Introduction
A common characteristic of modern applications of the Bayesian
Logistic Regression Model (BLRM) for model-based dose escalation is the
availability of relevant historical data on the study treatment(s). For
example, studies testing novel combination therapies may wish to
incorporate available dose-toxicity data on the individual agents from
single-agent dose escalation trials.
There are two broad categories for statistical approaches to
this:
- Meta-Analytic Combined (MAC) approaches use a hierarchical model to
jointly analyze both the historical data and data from the current
study. External data from concurrently running trials may also be
incorporated in such an approach. The introductory vignette illustrates this
approach.
- Meta-Analytic Predictive (MAP) approaches involve two stages. In the
first, an initial hierarchical model is estimated from the historical
data, and from this model one extracts a posterior prediction for the
trial-level parameters in a future trial. The posterior prediction is
then approximated in parametric form which is used as MAP prior in the
future trial.
These two approaches are statistically equivalent to each other
(provided the parametric representation of the MAP prior is exact), but
differ in the “modelling workflow.” While the MAC approach is a
prospective approach enabling to include even concurrent data to borrow
from, the MAP approach is retrospective in that the historical data is
summarized once and usually taken as immutable and static. The MAP
approach has the advantage when designing prospectively a future study
that we can pre-specify (and hence document in a statistical analysis
plan) the information borrowed from the historical data and the model
used for the future study becomes structurally simpler, since there is
often no more a need for a hierarchical model structure at that
stage.
The recommended reference for more details on these approaches, how
they relate to each other, and how they are applied in the dose-toxicity
setting is explained in Neuenschwander et al (2016).
In this vignette, we illustrate the implementation of the MAP
approach, and describe its usage for safety monitoring in a realistic
application.
Initial R session setup:
library(OncoBayes2)
library(RBesT)
library(posterior)
library(ggplot2)
library(dplyr)
library(tidyr)
library(knitr)
Derivation of MAP
priors
In the MAP approach, we begin with meta-analytic hierarchical
modelling of the historical data, accounting for the possibility of
between-trial heterogeneity. It is important to note that this is done
even for the case of a single historical study being available. The
information on the between-trial heterogeneity relies in this case on
the priors used for the heterogeneity. In practice log normal priors for
the heterogeneity parameters based on the following table have worked
well in many situations. The approach is to categorize the degree of
heterogeneity between trials in terms of anticipated differences in the
population and further factors which depend on a specific case:
Heterogeneity categorization for intercept \(\alpha\) and slope \(\beta\)
| Heterogeneity degree |
\(\tau_\alpha\) |
\(\tau_\beta\) |
| small |
0.125 |
0.0625 |
| moderate |
0.250 |
0.1250 |
| substantial |
0.500 |
0.2500 |
| large |
1.000 |
0.5000 |
| very large |
2.000 |
1.0000 |
Here we fit separate models to the historical drug-A and drug-B
single-agent data. Note, equivalently, a single hierarchical
double-combination model without any interaction could have been fit to
those data, and would have produced the same MAP priors for the
drug-level intercepts and slopes. A detailed specification of the models
being fit below is given in Appendix 7.1.
In this example we assume that the historical data for drug A comes
from a relatively different population (for example different line of
therapy) and the historical data for drug B stems from a very closely
related patient population. This translates into respective choices for
the between-trial heterogeneity priors according to the table above.
Drug A model
Implementation of this meta-analysis of single-agent data is done
using blrm_exnex(). In order to obtain directly a sample of
the MAP prior, we set the sample_map argument to
TRUE.
