MultiGroupSequential: Group-Sequential Procedures with Multiple Hypotheses

Description

It is often challenging to strongly control the family-wise type-1 error rate in the group-sequential trials with multiple endpoints (hypotheses). The inflation of type-1 error rate comes from two sources (S1) repeated testing individual hypothesis and (S2) simultaneous testing multiple hypotheses. The 'MultiGroupSequential' package is intended to help researchers to tackle this challenge. The procedures provided include the sequential procedures described in Luo and Quan (2023) doi:10.1080/19466315.2023.2191989 and the graphical procedure proposed by Maurer and Bretz (2013) doi:10.1080/19466315.2013.807748. Luo and Quan (2013) describes three procedures, and the functions to implement these procedures are (1) seqgspgx() implements a sequential graphical procedure based on the group-sequential p-values; (2) seqgsphh() implements a sequential Hochberg/Hommel procedure based on the group-sequential p-values; and (3) seqqvalhh() implements a sequential Hochberg/Hommel procedure based on the q-values. In addition, seqmbgx() implements the sequential graphical procedure described in Maurer and Bretz (2013).

Author(s)

Maintainer: Xiaodong Luo Xiaodong.Luo@sanofi.com

Other contributors:

• Hui Quan [contributor]

• Sanofi [copyright holder]

References

Xiaodong Luo & Hui Quan (2023) Some Multiplicity Adjustment Procedures for Clinical Trials with Sequential Design and Multiple Endpoints, Statistics in Biopharmaceutical Research, DOI: 10.1080/19466315.2023.2191989.

Willi Maurer & Frank Bretz (2013) Multiple Testing in Group Sequential Trials Using Graphical Approaches, Statistics in Biopharmaceutical Research, 5:4, 311-320, DOI: 10.1080/19466315.2013.807748

Calculate group-sequential p-values for one hypothesis

Description

calgsp1() calculates the group-sequential p-values for one hypothesis.

Usage

calgsp1(
  sx = qnorm(1 - c(0.03, 0.04, 0.01)),
  scrit = qnorm(1 - c(0.01, 0.02, 0.025)),
  salpha = c(0.01, 0.02, 0.025),
  smatrix = diag(3),
  sided = 1
)

Arguments

Value

List containing the group-sequential p-values.

Author(s)

Xiaodong Luo

Examples

calgsp1(
  sx = qnorm(1 - c(0.03, 0.04, 0.01)),
  scrit = qnorm(1 - c(0.01, 0.02, 0.025)),
  salpha = c(0.01, 0.02, 0.025),
  smatrix = diag(3),
  sided = 1
)

Calculate group-sequential p-values for multiple hypotheses

Description

calgspn() calculates the group-sequential p-values for multiple hypotheses.

Usage

calgspn(
  xm = qnorm(matrix(rep(c(0.03, 0.04, 0.01), times = 2), ncol = 3, nrow = 2)),
  alpham = matrix(rep(c(0.02, 0.03, 0.05), each = 2), ncol = 3, nrow = 2),
  critm = matrix(rep(qnorm(c(0.02, 0.03, 0.05)), each = 2), ncol = 3, nrow = 2),
  matrix.list = list(diag(3), diag(3)),
  sided = rep(-1, 2)
)

Arguments

Value

List with element pm containing the group-sequential p-values.

Author(s)

Xiaodong Luo

Examples

calgspn(
  xm = qnorm(matrix(rep(c(0.03,0.04,0.01),times=2),ncol=3,nrow=2)),
  alpham = matrix(rep(c(0.02,0.03,0.05),each=2),ncol=3,nrow=2),
  critm = matrix(rep(qnorm(c(0.02,0.03,0.05)),each=2),ncol=3,nrow=2),
  matrix.list = list(diag(3),diag(3)),
  sided = rep(-1,2)
)

Check critical values

Description

checkcrit() is a helper function that checks if the critical values are valid.

Usage

checkcrit(
  scrit = qnorm(c(0.01, 0.02, 0.025)),
  salpha = c(0.01, 0.02, 0.025),
  smatrix = diag(3),
  sided = 1
)

Arguments

Value

List with:

• crit.value: Critical values

• salpha: Cumulative alpha levels passed to salpha argument

Author(s)

Xiaodong Luo

Examples

checkcrit(
  scrit = qnorm(c(0.01, 0.02, 0.025)),
  salpha = c(0.01, 0.02, 0.025),
  smatrix = diag(3),
  sided = 1
)

Calculate critical values

Description

findcirt() calculates the critical values in the general correlation matrix

Usage

findcrit(
  salpha = c(0.01, 0.02, 0.025),
  smatrix = diag(3),
  sided = 1,
  tol = 1e-10,
  alpha.tol = 1e-11
)

Arguments

Value

List with element crit.value containing the obtained critical values.

