Package 'OncoBayes2'

Bind rows of multiple data frames with zero fill

Description

A version of bind_rows out of dplyr that fills non-common columns with zeroes instead of NA. Gives an error if any of the input data contains NA already.

Usage

bind_rows_0(...)

Arguments

...	Data frames to combine, passed into bind_rows (see dplyr documentation)

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
)

library(tibble)

dose_info_A <- tibble(
  group_id = "hist_A",
  drug_A = 1
)

dose_info_B <- tibble(
  group_id = "hist_B",
  drug_B = 100 * (1:2)
)

bind_rows_0(dose_info_A, dose_info_B)

## Recover user set sampling defaults
options(.user_mc_options)

---

Bayesian Logistic Regression Model for N-compounds with EXNEX

Description

Bayesian Logistic Regression Model (BLRM) for N compounds using EXchangability and NonEXchangability (EXNEX) modeling.

Usage

blrm_exnex(
  formula,
  data,
  prior_EX_mu_comp,
  prior_EX_mu_mean_comp,
  prior_EX_mu_sd_comp,
  prior_EX_tau_comp,
  prior_EX_tau_mean_comp,
  prior_EX_tau_sd_comp,
  prior_EX_corr_eta_comp,
  prior_EX_mu_inter,
  prior_EX_mu_mean_inter,
  prior_EX_mu_sd_inter,
  prior_EX_tau_inter,
  prior_EX_tau_mean_inter,
  prior_EX_tau_sd_inter,
  prior_EX_corr_eta_inter,
  prior_is_EXNEX_inter,
  prior_is_EXNEX_comp,
  prior_EX_prob_comp,
  prior_EX_prob_inter,
  prior_NEX_mu_comp,
  prior_NEX_mu_mean_comp,
  prior_NEX_mu_sd_comp,
  prior_NEX_mu_inter,
  prior_NEX_mu_mean_inter,
  prior_NEX_mu_sd_inter,
  prior_tau_dist,
  iter = getOption("OncoBayes2.MC.iter", 2000),
  warmup = getOption("OncoBayes2.MC.warmup", 1000),
  save_warmup = getOption("OncoBayes2.MC.save_warmup", TRUE),
  thin = getOption("OncoBayes2.MC.thin", 1),
  init = getOption("OncoBayes2.MC.init", 0.5),
  chains = getOption("OncoBayes2.MC.chains", 4),
  cores = getOption("mc.cores", 1L),
  control = getOption("OncoBayes2.MC.control", list()),
  backend = getOption("OncoBayes2.MC.backend", "rstan"),
  prior_PD = FALSE,
  verbose = FALSE
)

## S3 method for class 'blrmfit'
print(x, ..., prob = 0.95, digits = 2)

Arguments

formula	the model formula describing the linear predictors of the model. The lhs of the formula is a two-column matrix which are the number of occured events and the number of times no event occured. The rhs of the formula defines the linear predictors for the marginal models for each drug component, then the interaction model and at last the grouping and optional stratum factors of the models. These elements of the formula are separated by a vertical bar. The marginal models must follow a intercept and slope form while the interaction model must not include an interaction term. See the examples below for an example instantiation.
data	optional data frame containing the variables of the model. If not found in data, the variables are taken from environment(formula).
prior_EX_mu_comp	List of bivariate normal mixture priors for intercept and slope parameters $\boldsymbol\mu_i = (\mu_{\alpha i}, \mu_{\beta i})$ of each component. In case of a single drug model, then a mixture prior is accepted as well.
prior_EX_mu_mean_comp, prior_EX_mu_sd_comp	Please use prior_EX_mu_comp instead. Mean and sd for the prior on the mean parameters $\boldsymbol\mu_i = (\mu_{\alpha i}, \mu_{\beta i})$ of each component. Two column matrix (intercept, log-slope) with one row per component.
prior_EX_tau_comp	List of bivariate normal mixture priors for heterogeniety parameter $\boldsymbol\tau_{si} = (\tau_{\alpha s i}, \tau_{\beta s i})$ of each stratum. If no differential discounting is required (i.e. if there is only one stratum $s = 1$), then it suffices to provide a bivariate normal mixture prior instead of a list with just one element.
prior_EX_tau_mean_comp, prior_EX_tau_sd_comp	Please use prior_EX_tau_comp instead. Prior mean and sd for heterogeniety parameter $\boldsymbol\tau_{si} = (\tau_{\alpha s i}, \tau_{\beta s i})$ of each stratum. If no differential discounting is required (i.e. if there is only one stratum $s = 1$), then it is a two-column matrix (intercept, log-slope) with one row per component. Otherwise it is a three-dimensional array whose first dimension indexes the strata, second dimension indexes the components, and third dimension of length two for (intercept, log-slope).
prior_EX_corr_eta_comp	Prior LKJ correlation parameter for each component given as numeric vector. If missing, then a 1 is assumed corresponding to a marginal uniform prior of the correlation.
prior_EX_mu_inter	Multivariate normal mixture prior for interaction parameter vector $\boldsymbol{\mu_{\eta}}$. Dimension must correspond to the number of interactions.
prior_EX_mu_mean_inter, prior_EX_mu_sd_inter	Please use prior_EX_mu_inter instead. Prior mean and sd for population mean parameters $\mu_{\eta k}$ of each interaction parameter. Vector of length equal to the number of interactions.
prior_EX_tau_inter	List of multivariate normal mixture priors for heterogeniety interaction parameter vector $\boldsymbol{\tau_{\eta s}}$ of each stratum. If no differential discounting is required (i.e. if there is only one stratum $s = 1$), then it suffices to provide a mixture prior instead of a list with just one element.
prior_EX_tau_mean_inter, prior_EX_tau_sd_inter	Please use prior_EX_tau_inter instead. Prior mean and sd for heterogeniety parameter $\tau_{\eta s k}$ of each stratum. Matrix with one column per interaction and one row per stratum.
prior_EX_corr_eta_inter	Prior LKJ correlation parameter for interaction given as numeric. If missing, then a 1 is assumed corresponding to a marginal uniform prior of the correlations.
prior_is_EXNEX_inter	Defines if non-exchangability is admitted for a given interaction parameter. Logical vector of length equal to the number of interactions. If missing FALSE is assumed for all interactions.
prior_is_EXNEX_comp	Defines if non-exchangability is admitted for a given component. Logical vector of length equal to the number of components. If missing TRUE is assumed for all components.
prior_EX_prob_comp	Prior probability $p_{ij}$ for exchangability of each component per group. Matrix with one column per component and one row per group. Values must lie in $[0-1]$ range.
prior_EX_prob_inter	Prior probability $p_{\eta k j}$ for exchangability of each interaction per group. Matrix with one column per interaction and one row per group. Values must lie in $[0-1]$ range.
prior_NEX_mu_comp	List of bivariate normal mixture priors $\boldsymbol m_{ij}$ and $\boldsymbol s_{ij} = \mbox{diag}(\boldsymbol S_{ij})$ of each component for non-exchangable case. If missing set to the same prior as given for the EX part. It is required that the specification be the same across groups j.
prior_NEX_mu_mean_comp, prior_NEX_mu_sd_comp	Please use prior_NEX_mu_comp instead. Prior mean $\boldsymbol m_{ij}$ and sd $\boldsymbol s_{ij} = \mbox{diag}(\boldsymbol S_{ij})$ of each component for non-exchangable case. Two column matrix (intercept, log-slope) with one row per component. If missing set to the same prior as given for the EX part. It is required that the specification be the same across groups j.
prior_NEX_mu_inter	Multivariate normal mixture prior (mean $m_{\eta k j}$, sd $s_{\eta k j}$ and covariance) for the interaction parameter vector for non-exchangable case. Dimension must correspond to the number of interactions. If missing set to the same prior as given for the EX part.
prior_NEX_mu_mean_inter, prior_NEX_mu_sd_inter	Please use prior_NEX_mu_inter instead. Prior mean $m_{\eta k j}$ and sd $s_{\eta k j}$ for each interaction parameter for non-exchangable case. Vector of length equal to the number of interactions. If missing set to the same prior as given for the EX part.
prior_tau_dist	Defines the distribution used for heterogeniety parameters. Choices are 0=fixed to it's mean, 1=log-normal, 2=truncated normal.
iter	number of iterations (including warmup).
warmup	number of warmup iterations.
save_warmup	save warmup samples (TRUE / FALSE). Only if set to TRUE, then all random variables are saved in the posterior. This substantially increases the storage needs of the posterior.
thin	period of saving samples.
init	positive number to specify uniform range on unconstrained space for random initialization. See stan.
chains	number of Markov chains.
cores	number of cores for parallel sampling of chains.
control	additional sampler parameters for NuTS algorithm.
backend	sets Stan backend to be used. Possible choices are "rstan" (default) or "cmdstanr".
prior_PD	Logical flag (defaults to FALSE) indicating if to sample the prior predictive distribution instead of conditioning on the data.
verbose	Logical flag (defaults to FALSE) controlling if additional output like stan progress is reported.
x	blrmfit object to print
...	not used in this function
prob	central probability mass to report, i.e. the quantiles 0.5-prob/2 and 0.5+prob/2 are displayed. Multiple central widths can be specified.
digits	number of digits to show