# historical data for drug A
hist_A <- data.frame(
group_id = factor("trial_A"),
drug_A = c(12.5, 25, 50, 80, 100, 150),
num_patients = c(1, 1, 3, 9, 23, 3),
num_toxicities = c(0, 0, 0, 1, 4, 2)
)
# reference dose for drug A
dref_A <- 80
# bivariate normal prior for (intercept, log-slope)
prior_A <- mixmvnorm(c(1.0,
logit(0.2), 0,
diag(c(1, (log(4) / 1.96)^2 ))))
# between-trial heterogeneity is centered on "substantial"
tau_substantial_prior <- mixmvnorm(c(1.0,
log(c(0.5, 0.25)),
diag(rep(log(2) / 1.96, 2)^2)))
# fit the model to the historical data
map_model_A <- blrm_exnex(
cbind(num_toxicities, num_patients - num_toxicities) ~
1 + log(drug_A / dref_A) |
0 |
group_id,
data = hist_A,
prior_EX_mu_comp = list(prior_A),
prior_EX_tau_comp = tau_substantial_prior,
prior_EX_prob_comp = matrix(1, nrow = 1, ncol = 1),
prior_tau_dist = 1,
sample_map = TRUE
)
Drug B model
Similarly, for drug B:
# historical data for drug B
hist_B <- data.frame(
group_id = factor("trial_B"),
drug_B = c(0.125, 0.25, 0.5, 1, 2, 2.5, 3, 4),
num_patients = c(2, 1, 2, 2, 3, 7, 12, 3),
num_toxicities = c(0, 0, 0, 0, 1, 0, 0, 1)
)
# reference dose
dref_B <- 1
# prior for (intercept, log-slope)
prior_B <- prior_A
# between-trial heterogeneity is centered on "small", but with a
# greater degree of uncertainty on this classification
tau_small_prior <- mixmvnorm(c(1.0,
log(c(0.125, 0.0625)),
diag(rep(log(4) / 1.96, 2)^2)))
# fit model to historical data
map_model_B <- blrm_exnex(
cbind(num_toxicities, num_patients - num_toxicities) ~
1 + log(drug_B / dref_B) |
0 |
group_id,
data = hist_B,
prior_EX_mu_comp = list(prior_B),
prior_EX_tau_comp = tau_small_prior,
prior_EX_prob_comp = matrix(1, nrow = 1, ncol = 1),
prior_tau_dist = 1,
sample_map = TRUE
)
Combination model with
MAP priors for the new study
Finally, these mixtures become the prior distributions for the
combination dose-toxicity model in the new study. In this combination
model we now do not need any longer a hierarchical model structure such
we turn this part of the model off by setting
tau_prior_dist=NULL. Note that doing so merely setups the
model to use known heterogeneities \(\tau\) which are fixed to a zero mean,
which is what subsequent output will display. A detailed description of
this model is found in Appendix 7.2.
# one row of dummy data with number of patients set to zero in order
# to allow the model to be fit
dummy_data_AB <- data.frame(
group_id = factor("trial_AB", levels = c("trial_AB")),
drug_A = 1,
drug_B = 1,
num_patients = 0,
num_toxicities = 0
)
interaction_prior <- mixmvnorm(c(1, 0, (log(9)/1.96)^2))
# fit a double-combination model using the MAP priors
map_combo <- blrm_exnex(
cbind(num_toxicities, num_patients - num_toxicities) ~
1 + I(log(drug_A / dref_A)) |
1 + I(log(drug_B / dref_B)) |
0 + I(2 * (drug_A/dref_A * drug_B/dref_B) / (1 + drug_A/dref_A * drug_B/dref_B)) |
group_id,
data = dummy_data_AB,
prior_EX_mu_comp = list(map_mix_A, map_mix_B), # BVN mixtures for (log-alpha, log-beta) for drugs A and B
prior_EX_mu_inter = interaction_prior,
