getGroupSequentialProbabilities {rpact}

Get Group Sequential Probabilities

Description

Calculates probabilities in the group sequential setting.

Usage

getGroupSequentialProbabilities(decisionMatrix, informationRates)

Arguments

decisionMatrix

A matrix with either 2 or 4 rows and kMax = length(informationRates) columns, see details.

informationRates

The information rates t_1, ..., t_kMax (that must be fixed prior to the trial), default is (1:kMax) / kMax. For the weighted inverse normal design, the weights are derived through w_1 = sqrt(t_1), and w_k = sqrt(t_k - t_(k-1)). For the weighted Fisher's combination test, the weights (scales) are w_k = sqrt((t_k - t_(k-1)) / t_1) (see the documentation).

Details

Given a sequence of information rates (fixing the correlation structure), and decisionMatrix with either 2 or 4 rows and kMax = length(informationRates) columns, this function calculates a probability matrix containing, for two rows, the probabilities:
P(Z_1 <- l_1), P(l_1 <- Z_1 < u_1, Z_2 < l_1),..., P(l_kMax-1 <- Z_kMax-1 < u_kMax-1, Z_kMax < l_l_kMax)
P(Z_1 <- u_1), P(l_1 <- Z_1 < u_1, Z_2 < u_1),..., P(l_kMax-1 <- Z_kMax-1 < u_kMax-1, Z_kMax < u_l_kMax)
P(Z_1 <- Inf), P(l_1 <- Z_1 < u_1, Z_2 < Inf),..., P(l_kMax-1 <- Z_kMax-1 < u_kMax-1, Z_kMax < Inf)
with continuation matrix
l_1,...,l_kMax
u_1,...,u_kMax
For 4 rows, the continuation region contains of two regions and the probability matrix is obtained analogously (cf., Wassmer and Brannath, 2016).

Value

Returns a numeric matrix containing the probabilities described in the details section.

See Also

Other design functions: getDesignCharacteristics(), getDesignConditionalDunnett(), getDesignFisher(), getDesignGroupSequential(), getDesignInverseNormal(), getPowerAndAverageSampleNumber()

Examples

# Calculate Type I error rates in the two-sided group sequential setting when
# performing kMax interim stages with constant critical boundaries at level alpha:
alpha <- 0.05
kMax <- 10
decisionMatrix <- matrix(c(
    rep(-qnorm(1 - alpha / 2), kMax),
    rep(qnorm(1 - alpha / 2), kMax)
), nrow = 2, byrow = TRUE)
informationRates <- (1:kMax) / kMax
probs <- getGroupSequentialProbabilities(decisionMatrix, informationRates)
cumsum(probs[3, ] - probs[2, ] + probs[1, ])

# Do the same for a one-sided design without futility boundaries:
decisionMatrix <- matrix(c(
    rep(-Inf, kMax),
    rep(qnorm(1 - alpha), kMax)
), nrow = 2, byrow = TRUE)
informationRates <- (1:kMax) / kMax
probs <- getGroupSequentialProbabilities(decisionMatrix, informationRates)
cumsum(probs[3, ] - probs[2, ])

# Check that two-sided Pampallona and Tsiatis boundaries with binding 
# futility bounds obtain Type I error probabilities equal to alpha:
x <- getDesignGroupSequential(
    alpha = 0.05, beta = 0.1, kMax = 3, typeOfDesign = "PT",
    deltaPT0 = 0, deltaPT1 = 0.4, sided = 2, bindingFutility = TRUE
)
dm <- matrix(c(
    -x$criticalValues, -x$futilityBounds, 0,
    x$futilityBounds, 0, x$criticalValues
), nrow = 4, byrow = TRUE)
dm[is.na(dm)] <- 0
probs <- getGroupSequentialProbabilities(
    decisionMatrix = dm, informationRates = (1:3) / 3
)
sum(probs[5, ] - probs[4, ] + probs[1, ])

# Check the Type I error rate decrease when using non-binding futility bounds:
x <- getDesignGroupSequential(
    alpha = 0.05, beta = 0.1, kMax = 3, typeOfDesign = "PT",
    deltaPT0 = 0, deltaPT1 = 0.4, sided = 2, bindingFutility = FALSE
)
dm <- matrix(c(
    -x$criticalValues, -x$futilityBounds, 0,
    x$futilityBounds, 0, x$criticalValues
), nrow = 4, byrow = TRUE)
dm[is.na(dm)] <- 0
probs <- getGroupSequentialProbabilities(
    decisionMatrix = dm, informationRates = (1:3) / 3
)
sum(probs[5, ] - probs[4, ] + probs[1, ])