Details

blrm_exnex is a flexible function for Bayesian meta-analytic modeling of binomial count data. In particular, it is designed to model counts of the number of observed dose limiting toxicities (DLTs) by dose, for guiding dose-escalation studies in Oncology. To accommodate dose escalation over more than one agent, the dose may consist of combinations of study drugs, with any number of treatment components.

In the simplest case, the aim is to model the probability $\pi$ that a patient experiences a DLT, by complementing the binomial likelihood with a monotone logistic regression

$\mbox{logit}\,\pi(d) = \log\,\alpha + \beta \, t(d),$

where $\beta > 0$. Most typically, $d$ represents the dose, and $t(d)$ is an appropriate transformation, such as $t(d) = \log (d \big / d^*)$. A joint prior on $\boldsymbol \theta = (\log\,\alpha, \log\,\beta)$ completes the model and ensures monotonicity $\beta > 0$.

Many extensions are possible. The function supports general combination regimens, and also provides framework for Bayesian meta-analysis of dose-toxicity data from multiple historical and concurrent sources.

For an example of a single-agent trial refer to example-single-agent.

Value

The function returns a S3 object of type blrmfit.

Functions

• print(blrmfit): print function.

Combination of two treatments

For a combination of two treatment components, the basic modeling framework is that the DLT rate $\pi(d_1,d_2)$ is comprised of (1) a "no-interaction" baseline model $\tilde \pi(d_1,d_2)$ driven by the single-agent toxicity of each component, and (2) optional interaction terms $\gamma(d_1,d_2)$ representing synergy or antagonism between the drugs. On the log-odds scale,

$\mbox{logit} \,\pi(d_1,d_2) = \mbox{logit} \, \tilde \pi(d_1,d_2) + \eta \, \gamma(d_1,d_2).$

The "no interaction" part $\tilde \pi(d_1,d_2)$ represents the probability of a DLT triggered by either treatment component acting independently. That is,

$\tilde \pi(d_1,d_2) = 1- (1 - \pi_1(d_1))(1 - \pi_2(d_2)).$

In simple terms, P(no DLT for combination) = P(no DLT for drug 1) * P(no DLT from drug 2). To complete this part, the treatment components can then be modeled with monotone logistic regressions as before.

$\mbox{logit} \, \pi_i(d_i) = \log\, \alpha_i + \beta_i \, t(d_i),$

where $t(d_i)$ is a monotone transformation of the doses of the respective drug component $i$, such as $t(d_i) = \log (d_i \big / d_i^*)$.

The inclusion of an interaction term $\gamma(d_1,d_2)$ allows DLT rates above or below the "no-interaction" rate. The magnitude of the interaction term may also be made dependent on the doses (or other covariates) through regression. As an example, one could let

$\gamma(d_1, d_2) = \frac{d_1}{d_1^*} \frac{d_1}{d_2^*}.$

The specific functional form is specified in the usual notation for a design matrix. The interaction model must respect the constraint that whenever any dose approaches zero, then the interaction term must vanish as well. Therefore, the interaction model must not include an intercept term which would violate this consistency requirement. A dual combination example can be found in example-combo2.

General combinations

The model is extended to general combination treatments consisting of $N$ components by expressing the probability $\pi$ on the logit scale as

$\mbox{logit} \, \pi(d_1,\ldots,d_N) = \mbox{logit} \Bigl( 1 - \prod_{i = 1}^N ( 1 - \pi_i(d_i) ) \Bigr) + \sum_{k=1}^K \eta_k \, \gamma_k(d_1,\ldots,d_N),$

Multiple drug-drug interactions among the $N$ components are now possible, and are represented through the $K$ interaction terms $\gamma_k$.

Regression models can be again be specified for each $\pi_i$ and $\gamma_k$, such as

$\mbox{logit}\, \pi_i(d_i) = \log\, \alpha_i + \beta_i \, t(d_i)$

Interactions for some subset $I(k) \subset \{1,\ldots,N \}$ of the treatment components can be modeled with regression as well, for example on products of doses,

$\gamma_k(d_1,\ldots,d_N) = \prod_{i \in I(k)} \frac{d_i}{d_i^*}.$

For example, $I(k) = \{1,2,3\}$ results in the three-way interaction term

$\frac{d_1}{d_1^*} \frac{d_2}{d_2^*} \frac{d_3}{d_3^*}$

for drugs 1, 2, and 3.

For a triple combination example please refer to example-combo3.

