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Introduction

In confirmatory clinical trials, regulatory guidelines mandate the strong control of the family-wise error rate at a prespecified level α\alpha. Many multiple comparison procedures (MCPs) have been proposed for this purpose. The graphical approaches are a general framework that include many common MCPs as special cases. In this vignette, we illustrate how to use graphicalMCP to perform some common MCPs.

To test mm hypotheses using a graphical MCP, each hypothesis HiH_i receives a weight 0wi10\leq w_i\leq 1 (called hypothesis weight), where i=1mwi1\sum_{i=1}^{m}w_i\leq 1. From HiH_i to HjH_j, there could be a directed and weighted edge 0gij10\leq g_{ij}\leq 1, which means that when HiH_i is rejected, its hypothesis weight will be propagated (or transferred) to HjH_j and gijg_{ij} determines how much of the propagation. We also require j=1mgij1\sum_{j=1}^{m}g_{ij}\leq 1 and gii=0g_{ii}=0.

Bonferroni-based procedures

Weighted Bonferroni test

A weighted Bonferroni test splits α\alpha among hypotheses by testing every hypothesis at a significance level of wiαw_i\alpha. Thus it rejects a hypothesis if its p-value is less than or equal to its significance level. When wi=wjw_i=w_j for all i,ji,j, this means an equal split and the test is the Bonferroni test. There is no propagation between any pair of hypothesis.

set.seed(1234)
alpha <- 0.025
m <- 3
bonferroni_graph <- bonferroni(rep(1 / m, m))
# transitions <- matrix(0, m, m)
# bonferroni_graph <- graph_create(rep(1 / m, m), transitions)

plot(
  bonferroni_graph,
  layout = igraph::layout_in_circle(
    as_igraph(bonferroni_graph),
    order = c(2, 1, 3)
  ),
  vertex.size = 70
)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    bonferroni_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>    H1    H2    H3 
#>  TRUE FALSE FALSE

Holm Procedure

Holm (or Bonferroni-Holm) procedures improve over Bonferroni tests by allowing propagation (Holm 1979). In other words, transition weights between hypotheses may not be zero. So it is uniformly more powerful than Bonferroni tests.

set.seed(1234)
alpha <- 0.025
m <- 3
holm_graph <- bonferroni_holm(rep(1 / m, m))
# transitions <- matrix(1 / (m - 1), m, m)
# diag(transitions) <- 0
# holm_graph <- graph_create(rep(1 / m, m), transitions)

plot(holm_graph,
  layout = igraph::layout_in_circle(
    as_igraph(holm_graph),
    order = c(2, 1, 3)
  ),
  vertex.size = 70
)


p_values <- runif(m, 0, alpha)

test_results <- graph_test_shortcut(holm_graph, p = p_values, alpha = alpha)

test_results$outputs$rejected
#>    H1    H2    H3 
#>  TRUE FALSE FALSE

Fixed sequence procedure

Fixed sequence (or hierarchical) procedures pre-specify an order of testing [Maurer, Hothorn, and Lehmacher (1995);westfall-2001-optimally]. For example, the procedure will test H1H_1 first. If it is rejected, it will test H2H_2; otherwise the testing stops. If H2H_2 is rejected, it will test H3H_3; otherwise the testing stops. For each hypothesis, it will be tested at the full α\alpha level, when it can be tested.

set.seed(1234)
alpha <- 0.025
m <- 3
fixed_sequence_graph <- fixed_sequence(m)
# transitions <- rbind(
#   c(0, 1, 0),
#   c(0, 0, 1),
#   c(0, 0, 0)
# )
# fixed_sequence_graph <- graph_create(c(1, 0, 0), transitions)

plot(fixed_sequence_graph, nrow = 1, asp = 0.05, vertex.size = 40)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    fixed_sequence_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>   H1   H2   H3 
#> TRUE TRUE TRUE

Fallback procedure

Fallback procedures have one-way propagation (like fixed sequence procedures) but allow hypotheses to be tested at different significance levels (Wiens 2003).

set.seed(1234)
alpha <- 0.025
m <- 3
fallback_graph <- fallback(rep(1 / 3, 3))
# transitions <- rbind(
#   c(0, 1, 0),
#   c(0, 0, 1),
#   c(0, 0, 0)
# )
# fallback_graph <- graph_create(rep(1 / 3, 3), transitions)

plot(fallback_graph, nrow = 1, asp = 0.05, vertex.size = 40)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    fallback_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>   H1   H2   H3 
#> TRUE TRUE TRUE

