ISS

Description

The function implements the combination of p-value calculation and familywise error rate control through DAG testing procedures described in Müller et al. (2023).

Usage

ISS(
  X,
  y,
  tau,
  alpha = 0.05,
  m = nrow(X),
  p_value = c("sub-Gaussian-normalmixture", "sub-Gaussian", "Gaussian", "classification",
    "quantile"),
  sigma2,
  rho = 1/2,
  FWER_control = c("ISS", "Holm", "MG all", "MG any", "split", "split oracle"),
  minimal = FALSE,
  split_proportion = 1/2,
  eta = NA,
  theta = 1/2
)

Arguments

Value

A numeric matrix giving the points in X determined to lie in the tau-superlevel set of the regression function with probability at least 1 - alpha or, if minimal == TRUE, a subset of points thereof that have the same upper hull.

References

Meijer RJ, Goeman JJ (2015). “A multiple testing method for hypotheses structured in a directed acyclic graph.” Biometrical Journal, 57(1), 123–143.

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852v2.

Examples

d <- 2
n <- 1000
m <- 100
sigma2 <- (1 / 4)^2
tau <- 0.5
alpha <- 0.05

X <- matrix(runif(n * d), nrow = n)
eta_X <- apply(X, MARGIN = 1, max)
y <- eta_X + rnorm(n, sd = sqrt(sigma2))
X_rej <- ISS(X = X, y = y, tau = tau, alpha = alpha, m = m, sigma2 = sigma2)

if (d == 2) {
  plot(0, type = "n", xlim = c(0, 1), ylim = c(0, 1), xlab = NA, ylab = NA)
  for (i in 1:nrow(X_rej)) {
    rect(
      xleft = X_rej[i, 1], xright = 1, ybottom = X_rej[i, 2], ytop = 1,
      border = NA, col = "indianred"
    )
  }

  points(X, pch = 16, cex = 0.5, col = "gray")
  points(X[1:m, ], pch = 16, cex = 0.5, col = "black")
  lines(x = c(0, tau), y = c(tau, tau), lty = 2)
  lines(x = c(tau, tau), y = c(tau, 0), lty = 2)

  legend(
    x = "bottomleft",
    legend = c(
      "superlevel set boundary",
      "untested covariate points",
      "tested covariate points",
      "selected set"
    ),
    col = c("black", "gray", "black", "indianred"),
    lty = c(2, NA, NA, NA),
    lwd = c(1, NA, NA, NA),
    pch = c(NA, 16, 16, NA),
    fill = c(NA, NA, NA, "indianred"),
    border = c(NA, NA, NA, "indianred")
  )
}

dag_test_FS

Description

Implements the fixed sequence testing procedure of familywise error rate control. The sequence is given through ordering elements of p_order increasingly.

Usage

dag_test_FS(p_order, p, alpha, decreasing = FALSE)

Arguments

Value

A boolean vector of the same length as p with each element being TRUE if the corresponding hypothesis is rejected and FALSE otherwise.

Examples

p_order <- c(0.5, 0, 1)
p <- c(0.01, 0.1, 0.05)
alpha <- 0.05
dag_test_FS(p_order, p, alpha, decreasing = TRUE)

dag_test_Holm

Description

Given a vector of p-values, each concerning a row in the matrix X0, dag_test_Holm() first applies Holm's method to the p-values and then also rejects hypotheses corresponding to points coordinate-wise greater or equal to any point whose hypothesis has been rejected.

Usage

dag_test_Holm(X0, p, alpha)

Arguments

Value

A boolean vector of the same length as p with each element being TRUE if the corresponding hypothesis is rejected and FALSE otherwise.

Examples

X0 <- rbind(c(0.5, 0.5), c(0.8, 0.9), c(0.4, 0.6))
p <- c(0.01, 0.1, 0.05)
alpha <- 0.05
dag_test_Holm(X0, p, alpha)

dag_test_ISS

Description

Implements the DAG testing procedure given in Algorithm 1 by Müller et al. (2023).

Usage

dag_test_ISS(X0, p, alpha)

Arguments

Value

A boolean vector of the same length as p with each element being TRUE if the corresponding hypothesis is rejected and FALSE otherwise.

References

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852.

Examples

X0 <- rbind(c(0.5, 0.6), c(0.8, 0.9), c(0.9, 0.8))
p <- c(0.02, 0.025, 0.1)
alpha <- 0.05
dag_test_ISS(X0, p, alpha)

dag_test_MG

Description

Implements the graph-testing procedures proposed by Meijer and Goeman (2015) for one-way logical relationships. Here implemented for the specific application to isotonic subgroup selection.

Usage

dag_test_MG(
  X0,
  p,
  alpha,
  version = c("all", "any"),
  leaf_weights,
  sparse = FALSE
)

Arguments

Value

A boolean vector of the same length as p with each element being TRUE if the corresponding hypothesis is rejected and FALSE otherwise.

References

Meijer RJ, Goeman JJ (2015). “A multiple testing method for hypotheses structured in a directed acyclic graph.” Biometrical Journal, 57(1), 123–143.

Examples

X0 <- rbind(c(0.5, 0.6), c(0.8, 0.9), c(0.9, 0.8))
p <- c(0.02, 0.025, 0.1)
alpha <- 0.05
dag_test_MG(X0, p, alpha)
dag_test_MG(X0, p, alpha, version = "any")
dag_test_MG(X0, p, alpha, sparse = TRUE)

get_DAG

Description

This function is used to construct the induced DAG, induced polyforest and reverse topological orderings thereof from a numeric matrix X0. See Definition 2 in Müller et al. (2023).