Meta-analytic framework

Information on the toxicity of a drug may be available from multiple studies or sources. Furthermore, one may wish to stratify observations within a single study (for example into groups of patients corresponding to different geographic regions, or multiple dosing dose_info corresponding to different schedules).

blrm_exnex provides tools for robust Bayesian hierarchical modeling to jointly model data from multiple sources. An additional index $j=1, \ldots, J$ on the parameters and observations denotes the $J$ groups. The resulting model allows the DLT rate to differ across the groups. The general $N$-component model becomes

$\mbox{logit} \, \pi_j(d_1,\ldots,d_N) = \mbox{logit} \Bigl( 1 - \prod_{i = 1}^N ( 1 - \pi_{ij}(d_i) ) \Bigr) + \sum_{k=1}^K \eta_{kj} \, \gamma_{k}(d_1,\ldots,d_N),$

for groups $j = 1,\ldots,J$. The component toxicities $\pi_{ij}$ and interaction terms $\gamma_{k}$ are modelled, as before, through regression. For example, $\pi_{ij}$ could be a logistic regression on $t(d_i) = \log(d_i/d_i^*)$ with intercept and log-slope $\boldsymbol \theta_{ij}$, and $\gamma_{k}$ regressed with coefficient $\eta_{kj}$ on a product $\prod_{i\in I(k)} (d_i/d_i^*)$ for some subset $I(k)$ of components.

Thus, for $j=1,\ldots,J$, we now have group-specific parameters $\boldsymbol\theta_{ij} = (\log\, \alpha_{ij}, \log\, \beta_{ij})$ and $\boldsymbol\nu_{j} = (\eta_{1j}, \ldots, \eta_{Kj})$ for each component $i=1,\ldots,N$ and interaction $k=1,\ldots,K$.

The structure of the prior on $(\boldsymbol\theta_{i1},\ldots,\boldsymbol\theta_{iJ})$ and $(\boldsymbol\nu_{1}, \ldots, \boldsymbol\nu_{J})$ determines how much information will be shared across groups $j$. Several modeling choices are available in the function.

• EX (Full exchangeability): One can assume the parameters are conditionally exchangeable given hyperparameters

  $\boldsymbol \theta_{ij} \sim \mbox{N}\bigl( \boldsymbol \mu_{\boldsymbol \theta i}, \boldsymbol \Sigma_{\boldsymbol \theta i} \bigr),$

  independently across groups $j = 1,\ldots, J$ and treatment components $i=1,\ldots,N$. The covariance matrix $\boldsymbol \Sigma_{\boldsymbol \theta i}$ captures the patterns of cross-group heterogeneity, and is parametrized with standard deviations $\boldsymbol \tau_{\boldsymbol\theta i} = (\tau_{\alpha i}, \tau_{\beta i})$ and the correlation $\rho_i$. Similarly for the interactions, the fully-exchangeable model is

  $\boldsymbol \nu_{j} \sim \mbox{N}\bigl( \boldsymbol \mu_{\boldsymbol \nu}, \boldsymbol \Sigma_{\boldsymbol \nu} \bigr)$

  for groups $j = 1,\ldots, J$ and interactions $k=1,\ldots,K$, and the prior on the covariance matrix $\boldsymbol \Sigma_{\boldsymbol \nu}$ captures the amount of heterogeneity expected in the interaction terms a-priori. The covariance is again parametrized with standard deviations $(\tau_{\eta 1}, \ldots, \tau_{\eta K})$ and its correlation matrix.

• Differential discounting: For one or more of the groups $j=1,\ldots,J$, larger deviations of $\boldsymbol\theta_{ij}$ may be expected from the mean $\boldsymbol\mu_i$, or of the interactions $\eta_{kj}$ from the mean $\mu_{\eta,k}$. Such differential heterogeneity can be modeled by mapping the groups $j = 1,\ldots,J$ to strata through $s_j \in \{1,\ldots,S\}$, and modifying the model specification to

  $\boldsymbol \theta_{ij} \sim \mbox{N}\bigl( \boldsymbol \mu_{\boldsymbol \theta i}, \boldsymbol \Sigma_{\boldsymbol \theta ij} \bigr),$

  where

  $\boldsymbol \Sigma_{\boldsymbol \theta ij} = \left( \begin{array}{cc} \tau^2_{\alpha s_j i} & \rho_i \tau_{\alpha s_j i} \tau_{\beta s_j i}\\ \rho_i \tau_{\alpha s_j i} \tau_{\beta s_j i} & \tau^2_{\beta s_j i} \end{array} \right).$

  For the interactions, the model becomes

  $\boldsymbol \nu_{j} \sim \mbox{N}\bigl( \boldsymbol \mu_{\boldsymbol \nu}, \boldsymbol \Sigma_{\boldsymbol \nu j} \bigr),$

  where the covariance matrix $\boldsymbol \Sigma_{\boldsymbol \nu j}$ is modelled as stratum specific standard deviations $(\tau_{\eta 1 s_j}, \ldots, \tau_{\eta K s_j})$ and a stratum independent correlation matrix. Each stratum $s=1,\ldots,S$ then corresponds to its own set of standard deviations $\tau$ leading to different discounting per stratum. Independent priors are specified for the component parameters $\tau_{\alpha s i}$ and $\tau_{\beta s i}$ and for the interaction parameters $\tau_{\eta s k}$ for each stratum $s=1,\ldots,S$. Inference for strata $s$ where the prior is centered on larger values of the $\tau$ parameters will exhibit less shrinkage towards the the means, $\boldsymbol\mu_{\boldsymbol \theta i}$ and $\boldsymbol \mu_{\boldsymbol \nu}$ respectively.

• EXNEX (Partial exchangeability): Another mechansim for increasing robustness is to introduce mixture priors for the group-specific parameters, where one mixture component is shared across groups, and the other is group-specific. The result, known as an EXchangeable-NonEXchangeable (EXNEX) type prior, has a form

  $\boldsymbol \theta_{ij} \sim p_{\boldsymbol \theta ij}\, \mbox{N}\bigl( \boldsymbol \mu_{\boldsymbol \theta i}, \boldsymbol \Sigma_{\boldsymbol \theta i} \bigr) +(1-p_{\boldsymbol \theta ij})\, \mbox{N}\bigl(\boldsymbol m_{\boldsymbol \theta ij}, \boldsymbol S_{\boldsymbol \theta ij}\bigr)$

  when applied to the treatment-component parameters, and

  $\boldsymbol \nu_{kj} \sim p_{\boldsymbol \nu_{kj}} \,\mbox{N}\bigl(\mu_{\boldsymbol \nu}, \boldsymbol \Sigma_{\boldsymbol \nu}\bigr)_k + (1-p_{\boldsymbol \nu_{kj}})\, \mbox{N}(m_{\boldsymbol \nu_{kj}}, s^2_{\boldsymbol \nu_{kj}})$

  when applied to the interaction parameters. The exchangeability weights $p_{\boldsymbol \theta ij}$ and $p_{\boldsymbol \nu_{kj}}$ are fixed constants in the interval $[0,1]$ that control the degree to which inference for group $j$ is informed by the exchangeable mixture components. Larger values for the weights correspond to greater exchange of information, while smaller values increase robustness in case of outlying observations in individual groups $j$.

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.

Neuenschwander, B., Wandel, S., Roychoudhury, S., & Bailey, S. (2016). Robust exchangeability designs for early phase clinical trials with multiple strata. Pharmaceutical statistics, 15(2), 123-134.

Neuenschwander, B., Branson, M., & Gsponer, T. (2008). Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in medicine, 27(13), 2420-2439.