Further they can be improved to allow propagation from later hypotheses to earlier hypotheses, because it is possible that a later hypothesis is rejected before an earlier hypothesis can be rejected. There are two versions of improvement: fallback_improved_1 due to (Wiens and Dmitrienko 2005) and fallback_improved_2 due to (Bretz et al. 2009) respectively.

set.seed(1234)
alpha <- 0.025
m <- 3
fallback_improved_1_graph <- fallback_improved_1(rep(1 / 3, 3))
# hypotheses <- rep(1 / 3, 3)
# transitions <- rbind(
#   c(0, 1, 0),
#   c(0, 0, 1),
#   c(hypotheses[seq_len(m - 1)] / sum(hypotheses[seq_len(m - 1)]), 0)
# )
# fallback_improved_1_graph <- graph_create(rep(1 / 3, 3), transitions)

plot(
  fallback_improved_1_graph,
  nrow = 1,
  asp = 0.05,
  vertex.size = 40,
  edge_curves = c("pairs" = 7, "H3|H1" = -10)
)


epsilon <- 0.0001
fallback_improved_2_graph <- fallback_improved_2(rep(1 / 3, 3), epsilon)
# hypotheses <- rep(1 / 3, 3)
# transitions <- rbind(
#   c(0, 1, 0),
#   c(1 - epsilon, 0, epsilon),
#   c(1, 0, 0)
# )
# fallback_improved_2_graph <- graph_create(rep(1 / 3, 3), transitions)

plot(
  fallback_improved_2_graph,
  nrow = 1,
  asp = 0.05,
  eps = 0.0001,
  edge_curves = c("pairs" = 7, "H3|H1" = -10),
  vertex.size = 40
)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    fallback_improved_2_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>   H1   H2   H3 
#> TRUE TRUE TRUE

Serial gatekeeping procedure

Serial gatekeeping procedures involve ordered multiple families of hypotheses, where all hypotheses of a family of hypotheses must be rejected before proceeding in the test sequence. The example below considers a primary family consisting of two hypotheses H1H_1 and H2H_2 and a secondary family consisting of a single hypothesis H3H_3. In the primary family, the Holm procedure is applied. If both H1H_1 and H2H_2 are rejected, H3H_3 can be tested at level α\alpha; otherwise H3H_3 cannot be rejected. To allow the conditional propagation to H3H_3, an ε\varepsilon edge is used from H2H_2 to H3H_3. It has a very small transition weight so that H2H_2 propagates most of its hypothesis weight to H1H_1 (if not already rejected) and retains a small (non-zero) weight for H3H_3 so that if H1H_1 has been rejected, all hypothesis weight of H2H_2 will be propagated to H3H_3. Here ε\varepsilon is assigned to be 0.0001 and in practice, the value could be adjusted but it should be much smaller than the smallest p-value observed.

set.seed(1234)
alpha <- 0.025
m <- 3
epsilon <- 0.0001

transitions <- rbind(
  c(0, 1, 0),
  c(1 - epsilon, 0, epsilon),
  c(0, 0, 0)
)

serial_gatekeeping_graph <- graph_create(c(0.5, 0.5, 0), transitions)

plot(
  serial_gatekeeping_graph,
  nrow = 1,
  asp = 0.05,
  eps = 0.0001,
  edge_curves = c("pairs" = 7, "H3|H1" = -10),
  vertex.size = 40
)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    serial_gatekeeping_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>   H1   H2   H3 
#> TRUE TRUE TRUE

Parallel gatekeeping procedure

Parallel gatekeeping procedures also involve multiple ordered families of hypotheses, where any null hypotheses of a family of hypotheses must be rejected before proceeding in the test sequence (Dmitrienko, Offen, and Westfall 2003). The example below considers a primary family consisting of two hypotheses H1H_1 and H2H_2 and a secondary family consisting of two hypotheses H3H_3 and H4H_4. In the primary family, the Bonferroni test is applied. If any of H1H_1 and H2H_2 is rejected, H3H_3 and H4H_4 can be tested at level α/2\alpha/2 using the Holm procedure; if both H1H_1 and H2H_2 are rejected, H3H_3 and H4H_4 can be tested at level α\alpha using the Holm procedure; otherwise H3H_3 and H4H_4 cannot be rejected.

