example-combo2 {OncoBayes2}

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,

\mbox{logit}\, \pi_{1j}(d_1) = \log\, \alpha_{1j} + \beta_{1j} \, \log\, \Bigl(\frac{d_1}{d_1^*}\Bigr),

and

\mbox{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,

\mbox{logit} \, \pi_{j}(d_1, d_2) = \mbox{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 \mbox{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 \mbox{BVN}(\boldsymbol m_i, \boldsymbol S_i),

while in the manuscript the prior \boldsymbol m_i = (\mbox{logit}\, 0.1, \log 1) and \boldsymbol S_i = \mbox{diag}(3.33^2, 1^2) for each i = 1,2 is used, we deviate here and use instead \boldsymbol m_i = (\mbox{logit}\, 0.2, \log 1) and \boldsymbol S_i = \mbox{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 \mbox{Log-Normal}(\log\, 0.25, ((\log 4) / 1.96)^2),

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

For the interaction parameters \eta_j in each group, the hierarchical model has

\eta_j \sim \mbox{N}(\mu_\eta, \tau^2_\eta),

for j = 1,\ldots, 4, with \mu_\eta \sim \mbox{N}(0, 1.121^2) and \tau_\eta \sim \mbox{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_mean_comp = matrix(
     c(logit(0.2), 0,  # hyper-mean of (intercept, log-slope) for drug A
       logit(0.2), 0), # hyper-mean of (intercept, log-slope) for drug B
       nrow = num_comp,
       ncol = 2,
       byrow = TRUE
      ),
  prior_EX_mu_sd_comp = matrix(
     c(2.0, 1,  # hyper-sd of mean mu for (intercept, log-slope) for drug A
       2.0, 1), # hyper-sd of mean mu for (intercept, log-slope) for drug B
       nrow = num_comp,
       ncol = 2,
       byrow = TRUE
      ),
  prior_EX_tau_mean_comp = matrix(
     c(log(0.25), log(0.125),
       log(0.25), log(0.125)),
       nrow = num_comp,
       ncol = 2,
       byrow = TRUE
      ),
  prior_EX_tau_sd_comp = matrix(
     c(log(4) / 1.96, log(4) / 1.96,
       log(4) / 1.96, log(4) / 1.96),
       nrow = num_comp,
       ncol = 2,
       byrow = TRUE
      ),
  prior_EX_mu_mean_inter = 0,
  prior_EX_mu_sd_inter = 1.121,
  prior_EX_tau_mean_inter = matrix(log(0.125), nrow = num_strata, ncol = num_inter),
  prior_EX_tau_sd_inter = matrix(log(4) / 1.96, nrow = num_strata, ncol = num_inter),
  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)