Neuenschwander, B., Matano, A., Tang, Z., Roychoudhury, S., Wandel, S. Bailey, Stuart. (2014). A Bayesian Industry Approach to Phase I Combination Trials in Oncology. In Statistical methods in drug combination studies (Vol. 69). CRC Press.

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
)

# fit an example model. See documentation for "combo3" example
example_model("combo3")

# print a summary of the prior
prior_summary(blrmfit, digits = 3)

# print a summary of the posterior (model parameters)
print(blrmfit)

# summary of posterior for DLT rate by dose for observed covariate levels
summ <- summary(blrmfit, interval_prob = c(0, 0.16, 0.33, 1))
print(cbind(hist_combo3, summ))

# summary of posterior for DLT rate by dose for new set of covariate levels
newdata <- expand.grid(
  stratum_id = "BID", group_id = "Combo",
  drug_A = 400, drug_B = 800, drug_C = c(320, 400, 600, 800),
  stringsAsFactors = FALSE
)
summ_pred <- summary(blrmfit, newdata = newdata, interval_prob = c(0, 0.16, 0.33, 1))
print(cbind(newdata, summ_pred))

# update the model after observing additional data
newdata$num_patients <- rep(3, nrow(newdata))
newdata$num_toxicities <- c(0, 1, 2, 2)
library(dplyr)
blrmfit_new <- update(blrmfit,
  data = rbind(hist_combo3, newdata) %>%
    arrange(stratum_id, group_id)
)

# updated posterior summary
summ_upd <- summary(blrmfit_new, newdata = newdata, interval_prob = c(0, 0.16, 0.33, 1))
print(cbind(newdata, summ_upd)) 
## Recover user set sampling defaults
options(.user_mc_options)

---

Build a BLRM formula with linear interaction term in logit-space

Description

blrm_formula_linear is a convenience function for generating a formula for blrm_trial and blrm_exnex with an interaction of the form:

$\eta \, \prod_{i=1}^N (d_i \big / d_i^*))$

Usage

blrm_formula_linear(
  ref_doses,
  max_interaction_level = 2,
  specific_interaction_terms = NULL
)

Arguments

ref_doses	Numeric vector of reference doses with names corresponding to drug names
max_interaction_level	Highest interaction order to consider $[1 - Inf]$. Default: 2
specific_interaction_terms	List of custom interaction terms to generate (e.g. list(c("drug1", "drug2"), c("drug1", "drug3"))). Default: NULL

Value

The function returns an object of class blrm_formula.

Examples

ref_doses <- c(drug_A = 10, drug_B = 20)

# can be used with blrm_trial
blrm_formula_linear(ref_doses)

---

Build a BLRM formula with saturating interaction term in logit-space

Description

blrm_formula_saturating is a convenience function for generating a formula for blrm_trial and blrm_exnex with an interaction of the form:

$2 \eta \, \frac{\prod_{i=1}^N (d_i \big / d_i^*)}{1 + \prod_{i=1}^N (d_i \big / d_i^*)}$

Usage

blrm_formula_saturating(
  ref_doses,
  max_interaction_level = 2,
  specific_interaction_terms = NULL
)

Arguments

ref_doses	Numeric vector of reference doses with names corresponding to drug names
max_interaction_level	Highest interaction order to consider $[1 - Inf]$. Default: 2
specific_interaction_terms	List of custom interaction terms to generate (e.g. list(c("drug1", "drug2"), c("drug1", "drug3"))).

Value

The function returns an object of class blrm_formula.

References

Widmer, L.A., Bean, A., Ohlssen, D., Weber, S., Principled Drug-Drug Interaction Terms for Bayesian Logistic Regression Models of Drug Safety in Oncology Phase I Combination Trials arXiv pre-print, 2023, doi:10.48550/arXiv.2302.11437

Examples

ref_doses <- c(drug_A = 10, drug_B = 20)

# can be used with blrm_trial
blrm_formula_saturating(ref_doses)

---

Dose-Escalation Trials guided by Bayesian Logistic Regression Model

Description

blrm_trial facilitates the conduct of dose escalation studies guided by Bayesian Logistic Regression Models (BLRM). While the blrm_exnex only fits the BLRM model to data, the blrm_trial function standardizes the specification of the entire trial design and provides various standardized functions for trial data accrual and derivation of model summaries needed for dose-escalation decisions.

Usage

blrm_trial(
  data,
  dose_info,
  drug_info,
  simplified_prior = FALSE,
  EXNEX_comp = FALSE,
  EX_prob_comp_hist = 1,
  EX_prob_comp_new = 0.8,
  EXNEX_inter = FALSE,
  EX_prob_inter = 1,
  formula_generator = blrm_formula_saturating,
  interval_prob = c(0, 0.16, 0.33, 1),
  interval_max_mass = c(prob_underdose = 1, prob_target = 1, prob_overdose = 0.25),
  ...
)

## S3 method for class 'blrm_trial'
print(x, ...)

Arguments

data	dose-toxicity data available at design stage of trial
dose_info	specificaion of the dose levels as planned for the ongoing trial arms.
drug_info	specification of drugs used in trial arms.
simplified_prior	logical (defaults to FALSE) indicating whether a simplified prior should be employed based on the reference_p_dlt values provided in drug_info. Warning: The simplified prior will change between releases. Please read instructions below in the respective section for the simplified prior.
EXNEX_comp	logical (default to TRUE) indicating whether EXchangeable-NonEXchangeable priors should be employed for all component parameters
EX_prob_comp_hist	prior weight ($[0,1]$, default to 1) on exchangeability for the component parameters in groups representing historical data
EX_prob_comp_new	prior weight ($[0,1]$, default to 0.8) on exchangeability for the component parameters in groups representing new or concurrent data
EXNEX_inter	logical (default to FALSE) indicating whether EXchangeable-NonEXchangeable priors should be employed for all interaction parameters
EX_prob_inter	prior weight ($[0,1]$, defaults to 0.8) on exchangeability for the interaction parameters
formula_generator	formula generation function (see for example blrm_formula_linear or blrm_formula_saturating). The formula generator defines the employed interaction model.
interval_prob	defines the interval probabilities reported in the standard outputs. Defaults to c(0, 0.16, 0.33, 1).
interval_max_mass	named vector defining for each interval of the interval_prob vector a maxmimal admissable probability mass for a given dose level. Whenever the posterior probability mass in a given interval exceeds the threshold, then the Escalation With Overdose Control (EWOC) criterion is considered to be not fullfilled. Dose levels not fullfilling EWOC are ineligible for the next cohort of patients. The default restricts the overdose probability to less than 0.25.
...	Additional arguments are forwarded to blrm_exnex, i.e. for the purpose of prior specification.
x	blrm_trial object to print

Details

blrm_trial constructs an object of class blrm_trial which stores the compelte information about the study design of a dose-escalation trial. The study design is defined through the data sets (see sections below for a definition of the columns):

data (historical data)

The data argument defines available dose-toxicity data at the design stage of the trial. Together with the prior of model (without any data) this defines the prior used for the trial conduct.

dose_info

Definition of the pre-specified dose levels explored in the ongoing trial arms. Thus, all dose-toxcitiy trial data added to the object is expected correspond to one of the dose levels in the pre-defined set of dose_info.

drug_info

Determines the drugs used in the trial, their units, reference dose level and optionally defines the expected probability for a toxicity at the reference dose.

