Procedure to compute p-value boundaries by weighted Bonferroni

library(tibble)
library(gt)
library(gsDesign)
library(dplyr)
library(wpgsd)

Example overview

In a 2-arm controlled clinical trial example with one primary endpoint, there are 3 patient populations defined by the status of two biomarkers A and B:

biomarker A positive,
biomarker B positive,
overall population.

The 3 primary elementary hypotheses are:

\(H_1\): the experimental treatment is superior to the control in the biomarker A positive population;
\(H_2\): the experimental treatment is superior to the control in the biomarker B positive population;
\(H_3\): the experimental treatment is superior to the control in the overall population.

Assume an interim analysis and a final analysis are planned for the study and the number of events are listed as

k <- 2 # Number of total analysis
n_hypotheses <- 3 # Number of hypotheses

Observed p-values

obs_tbl <- tribble(
  ~hypothesis, ~analysis, ~obs_p,
  "H1", 1, 0.02,
  "H2", 1, 0.01,
  "H3", 1, 0.006,
  "H1", 2, 0.015,
  "H2", 2, 0.012,
  "H3", 2, 0.004
) %>%
  mutate(obs_Z = -qnorm(obs_p))

obs_tbl %>%
  gt() %>%
  tab_header(title = "Nominal p-values")

hypothesis	analysis	obs_p	obs_Z
Nominal p-values
H1	1	0.020	2.053749
H2	1	0.010	2.326348
H3	1	0.006	2.512144
H1	2	0.015	2.170090
H2	2	0.012	2.257129
H3	2	0.004	2.652070

p_obs_IA <- (obs_tbl %>% filter(analysis == 1))$obs_p
p_obs_FA <- (obs_tbl %>% filter(analysis == 2))$obs_p

Information fraction

alpha <- 0.025
event_tbl <- tribble(
  ~population, ~analysis, ~event,
  "A positive", 1, 80,
  "B positive", 1, 88,
  "AB positive", 1, 64,
  "overall", 1, 180,
  "A positive", 2, 160,
  "B positive", 2, 176,
  "AB positive", 2, 128,
  "overall", 2, 360,
)

The information fraction of \(H_1\), \(H_2\), \(H_3\) at IA is

IF_IA <- c(
  ((event_tbl %>% filter(analysis == 1, population == "A positive"))$event + (event_tbl %>% filter(analysis == 1, population == "overall"))$event) /
    ((event_tbl %>% filter(analysis == 2, population == "A positive"))$event + (event_tbl %>% filter(analysis == 2, population == "overall"))$event),
  ((event_tbl %>% filter(analysis == 1, population == "B positive"))$event + (event_tbl %>% filter(analysis == 1, population == "overall"))$event) /
    ((event_tbl %>% filter(analysis == 2, population == "B positive"))$event + (event_tbl %>% filter(analysis == 2, population == "overall"))$event),
  ((event_tbl %>% filter(analysis == 1, population == "AB positive"))$event + (event_tbl %>% filter(analysis == 1, population == "overall"))$event) /
    ((event_tbl %>% filter(analysis == 2, population == "AB positive"))$event + (event_tbl %>% filter(analysis == 2, population == "overall"))$event)
)

IF_IA

## [1] 0.5 0.5 0.5

Initial weight and transition matrix

We assign the initial weights of \(H_1\), \(H_2\), \(H_3\) as \(\left(w_1(I), w_2(I), w_3(I) \right) = (0.3, 0.3, 0.4).\) And its multiplicity strategy is visualized in below. If \(H_1\) is rejected, then \(3/7\) local significance level \(\alpha_1\) will be propagated to \(H_2\), and \(4/7\) will go to \(H_3\). If \(H_3\) is rejected, then half of \(\alpha_3\) goes to \(H_1\), and half goes to \(H_2\).

m <- matrix(c( # Transition matrix
  0, 3 / 7, 4 / 7,
  3 / 7, 0, 4 / 7,
  1 / 2, 1 / 2, 0
), nrow = 3, byrow = TRUE)

w <- c(0.3, 0.3, 0.4) # Initial weights

name_hypotheses <- c(
  "H1: Biomarker A positive",
  "H2: Biomarker B positive",
  "H3: Overall Population"
)

hplot <- gMCPLite::hGraph(
  3,
  alphaHypotheses = w, m = m,
  nameHypotheses = name_hypotheses, trhw = .2, trhh = .1,
  digits = 5, trdigits = 3, size = 5, halfWid = 1, halfHgt = 0.5,
  offset = 0.2, trprop = 0.4,
  fill = as.factor(c(2, 3, 1)),
  palette = c("#BDBDBD", "#E0E0E0", "#EEEEEE"),
  wchar = "w"
)
hplot

