blrm_exnex {OncoBayes2} | R Documentation |
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_mean_comp,
prior_EX_mu_sd_comp,
prior_EX_tau_mean_comp,
prior_EX_tau_sd_comp,
prior_EX_corr_eta_comp,
prior_EX_mu_mean_inter,
prior_EX_mu_sd_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_mean_comp,
prior_NEX_mu_sd_comp,
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 |
prior_EX_mu_mean_comp , prior_EX_mu_sd_comp |
Mean and sd for
the prior on the mean parameters |
prior_EX_tau_mean_comp , prior_EX_tau_sd_comp |
Prior mean and
sd for heterogeniety parameter |
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_mean_inter , prior_EX_mu_sd_inter |
Prior mean and
sd for population mean parameters |
prior_EX_tau_mean_inter , prior_EX_tau_sd_inter |
Prior mean and
sd for heterogeniety parameter |
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
|
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 |
prior_EX_prob_comp |
Prior probability |
prior_EX_prob_inter |
Prior probability |
prior_NEX_mu_mean_comp , prior_NEX_mu_sd_comp |
Prior mean
|
prior_NEX_mu_mean_inter , prior_NEX_mu_sd_inter |
Prior mean
|
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 ( |
thin |
period of saving samples. |
init |
positive number to specify uniform range on
unconstrained space for random initialization. See
|
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
|
prior_PD |
Logical flag (defaults to |
verbose |
Logical flag (defaults to |
x |
|
... |
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 that
a patient experiences a DLT, by complementing the binomial likelihood with
a monotone logistic regression
where . Most typically,
represents the dose, and
is an appropriate transformation, such as
. A
joint prior on
completes
the model and ensures monotonicity
.
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
.
Methods (by generic)
-
print
: print function.
Combination of two treatments
For a combination of two treatment components, the basic modeling framework
is that the DLT rate is comprised of (1) a "no-interaction"
baseline model
driven by the single-agent toxicity of
each component, and (2) optional interaction terms
representing synergy or antagonism between the drugs. On the log-odds scale,
The "no interaction" part represents the probability
of a DLT triggered by either treatment component acting independently.
That is,
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.
where is a monotone transformation of the doses, such as
.
The inclusion of an interaction term 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
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 components by expressing the probability
on
the logit scale as
Multiple drug-drug interactions among the components are
now possible, and are represented through the
interaction
terms
.
Regression models can be again be specified for each
and
, such as
Interactions for some subset of
the treatment components can be modeled with regression as well,
for example on products of doses,
For example, results in the three-way
interaction term
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 on the parameters and observations
denotes the
groups. The resulting model allows the DLT rate
to differ across the groups. The general
-component model
becomes
for groups . The component toxicities
and interaction terms
are
modelled, as before, through regression. For example,
could be a logistic regression on
with intercept and log-slope
, and
regressed with coefficient
on a product
for some subset
of components.
Thus, for , we now have group-specific parameters
and
for each component
and interaction
.
The structure of the prior on
and
determines how much
information will be shared across groups
. Several modeling
choices are available in the function.
-
EX (Full exchangeability): One can assume the parameters are conditionally exchangeable given hyperparameters
independently across groups
and treatment components
. The covariance matrix
captures the patterns of cross-group heterogeneity, and is parametrized with standard deviations
and the correlation
. Similarly for the interactions, the fully-exchangeable model is
for groups
and interactions
, and the prior on the covariance matrix
captures the amount of heterogeneity expected in the interaction terms a-priori. The covariance is again parametrized with standard deviations
and its correlation matrix.
-
Differential discounting: For one or more of the groups
, larger deviations of
may be expected from the mean
, or of the interactions
from the mean
. Such differential heterogeneity can be modeled by mapping the groups
to strata through
, and modifying the model specification to
where
For the interactions, the model becomes
where the covariance matrix
is modelled as stratum specific standard deviations
and a stratum independent correlation matrix. Each stratum
then corresponds to its own set of standard deviations
leading to different discounting per stratum. Independent priors are specified for the component parameters
and
and for the interaction parameters
for each stratum
. Inference for strata
where the prior is centered on larger values of the
parameters will exhibit less shrinkage towards the the means,
and
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
when applied to the treatment-component parameters, and
when applied to the interaction parameters. The exchangeability weights
and
are fixed constants in the interval
that control the degree to which inference for group
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
.
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)