Once the blrm_trial object is setup the complete trial design is specified and the model is fitted to the given data. This allows evaluation of the pre-specified dose levels of the trial design wrt. to safety, i.e. whether the starting dose of the trial fullfills the escalate with overdose criterion (EWOC) condition.

The blrm_trial trial can also be constructed in a 2-step process which allows for a more convenient specification of the prior since meta data like number of drugs and the like can be used. See the example section for details.

After setup of the initial blrm_trial object additional data is added through the use of the update method which has a add_data argument intended to add data from the ongoing trial. The summary function finally allows to extract various model summaries. In particular, the EWOC criterion can be calculated for the pre-defined dose levels of the trial.

Value

The function returns an object of class blrm_trial.

Methods (by generic)

• print(blrm_trial): print function.

Simplified prior

As a convenience for the user, a simplified prior can be specifed whenever the reference_p_dlt column is present in the drug_info data set. However, the user is warned that the simplified prior will change in future releases of the package and thus we strongly discourage the use of the simplified prior for setting up trial designs. The functionality is intended to provide the user a quick start and as a starting point. The actually instantiated prior can be seen as demonstrated below in the examples.

Input data

The data given to the data argument of blrm_trial is considered as the available at design stage of the trial. The collected input data thus does not necessarily need to have the same dose levels as the pre-specified dose_info for the ongoing trial(s). It's data columns must include, but are not limited to:

group_id

study

stratum_id

optional, only required for differential discounting of groups

num_patients

number of patients

num_toxicities

number of toxicities

drug_A

Columns for the dose of each treatment component, with column names matching the drug_name values specified in the drug_info argument of blrm_trial

Drug info data

The drug information data-set defines drug properties. The fields included are:

drug_name

name of drug which is also used as column name for the dose

dose_ref

reference dose

dose_unit

units used for drug amounts

reference_p_dlt

optional; if provided, allows setup of a simplified prior

Dose info data

The drug_info data-set pre-specifies the dose levels of the ongoing trial. Thus, all data added to the blrm_trial through the update command must be consistent with the pre-defined dose levels as no other than those pre-specified ones can be explored in an ongoing trial.

dose_id

optional column which assigns a unique id to each group_id/dose combination. If not specified the column is internally generated.

group_id

study

drug_A

Columns for the dose of each treatment component, with column names matching the drug_name values specified in the drug_info argument of blrm_trial

References

Babb, J., Rogatko, A., & Zacks, S. (1998). Cancer phase I clinical trials: efficient dose escalation with overdose control. Statistics in medicine, 17(10), 1103-1120.

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

See Also

Other blrm_trial combo2 example: dose_info_combo2, drug_info_combo2, example-combo2_trial

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
)


# construct initial blrm_trial object from built-in example datasets
combo2_trial_setup <- blrm_trial(
  data = hist_combo2,
  dose_info = dose_info_combo2,
  drug_info = drug_info_combo2,
  simplified_prior = TRUE
)

# extract blrm_call to see setup of the prior as passed to blrm_exnex
summary(combo2_trial_setup, "blrm_exnex_call")

# Warning: The simplified prior will change between releases!
# please refer to the combo2_trial example for a complete
# example. You can obtain this example with
# ?example-combo2_trial
# or by running
# example_model("combo2_trial")

## Recover user set sampling defaults
options(.user_mc_options)

---

Dataset: historical and concurrent data on a two-way combination

Description

One of two datasets from the application described in Neuenschwander et al (2016). 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 is available from single agent trials trial_A and trial_B. Another study IIT was run concurrently to trial_AB, and studies the same combination. A second dataset hist_combo2 is available from this example, which includes only the data from the single agent studies, prior to the initiation of trial_AB and IIT.

Usage

codata_combo2

Format

A data frame with 20 rows and 5 variables:

group_id

study

drug_A

dose of Drug A

drug_B

dose of Drug B

num_patients

number of patients

num_toxicities

number of DLTs

cohort_time

cohort number of patients

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)

---

Critical quantile

Description

Calculates the critical value of a model covariate for which a fixed quantile of the crude rate is equal to the specified (tail or interval) probability. The specified quantile can relate to the posterior or the predictive distribution of the response rate.

Usage

critical_quantile(object, ...)

## S3 method for class 'blrmfit'
critical_quantile(
  object,
  newdata,
  x,
  p,
  qc,
  lower.tail,
  interval.x,
  extendInt.x = c("auto", "no", "yes", "downX", "upX"),
  log.x,
  predictive = FALSE,
  maxiter = 100,
  ...
)

## S3 method for class 'blrm_trial'
critical_quantile(
  object,
  newdata,
  x,
  p,
  qc,
  lower.tail,
  interval.x,
  extendInt.x = c("auto", "no", "yes", "downX", "upX"),
  log.x,
  predictive = FALSE,
  maxiter = 100,
  ...
)

Arguments

object	fitted model object
...	not used in this function
newdata	optional data frame specifying for what to predict; if missing, then the data of the input model object is used
x	character giving the covariate name which is being search for to meet the critical conditions. This also supports 'tidy' parameter selection by specifying x = vars(...), where ... is specified the same way as in dplyr::select() and similar functions. Examples of using x in this way can be found in the examples. For blrm_trial methods, it defaults to the first entry in summary(blrm_trial, "drug_info")$drug_name.
p	cumulative probability corresponding to the critical quantile range.
qc	if given as a sorted vector of length 2, then the two entries define the interval on the response rate scale which must correspond to the cumulative probability p. In this case lower.tail must be NULL or left missing. If qc is only a single number, then lower.tail must be set to complete the specification of the tail area.
lower.tail	defines if probabilities are lower or upper tail areas. Must be defined if qc is a single number and must not be defined if qc denotes an interval.
interval.x	interval the covariate x is searched. Defaults to the minimal and maximal value of x in the data set used. Whenever lower.tail is set such that it is known that a tail area is used, then the function will automatically attempt to enlarge the search interval since the direction of the inverted function is determined around the critical value.
extendInt.x	controls if the interval given in interval.x should be extended during the root finding. The default "auto" attempts to guess an adequate setting in cases thats possible. Other possible values are the same as for the extendInt argument of the stats::uniroot() function.
log.x	determines if during the numerical search for the critical value the covariate space is logarithmized. By default x is log transformed whenever all values are positive.
predictive	logical indicates if the critical quantile relates to the posterior (FALSE) or the predictive (TRUE) distribution. Defaults to FALSE.
maxiter	maximal number of iterations for root finding. Defaults to 100.

