Fit a Stan MAP prior generation model to summary-level data.
Source:R/stan_map_summary_data.R
stan_map_summary_data.RdFit a Stan MAP prior generation model to summary-level data.
Arguments
- y
integer vector, number of events in the trials.
- n
integer vector, number of patients in the trials.
- time,
numeric vector, strictly positive, time at risk in the trials.
- a_alpha
numeric, first shape param of beta prior on a. See Details.
- a_beta
numeric, second shape param of beta prior on a. See Details.
- b_alpha
numeric, first shape param of beta prior on b. See Details.
- b_beta
numeric, second shape param of beta prior on b. See Details.
- ...
extra parameters are passed to
sampling.
Value
object of type bssd_binary_map.
Details
This routine fits the meta-analytical model equation:
$$ log(-log(1 - p_{i})) = \alpha_{i} + log(T_{i}) $$
where \(p_{i}\) is the probability of event in trial i, adjusted for the
intended or average time-at-risk, \(T_{i}\).
The model is equivalent to:
$$ log(\lambda_{i}) = \alpha_{i} $$
where
$$ \lambda_{i} = -log(1 - p_{i}) / T_{i} $$
is the hazard of the event, assumed constant through time, and \(p_{i}\) is
the probability of event in unit time in trial i.
Thus this is a random-effects model that assumes the log-hazard terms are
exchangeable across trials, and the \( \alpha_{i} \) terms are
trial-specific intercepts.
To complete the meta-analytical model, we apply the hierarchical distribution
$$ p_{j} \sim Beta(abN, (1-a)bN) $$
where N is the total number of patients in the historical trials, and a and b
are hyperparameters.
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.
This method creates a prior from summary-level data, requiring only the
intended or average times-at-risk and total counts of patients with an
adverse event in each trial. It is likely aggregate statistics like this can
be derived from publications and/or filings. However, if you have
patient-level times-at-risk and adverse event statuses,
stan_map_patient_data provides a similar method to create a
prior from patient-level data. The patient-level approach is generally
preferable because it allows adjustment for how long each individual patient
spent at risk.
This model is effectively the same as that fit by
stan_map_patient_data with the notable exception that since
patient-level times-at-risk are not known, the intended trial follow-up
times, \( T_{j} \), are used instead.
Note: setting a seed in R does not affect the state of the
random generator in stan. If a reproducible output is desired, then the
optional parameter seed needs to be populated. For additional
information about this parameter, see sampling.
Examples
set.seed(2021)
H <- 10
time <- 0.5 + cumsum(rexp(H, 5))
# Use constant hazard model:
hazard <- 0.15
prob <- 1 - exp(- hazard * time)
n <- rpois(H, lambda = 100 * time)
y <- rbinom(H, size = n, prob = prob)
x <- stan_map_summary_data(y = y, n = n, time = time,
a_alpha = 1 / 3, a_beta = 1 / 3,
b_alpha = 5, b_beta = 5)
#>
#> SAMPLING FOR MODEL 'map_prior_summary_data' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 1.8e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.18 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1:
#> Chain 1: Elapsed Time: 0.074 seconds (Warm-up)
#> Chain 1: 0.079 seconds (Sampling)
#> Chain 1: 0.153 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'map_prior_summary_data' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 1e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.1 seconds.
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#> Chain 2: Elapsed Time: 0.069 seconds (Warm-up)
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#> Chain 2: 0.15 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'map_prior_summary_data' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 1e-05 seconds
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#> Chain 3: Elapsed Time: 0.072 seconds (Warm-up)
#> Chain 3: 0.076 seconds (Sampling)
#> Chain 3: 0.148 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'map_prior_summary_data' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 1e-05 seconds
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#> Chain 4: Elapsed Time: 0.068 seconds (Warm-up)
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#> Chain 4: 0.128 seconds (Total)
#> Chain 4:
x
#> Prob(Event): Mean 0.148 (95% HPDI 0.129, 0.168)
#> MAP Prob(Event): Mean 0.148 (95% HPDI 0.113, 0.183)