explore_q: Exploration of the Information Weights

In informativeSCI: Informative Simultaneous Confidence Intervals

Exploration of the Information Weights

Description

The function calculates various statistical quantities giving some information about the behavior of informative lower SCI-bounds (informSCI) and its induced test for a given graphical test procedure with m hypotheses. The simulation is done for different information weights of the hypotheses. These statistical quantities are intended to be used for determining information weights that represent the best possible trade-off between the number of rejections and the expected size of the informative lower informative SCI-bounds. The statistical quantities can also be calculated for the graphical test and the related compatible lower SCI-bounds, which allows a comparison between the two strategies.

Usage

explore_q(
  gMCP = NULL,
  g = NULL,
  weights = NULL,
  trueParam,
  sigma = NULL,
  qFixed = matrix(0, 0, 2),
  mu_0 = 0,
  alpha = 0.05,
  addHyp = matrix(0, 0, 3),
  allRej = NULL,
  atLeastOneRej = NULL,
  qGrid = NULL,
  qInterval = c(0, 1),
  qStepSize = 1/10,
  numSim = 1000,
  sampleSizes = NULL,
  sampleSizeControl = NULL,
  varObs = NULL,
  exploreGraph = TRUE,
  eps = 1/10^5,
  timesSmallerEps = 3,
  maxIterSCI = 1000,
  maxIterBisec = 1000,
  tolBisec = 1/10^3
)

Arguments

gMCP	An object of class graphMCP indicating the underlying graphical test.
g	Numeric square matrix of transition weights for the graphical test with m rows and m columns. The i-th row of the entered matrix defines the arrows starting from the i-th hypothesis. Each entry has to be between 0 and 1 and each row must sum to a number less than or equal to 1. The diagonal elements must be zero. Entering g and weights can be used as the input as an alternative to specifying gMCP.
weights	Numeric vector of weights of dimension m. It defines the initial proportion of significance level which is assigned to each null hypothesis. Entering g and weights can be used as the input as an alternative to specifying gMCP.
trueParam	A numeric vector of dimension m defining the assumed true parameters \vartheta_i, 1\leq i\leq m.
sigma	A covariance matrix of dimension m\times m. sigma indicates the covariance matrix of the point estimators for the parameter of interest. Can be missing in the case of a many-to-one comparison. Then, sampleSizes, sampleSizeControl and varObs must be specified.
qFixed	A numeric matrix with l rows and 2 columns, where l is an integer between 0 and m. The matrix describes the fixed information weights of the simulation. The first column indicates the indices of the hypothesis for which the information weight should be fixed during the simulation (i.e. the entries of the first column must be natural numbers between 1 and m). The second column contains the fixed values of their respective fixed information weights (i.e. the entries of the second column must be between 0 and 1 (inclusive)). It is permissible for all information weights to be fixed (i.e. qFixed has m rows) or none to be fixed (i.e. qFixed has 0 rows).
mu_0	A numeric vector of dimension 1 or m defining the bounds of the null hypotheses of the underlying graphical test. If mu_0 has dimension 1, the same value is used for each null hypothesis.
alpha	A numeric defining the overall significance level for the graphical test (i.e. SCIs will have coverage probability of at least 1-alpha. The parameter must be strictly between 0 and 1.
addHyp	A numeric matrix with k rows and 3 columns (k can be 0) The matrix indicates for which (further) shifted hypotheses the rejection probability is to be calculated. Every row describes one hypothesis. The first entry is a natural number greater than m identifying the hypothesis. The second entry of each row is the index of the corresponding parameter of interest. The third entry is the right border of the hypothesis.
allRej	A list of vectors. Each vector in the list contains the indices of subfamilies of the family of all hypotheses, including the addHyp. The indices of the null hypotheses of the underlying graph range from 1 to m. The indices for addHyp are given by the first column of addHyp. For each such family, the probability of rejecting all hypotheses at the same time is calculated.
atLeastOneRej	A list of vectors. Each vector in the list contains the indices of subfamilies of the family of all hypotheses, including the addHyp. The indices of the null hypotheses of the underlying graph range from 1 to m. The indices for addHyp are given by the first column of addHyp. For each such family, the probability of rejecting at least one hypothesis is calculated.
qGrid	A numeric vector indicating the values of the non-fixed information weights for the simulation. The entries must be between 0 and 1 (inclusive).
qInterval	A numeric vector of dimension 2 specifying the minimum and maximum values allowed for the varying information weights. qInterval and qStepsize can be used as the input as an alternative to specifying qGrid. If all are entered, qGrid is used and qInterval and qStepSize are ignored.
qStepSize	A positive numeric defining the step size for the varying information weights. qInterval and qStepsize can be used as the input as an alternative to specifying qGrid.
numSim	A natural number indicating how many simulations are to be performed.
sampleSizes	A numeric vector indicating the sample size of each non-control group, in the many-to-one case. Not required if sigma is entered.
sampleSizeControl	A numeric indicating the sample size of the control group, in the many-to-one case. Not required if sigma is entered.
varObs	A positive numeric indicating the variance of the individual observations, in the many-to-one case. Not required if sigma is entered.
exploreGraph	A boolean indicating whether the simulation should be also done for the underlying graphical test and the corresponding compatible lower SCI-bounds.
eps	A numeric for the informSCI-algorithm indicating the desired strict upper bound on the Chebyshev distance between two successive calculated approximations (the Chebyshev distance is induced by the maximum norm).
timesSmallerEps	A positive integer for the informSCI-algorithm indicating how many times the Chebyshev distance of two successive calculated approximations should be less than eps in succession. Here we use the convention -\infty- (-\infty):=0.
maxIterSCI	Maximum number of iterations for determining the lower informative SCI-bounds.
maxIterBisec	Maximum number of iterations of the bisection method which is used during the informSCI-algorithm for finding roots.
tolBisec	A non-negative numeric indicating the error tolerance of the bisection method which is used for finding roots in the informSCI-algorithm.