Author(s)

Xiaodong Luo

Examples

findcrit(
  salpha = c(0.01, 0.02, 0.025),
  smatrix = diag(3),
  sided = 1,
  tol = 1e-10,
  alpha.tol = 1e-11
)

Graphical procedure

Description

graphical() performs graphical procedure to test multiple hypotheses

Usage

graphical(
  p = c(0.01, 0.04, 0.03),
  W = c(0.5, 0.25, 0.25),
  G = rbind(c(0, 1, 0), c(0, 0, 1), c(1, 0, 0)),
  alpha = 0.05
)

Arguments

Value

A list with a single element containing a vector indicating whether hypotheses are rejected (1) or not (0).

Author(s)

Kaiyuan Hua, Xiaodong Luo

Examples

graphical(p = c(0.02, 0.03, 0.01))

Hochberg procedure

Description

hochbergd() computes the Hochberg procedure with different alphas for different endpoints.

Usage

hochbergd(pvalues, alpha, epsilon = 1e-10, precision = 10)

Arguments

Value

List with element named decisions containing an index of rejected hypotheses.

Author(s)

Xiaodong Luo

Examples

hochbergd(
  pvalues = runif(5),
  alpha = seq(0.01, 0.025, len = 5),
  epsilon = 1.0e-10,
  precision = 10
)

Hommel procedure

Description

hommeld() implement the Hommel procedure with different alphas for different endpoints.

Usage

hommeld(pvalues, alpha, epsilon = 1e-10, precision = 10)

Arguments

Details

The package hommel can handle Hommel procedure with different alpha's for different endpoints, the function hommeld() is just a wrapper of hommel::hommel().

Value

List with element named decisions containing an index of rejected hypotheses.

Author(s)

Xiaodong Luo

Examples

hommeld(
  pvalues = runif(5),
  alpha = seq(0.01, 0.025, len = 5),
  epsilon = 1.0e-10,
  precision = 10
)

Transform information fractions into correlation matrix

Description

inftocor() transforms information (fractions) into correlation matrix.

Usage

inftocor(ir = c(0.2, 0.5, 1))

Arguments

Value

List with an element named cor for the correlation matrix.

Author(s)

Xiaodong Luo

Examples

inftocor(ir = c(0.2, 0.5, 1.0))

Sequential graphical procedure based on group-sequential p-values

Description

seqgspgx() implements the sequential graphical procedure for multiple hypotheses based on group-sequential p-values.

Usage

seqgspgx(
  pm = matrix(rep(c(0.03, 0.04, 0.01), times = 2), ncol = 3, nrow = 2),
  alpha = 0.025,
  W = c(0.6, 0.4),
  G = rbind(c(0, 1), c(1, 0))
)

Arguments

Value

List with elements

• rejected: the index set of rejected hypotheses

• decisionsm: rejection decision for each endpoint (row) at each timepoint (column)

• cumdecisionsm: cumulative rejection decision for each endpoint (row) at each timepoint (column)

Author(s)

Xiaodong Luo

Examples

seqgspgx(
  pm = matrix(rep(c(0.03, 0.04, 0.01), times = 2), ncol = 3, nrow = 2),
  alpha = 0.025,
  W = c(0.6, 0.4),
  G = rbind(c(0, 1), c(1, 0))
)

Sequential generalized Hochberg and Hommel procedures based on group-sequential p-values

Description

seqgsphh() implements the sequential Generalized Hochberg and Hommel procedures based on group-sequential p-values.

Usage

seqgsphh(
  pm = matrix(rep(c(0.03, 0.04, 0.01), times = 2), ncol = 3, nrow = 2),
  alpha = 0.025,
  epsilon = 1e-10,
  precision = 10,
  method = "Hochberg"
)

Arguments

Value

List with elements

• rejected: the index set of rejected hypotheses

• decisionsm: rejection decision for each endpoint (row) at each timepoint (column)

• cumdecisionsm: cumulative rejection decision for each endpoint (row) at each timepoint (column)

Author(s)

Xiaodong Luo

Examples

pm <- matrix(rep(c(0.03, 0.04, 0.01), times = 2), ncol = 3, nrow = 2)
seqgsphh(pm = pm, alpha = 0.025, method = "Hochberg")
seqgsphh(pm = pm, alpha = 0.025, method = "Hommel")

Maurer-Bretz sequential graphical approach

Description

seqmbgx() conducts group-sequential testing for multiple hypotheses based on Maurer-Bretz approach.