Details

The function searches for a critical value $x_c$ of the covariate $x$ (x) at which the integral over the model response in the specified interval $\pi_1$ to $\pi_2$ (qc) is equal to the given probability $p$ (p). The given interval is considered a tail area when lower.tail is used such that $\pi_1 = 0$ for TRUE and $\pi_2=1$ for FALSE. At the critical value $x_c$ the equality holds

$p = \int_{\pi_1}^{\pi_2} p(\pi | x_c) \, d\pi .$

Note that a solution not guranteed to exist under all circumstances. However, for a single agent model and when requesting a tail probability then a solution does exist. In case an interval probability is specified, then the solution may not necessarily exist and the root finding is confined to the range specified in interval.x.

In case the solution is requested for the predictive distribution (predictive=TRUE), then the respective problem solved leads to the equality of

$p = \sum_{y= r_1 = \lceil n \, \pi_1 \rceil }^{r_2 = \lceil n \, \pi_2 \rceil } \int p(y|\pi, n) \, p(\pi | x_c) \, d\pi .$

Furthermore, the covariate space is log transformed by default whenever all values of the covariate $x$ are positive in the data set. Values of $0$ are shifted into the positive domain by the machine precision to avoid issues with an ill-defined $\log(0)$.

For blrm_trial objects the default arguments for p, qc and lower.tail are set to correspond to the highest interval of interval_prob to be constrained by the respective interval_max_mass (which are defined as part of the blrm_trial objects). This will in the default case correspond to the EWOC metric. These defaults are only applied if p, qc and lower.tail are missing.

Value

Vector of length equal to the number of rows of the input data set with the crticial value for the covariate specified as x which fullfills the requirements as detailled above. May return NA for cases where no solution is found on the specified interval interval.x.

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
)

# fit an example model. See documentation for "combo2" example
example_model("combo2", silent = TRUE)

# Find dose of drug_A at which EWOC criterium is just fulfilled
data_trial_ab <- subset(codata_combo2, group_id == "trial_AB")
drug_A_crit <- critical_quantile(blrmfit,
  newdata = data_trial_ab,
  x = "drug_A", p = 0.25, qc = 0.33,
  lower.tail = FALSE
)
data_trial_ab$drug_A <- drug_A_crit
summary(blrmfit, newdata = data_trial_ab, interval_prob = c(0, 0.16, 0.33, 1))

## Recover user set sampling defaults
options(.user_mc_options)

---

Extract Diagnostic Quantities of OncoBayes2 Models

Description

Extract quantities that can be used to diagnose sampling behavior of the algorithms applied by Stan at the back-end of OncoBayes2.

Usage

## S3 method for class 'blrmfit'
log_posterior(object, ...)

## S3 method for class 'blrmfit'
nuts_params(object, pars = NULL, ...)

## S3 method for class 'blrmfit'
rhat(object, pars = NULL, ...)

## S3 method for class 'blrmfit'
neff_ratio(object, pars = NULL, ...)

Arguments

object	A blrmfit or blrmtrial object.
...	Arguments passed to individual methods.
pars	An optional character vector of parameter names. For nuts_params these will be NUTS sampler parameter names rather than model parameters. If pars is omitted all parameters are included.

Details

For more details see bayesplot-extractors.

Value

The exact form of the output depends on the method.

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
)

example_model("single_agent", silent = TRUE)

head(log_posterior(blrmfit))

np <- nuts_params(blrmfit)
str(np)
# extract the number of divergence transitions
sum(subset(np, Parameter == "divergent__")$Value)

head(rhat(blrmfit))
head(neff_ratio(blrmfit))

## Recover user set sampling defaults
options(.user_mc_options)

---

Dataset: trial dose information for a dual-agent combination study

Description

The data set defines all pre-defined dose-levels which can be explored in the dual-agent example trial.

Usage

dose_info_combo2

Format

An object of class tbl_df (inherits from tbl, data.frame) with 42 rows and 4 columns.

Details

group_id

study

drug_A

drug A dose amount

drug_B

drug B dose amount

dose_id

unique id of record

See Also

Other blrm_trial combo2 example: blrm_trial(), drug_info_combo2, example-combo2_trial

---

Transform blrmfit to draws objects

Description

Transform a blrmfit object to a format supported by the posterior package.

Usage

## S3 method for class 'blrmfit'
as_draws(x, variable = NULL, regex = FALSE, inc_warmup = FALSE, ...)

## S3 method for class 'blrmfit'
as_draws_matrix(x, variable = NULL, regex = FALSE, inc_warmup = FALSE, ...)

## S3 method for class 'blrmfit'
as_draws_array(x, variable = NULL, regex = FALSE, inc_warmup = FALSE, ...)

## S3 method for class 'blrmfit'
as_draws_df(x, variable = NULL, regex = FALSE, inc_warmup = FALSE, ...)

## S3 method for class 'blrmfit'
as_draws_list(x, variable = NULL, regex = FALSE, inc_warmup = FALSE, ...)

## S3 method for class 'blrmfit'
as_draws_rvars(x, variable = NULL, regex = FALSE, inc_warmup = FALSE, ...)

Arguments

x	A blrmfit object.
variable	A character vector providing the variables to extract. By default, all variables are extracted.
regex	Logical; Should variable be treated as a (vector of) regular expressions? Any variable in x matching at least one of the regular expressions will be selected. Defaults to FALSE.
inc_warmup	Should warmup draws be included? Defaults to FALSE.
...	Arguments passed to individual methods (if applicable).

Details

To subset iterations, chains, or draws, use the subset_draws method after transforming the blrmfit to a draws object.

The function is experimental as the set of exported posterior variables are subject to updates.

See Also

draws subset_draws

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
)

# fit an example model. See documentation for "combo2" example
example_model("combo2")

post <- as_draws(blrmfit)

## Recover user set sampling defaults
options(.user_mc_options)

---

Dataset: drug information for a dual-agent combination study

Description

Data set describing the two drugs involved in the example for a dual-agent combination study.

Usage

drug_info_combo2

Format

A tibble with 2 rows (one per durg) and 4 columns:

drug_name

name of drug

dose_ref

reference dose

dose_unit

units used for drug amounts

reference_p_dlt

a-priori probability for a DLT at the reference dose

See Also

Other blrm_trial combo2 example: blrm_trial(), dose_info_combo2, example-combo2_trial

---

Runs example models

Description

Runs example models

Usage

example_model(topic, envir = parent.frame(), silent = FALSE)

Arguments

topic	example to run
envir	environment which the example is loaded into. Defaults to the caller environment.
silent	logical controlling if execution is run silently (defaults to FALSE)

Value

When topic is not specified a list of all possible topics is return. Whenever a valid topic is specified, the function inserts the example into the environment given and returns (invisibly) the updated environment.

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
)


## get a list of available examples
example_model()

## run 3 component example
example_model("combo3")

## Recover user set sampling defaults
options(.user_mc_options)

---

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_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)

---

Two-drug combination example using BLRM Trial

Description

Example using blrm_trial to guide the built-in two-drug combination study example.

Details

blrm_trial is used to collect and store all relevant design information for the example. Subsequent use of the update.blrm_trial command allows convenient model fitting via blrm_exnex. The summary.blrm_trial method allows exploration of the design and modeling results.

To run this example, use example_model("combo2_trial"). See example_model.