# Get weights for all intersection hypotheses
graph <- gMCPLite::matrix2graph(m)
graph <- gMCPLite::setWeights(graph, w)

# Set up hypothetical p-values (0 or 1) to obtain all combinations
pvals <- NULL
for (i in 1:n_hypotheses) {
  if (i == 1) {
    pvals <- data.frame(x = c(0, 1))
    names(pvals) <- paste("pval_H", i, sep = "")
  } else {
    tmp <- data.frame(x = c(0, 1))
    names(tmp) <- paste("pval_H", i, sep = "")
    pvals <- merge(pvals, tmp)
  }
}
# Get the weights for each intersection hypothesis
inter_weight <- NULL # Create an empty table to store the weight of interaction hypotheses
for (i in seq_len(nrow(pvals))) { # Each row in `pvals` is 1 possible interaction hypothesis
  pval_tmp <- as.numeric(pvals[i, ])
  graph_tmp <- gMCPLite::gMCP(graph = graph, pvalues = pval_tmp, alpha = alpha)
  weight_tmp <- gMCPLite::getWeights(graph_tmp)
  inter_weight <- dplyr::bind_rows(inter_weight, weight_tmp)
}

inter_weight <- replace(inter_weight, pvals == 0, NA) # Replace the empty hypothesis as NA
inter_weight <- inter_weight[-1, ] # Delete the first row since it is empty set

inter_weight %>%
  gt() %>%
  tab_header("Weight of all possible interaction hypothesis")

H1	H2	H3
Weight of all possible interaction hypothesis
1.0000000	NA	NA
NA	1.0000000	NA
0.5000000	0.5000000	NA
NA	NA	1.0000000
0.4285714	NA	0.5714286
NA	0.4285714	0.5714286
0.3000000	0.3000000	0.4000000

Correlations

The correlation of the 6 statistic (2 analyses \(\times\) 3 hypotheses) are

# Event count of intersection of paired hypotheses - Table 2
# H1, H2: Hypotheses intersected.
# (1, 1) represents counts for hypothesis 1
# (1, 2) for counts for the intersection of hypotheses 1 and 2
event <- tribble(
  ~H1, ~H2, ~Analysis, ~Event,
  1, 1, 1, event_tbl %>% filter(analysis == 1, population == "A positive") %>% select(event) %>% as.numeric(),
  2, 2, 1, event_tbl %>% filter(analysis == 1, population == "B positive") %>% select(event) %>% as.numeric(),
  3, 3, 1, event_tbl %>% filter(analysis == 1, population == "overall") %>% select(event) %>% as.numeric(),
  1, 2, 1, event_tbl %>% filter(analysis == 1, population == "AB positive") %>% select(event) %>% as.numeric(),
  1, 3, 1, event_tbl %>% filter(analysis == 1, population == "A positive") %>% select(event) %>% as.numeric(),
  2, 3, 1, event_tbl %>% filter(analysis == 1, population == "B positive") %>% select(event) %>% as.numeric(),
  1, 1, 2, event_tbl %>% filter(analysis == 2, population == "A positive") %>% select(event) %>% as.numeric(),
  2, 2, 2, event_tbl %>% filter(analysis == 2, population == "B positive") %>% select(event) %>% as.numeric(),
  3, 3, 2, event_tbl %>% filter(analysis == 2, population == "overall") %>% select(event) %>% as.numeric(),
  1, 2, 2, event_tbl %>% filter(analysis == 2, population == "AB positive") %>% select(event) %>% as.numeric(),
  1, 3, 2, event_tbl %>% filter(analysis == 2, population == "A positive") %>% select(event) %>% as.numeric(),
  2, 3, 2, event_tbl %>% filter(analysis == 2, population == "B positive") %>% select(event) %>% as.numeric()
)
event