# shut off hierarchical part of model --
prior_tau_dist = NULL
)
This defines the prior distribution for the dose-toxicity
relationship in the novel combination. We can visualize the prior
distributions for the DLT rates:
doses <- expand_grid(
group_id = "trial_AB",
drug_A = c(25, 50, 80, 100),
drug_B = c(0.5, 1, 3)
) %>%
arrange(drug_B, drug_A)
plot_toxicity_curve(map_combo,
newdata = doses,
x = vars(drug_A),
group = ~ drug_B) +
theme(legend.position="bottom")
We also now have an understanding of which combination doses may have
a high risk of DLTs in terms of the posterior mean,
map_summ <- summary(map_combo, newdata = doses, interval_prob = c(0, 0.16, 0.33, 1))
kable(
cbind(doses[c("drug_A", "drug_B")], map_summ[c("mean", "sd", "(0.33,1]")]),
col.names = c("Drug A", "Drug B", "mean", "sd", "P(DLT rate > 0.33)"),
digits = 2,
caption = "Posterior summary statistics for P(DLT) by dose"
)
Posterior summary statistics for P(DLT) by dose
| Drug A |
Drug B |
mean |
sd |
P(DLT rate > 0.33) |
| 25 |
0.5 |
0.10 |
0.08 |
0.01 |
| 50 |
0.5 |
0.16 |
0.12 |
0.08 |
| 80 |
0.5 |
0.24 |
0.17 |
0.26 |
| 100 |
0.5 |
0.30 |
0.20 |
0.37 |
| 25 |
1.0 |
0.13 |
0.10 |
0.05 |
| 50 |
1.0 |
0.19 |
0.16 |
0.19 |
| 80 |
1.0 |
0.28 |
0.22 |
0.33 |
| 100 |
1.0 |
0.33 |
0.25 |
0.43 |
| 25 |
3.0 |
0.23 |
0.19 |
0.25 |
| 50 |
3.0 |
0.29 |
0.25 |
0.36 |
| 80 |
3.0 |
0.35 |
0.29 |
0.45 |
| 100 |
3.0 |
0.39 |
0.31 |
0.49 |
and the posterior predictive distribution when assuming a specific
cohort size (6 used as example here):
map_pred_summ <- summary(map_combo,
newdata = mutate(doses, num_toxicities=0, num_patients=6),
predictive=TRUE, interval_prob = c(-1, 0, 1, 6))
kable(
cbind(doses[c("drug_A", "drug_B")], map_pred_summ[c("(-1,0]", "(0,1]", "(1,6]")]),
col.names = c("Drug A", "Drug B", "Pr(0 of 6 DLT)", "Pr(1 of 6 DLT)", "Pr(>=2 of 6 DLT)"),
digits = 2,
caption = "Posterior predictive summary for Pr(DLT) by dose"
)
Posterior predictive summary for Pr(DLT) by dose
| Drug A |
Drug B |
Pr(0 of 6 DLT) |
Pr(1 of 6 DLT) |
Pr(>=2 of 6 DLT) |
| 25 |
0.5 |
0.58 |
0.29 |
0.13 |
| 50 |
0.5 |
0.45 |
0.31 |
0.24 |
| 80 |
0.5 |
0.32 |
0.29 |
0.40 |
| 100 |
0.5 |
0.26 |
0.26 |
0.49 |
| 25 |
1.0 |
0.51 |
0.30 |
0.19 |
| 50 |
1.0 |
0.40 |
0.28 |
0.31 |
| 80 |
1.0 |
0.31 |
0.24 |
0.44 |
| 100 |
1.0 |
0.27 |
0.22 |
0.51 |
| 25 |
3.0 |
0.37 |
0.26 |
0.37 |
| 50 |
3.0 |
0.34 |
0.22 |
0.45 |
| 80 |
3.0 |
0.29 |
0.19 |
0.52 |
| 100 |
3.0 |
0.26 |
0.17 |
0.56 |
One can see that under the combination one has to decrease the
starting doses well below the declared MTDs for each drug. This is a
consequence of their combination in a new trial such that the
between-trial heterogeneity leads to an increased uncertainty in the
dose-toxicity. The uncertainty is additionally increased through the
introduction of a possible interaction between the two drugs. With the
increased uncertainty in the dose-toxicity curve, the escalation with
overdose criterion EWOC automatically penalizes this lack of knowledge
and requires a conservative starting dose for escalation.