Usage

get_DAG(X0, sparse = FALSE, twoway = FALSE)

Arguments

Value

A list with named elements giving the leaves, parents, ancestors and reverse topological ordering and additionally, if twoway == TRUE, the roots, children and descendants, of the constructed graph.

References

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852.

Examples

X <- rbind(
  c(0.2, 0.8), c(0.2, 0.8), c(0.1, 0.7),
  c(0.2, 0.1), c(0.3, 0.5), c(0.3, 0)
)
get_DAG(X0 = X)
get_DAG(X0 = X, sparse = TRUE, twoway = TRUE)

get_boundary_points

Description

Given a set of points, returns the minimal subset with the same upper hull.

Usage

get_boundary_points(X)

Arguments

Value

A numeric matrix of the same number of columns as X.

Examples

X <- rbind(c(0, 1), c(1, 0), c(1, 0), c(1, 1))
get_boundary_points(X)

get_p_Gaussian

Description

Calculate the p-value in Definition 19 of Müller et al. (2023).

Usage

get_p_Gaussian(X, y, x0, tau)

Arguments

Value

A single numeric value in (0, 1].

References

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852.

Examples

set.seed(123)
n <- 100
d <- 2
X <- matrix(runif(d * n), ncol = d)
eta <- function(x) sum(x)
y <- apply(X, MARGIN = 1, FUN = eta) + rnorm(n, sd = 1)
get_p_Gaussian(X, y, x0 = c(1, 1), tau = 1)
get_p_Gaussian(X, y, x0 = c(1, 1), tau = -1)

get_p_classification

Description

Calculate the p-value in Definition 21 of Müller et al. (2023).

Usage

get_p_classification(X, y, x0, tau)

Arguments

Value

A single numeric value in (0, 1].

References

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852.

Examples

set.seed(123)
n <- 100
d <- 2
X <- matrix(runif(d * n), ncol = d)
eta <- function(x) sum(x)
X_eta <- apply(X, MARGIN = 1, FUN = function(x) 1 / (1 + exp(-eta(x))))
y <- as.numeric(runif(n) < X_eta)
get_p_classification(X, y, x0 = c(1, 1), tau = 0.6)
get_p_classification(X, y, x0 = c(1, 1), tau = 0.9)

get_p_subGaussian

Description

Calculate the p-value in Definition 1 of Müller et al. (2023).

Usage

get_p_subGaussian(X, y, x0, sigma2, tau)

Arguments

Value

A single numeric value in (0, 1].

References

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852.

Examples

set.seed(123)
n <- 100
d <- 2
X <- matrix(runif(d*n), ncol = d)
eta <- function(x) sum(x)
y <- apply(X, MARGIN = 1, FUN = eta) + rnorm(n, sd = 0.5)
get_p_subGaussian(X, y, x0 = c(1, 1), sigma2 = 0.25, tau = 1)
get_p_subGaussian(X, y, x0 = c(1, 1), sigma2 = 0.25, tau = 3)

get_p_subGaussian_NM

Description

Calculate the p-value in Definition 18 of Müller et al. (2023).

Usage

get_p_subGaussian_NM(X, y, x0, sigma2, tau, rho = 0.5)

Arguments

Value

A single numeric value in (0, 1].

References

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852.

Examples

set.seed(123)
n <- 100
d <- 2
X <- matrix(runif(d * n), ncol = d)
eta <- function(x) sum(x)
y <- apply(X, MARGIN = 1, FUN = eta) + rnorm(n, sd = 0.5)
get_p_subGaussian_NM(X, y, x0 = c(1, 1), sigma2 = 0.25, tau = 3)
get_p_subGaussian_NM(X, y, x0 = c(1, 1), sigma2 = 0.25, tau = 1)
get_p_subGaussian_NM(X, y, x0 = c(1, 1), sigma2 = 0.25, tau = 1, rho = 2)

get_p_value

Description

A wrapper function used to call the correct function for calculating the p-value.

Usage

get_p_value(
  p_value_method = c("sub-Gaussian-normalmixture", "sub-Gaussian", "Gaussian",
    "classification", "quantile"),
  X,
  y,
  x0,
  tau,
  sigma2,
  rho = 1/2,
  theta = 1/2
)

Arguments

Value

A single numeric value in (0, 1].

References

Müller MM, Reeve HWJ, Cannings TI, Samworth RJ (2023). “Isotonic subgroup selection.” arXiv preprint arXiv:2305.04852.

Examples

set.seed(123)
n <- 100
d <- 2
X <- matrix(runif(d * n), ncol = d)
eta <- function(x) sum(x)
X_eta <- apply(X, MARGIN = 1, FUN = function(x) 1 / (1 + exp(-eta(x))))
y <- as.numeric(runif(n) < X_eta)
get_p_value(p_value_method = "classification", X, y, x0 = c(1, 1), tau = 0.6)
get_p_value(p_value_method = "classification", X, y, x0 = c(1, 1), tau = 0.9)

X_eta <- apply(X, MARGIN = 1, FUN = eta)
y <- X_eta + rcauchy(n)
get_p_value(p_value_method = "quantile", X, y, x0 = c(1, 1), tau = 1/2)
get_p_value(p_value_method = "quantile", X, y, x0 = c(1, 1), tau = 3)
get_p_value(p_value_method = "quantile", X, y, x0 = c(1, 1), tau = 3, theta = 0.95)