## # A tibble: 12 × 4
##       H1    H2 Analysis Event
##    <dbl> <dbl>    <dbl> <dbl>
##  1     1     1        1    80
##  2     2     2        1    88
##  3     3     3        1   180
##  4     1     2        1    64
##  5     1     3        1    80
##  6     2     3        1    88
##  7     1     1        2   160
##  8     2     2        2   176
##  9     3     3        2   360
## 10     1     2        2   128
## 11     1     3        2   160
## 12     2     3        2   176

# Generate correlation from events
corr <- wpgsd::generate_corr(event)
corr %>% round(2)

##      H1_A1 H2_A1 H3_A1 H1_A2 H2_A2 H3_A2
## [1,]  1.00  0.76  0.67  0.71  0.54  0.47
## [2,]  0.76  1.00  0.70  0.54  0.71  0.49
## [3,]  0.67  0.70  1.00  0.47  0.49  0.71
## [4,]  0.71  0.54  0.47  1.00  0.76  0.67
## [5,]  0.54  0.71  0.49  0.76  1.00  0.70
## [6,]  0.47  0.49  0.71  0.67  0.70  1.00

Boundary calculation

Boundary of \(H_1\)

For the elementary hypothesis \(H_1\), its weight is 1, namely,

w_H1 <- 1

# Index to select from the correlation matrix
indx <- grep("H1", colnames(corr))
corr_H1 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H1 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[1],
  n.I = corr_H1[, ncol(corr_H1)]^2,
  alpha = alpha * w_H1[1],
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

ans <- tibble(
  Analysis = 1:2,
  `Interaction/Elementary hypotheses` = "H1",
  `H1 p-value boundary` = pval_H1,
  `H2 p-value boundary` = NA,
  `H3 p-value boundary` = NA
)
ans %>% gt()

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary
1	H1	0.002980073	NA	NA
2	H1	0.023788266	NA	NA

Boundary of \(H_2\)

For the elementary hypothesis \(H_2\), its weight is 1, namely,

w_H2 <- 1

# Index to select from the correlation matrix
indx <- grep("H2", colnames(corr))
corr_H2 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H2 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[2],
  n.I = corr_H2[, ncol(corr_H2)]^2,
  alpha = alpha * w_H2[1],
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

ans_new <- tibble(
  Analysis = 1:2,
  `Interaction/Elementary hypotheses` = "H2",
  `H1 p-value boundary` = NA,
  `H2 p-value boundary` = pval_H2,
  `H3 p-value boundary` = NA
)
ans_new %>% gt()

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary
1	H2	NA	0.002980073	NA
2	H2	NA	0.023788266	NA

ans <- rbind(ans, ans_new)

Boundary of \(H_3\)

For the elementary hypothesis \(H_3\), its weight is 1, namely,

w_H3 <- 1

# Index to select from the correlation matrix
indx <- grep("H3", colnames(corr))
corr_H3 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H3 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[3],
  n.I = corr_H3[, ncol(corr_H3)]^2,
  alpha = alpha * w_H3[1],
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

ans_new <- tibble(
  Analysis = 1:2,
  `Interaction/Elementary hypotheses` = "H3",
  `H1 p-value boundary` = NA,
  `H2 p-value boundary` = NA,
  `H3 p-value boundary` = pval_H1
)
ans_new %>% gt()

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary
1	H3	NA	NA	0.002980073
2	H3	NA	NA	0.023788266

ans <- rbind(ans, ans_new)

Boundary of \(H_1 \cap H_2\)

For the interaction hypothesis \(H_1 \cap H_2\), its weight is

w_H12 <- inter_weight %>% filter(!is.na(H1), !is.na(H2), is.na(H3))
w_H12 <- w_H12[(!is.na(w_H12))] # Remove NA from weight
w_H12

## [1] 0.5 0.5

And the boundary for \(H_1\) and \(H_2\) are

# -------------#
#      H1      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H1", colnames(corr))
corr_H1 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H1 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[1],
  n.I = corr_H1[, ncol(corr_H1)]^2,
  alpha = alpha * w_H12[1], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

# -------------#
#      H2      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H2", colnames(corr))
corr_H2 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H2 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[2],
  n.I = corr_H2[, ncol(corr_H2)]^2,
  alpha = alpha * w_H12[2], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

ans_new <- tibble(
  Analysis = 1:2,
  `Interaction/Elementary hypotheses` = "H1, H2",
  `H1 p-value boundary` = pval_H1,
  `H2 p-value boundary` = pval_H2,
  `H3 p-value boundary` = NA
)
ans_new %>% gt()

