OncoBayes2 (version 0.9-3)

example-combo2: Two-drug combination example

Description

Example using a combination of two experimental drugs.

Details

The following example is described in the reference Neuenschwander, B. et al (2016). The data are described in the help page for codata_combo2. In the study trial_AB, the risk of DLT was studied as a function of dose for two drugs, drug A and drug B. Historical information on the toxicity profiles of these two drugs was available from single agent trials trial_A and trial_B. Another study IIT was run concurrently to trial_AB, and studies the same combination.

The model described in Neuenschwander, et al (2016) is adapted as follows. For groups \(j = 1,\ldots, 4\) representing each of the four sources of data mentioned above, $$\text{logit}\, \pi_{1j}(d_1) = \log\, \alpha_{1j} + \beta_{1j} \, \log\, \Bigl(\frac{d_1}{d_1^*}\Bigr),$$ and $$\text{logit}\, \pi_{2j}(d_2) = \log\, \alpha_{2j} + \beta_{2j} \, \log\, \Bigl(\frac{d_2}{d_2^*}\Bigr),$$ are logistic regressions for the single-agent toxicity of drugs A and B, respectively, when administered in group \(j\). Conditional on the regression parameters \(\boldsymbol\theta_{1j} = (\log \, \alpha_{1j}, \log \, \beta_{1j})\) and \(\boldsymbol\theta_{2j} = (\log \, \alpha_{2j}, \log \, \beta_{2j})\), the toxicity \(\pi_{j}(d_1, d_2)\) for the combination is modeled as the "no-interaction" DLT rate, $$\tilde\pi_{j}(d_1, d_2) = 1 - (1-\pi_{1j}(d_1) )(1- \pi_{2j}(d_2))$$ with a single interaction term added on the log odds scale, $$\text{logit} \, \pi_{j}(d_1, d_2) = \text{logit} \, \tilde\pi_{j}(d_1, d_2) + \eta_j \frac{d_1}{d_1^*}\frac{d_2}{d_2^*}.$$ A hierarchical model across the four groups \(j\) allows dose-toxicity information to be shared through common hyperparameters.

For the component parameters \(\boldsymbol\theta_{ij}\), $$\boldsymbol\theta_{ij} \sim \text{BVN}(\boldsymbol \mu_i, \boldsymbol\Sigma_i).$$ For the mean, a further prior is specified as $$\boldsymbol\mu_i = (\mu_{\alpha i}, \mu_{\beta i}) \sim \text{BVN}(\boldsymbol m_i, \boldsymbol S_i),$$ while in the manuscript the prior \(\boldsymbol m_i = (\text{logit}\, 0.1, \log 1)\) and \(\boldsymbol S_i = \text{diag}(3.33^2, 1^2)\) for each \(i = 1,2\) is used, we deviate here and use instead \(\boldsymbol m_i = (\text{logit}\, 0.2, \log 1)\) and \(\boldsymbol S_i = \text{diag}(2^2, 1^2)\). For the standard deviations and correlation parameters in the covariance matrix, $$\boldsymbol\Sigma_i = \left( \begin{array}{cc} \tau^2_{\alpha i} & \rho_i \tau_{\alpha i} \tau_{\beta i}\\ \rho_i \tau_{\alpha i} \tau_{\beta i} & \tau^2_{\beta i} \end{array} \right), $$ the specified priors are \(\tau_{\alpha i} \sim \text{Log-Normal}(\log\, 0.25, ((\log 4) / 1.96)^2)\),

\(\tau_{\beta i} \sim \text{Log-Normal}(\log\, 0.125, ((\log 4) / 1.96)^2)\), and \(\rho_i \sim \text{U}(-1,1)\) for \(i = 1,2\).

For the interaction parameters \(\eta_j\) in each group, the hierarchical model has $$\eta_j \sim \text{N}(\mu_\eta, \tau^2_\eta),$$ for \(j = 1,\ldots, 4\), with \(\mu_\eta \sim \text{N}(0, 1.121^2)\) and \(\tau_\eta \sim \text{Log-Normal}(\log\, 0.125, ((\log 4) / 1.96)^2).\)

Below is the syntax for specifying this fully exchangeable model in blrm_exnex.

References

Neuenschwander, B., Roychoudhury, S., & Schmidli, H. (2016). On the use of co-data in clinical trials. Statistics in Biopharmaceutical Research, 8(3), 345-354.

Examples

## Setting up dummy sampling for fast execution of example
## Please use 4 chains and 100x more warmup & iter in practice
.user_mc_options <- options(
  OncoBayes2.MC.warmup = 10, OncoBayes2.MC.iter = 20, OncoBayes2.MC.chains = 1,
  OncoBayes2.MC.save_warmup = FALSE
)

dref <- c(6, 960)

num_comp   <- 2 # two investigational drugs
num_inter  <- 1 # one drug-drug interaction needs to be modeled
num_groups <- nlevels(codata_combo2$group_id) # no stratification needed
num_strata <- 1 # no stratification needed

blrmfit <- blrm_exnex(
  cbind(num_toxicities, num_patients - num_toxicities) ~
    1 + I(log(drug_A / dref[1])) |
      1 + I(log(drug_B / dref[2])) |
      0 + I(drug_A / dref[1] * drug_B / dref[2]) |
      group_id,
  data = codata_combo2,
  prior_EX_mu_comp  = list(mixmvnorm(c(1, logit(0.2), 0, diag(c(2^2, 1)))),
                           mixmvnorm(c(1, logit(0.2), 0, diag(c(2^2, 1))))),
  prior_EX_tau_comp = list(mixmvnorm(c(1,
                                       log(0.250), log(0.125),
                                       diag(c(log(4)/1.96, log(4)/1.96)^2))),
                           mixmvnorm(c(1,
                                       log(0.250), log(0.125),
                                       diag(c(log(4)/1.96, log(4)/1.96)^2)))),
  prior_EX_mu_inter = mixmvnorm(c(1, 0, 1.121^2)),
  prior_EX_tau_inter = mixmvnorm(c(1, log(0.125), (log(4) / 1.96)^2)),
  prior_is_EXNEX_comp = rep(FALSE, num_comp),
  prior_is_EXNEX_inter = rep(FALSE, num_inter),
  prior_EX_prob_comp = matrix(1, nrow = num_groups, ncol = num_comp),
  prior_EX_prob_inter = matrix(1, nrow = num_groups, ncol = num_inter),
  prior_tau_dist = 1
)
## Recover user set sampling defaults
options(.user_mc_options)