set.seed(1234)
alpha <- 0.025
m <- 4

transitions <- rbind(
  c(0, 0, 0.5, 0.5),
  c(0, 0, 0.5, 0.5),
  c(0, 0, 0, 1),
  c(0, 0, 1, 0)
)

parallel_gatekeeping_graph <- graph_create(c(0.5, 0.5, 0, 0), transitions)

plot(parallel_gatekeeping_graph, vertex.size = 70)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    parallel_gatekeeping_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>    H1    H2    H3    H4 
#>  TRUE FALSE FALSE FALSE

The above parallel gatekeeping procedure can be improved by adding ε\varepsilon edges from secondary hypotheses to primary hypotheses, because it is possible that both secondary hypotheses are rejected but there is still a remaining primary hypothesis not rejected (Bretz et al. 2009).

set.seed(1234)
alpha <- 0.025
m <- 4
epsilon <- 0.0001

transitions <- rbind(
  c(0, 0, 0.5, 0.5),
  c(0, 0, 0.5, 0.5),
  c(epsilon, 0, 0, 1 - epsilon),
  c(0, epsilon, 1 - epsilon, 0)
)

parallel_gatekeeping_improved_graph <-
  graph_create(c(0.5, 0.5, 0, 0), transitions)

plot(parallel_gatekeeping_improved_graph, eps = 0.0001, vertex.size = 70)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    parallel_gatekeeping_improved_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>    H1    H2    H3    H4 
#>  TRUE FALSE FALSE FALSE

Successive procedure

Successive procedures incorporate successive relationships between hypotheses. For example, the secondary hypothesis is not tested until the primary hypothesis has been rejected. This is similar to using the fixed sequence procedure as a component of a graph. The example below considers two primary hypotheses H1H_1 and H2H_2 and two secondary hypotheses H3H_3 and H4H_4. Primary hypotheses H1H_1 and H2H_2 receive the equal hypothesis weight of 0.5; secondary hypotheses H3H_3 and H4H_4 receive the hypothesis weight of 0. A secondary hypothesis H3(H4)H_3 (H_4) can be tested only if the corresponding primary hypothesis H1(H2)H_1 (H_2) has been rejected. This represents the successive relationships between H1H_1 and H3H_3, and H2H_2 and H4H_4, respectively (Maurer, Glimm, and Bretz 2011). If both H1H_1 and H3H_3 are rejected, their hypothesis weights are propagated to H2H_2 and H4H_4, and vice versa.

set.seed(1234)
alpha <- 0.025
m <- 4
simple_successive_graph <- simple_successive_1()
# transitions <- rbind(
#   c(0, 0, 1, 0),
#   c(0, 0, 0, 1),
#   c(0, 1, 0, 0),
#   c(1, 0, 0, 0)
# )
# simple_successive_graph <- graph_create(c(0.5, 0.5, 0, 0), transitions)

plot(simple_successive_graph, layout = "grid", nrow = 2, vertex.size = 70)


p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    simple_successive_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>    H1    H2    H3    H4 
#>  TRUE FALSE FALSE FALSE

The above graph could be generalized to allow propagation between primary hypotheses (Maurer, Glimm, and Bretz 2011). A general successive graph is illustrate below with a variable to determine the propagation between H1H_1 and H2H_2.

set.seed(1234)
alpha <- 0.025
m <- 4

successive_var <- simple_successive_var <- function(gamma) {
  graph_create(
    c(0.5, 0.5, 0, 0),
    rbind(
      c(0, gamma, 1 - gamma, 0),
      c(gamma, 0, 0, 1 - gamma),
      c(0, 1, 0, 0),
      c(1, 0, 0, 0)
    )
  )
}

successive_var_graph <- successive_var(0.5)
plot(successive_var_graph, layout = "grid", nrow = 2, vertex.size = 70)

p_values <- runif(m, 0, alpha)

test_results <-
  graph_test_shortcut(
    successive_var_graph,
    p = p_values,
    alpha = alpha
  )

test_results$outputs$rejected
#>    H1    H2    H3    H4 
#>  TRUE  TRUE FALSE FALSE

Simes-based procedures

Hommel procedure

Hommel procedure (Hommel 1988) is a closed test procedure which uses Simes tests for every intersection hypothesis. According to Xi and Bretz (2019), the graph for Hommel procedures is the same as the graph for Holm procedures. Thus to perform Hommel procedure, we just need to specify test_type to be simes.