See Also

Other blrm_trial combo2 example: blrm_trial(), dose_info_combo2, drug_info_combo2

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
)

library(tibble)
library(dplyr)
library(tidyr)

# Combo2 example using blrm_trial functionality

# construct initial blrm_trial object from built-in example datasets
combo2_trial_setup <- blrm_trial(
  data = hist_combo2,
  dose_info = dose_info_combo2,
  drug_info = drug_info_combo2,
  simplified_prior = FALSE
)

# summary of dimensionality of data structures
dims <- summary(combo2_trial_setup, "dimensionality")

# Fit the initial model with the historical data and fully specified prior


combo2_trial_start <- update(
   combo2_trial_setup,
   ## bivariate normal prior for drug A and drug B of intercept and
   ## log-slope
   prior_EX_mu_comp =
     replicate(2,
               mixmvnorm(c(1,
                           logit(0.2), 0,
                           diag(c(2^2, 1))))
             , FALSE),
   prior_EX_tau_comp =
     replicate(2,
               mixmvnorm(c(1,
                           log(0.25), log(0.125),
                           diag(c(log(4)/1.96, log(4)/1.96)^2)))
             , FALSE),
   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 = c(FALSE, FALSE),
   prior_is_EXNEX_inter = FALSE,
   prior_EX_prob_comp = matrix(1,
     nrow = dims$num_groups,
     ncol = 2
   ),
   prior_EX_prob_inter = matrix(1,
     nrow = nlevels(dose_info_combo2$group_id),
     ncol = 1
   ),
   prior_tau_dist = 1
 )

# print summary of prior specification
prior_summary(combo2_trial_start)

# summarize inference at observed dose levels
summary(combo2_trial_start, "data_prediction")

# summarize inference at specified dose levels
summary(combo2_trial_start, "dose_prediction")


# Update again with new data

# using update() with data argument supplied
# dem <- update(combo2_trial_start, data = codata_combo2)

# alternate way using update() with add_data argument for
# new observations only (those collected after the trial
# design stage).
new_data <- filter(codata_combo2, cohort_time > 0)

combo2_trial <- update(combo2_trial_start, add_data = new_data)

summary(combo2_trial, "data") # cohort_time is tracked
summary(combo2_trial, "data_prediction")
summary(combo2_trial, "dose_prediction")

rm(dims, new_data)

## Recover user set sampling defaults
options(.user_mc_options)

---

Three-drug combination example

Description

Example using a combination of two experimental drugs, with EXNEX and differential discounting.

Details

This dataset involves a hypothetical dose-escalation study of combination therapy with three treatment components. From two previous studies HistAgent1 and HistAgent2, historical data is available on each of the treatments as single-agents, as well as two of the two-way combinations. However, due to a difference in treatment schedule between the Combo study and the historical studies, a stratification (through stratum_id) is made between the groups to allow differential discounting of the alternate-schedule data. The association is as below.

group_id (j):	stratum_id (s_j):
Combo (1)	BID (1)
HistAgent1 (2)	QD (2)
HistAgent2 (3)	QD (2)

For additional robustness, EXNEX priors are used for all group-level treatment components while not for the interaction parameters. This is to limit the amount of borrowing in case of significant heterogeneity across groups.

The complete model is as follows. As a function of doses $d_1,d_2,d_3$, the DLT rate in group $j$ is, for $j = 1,\ldots,3$,

$\mbox{logit}\, \pi_j(d_1,d_2,d_3) = \mbox{logit}\Bigl( 1 - \prod_{i=1}^3 (1-\pi_{ij}(d_i))\Bigr) + \eta_{j}^{(12)}\frac{d_1}{d_1^*}\frac{d_2}{d_2^*} + \eta_{j}^{(13)}\frac{d_1}{d_1^*}\frac{d_3}{d_3^*} + \eta_{j}^{(23)}\frac{d_2}{d_2^*}\frac{d_3}{d_3^*} + \eta_{j}^{(123)}\frac{d_1}{d_1^*}\frac{d_2}{d_2^*}\frac{d_3}{d_3^*}.$

In group $j$ each treatment component $i$ toxicity is modeled with logistic regression,

$\mbox{logit}\, \pi_{ij}(d_i) = \log\, \alpha_{ij} + \beta_{ij} \, \log\, \Bigl(\frac{d_i}{d_i^*}\Bigr).$

The intercept and log-slope parameters $\boldsymbol\theta_{ij} = (\log\, \alpha_{ij}, \log\, \beta_{ij})$ are are given an EXNEX prior

$\boldsymbol\theta_{ij} \sim p_{ij} \mbox{BVN}(\boldsymbol\mu_i, \boldsymbol\Sigma_{ij}) + (1-p_{ij}) \mbox{BVN}(\boldsymbol m_{ij}, \boldsymbol S_{ij}),$

where the exchangeability weights are all $p_{ij} = 0.9$. The NEX parameters are set to $\boldsymbol m_{ij} = (\mbox{logit}(1/3), \log\, 1)$, $\boldsymbol S_{ij} = \mbox{diag}(2^2, 1^2)$ for all components $i=1,2,3$ and groups $j = 1,2,3$, and the EX parameters are modeled hierarchically. The mean of the exchangeable part has the distribution

$\boldsymbol\mu_i = (\mu_{\alpha i}, \mu_{\beta i}) \sim \mbox{BVN}(\boldsymbol m_i, \boldsymbol S_i),$

with $\boldsymbol m_i = (\mbox{logit}(1/3), \log 1)$ and $\boldsymbol S_i = \mbox{diag}(2^2, 1^2)$ for each component $i = 1,2,3$. For differentially discounting data from each schedule (QD and BID), the covariance parameters for the exchangeable part

$\Sigma_{ij} = \left( \begin{array}{cc} \tau^2_{\alpha s_j i} & \rho_i \tau_{\alpha s_j i} \tau_{\beta s_j i}\\ \rho_i \tau_{\alpha s_j i} \tau_{\beta s_j i} & \tau^2_{\beta s_j i} \end{array} \right).$

are allowed to vary across groups $j$ depending on their mapping to strata $s(j)$ as described above. For stratum $s=1$ (BID, which contains only the group $j = 1$ (Combo)), the standard deviations are modeled as

$\tau_{\alpha 1 i} \sim \mbox{Log-Normal}(\log\,0.25, (\log 4 / 1.96)^2)$

$\tau_{\beta 1 i} \sim \mbox{Log-Normal}(\log\,0.125, (\log 4 / 1.96)^2).$

Whereas in stratum $s=2$ (QD, which contains the historical groups $j=2,3$ (HistData1, HistData2)), the standard deviations are

$\tau_{\alpha 2 i} \sim \mbox{Log-Normal}(\log\,0.5, (\log 4 / 1.96)^2)$

$\tau_{\beta 2 i} \sim \mbox{Log-Normal}(\log\,0.25, (\log 4 / 1.96)^2).$

For all interaction parameters $\eta_{j}^{(12)}$, $\eta_{j}^{(13)}$, $\eta_{j}^{(23)}$, and $\eta_{j}^{(123)}$ ($j = 1,2,3$), the following prior is assumed:

$\eta_{j}^{(\cdot)} \sim \mbox{N}(\mu_{\eta}^{(\cdot)},{\tau_{\eta s_j}^{(\cdot)}}^2).$

The exchangeability weights are $p_{\eta j}^{(\cdot)} = 0.9$ for all parameters with EXNEX. Here, for each $\mu_{\eta}^{(12)}$, $\mu_{\eta}^{(13)}$, $\mu_{\eta}^{(23)}$, and $\mu_{\eta}^{(123)}$, we take

$\mu_{\eta}^{(\cdot)} \sim \mbox{N}(0, 1/2),$

and for each $\tau_{\eta s}^{(12)}$, $\tau_{\eta s}^{(13)}$, $\tau_{\eta s}^{(23)}$, and $\tau_{\eta s}^{(123)}$,

$\tau_{\eta s}^{(\cdot)} \sim \mbox{Log-Normal}(\log(0.25), (\log 2 / 1.96)^2),$

for both strata $s = 1,2$.