Usage

seqmbgx(
  xm = qnorm(matrix(rep(c(0.03, 0.04, 0.01), times = 4), ncol = 3, nrow = 4)),
  informationm = matrix(rep(c(0.4, 0.8, 1), each = 4), ncol = 3, nrow = 4),
  spending = rep("OBF", 4),
  param.spending = rep(1, 4),
  alpha = 0.025,
  sided = -1,
  W = c(0.5, 0.5, 0, 0),
  G = rbind(c(0, 0, 1, 0), c(0, 0, 0, 1), c(0, 1, 0, 0), c(1, 0, 0, 0)),
  tol = 1e-10,
  retrospective = 0
)

Arguments

Value

List with elements

• Hrej: rejected hypotheses

• rejected: the index set of rejected hypotheses

• decisionsm: rejection decision for each endpoint (row) at each timepoint (column)

• cumdecisionsm: cumulative rejection decision for each endpoint (row) at each timepoint (column)

Author(s)

Xiaodong Luo

Examples

seqmbgx(
  xm = qnorm(matrix(rep(c(0.03, 0.04, 0.01), times = 4), ncol = 3, nrow = 4)),
  informationm = matrix(rep(c(0.4, 0.8, 1), each = 4), ncol = 3, nrow = 4),
  spending = rep("OBF", 4),
  param.spending = rep(1, 4),
  alpha = 0.025,
  W = c(0.5, 0.5, 0, 0),
  G = rbind(c(0, 0, 1, 0), c(0, 0, 0, 1), c(0, 1, 0, 0), c(1, 0, 0, 0)),
  retrospective = 0
)

Sequential generalized Hochberg and Hommel procedures based on q-values

Description

Sequential generalized Hochberg and Hommel procedures based on q-values

Usage

seqqvalhh(
  pm = matrix(rep(c(0.03, 0.04, 0.01), times = 2), ncol = 3, nrow = 2),
  alpham = matrix(rep(c(0.02, 0.03, 0.05), each = 2), ncol = 3, nrow = 2),
  epsilon = 1e-10,
  precision = 10,
  method = "Hochberg"
)

Arguments

Value

List with elements

• rejected: the index set of rejected hypotheses

• decisionsm: rejection decision for each endpoint (row) at each timepoint (column)

• cumdecisionsm: cumulative rejection decision for each endpoint (row) at each timepoint (column);

• alphaused: alpha levels actually used for each endpoint (row) at each timepoint (column).

Author(s)

Xiaodong Luo

Examples

pm <- matrix(rep(c(0.03, 0.04, 0.01), times = 2), ncol = 3, nrow = 2)
alpham <- matrix(rep(c(0.02, 0.03, 0.05), each = 2), ncol = 3, nrow = 2)
seqqvalhh(pm = pm, alpham = alpham, method = "Hochberg")
seqqvalhh(pm = pm, alpham = alpham, method = "Hommel")

Calculate alpha spending function

Description

spendingfun() calculates the alpha spending function.

Usage

spendingfun(alpha, fractions = seq(0.2, 1, by = 0.2), family = "OBF", rho = 1)

Arguments

Details

• "OBF": O'Brien-Fleming family; 2\{1-\Phi(\Phi^{-1}(1-\alpha/2)/t^{\rho/2})\};

• "pocock": Pocock family; \alpha \log\{1+(e-1)*t\};

• "power": Power family; \alpha*t^{\rho}

Note that the OBF and Pocock spending functions are not the originally proposed ones, they are the modified ones that closely resemble the original versions. That being said, you might still see some differences.

Value

List with an element named aseq for the alpha spending sequence.

Author(s)

Xiaodong Luo

Examples

spendingfun(
  alpha = 0.025,
  fractions = seq(0.2, 1, by = 0.2),
  family = "OBF",
  rho = 1
)

Update graph

Description

updategraph() updates the graph when only a subset of original hypotheses is concerned.

Usage

updategraph(
  S1 = c(2, 3),
  W0 = c(0.5, 0.5, 0, 0),
  G0 = rbind(c(0, 0, 1, 0), c(0, 0, 0, 1), c(0, 1, 0, 0), c(1, 0, 0, 0)),
  S0 = seq(1, length(W0), by = 1)
)

Arguments

Value

List with the following elements

• S1: Integer indices the same as the input S1.

• W1: Numeric vector for weights of the updated graph.

• G1: Numeric transition of the updated graph.

Author(s)

Xiaodong Luo

Examples

## We can use the function to produce a closed testing tree
## A function to create power set
powerset <- function(x) {
  sets <- lapply(1:(length(x)), function(i) combn(x, i, simplify = FALSE))
  unlist(sets, recursive = FALSE)
}

n <- 3    # number of hypotheses
pn <- 2^n-1
pset <- powerset(seq(1, n, by = 1))    # create the power set
df <- data.frame(matrix(ncol = 1+n, nrow = 0))    # create the dataset
colnames(df) <- c("Test", paste0("H", seq(1, n, by = 1), sep = ""))

W0 <- c(1/3, 1/3, 1/3)    # the weights of the graph
m <- rbind(H1 = c(0, 1/2, 1/2),
           H2 = c(1/2, 0, 1/2),
           H3 = c(1/2, 1/2, 0))
G0 <- matrix(m, nrow = 3, ncol = 3)    # the transition matrix of the graph

for (j in 1:pn){
    abc <- updategraph(S1 = pset[[j]], W0 = W0, G0 = G0)
    temp <- rep("-", n)
    temp[pset[[j]]] <- abc$W1
    temp <- c(paste(pset[[j]], collapse = ""), temp)
    df[j, ] <- temp
}
df    # the dataframe lists the closed testing tree