Robustification
Direct use of the MAP priors assumes ex-changeability of the
historical and the new trial data. However, this assumption can be
violated such that we may wish to robustify the analysis. This can be
achieved by complementing the MAP prior by a weakly informative mixture
component with some non-zero weight. While the hierarchical model itself
down-weights the historical data already, that is we admit through the
interchangeability differences between trials, adding in this way an
additional weakly informative mixture component leads to an even greater
robustness against a prior-data conflict. This process is sometimes
referred to as “robustification”, and the resulting prior known as a
robust MAP (rMAP) prior. The robustification step is conceptually
equivalent to the use of the EXNEX model under a MAC approach. However,
the two techniques - robustification and EXNEX - differ in that
robustification is a step done after derivation of the MAP prior while
the EXNEX model is a joint model fit.
The robust MAP approach is implemented as:
# weakly-informative mixture component is chosen here to be equal to
# the priors used in the first place
weak_A <- prior_A
weak_B <- prior_B
# adding this to the drug-A and drug-B MAP priors with weight 0.1
mix_A_robust <- mixcombine(map_mix_A, weak_A, weight = c(0.9, 0.1))
mix_B_robust <- mixcombine(map_mix_B, weak_B, weight = c(0.9, 0.1))
# robust MAP priors
mix_robust <- list(mix_A_robust, mix_B_robust)
# fit a double-combination model using the rMAP priors
robust_map_combo <- update(map_combo, prior_EX_mu_comp = mix_robust)
# disabled plot here
plot_toxicity_curve(robust_map_combo,
newdata = doses,
x = vars(drug_A),
group = ~ drug_B) +
theme(legend.position="bottom")
Equivalence of MAP and
MAC (advanced)
To demonstrate the equivalence of MAP and MAC, we will illustrate
implementation of the latter approach.
First, we combine the data from all three sources.
groups <- c("trial_A", "trial_B", "trial_AB")
mac_data <- bind_rows_0(hist_A, hist_B)
mac_data$group_id <- factor(as.character(mac_data$group_id), levels = groups)
Next, we jointly model all available data using a hierarchical
double-combination BLRM, whose detailed specification can be found in
Appendix 7.3. Note that the hierarchical model on the interaction
parameter is not present in the combination model. This cannot be turned
off here such that we assign a negligibly small mean value for the \(\tau\) interaction heterogeneity.
dummy_tau_inter <- mixmvnorm(c(1.0, log(1E-4), (log(2) / 1.96)^2))
num_comp <- 2 # number of treatment components
num_inter <- 1
num_groups <- length(groups)
mac_combo <- blrm_exnex(
cbind(num_toxicities, num_patients - num_toxicities) ~
1 + I(log(drug_A / dref_A)) |
1 + I(log(drug_B / dref_B)) |
0 + I(2 * (drug_A/dref_A * drug_B/dref_B) / (1 + drug_A/dref_A * drug_B/dref_B)) |
group_id,
data = mac_data,
prior_EX_mu_comp = list(prior_A, prior_B),
prior_EX_tau_comp = list(tau_substantial_prior, tau_small_prior),
prior_EX_mu_inter = interaction_prior,
prior_EX_tau_inter = dummy_tau_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
)