Below is the syntax for specifying this 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
)

## example combo3

library(abind)

dref <- c(500, 500, 1000)
num_comp <- 3
num_inter <- choose(3, 2) + 1
num_strata <- nlevels(hist_combo3$stratum_id)
num_groups <- nlevels(hist_combo3$group_id)

blrmfit <- blrm_exnex(
  cbind(num_toxicities, num_patients - num_toxicities) ~
    1 + I(log(drug_A / dref[1])) |
      1 + I(log(drug_B / dref[2])) |
      1 + I(log(drug_C / dref[3])) |
      0
      + I(drug_A / dref[1] * drug_B / dref[2])
        + I(drug_A / dref[1] * drug_C / dref[3])
        + I(drug_B / dref[2] * drug_C / dref[3])
        + I(drug_A / dref[1] * drug_B / dref[2] * drug_C / dref[3]) |
      stratum_id / group_id,
  data = hist_combo3,
  prior_EX_mu_comp = replicate(num_comp, mixmvnorm(c(1, logit(1/3), 0, diag(c(2^2, 1)))), FALSE),
  prior_EX_tau_comp = list(replicate(num_comp,
                                     mixmvnorm(c(1, log(c(0.25, 0.125)),
                                               diag(c(log(4)/1.96, log(4)/1.96)^2))), FALSE),
                           replicate(num_comp,
                                     mixmvnorm(c(1, log(2 * c(0.25, 0.125)),
                                               diag(c(log(4)/1.96, log(4)/1.96)^2))), FALSE)),
  prior_EX_mu_inter = mixmvnorm(c(1, rep.int(0, num_inter),
                                     diag((rep.int(sqrt(2) / 2, num_inter))^2))),
  prior_EX_tau_inter = replicate(num_strata,
                                 mixmvnorm(c(1, rep.int(log(0.25), num_inter),
                                             diag((rep.int(log(2) / 1.96, num_inter))^2))), FALSE),
  prior_EX_prob_comp = matrix(0.9, nrow = num_groups, ncol = num_comp),
  prior_EX_prob_inter = matrix(1.0, nrow = num_groups, ncol = num_inter),
  prior_is_EXNEX_comp = rep(TRUE, num_comp),
  prior_is_EXNEX_inter = rep(FALSE, num_inter),
  prior_tau_dist = 1,
  prior_PD = FALSE
)
## Recover user set sampling defaults
options(.user_mc_options)

---

Single Agent Example

Description

Example using a single experimental drug.

Details

The single agent example is described in the reference Neuenschwander, B. et al (2008). The data are described in the help page for hist_SA. In this case, the data come from only one study, with the treatment being only single agent. Hence the model specified does not involve a hierarchical prior for the intercept and log-slope parameters. The model described in Neuenschwander, et al (2008) is adapted as follows:

$\mbox{logit}\, \pi(d) = \log\, \alpha + \beta \, \log\, \Bigl(\frac{d}{d^*}\Bigr),$

where $d^* = 250$, and the prior for $\boldsymbol\theta = (\log\, \alpha, \log\, \beta)$ is

$\boldsymbol\theta \sim \mbox{N}(\boldsymbol m, \boldsymbol S),$

and $\boldsymbol m = (\mbox{logit}\, 0.5, \log\, 1)$ and $\boldsymbol S = \mbox{diag}(2^2, 1^2)$ are constants.

In the blrm_exnex framework, in which the prior must be specified as a hierarchical model $\boldsymbol\theta \sim \mbox{N}(\boldsymbol \mu, \boldsymbol \Sigma)$ with additional priors on $\boldsymbol\mu$ and $\boldsymbol\Sigma$, the simple prior distribution above is accomplished by fixing the diagonal elements $\tau^2_\alpha$ and $\tau^2_\beta$ of $\boldsymbol\Sigma$ to zero, and taking

$\boldsymbol\mu \sim \mbox{N}(\boldsymbol m, \boldsymbol S).$

The arguments prior_tau_dist and prior_EX_tau_mean_comp as specified below ensure that the $\tau$'s are fixed at zero.

References

Neuenschwander, B., Branson, M., & Gsponer, T. (2008). Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in medicine, 27(13), 2420-2439.

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
)

## Example from Neuenschwander, B., et al. (2009). Stats in Medicine

num_comp <- 1 # one investigational drug
num_inter <- 0 # no drug-drug interactions need to be modeled
num_groups <- nlevels(hist_SA$group_id) # no stratification needed
num_strata <- 1 # no stratification needed


dref <- 50

## Since there is no prior information the hierarchical model
## is not used in this example by setting tau to (almost) 0.
blrmfit <- blrm_exnex(
  cbind(num_toxicities, num_patients - num_toxicities) ~
    1 + log(drug_A / dref) |
      0 |
      group_id,
  data = hist_SA,
  prior_EX_mu_comp = mixmvnorm(c(1, logit(1 / 2), log(1), diag(c(2^2, 1)))),
  ## Here we take tau as known and as zero.
  ## This disables the hierarchical prior which is
  ## not required in this example as we analyze a
  ## single trial.
  prior_EX_tau_comp = mixmvnorm(c(1, 0, 0, diag(c(1, 1)))),
  prior_EX_prob_comp = matrix(1, nrow = num_comp, ncol = 1),
  prior_tau_dist = 0,
  prior_PD = FALSE
)
## Recover user set sampling defaults
options(.user_mc_options)

---

Dataset: historical data on two single-agents to inform a combination study

Description

One of two datasets from the application described in Neuenschwander et al (2016). The risk of DLT is to be studied as a function of dose for two drugs, drug A and drug B. Historical information on the toxicity profiles of these two drugs is available from single agent trials trial_A and trial_B. A second dataset codata_combo2 is available from this application, which includes additional dose-toxicity data from trial_AB and IIT of the combination of Drugs A and B.

Usage

hist_combo2

Format

A tibble with 11 rows and 5 variables:

group_id

study

drug_A

dose of Drug A

drug_B

dose of Drug B

num_patients

number of patients

num_toxicities

number of DLTs

cohort_time

cohort number of patients

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.

---