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary
1	H1, H2	0.001490037	0.001490037	NA
2	H1, H2	0.011782800	0.011782800	NA

ans <- rbind(ans, ans_new)

Boundary of \(H_1 \cap H_3\)

For the interaction hypothesis \(H_1 \cap H_2\), its weight is

w_H13 <- inter_weight %>% filter(!is.na(H1), is.na(H2), !is.na(H3))
w_H13 <- w_H13[(!is.na(w_H13))] # Remove NA from weight
w_H13

## [1] 0.4285714 0.5714286

And the boundary for \(H_1\) and \(H_3\) are

# -------------#
#      H1      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H1", colnames(corr))
corr_H1 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H1 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[1],
  n.I = corr_H1[, ncol(corr_H1)]^2,
  alpha = alpha * w_H13[1], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

# -------------#
#      H3      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H3", colnames(corr))
corr_H3 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H3 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[3],
  n.I = corr_H3[, ncol(corr_H3)]^2,
  alpha = alpha * w_H13[2], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

ans_new <- tibble(
  Analysis = 1:2,
  `Interaction/Elementary hypotheses` = "H1, H3",
  `H1 p-value boundary` = pval_H1,
  `H2 p-value boundary` = NA,
  `H3 p-value boundary` = pval_H3
)
ans_new %>% gt()

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary
1	H1, H3	0.001277174	NA	0.001702899
2	H1, H3	0.010079863	NA	0.013489389

ans <- rbind(ans, ans_new)

Boundary of \(H_2 \cap H_3\)

For the interaction hypothesis \(H_2 \cap H_3\), its weight is

w_H23 <- inter_weight %>% filter(is.na(H1), !is.na(H2), !is.na(H3))
w_H23 <- w_H23[(!is.na(w_H23))] # Remove NA from weight
w_H23

## [1] 0.4285714 0.5714286

And the boundary for \(H_2\) and \(H_3\) are

# -------------#
#      H2      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H2", colnames(corr))
corr_H2 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H2 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[2],
  n.I = corr_H2[, ncol(corr_H2)]^2,
  alpha = alpha * w_H23[1], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

# -------------#
#      H3      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H3", colnames(corr))
corr_H3 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H3 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[3],
  n.I = corr_H3[, ncol(corr_H3)]^2,
  alpha = alpha * w_H23[2], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

ans_new <- tibble(
  Analysis = 1:2,
  `Interaction/Elementary hypotheses` = "H2, H3",
  `H1 p-value boundary` = NA,
  `H2 p-value boundary` = pval_H2,
  `H3 p-value boundary` = pval_H3
)
ans_new %>% gt()

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary
1	H2, H3	NA	0.001277174	0.001702899
2	H2, H3	NA	0.010079863	0.013489389

ans <- rbind(ans, ans_new)

Boundary of \(H1 \cap H_2 \cap H_3\)

For the interaction hypothesis \(H_1 \cap H_2\), its weight is

w_H123 <- inter_weight %>% filter(!is.na(H1), !is.na(H2), !is.na(H3))
w_H123 <- w_H123[(!is.na(w_H123))] # Remove NA from weight
w_H123

## [1] 0.3 0.3 0.4

And the boundary for \(H_1\), \(H_2\), and \(H_3\) are

# -------------#
#      H1      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H1", colnames(corr))
corr_H1 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H1 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[1],
  n.I = corr_H1[, ncol(corr_H1)]^2,
  alpha = alpha * w_H123[1], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

# -------------#
#      H2      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H2", colnames(corr))
corr_H2 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H2 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[2],
  n.I = corr_H2[, ncol(corr_H2)]^2,
  alpha = alpha * w_H123[1], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

# -------------#
#      H3      #
# -------------#
# Index to select from the correlation matrix
indx <- grep("H3", colnames(corr))
corr_H3 <- corr[indx, indx]

