Design with two continuous endpoints

Specifies a proof-of-concept study with two continuous endpoints: the sample sizes, the prior distributions, the external data, the form of the control group, the decision criterion for two endpoints and the method used to compute the probabilities. The design is used by qdm_decide, qdm_oc and qdm_calibrate.

Usage

design_continuous2(
  n_t,
  n_c,
  prior_t,
  prior_c = NULL,
  criterion,
  external_t = NULL,
  external_c = NULL,
  hypothetical_c = NULL,
  fixed_c = NULL,
  method = c("MM", "MC"),
  nMC = 10000L
)

Arguments

n_t: Number of subjects in the treatment group (a whole number of at least 3).
n_c: Number of subjects in the concurrent control group: 0 or a whole number of at least 3.
prior_t, prior_c: Prior distributions of the treatment and control groups, created by prior_niw or prior_vague. For a hypothetical or fixed control, prior_c must be NULL (the default).
criterion: Decision criterion for two endpoints, created by criterion_posterior or criterion_predictive. Its argument direction sets whether larger or smaller effects are favorable on each endpoint.
external_t, external_c: External data of the treatment and control groups, created by borrow with data_continuous2, or NULL (no external data).
hypothetical_c: Hypothetical control, created by hypothetical with mean of length 2 and r, or NULL. It can be used only when n_c = 0.
fixed_c: Fixed control means of endpoints 1 and 2, a numeric vector of length 2, or NULL. It can be used only when n_c = 0.
method: Method used to compute the probabilities: "MM" (default) or "MC".
nMC: Number of Monte Carlo draws for method = "MC".

Value

An object of class c("qdm_design_continuous2", "qdm_design"), a list with the arguments and the element control ("concurrent", "external", "hypothetical" or "fixed").

Details

The responses of a subject on the two endpoints have a bivariate normal distribution with mean vector \(\mu_k\) and covariance matrix \(\Sigma_k\) in group \(k\) (Yamaguchi et al., submitted, Appendix A.2). The prior distribution of \((\mu_k, \Sigma_k)\) is Normal-Inverse-Wishart (prior_niw) or vague (prior_vague, the density \(|\Sigma_k|^{-3/2}\)). External data enter through a power prior with weight \(a_0\), which has the conjugate form, and the posterior parameters are updated with the external data and then with the current data. The vague prior corresponds to \(\kappa_0 = 0\), \(\nu_0 = -1\) and \(\Lambda_0 = 0\), so the posterior distribution of the mean is \(t_{n - 2}(\bar y, S / (n (n - 2)))\) without external data, where \(S\) is the matrix of sums of squares and cross-products, \(n - 1\) times the sample covariance matrix (Yamaguchi et al., submitted, equation A.9).

The treatment effects are the differences of the means, \(\theta_k = \mu_{tk} - \mu_{ck}\). The mean of each group has a bivariate t distribution with \(\nu_n - 1\) degrees of freedom, location \(\mu_n\) and scale matrix \(\Lambda_n / (\kappa_n (\nu_n - 1))\). For a predictive criterion, the mean of \(m\) future subjects has the same distribution with the scale matrix \((1 / \kappa_n + 1 / m) \Lambda_n / (\nu_n - 1)\), because the future subjects share the same mean and covariance matrix (version 0.1.0 and Yamaguchi et al. used the scale matrix \((1 + \kappa_n) \Lambda_n / (\kappa_n (\nu_n - 1) m)\), which treats them as independent draws from the predictive distribution of one subject). A hypothetical control (hypothetical(mean = c(m1, m2), r = r)) gives the control mean the distribution of the treatment mean with location mean and the scale matrix multiplied by r (Yamaguchi et al., Section A.2.6, with the hyperparameters of the treatment group). With external control data only (n_c = 0 and external_c) and the vague prior, \(a_0 n_e\) must be greater than 2. A fixed control (n_c = 0 and fixed_c, the known control means of the two endpoints) shifts the distribution of the treatment mean, so the effect has a bivariate t distribution and the probabilities of the regions are computed exactly from its distribution function; method, nMC and m_c are then not used, and prior_c must be NULL.

The probabilities of the regions are computed by one of two methods:

"MM": moment-matching approximation: the effect is approximated by a bivariate t distribution with the same mean, covariance matrix and radial fourth moment (Yamaguchi, Homma, Takeda and Maruo, submitted, Theorem 3), whose rectangle probabilities are computed by numerical integration for real degrees of freedom. It requires more than 4 degrees of freedom for both groups and is the fastest method.
"MC": Monte Carlo simulation with nMC draws from each distribution; the probabilities of all regions are estimated from the same draws. The computing time is proportional to nMC, so "MM" is much faster for operating characteristics, which need the probabilities for many simulated studies.

The operating characteristics are computed by simulating the sample means and the sums of squares and cross-products of nsim studies (see qdm_oc).

References

Yamaguchi, Y., Hu, H., Huang, Y., Han, C., Homma, G. and Wang, A. (submitted). Bivariate Bayesian quantitative decision-making for two endpoints in rare disease drug development.

Yamaguchi, Y., Homma, G., Takeda, K. and Maruo, K. (submitted). Moment-matching approximation for t-differences in clinical trials with continuous endpoints.

Examples

# Case study 2 of Yamaguchi et al. (submitted)
L0 <- matrix(c(80^2, 15, 15, 6^2), 2, 2)
des <- design_continuous2(
  n_t = 30, n_c = 30,
  prior_t = prior_niw(mu0 = c(0, 1), kappa0 = 0.1, nu0 = 4, Lambda0 = L0),
  prior_c = prior_niw(mu0 = c(-10, -1), kappa0 = 0.1, nu0 = 4,
                      Lambda0 = L0),
  criterion = criterion_posterior(theta_tv = c(10, 2),
                                  theta_mav = c(5, 1),
                                  gamma_go = 0.296, gamma_nogo = 0.244)
)
des
#> Design: two continuous endpoints
#>   Treatment: n = 30, prior N-Inv-Wishart(mu0 = (0, 1), kappa0 = 0.1, nu0 = 4, Lambda0 = (6400, 15; 15, 36))
#>   Control: concurrent, n = 30, prior N-Inv-Wishart(mu0 = (-10, -1), kappa0 = 0.1, nu0 = 4, Lambda0 = (6400, 15; 15, 36))
#> Posterior probability criterion
#>   Go:   P(regions {1} | data) >= 0.296
#>   NoGo: P(regions {9} | data) >= 0.244
#>   TV = (10, 2), MAV = (5, 1)
#> Probabilities: moment-matching approximation
qdm_decide(des,
           data_t = data_continuous2(30, c(5, 2),
                                     matrix(c(6400, 190, 190, 36), 2, 2)),
           data_c = data_continuous2(30, c(-10, -1),
                                     matrix(c(5600, 150, 150, 42), 2, 2)))
#> Decision: Go
#>   Go:   P(regions {1} | data) = 0.4884 (threshold 0.296, met)
#>   NoGo: P(regions {9} | data) = 0.05469 (threshold 0.244, not met)
#> Region probabilities: R1 = 0.4884, R2 = 0.07602, R3 = 0.03684, R4 = 0.06708, R5 = 0.01735, R6 = 0.01054, R7 = 0.1825, R8 = 0.06663, R9 = 0.05469
#> Posterior distributions
#>   Treatment: mean ~ t(df = 33, location = (4.983, 1.997), scale = (193.298, 5.563; 5.563, 1.087))
#>   Control:   mean ~ t(df = 33, location = (-10, -1), scale = (169.939, 4.394; 4.394, 1.262))