MAP method for generating a prior from historical data
As detailed in Brock et al. (2022), the core model used in this package is:
\log{(-\log{(1 - \pi_{ij})})} = \alpha_{j} + \log{(T_{ij})}
where \pi_{ij} is the probability of event in study j, adjusted for the time-at-risk of patient i, T_{ij}. This model is equivalent to:
\log(\lambda_{j}) = \alpha_{j}
where
\lambda_{j} = -\log(1 - \pi_{ij}) / T_{ij}
is the hazard of event, assumed constant through time. We have a regression model, where the values \lambda_j must be best derived to fit the aggregate patients in each study.
Let \pi_{j} be the probability of event in unit time in study j. Then the relation \lambda_j = -\log(1 - \pi_j) yields \pi_j = 1 - \exp(-\lambda_j).
If we apply the hierarchical distribution:
\pi_{j} \sim Beta(abN, (1-a)bN)
where N is the total number of patients in the historical studies, and a and b are hyperparameters, we have a random-effects model that assumes the event probabilities in unit time are exchangeable across studies.
To illustrate the role of a and b, consider that the effective sample size of this distribution is abN + (1-a)bN = bN, i.e., b is a discount to the historical data, taking values on (0, 1). Furthermore, the mean of this distribution is abN / bN = a. Thus, a is the expected probability of event in unit time, and once again a is bound to the range (0, 1). Hyperpriors a \sim Beta(a_\alpha, a_\beta) and b \sim Beta(b_\alpha, b_\beta) are natural choices to enforce these ranges.
The parameter of interest is \pi^* \sim Beta(abN, (1-a)bN). We then use p(a) and p(b) (the distributions of a and b) to derive unconditional distribution of \pi^*:
p(\pi^*) = \int \int p(\pi^* |\, a, b) \, p(a) \, p(b) \, da \, db
This distribution will serve as an informative prior on the event probability in unit time under placebo in the current study. In practice, this distribution is observable only as MCMC samples, to which a beta mixture distribution is fit to obtain a tractable analytical prior to be used in further analysis.
If data is available, an analogous method can be used to create a prior for the active arm in the current study. Alternatively, an uninformative prior could be used for the active arm.
Chi-squared method for generating a prior from historical data
We might wish to use an alternative method to specify a prior. In the so-called chi-squared method, we iteratively use a chi-squared test to detect heterogeneity in the adverse event rates in the historical studies.
A complication to using this method is that the historical studies will generally have been conducted for different lengths of time. To test for heterogeneity in the historical studies, we require that the adverse event rates are standardized in some sense, to ensure valid comparability.
Let the number of patients in historical study j be n_j, with y_j experiencing at least one instance of an adverse event of interest. The naive event rates r_j = y_j / n_j will be affected by the length of time patients received therapy (i.e. were at risk) in trial j, t_j. To adjust for this, we calculate the standardized event counts that would be expected had each trial used time-at-risk, t^*:
y^*_j = n_j \left(1 - \left(1 - \left( \frac{y_j}{n_j} \right) \right)^{t^* / t_j} \right)
After this transformation, the standardized event rates r_j^* = y_j^* / n_j will be comparable because they each express a rate over a period t^*. We then test for a heterogeneity in the y_j^* and n_j pairs using a chi-squared test. If this test rejects the null hypothesis of homogeneity, we remove the trial j that appears to be the largest outlier (i.e. the number of events is furthest from that which is expected), and re-run the test. If this test does not reject the null, we effectively infer homogeneity and take \sum_{j} y_j^* and \sum_{j} n_j - y_j^* as the two shape parameters for a Beta prior distribution on the event rate in the ongoing study.
As above, we generally use this method to create a prior on the event rate in the placebo arm but the same approach could be used to create a prior on the experimental arm.
BSSD analysis for an ongoing study
We use equivalent model forms to those introduced above to conduct the Bayesian safety signal analysis in an ongoing or current study:
\log{(-\log{(1 - \pi_A)})} = \alpha_A + \log{(T_{j})}
and
\log{(-\log{(1 - \pi_C)})} = \alpha_C + \log{(T_{j})}
where \pi_A is the probability of adverse event in unit time under the active therapy, \pi_C is the same under placebo therapy, and T_j is time-at-risk for patient j (allocated to a possibly unknown therapy). We can place priors on \pi_A and \pi_C using the MAP method above if data is available and an informative prior is desirable. Naturally, other methods are available. For the active arm, we might prefer a traditional uninformative prior.
Fitting this model is relatively simple when we know the treatment
allocations. It is more complex, however, to fit this model in a blinded
setting, where the treatment allocations are unknown. To overcome this,
we use a mixture likelihood where each patient is considered to have
been potentially allocated to the active and placebo treatments,
weighted by the known randomisation probabilities. The bssd
package handles this complication for you.
From a Bayesian perspective, we conclude a safety signal exists if
Prob(\pi_A > \pi_C | X) > \gamma_0
where X represents the interim study data, and \gamma_0 is a critical threshold between 0 and 1. Greater values of \gamma_0 reflect that greater certainty is required to conclude a safety signal exists.