# Boundary for a single hypothesis across k for the intersection hypothesis
pval_H3 <- 1 - pnorm(gsDesign::gsDesign(
  k = k,
  test.type = 1,
  usTime = IF_IA[3],
  n.I = corr_H3[, ncol(corr_H3)]^2,
  alpha = alpha * w_H123[3], # alpha is different since the weight is updated
  sfu = sfHSD,
  sfupar = -4
)$upper$bound)

ans_new <- tibble(
  Analysis = 1:2,
  `Interaction/Elementary hypotheses` = "H1, H2, H3",
  `H1 p-value boundary` = pval_H1,
  `H2 p-value boundary` = pval_H2,
  `H3 p-value boundary` = pval_H3
)
ans_new %>% gt()

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary
1	H1, H2, H3	0.0008940219	0.0008940219	0.001192029
2	H1, H2, H3	0.0070254979	0.0070254979	0.009399818

ans <- rbind(ans, ans_new)

Summary

With the p-value boundaries, one can get the Z-statistics boundaries by qnorm().

ans %>%
  mutate(
    `H1 Z-statistics boundary` = -qnorm(`H1 p-value boundary`),
    `H1 Z-statistics boundary` = -qnorm(`H2 p-value boundary`),
    `H1 Z-statistics boundary` = -qnorm(`H3 p-value boundary`)
  ) %>%
  arrange(Analysis, `Interaction/Elementary hypotheses`) %>%
  gt() %>%
  tab_header("p-values/Z-statistics boundaries of weighted Bonferroni")

Analysis	Interaction/Elementary hypotheses	H1 p-value boundary	H2 p-value boundary	H3 p-value boundary	H1 Z-statistics boundary
p-values/Z-statistics boundaries of weighted Bonferroni
1	H1	0.0029800731	NA	NA	NA
1	H1, H2	0.0014900365	0.0014900365	NA	NA
1	H1, H2, H3	0.0008940219	0.0008940219	0.001192029	3.037681
1	H1, H3	0.0012771742	NA	0.001702899	2.928520
1	H2	NA	0.0029800731	NA	NA
1	H2, H3	NA	0.0012771742	0.001702899	2.928520
1	H3	NA	NA	0.002980073	2.749966
2	H1	0.0237882657	NA	NA	NA
2	H1, H2	0.0117828003	0.0117828003	NA	NA
2	H1, H2, H3	0.0070254979	0.0070254979	0.009399818	2.349480
2	H1, H3	0.0100798631	NA	0.013489389	2.211825
2	H2	NA	0.0237882657	NA	NA
2	H2, H3	NA	0.0100798631	0.013489389	2.211825
2	H3	NA	NA	0.023788266	1.981131

Implementation in wpgsd

The above results can be computed in one function call in wpgsd by using the generate_bounds() function as

generate_bounds(
  type = 0,
  k = 2,
  w = w,
  m = m,
  corr = corr,
  alpha = 0.025,
  sf = list(sfHSD, sfHSD, sfHSD),
  sfparm = list(-4, -4, -4),
  t = list(c(0.5, 1), c(0.5, 1), c(0.5, 1))
) %>% gt()

Analysis	Hypotheses	H1	H2	H3
1	H1	0.0029800731	NA	NA
1	H1, H2	0.0014900365	0.0014900365	NA
1	H1, H2, H3	0.0008940219	0.0008940219	0.001192029
1	H1, H3	0.0012771742	NA	0.001702899
1	H2	NA	0.0029800731	NA
1	H2, H3	NA	0.0012771742	0.001702899
1	H3	NA	NA	0.002980073
2	H1	0.0237882657	NA	NA
2	H1, H2	0.0117828003	0.0117828003	NA
2	H1, H2, H3	0.0070254979	0.0070254979	0.009399818
2	H1, H3	0.0100798631	NA	0.013489389
2	H2	NA	0.0237882657	NA
2	H2, H3	NA	0.0100798631	0.013489389
2	H3	NA	NA	0.023788266

Yujie Zhao

Example overview

Observed p-values

Information fraction

Initial weight and transition matrix

Correlations

Boundary calculation

Boundary of \(H_1\)

Boundary of \(H_2\)

Boundary of \(H_3\)

Boundary of \(H_1 \cap H_2\)

Boundary of \(H_1 \cap H_3\)

Boundary of \(H_2 \cap H_3\)

Boundary of \(H1 \cap H_2 \cap H_3\)

Summary

Implementation in wpgsd