specified 
  # by their index or name. 
  DAGpick(DAG, 5:8)
  DAGpick(DAG, c("1","2","3","4")) #as above, but while using names