## Info: The group(s) trial_AB have undefined strata. Assigning first stratum 1.
Now we compare posterior inference for the DLT rates:
mac_summ <- summary(mac_combo, newdata = doses, interval_prob = c(0, 0.16, 0.33, 1))
kable(
cbind(doses[c("drug_A", "drug_B")],
map_summ$mean, mac_summ$mean,
abs(map_summ$mean - mac_summ$mean),
map_summ[["(0.33,1]"]], mac_summ[["(0.33,1]"]]),
col.names = c("Drug A", "Drug B",
"MAP mean", "MAC mean",
"|MAP mean - MAC mean|",
"MAP P(DLT rate > 0.33)", "MAC P(DLT rate > 0.33)"),
digits = 2,
caption = "Posterior summary statistics for P(DLT) by dose"
)
Posterior summary statistics for P(DLT) by dose
| Drug A |
Drug B |
MAP mean |
MAC mean |
|MAP mean - MAC mean| |
MAP P(DLT rate > 0.33) |
MAC P(DLT rate > 0.33) |
| 25 |
0.5 |
0.10 |
0.10 |
0 |
0.01 |
0.02 |
| 50 |
0.5 |
0.16 |
0.16 |
0 |
0.08 |
0.09 |
| 80 |
0.5 |
0.24 |
0.24 |
0 |
0.26 |
0.25 |
| 100 |
0.5 |
0.30 |
0.30 |
0 |
0.37 |
0.37 |
| 25 |
1.0 |
0.13 |
0.13 |
0 |
0.05 |
0.05 |
| 50 |
1.0 |
0.19 |
0.19 |
0 |
0.19 |
0.17 |
| 80 |
1.0 |
0.28 |
0.28 |
0 |
0.33 |
0.33 |
| 100 |
1.0 |
0.33 |
0.33 |
0 |
0.43 |
0.42 |
| 25 |
3.0 |
0.23 |
0.23 |
0 |
0.25 |
0.23 |
| 50 |
3.0 |
0.29 |
0.29 |
0 |
0.36 |
0.35 |
| 80 |
3.0 |
0.35 |
0.35 |
0 |
0.45 |
0.44 |
| 100 |
3.0 |
0.39 |
0.39 |
0 |
0.49 |
0.51 |
Appendix: Model
specification
Combination BLRM (new
study model)
Let \(d_A\) and \(d_B\) represent the doses of drugs A and B,
respectively. The dose-toxicity model for the combination expresses the
DLT probability \(\pi_{AB}(d_A, d_B)\)
as a function of the doses, through the following model for the DLT
odds
\[ \frac{\pi_{AB}(d_A, d_B)}{1 -
\pi_{AB}(d_A, d_B)} =
\frac{\pi_\perp(d_A, d_B)}{1 - \pi_\perp(d_A, d_B)} \cdot \exp\left\{
2 \eta \frac{(d_A d_B) / (d_A^* d_B^*) }{1 + (d_A d_B) / (d_A^*
d_B^*)}\right\}. \]
The two terms on the right-hand side are, respectively, the odds of
DLT under assumed independence of action for the two drugs, i.e.
\[\pi_\perp(d_A, d_B) = 1 - (1 -
\pi_A(d_A))(1 - \pi_B(d_B)) \]
and an interaction term which allows for modelling drug toxicity
interactions. See Widmer et al (2023) for a discussion of the
interaction model.
The terms \(\pi_A(d_A)\) and \(\pi_B(d_B)\) represent the toxicity risk
under each single-agent, respectively, and are modelled with logistic
regression as in Neuenschwander et al (2008):
\[ \pi_A(d_A) = \log(\alpha_A) + \beta_A
\log(d_A / d_A^*)\]
\[ \pi_B(d_B) = \log(\alpha_B) + \beta_B
\log(d_B / d_B^*)\]
The model parameters are hence \(\alpha_A,
\beta_A, \alpha_B\) and \(\beta_B\) (the intercepts and slopes
determining the dose-toxicity curve for the two single agents), and
\(\eta\) (controlling the magnitude of
drug toxicity interactions), while \(d_A^*\) and \(d_B^*\) are fixed pre-specified reference
doses for each drug.
This parametrization is convenient because in many such situations,
relevant information is available to inform prior distributions for the
intercepts and slopes. Specifically, in this example, we have used MAP
priors based on analysis of the historical data under meta-analytic
models.