set.seed(1234)
alpha <- 0.025
m <- 3
hommel_graph <- bonferroni_holm(rep(1 / m, m))

plot(
  hommel_graph,
  layout = igraph::layout_in_circle(
    as_igraph(hommel_graph),
    order = c(2, 1, 3)
  ),
  vertex.size = 70
)


p_values <- runif(m, 0, alpha)

test_results <- graph_test_closure(
  hommel_graph,
  p = p_values,
  alpha = alpha,
  test_types = "simes"
)

test_results$outputs$rejected
#>   H1   H2   H3 
#> TRUE TRUE TRUE

Parametric procedures

Dunnett procedure

Step-down Dunnett procedures are a closed test procedure and an improvement from Holm procedures by incorporating the correlation structure between test statistics (Dunnett and Tamhane 1991). Thus they are the same graph as Holm procedures. Assume an equi-correlated case, where the correlation between any pair of test statistics is the same, e.g., 0.5. Then we can perform the step-down Dunnett procedure by specifying test_type to be parametric and providing the correlation matrix.

set.seed(1234)
alpha <- 0.025
m <- 3
dunnett_graph <- bonferroni_holm(rep(1 / m, m))

plot(
  dunnett_graph,
  layout = igraph::layout_in_circle(
    as_igraph(dunnett_graph),
    order = c(2, 1, 3)
  ),
  vertex.size = 70
)


p_values <- runif(m, 0, alpha)
corr <- matrix(0.5, m, m)
diag(corr) <- 1

test_results <- graph_test_closure(
  dunnett_graph,
  p = p_values, alpha = alpha,
  test_types = "parametric",
  test_corr = list(corr)
)

test_results$outputs$rejected
#>    H1    H2    H3 
#>  TRUE FALSE FALSE

Reference

Bretz, Frank, Willi Maurer, Werner Branath, and Martin Posch. 2009. “A Graphical Approach to Sequentially Rejective Multiple Test Procedures.” Statistics in Medicine 53 (4): 586–604. https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.3495.
Dmitrienko, Alexei, Walter W. Offen, and Peter H. Westfall. 2003. “Gatekeeping Strategies for Clinical Trials That Do Not Require All Primary Effects to Be Significant.” Statistics in Medicine 22 (15): 2387–2400. https://onlinelibrary.wiley.com/doi/pdf/10.1002/sim.1526.
Dunnett, Charles W., and Ajit C. Tamhane. 1991. “Step-down Multiple Tests for Comparing Treatments with a Control in Unbalanced One-Way Layouts.” Statistics in Medicine 10 (6): 939–47. https://onlinelibrary.wiley.com/doi/10.1002/sim.4780100614.
Holm, Sture. 1979. “A Simple Sequentially Rejective Multiple Test Procedure.” Scandinavian Journal of Statistics 75: 65–70.
Hommel, Gerhard. 1988. “A Stagewise Rejective Multiple Test Procedure Based on a Modified Bonferroni Test.” Biometrika 75 (2): 383–86. https://doi.org/10.1093/biomet/75.2.383.
Maurer, Willi, Ekkehard Glimm, and Frank Bretz. 2011. “Multiple and Repeated Testing of Primary, Coprimary, and Secondary Hypotheses.” Statistics in Biopharmaceutical Research 3 (2): 336–52. https://www.tandfonline.com/doi/abs/10.1198/sbr.2010.10010.
Maurer, Willi, Ludwig Hothorn, and Walter Lehmacher. 1995. “Multiple Comparisons in Drug Clinical Trials and Preclinical Assays: A-Priori Ordered Hypotheses.” Biometrie in Der Chemisch-Pharmazeutischen Industrie 6: 3–18.
Wiens, Brian L. 2003. “A Fixed Sequence Bonferroni Procedure for Testing Multiple Endpoints.” Pharmaceutical Statistics 2 (3): 211–15. https://onlinelibrary.wiley.com/doi/10.1002/pst.64.
Wiens, Brian L., and Alexei Dmitrienko. 2005. “The Fallback Procedure for Evaluating a Aingle Family of Hypotheses.” Journal of Biopharmaceutical Statistics 15 (6): 929–42. https://www.tandfonline.com/doi/full/10.1080/10543400500265660.
Xi, Dong, and Frank Bretz. 2019. “Symmetric Graphs for Equally Weighted Tests, with Application to the Hochberg Procedure.” Statistics in Medicine 38 (27): 5268–82. https://onlinelibrary.wiley.com/doi/10.1002/sim.8375.