Details

It is assumed that there are m parameters of interest \vartheta_1,\dots,\vartheta_m. For each parameter there is a null hypothesis defined as H_i^{{\mu_0}_i}:\vartheta_i\leq{\mu_0}_i. The bounds {\mu_0} correspond to mu_0. The underlying graphical test (specified by gMCP or g and weights) is based on these hypotheses.

The function simulates estimations of point estimators for the parameter of interest \vartheta_1,\dots, \vartheta_m. The estimators follow a multivariate normal distribution with mean trueParam and covariance matrix sigma. The function repeatedly calls the informSCI-function.

The algorithm only optimizes for a single parameter, which is used for all non-fixed information weights. The parameter is chosen from a grid specified by qInterval and qStepsize. The constructed grid contains all values which are between qInterval[1] and qInterval[2] and can be written as qInterval[1]+k\cdotqStepsize where k is a natural number. Alternatively, the parameter is chosen directly from qGrid.

Value

The function returns a list containing several statistical quantities to use for the informative lower SCI-bounds to find the best possible trade-off between the number of rejections and the expected size of the informative lower SCI-bounds. In the case that exploreGraph=TRUE, the returned list also contains the same quantities for the (original) graphical test and related compatible bounds. This allows a comparison.

• rejecHyp: A matrix containing for several hypotheses the empirical rejection probability by the informative confidence bounds. The first m rows correspond to the hypotheses of the graph. The other rows correspond to the hypotheses specified by addHyp. Each row indicates the rejection probability for different values of the information weights.

• meanISCI: A matrix containing in its columns the empirical mean of the lower informative confidence bounds for different information weights. Only the lower bounds which are greater than -Inf are used for the empirical mean.

• impISCI: A matrix containing in its columns the empirical average distance between the lower informative confidence bounds and mu_0 for different information weights. Only the lower bounds which are greater than -Inf are used for the empirical average distance.

• biasISCI: A matrix containing in its columns the empirical average distance between the lower informative confidence bounds and the true parameters trueParam for different information weights. Only the lower bounds which are greater than -Inf are used for the empirical average distance.

• numISCIfinite: A matrix containing in its columns how many times the lower informative confidence bounds were each greater than -Inf for different information weights.

• rejecAllHyp: A matrix containing in its columns for each family from allRej the empirical probability of rejecting all of the hypotheses from the family with the induced test at the same time for different information weights.

• rejecAtLeastHyp: A matrix containing in its columns for each family from atLeastOneRej the empirical probability of rejecting at least one of the hypotheses from the family with the induced test for different information weights.

If exploreGraph=TRUE:

• rejecHypGraph: A vector containing for each of the null hypotheses of the graph and of the additional hypotheses (specified by addHyp) its empirical rejection probability by the original graph.

• meanCSCI: A vector containing, for each parameter \vartheta_i, 1\leq i\leq m the empirical mean of the lower compatible confidence bounds. Only the lower bounds which are greater than -Inf are used for the empirical mean.

• impCSCI: A vector containing, for each parameter, the empirical average distance between the lower compatible confidence bounds and mu_0. Only the lower bounds which are greater than -Inf are used.

• biasCSCI: A vector containing, for each parameter, the empirical average distance between the lower compatible confidence bounds and the true parameters trueParam. Only the lower bounds which are greater than -Inf are used.

• numCSCIfinite: A vector containing, for each parameter, how many times the compatible lower confidence bounds were each greater than -Inf.

• rejecAllHypCSCI: A vector containing, for each family from allRej, the empirical probability of rejecting all of the hypotheses from the family with the (original) graphical test.

• rejecAtLeastHypCSCI: A vector containing, for each family from atLeastOneRej, the empirical probability of rejecting at least one of the hypotheses from the family with the (original) graphical test.

References

S. Schmidt, W. Brannath: Informative simultaneous confidence intervals for the fallback procedure. Biometrical Journal 57.4 (2015), pp. 712–719.

See Also

informSCI gMCP simConfint

Examples

explore_q(gMCP=BonferroniHolm(3), trueParam=c(1.5,1,0.2),
sigma=diag(3)*0.2, qFixed=matrix(c(2,3,0.3,0.3),2,2), mu_0=c(-0.5,0,0),
addHyp=matrix(c(4,1,0),1,3),allRej =list(c(1,2), c(4,2)), 
atLeastOneRej=list(c(2,3)),numSim=100)
explore_q(g=matrix(c(0,0,1,0),2,2), weights=c(1,0), trueParam=c(0.5,2), 
mu_0=c(-1,0), alpha=0.025, qGrid=c(1/10*c(1:10),c(0.97,0.98,0.99)), 
numSim=100, sampleSizes=c(89,95), sampleSizeControl=77, varObs=10)

informativeSCI documentation built on June 22, 2024, 10:01 a.m.