Prior
specification
For the intercepts and slopes, the MAP priors have the form
\[ (\log \alpha_A, \log \beta_A ) \sim
\sum_{k = 1}^{K_A} w_{Ak} \mbox{BVN}( \mathbf m_{Ak}, \mathbf S_{Ak} ),
\]
and
\[ (\log \alpha_B, \log \beta_B ) \sim
\sum_{k = 1}^{K_B} w_{Bk} \mbox{BVN}( \mathbf m_{Bk}, \mathbf S_{Bk} ),
\]
where the bivariate normal mixtures have been chosen to approximate
the MAP distribution, i.e.
## EM for Multivariate Normal Mixture Model
## Log-Likelihood = -4557.01
##
## Multivariate normal mixture
## Outcome dimension: 2
## Mixture Components:
## comp1 comp2
## w 0.56206771 0.43793229
## m[intercept] -1.88227571 -1.52601780
## m[log_slope] 0.59136954 0.01259954
## s[intercept] 0.68030368 0.92524450
## s[log_slope] 0.59829373 0.73241598
## rho[log_slope,intercept] -0.29870546 0.12832698
and
## EM for Multivariate Normal Mixture Model
## Log-Likelihood = -3252.692
##
## Multivariate normal mixture
## Outcome dimension: 2
## Mixture Components:
## comp1 comp2 comp3
## w 0.37748737 0.37130812 0.25120451
## m[intercept] -2.62052660 -2.62077157 -3.09269497
## m[log_slope] -0.36785969 -0.78173676 -0.08761646
## s[intercept] 0.42385373 0.66468212 0.66682656
## s[log_slope] 0.40861667 0.51156577 0.40148441
## rho[log_slope,intercept] -0.39229212 -0.05739021 -0.50121005
For the interaction, the prior distribution \(\eta \sim \mbox{N}(0,
(\log(9)/1.96)^2)\) was used. This prior is centered at zero
interaction, and 95% of the probability mass falls between DLT odds
multipliers of \(1/9\) and \(9\) (relative to the no-interaction
model).
Meta-analytic models
for the historical data
The MAP approach begins with hierarchical meta-analytic modelling of
the historical data, allowing for the possibility of between-study
heterogeneity.
We describe this model for drug A (drug B being analogous). For
historical studies \(h =1 ,\ldots, H\)
(in this case \(H=1\)), the DLT
probability for dose \(d_A\) is
modelled as
\[ \pi_{Ah}(d_A) = \log \alpha_{Ah} +
\beta_{Ah} \log(d_A / d_A^*). \]
Model parameters are assumed to be exchangeable across studies:
\[ (\log \alpha_{Ah}, \log \beta_{Ah}) \,
\, \sim \, \,
\mbox{BVN}\bigl( (\mu_{\alpha A}, \mu_{\beta A}), \boldsymbol \Sigma_A
\bigr) \hspace{20pt} \text{for }h = 1,\ldots, H, \]
and to facilitate prediction of the dose-toxicity curve in a new
study, we additionally assume that the study-level parameters for the
new study are exchangeable with those of the historical studies:
\[ (\log \alpha_{A\star}, \log
\beta_{A\star}) \, \, \sim \, \,
\mbox{BVN}\bigl( (\mu_{\alpha A}, \mu_{\beta A}), \boldsymbol \Sigma_A
\bigr). \]
Here, \(\boldsymbol \Sigma_A\) is a
\(2 \times 2\) covariance matrix with
standard deviations \(\tau_{\alpha A}\)
and \(\tau_{\beta A}\), and correlation
\(\rho_A\).
