Obtains the maximum information and stopping boundaries for a generic group sequential equivalence design assuming a constant treatment effect, or obtains the power given the maximum information and stopping boundaries.
getDesignEquiv(
beta = NA_real_,
IMax = NA_real_,
thetaLower = NA_real_,
thetaUpper = NA_real_,
theta = 0,
kMax = 1L,
informationRates = NA_real_,
criticalValues = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_
)An S3 class designEquiv object with three components:
overallResults: A data frame containing the following variables:
overallReject: The overall rejection probability.
alpha: The overall significance level.
attainedAlphaH10: The attained significance level under H10.
attainedAlphaH20: The attained significance level under H20.
kMax: The number of stages.
thetaLower: The parameter value at the lower equivalence
limit.
thetaUpper: The parameter value at the upper equivalence
limit.
theta: The parameter value under the alternative hypothesis.
information: The maximum information.
expectedInformationH1: The expected information under H1.
expectedInformationH10: The expected information under H10.
expectedInformationH20: The expected information under H20.
byStageResults: A data frame containing the following variables:
informationRates: The information rates.
efficacyBounds: The efficacy boundaries on the Z-scale for
each of the two one-sided tests.
rejectPerStage: The probability for efficacy stopping.
cumulativeRejection: The cumulative probability for efficacy
stopping.
cumulativeAlphaSpent: The cumulative alpha for each of
the two one-sided tests.
cumulativeAttainedAlphaH10: The cumulative probability for
efficacy stopping under H10.
cumulativeAttainedAlphaH20: The cumulative probability for
efficacy stopping under H20.
efficacyThetaLower: The efficacy boundaries on the
parameter scale for the one-sided null hypothesis at the
lower equivalence limit.
efficacyThetaUpper: The efficacy boundaries on the
parameter scale for the one-sided null hypothesis at the
upper equivalence limit.
efficacyP: The efficacy bounds on the p-value scale for
each of the two one-sided tests.
information: The cumulative information.
settings: A list containing the following components:
typeAlphaSpending: The type of alpha spending.
parameterAlphaSpending: The parameter value for alpha
spending.
userAlphaSpending: The user defined alpha spending.
spendingTime: The error spending time at each analysis.
The type II error.
The maximum information. Either beta or IMax
should be provided while the other one should be missing.
The parameter value at the lower equivalence limit.
The parameter value at the upper equivalence limit.
The parameter value under the alternative hypothesis.
The maximum number of stages.
The information rates. Fixed prior to the trial.
Defaults to (1:kMax) / kMax if left unspecified.
Upper boundaries on the z-test statistic scale for stopping for efficacy.
The significance level for each of the two one-sided tests, e.g., 0.05.
The type of alpha spending. One of the following: "OF" for O'Brien-Fleming boundaries, "P" for Pocock boundaries, "WT" for Wang & Tsiatis boundaries, "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early efficacy stopping. Defaults to "sfOF".
The parameter value for the alpha spending. Corresponds to Delta for "WT", rho for "sfKD", and gamma for "sfHSD".
The user defined alpha spending. Cumulative alpha spent up to each stage.
A vector of length kMax for the error spending
time at each analysis. Defaults to missing, in which case, it is the
same as informationRates.
Kaifeng Lu, kaifenglu@gmail.com
Consider the equivalence design with two one-sided hypotheses: $$H_{10}: \theta \leq \theta_{10},$$ $$H_{20}: \theta \geq \theta_{20}.$$ We reject \(H_{10}\) at or before look \(k\) if $$Z_{1j} = (\hat{\theta}_j - \theta_{10})\sqrt{I_j} \geq b_j$$ for some \(j=1,\ldots,k\), where \(\{b_j:j=1,\ldots,K\}\) are the critical values associated with the specified alpha-spending function, and \(I_j\) is the information for \(\theta\) (inverse variance of \(\hat{\theta}\) under the alternative hypothesis) at the \(j\)th look. For example, for estimating the risk difference \(\theta = \pi_1 - \pi_2\), $$I_j = \left\{\frac{\pi_1 (1-\pi_1)}{n_{1j}} + \frac{\pi_2(1-\pi_2)}{n_{2j}}\right\}^{-1}.$$ It follows that $$(Z_{1j} \geq b_j) = (Z_j \geq b_j + (\theta_{10}-\theta)\sqrt{I_j}),$$ where \(Z_j = (\hat{\theta}_j - \theta)\sqrt{I_j}\).
Similarly, we reject \(H_{20}\) at or before look \(k\) if $$Z_{2j} = (\hat{\theta}_j - \theta_{20})\sqrt{I_j} \leq -b_j$$ for some \(j=1,\ldots,k\). We have $$(Z_{2j} \leq -b_j) = (Z_j \leq - b_j + (\theta_{20}-\theta)\sqrt{I_j}).$$
Let \(l_j = b_j + (\theta_{10}-\theta)\sqrt{I_j}\), and \(u_j = -b_j + (\theta_{20}-\theta)\sqrt{I_j}\). The cumulative probability to reject \(H_0 = H_{10} \cup H_{20}\) at or before look \(k\) under the alternative hypothesis \(H_1\) is given by $$P_\theta\left(\cup_{j=1}^{k} (Z_{1j} \geq b_j) \cap \cup_{j=1}^{k} (Z_{2j} \leq -b_j)\right) = p_1 + p_2 + p_{12},$$ where $$p_1 = P_\theta\left(\cup_{j=1}^{k} (Z_{1j} \geq b_j)\right) = P_\theta\left(\cup_{j=1}^{k} (Z_j \geq l_j)\right),$$ $$p_2 = P_\theta\left(\cup_{j=1}^{k} (Z_{2j} \leq -b_j)\right) = P_\theta\left(\cup_{j=1}^{k} (Z_j \leq u_j)\right),$$ and $$p_{12} = P_\theta\left(\cup_{j=1}^{k} (Z_j \geq l_j) \cup (Z_j \leq u_j)\right).$$ Of note, both \(p_1\) and \(p_2\) can be evaluated using one-sided exit probabilities for group sequential designs. If there exists \(j\leq k\) such that \(l_j \leq u_j\), then \(p_{12} = 1\). Otherwise, \(p_{12}\) can be evaluated using two-sided exit probabilities for group sequential designs.
Since the equivalent hypothesis is tested using two one-sided tests, the type I error is controlled. To evaluate the attained type I error of the equivalence trial under \(H_{10}\) (or \(H_{20}\)), we simply fix the control group parameters, update the active treatment group parameters according to the null hypothesis, and use the parameters in the power calculation outlined above.
# Example 1: obtain the maximum information given power
(design1 <- getDesignEquiv(
beta = 0.2, thetaLower = log(0.8), thetaUpper = log(1.25),
kMax = 2, informationRates = c(0.5, 1),
alpha = 0.05, typeAlphaSpending = "sfOF"))
# Example 2: obtain power given the maximum information
(design2 <- getDesignEquiv(
IMax = 72.5, thetaLower = log(0.7), thetaUpper = -log(0.7),
kMax = 3, informationRates = c(0.5, 0.75, 1),
alpha = 0.05, typeAlphaSpending = "sfOF"))
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