This exchangeability assumption allows for posterior estimation of
the MAP distribution for a study-level parameters in a new study, given
historical data:
\[ f(\alpha_{A\star}, \beta_{A\star} |
\mathbf x_{\text{hist}}) = \int f(\alpha_{A\star}, \beta_{A\star} |
\mu_{\alpha A}, \mu_{\beta A}, \tau_{\alpha A}, \tau_{\beta A}, \rho) \,
\, dF(\mu_{\alpha A}, \mu_{\beta A}, \tau_{\alpha A}, \tau_{\beta A} |
\mathbf x_{\text{hist}}). \]
The model is completed with normal priors for the means
\[ \mu_{\alpha A} \sim \mbox{N} (
m_{\alpha A}, s^2_{\alpha A}) \] \[
\mu_{\beta A} \sim \mbox{N} ( m_{\beta A}, s^2_{\beta A}) \]
lognormal priors for the standard deviations,
\[ \log \tau_{\alpha A} \sim \mbox{N}(
m_{\tau \alpha A}, s^2_{\tau \alpha A}) \] \[ \log \tau_{\beta A} \sim \mbox{N}( m_{\tau \beta
A}, s^2_{\tau \beta A}) \]
and for the correlation,
\[ \rho_A \sim \mbox{U}(-1, 1).
\]
Prior
specification
Both models, for drugs A and B, used the same set of
hyper-parameters:
- Intercept mean and standard deviation:
\[m_{\alpha A} = m_{\alpha B} =
\mbox{logit}(0.2)\] \[s_{\alpha A} =
s_{\alpha B} = 1\]
- Slope mean and standard deviation:
\[m_{\beta A} = m_{\beta B} =
\log(1)\] \[s_{\beta A} = s_{\beta B}
= (\log 4) / 1.96\]
- Heterogeneity standard deviations:
\[m_{\tau \alpha A} = m_{\tau \alpha B} =
\log(0.25)\] \[m_{\tau \beta A} =
m_{\tau \beta B} = \log(0.125)\] \[s_{\tau \alpha A} = s_{\tau \beta A} = s_{\tau
\alpha B} = s_{\tau \beta B} = (\log 2) / 1.96\]
Hierarchical model
for combined data (MAC)
The MAC model is a combined model for data from all sources (both
single-agent trials and the planned combination study). Letting \(i\) index these sources, the model covers
toxicity of any combination of doses of the two agents:
\[ \frac{\pi_i(d_A, d_B)}{1 - \pi_i(d_A,
d_B)} = \frac{\pi_{\perp, i}(d_A, d_B)}{1 - \pi_{\perp, i}(d_A, d_B)}
\cdot \exp\left\{ 2 \eta_i \frac{(d_A d_B) / (d_A^* d_B^*) }{1 + (d_A
d_B) / (d_A^* d_B^*)}\right\}. \] With, as before
\[\pi_{\perp,i}(d_A, d_B) = 1 - (1 -
\pi_{Ai}(d_A))(1 - \pi_{Bi}(d_B)) \]
and \[ \pi_{Ai}(d_A) = \log(\alpha_{Ai}) +
\beta_{Ai} \log(d_A / d_A^*)\] \[
\pi_{Bi}(d_B) = \log(\alpha_{Bi}) + \beta_{Bi} \log(d_B /
d_B^*)\]
And as before, ex-changeability assumptions facilitate information
borrowing across data sources.
\[ (\log \alpha_{Ai}, \log \beta_{Ai}) \,
\, \sim \, \, \mbox{BVN}\bigl( (\mu_{\alpha A}, \mu_{\beta A}),
\boldsymbol \Sigma_A \bigr) \hspace{20pt} \text{for }i = 1,\ldots, I
\] \[ (\log \alpha_{Bi}, \log
\beta_{Bi}) \, \, \sim \, \, \mbox{BVN}\bigl( (\mu_{\alpha B},
\mu_{\beta B}), \boldsymbol \Sigma_B \bigr) \hspace{20pt} \text{for }i =
1,\ldots, I \] and
\[ \eta_i \sim \mbox{N}(\mu_\eta,
\tau^2_\eta). \]
Priors for the exchangeable means \(\mu\) and variances \(\tau\) are identical to the meta-analytic
models for the historical single-agent data in section 7.2.
