Beyond the elicited informative prior, regulatory submissions typically require one or more alternative prior specifications to demonstrate robustness. bayprior provides three well-established alternatives:
| Prior type | Function | Method | Reference |
|---|---|---|---|
| Robust mixture | robust_prior() |
Mixes informative + vague | Schmidli et al. (2014) |
| Sceptical | sceptical_prior() |
Centred at null effect | Spiegelhalter et al. (1994) |
| Power prior | calibrate_power_prior() |
Down-weights historical data | Ibrahim & Chen (2000) |
A robust prior protects against prior misspecification by mixing the informative elicited prior with a vague (diffuse) component:
\[\pi_{\text{robust}}(\theta) = (1 - w) \cdot \pi_{\text{informative}}(\theta) + w \cdot \pi_{\text{vague}}(\theta)\]
The vague component — a wide Normal centred at the informative prior mean — ensures the posterior is never completely dominated by a conflicting prior. The default weight is \(w = 0.20\) (80% informative, 20% vague).
informative <- elicit_beta(
mean = 0.30,
sd = 0.08,
method = "moments",
label = "Response rate"
)
rob <- robust_prior(
informative = informative,
vague_weight = 0.20,
label = "Robust mixture prior"
)
plot(rob)
cat("Informative component weight:", 1 - rob$vague_weight, "\n")
#> Informative component weight: 0.8
cat("Vague component weight: ", rob$vague_weight, "\n")
#> Vague component weight: 0.2
cat("Mixture mean:", round(rob$fit_summary$mean, 4), "\n")
#> Mixture mean: 0.3
cat("Mixture SD: ", round(rob$fit_summary$sd, 4), "\n")
#> Mixture SD: 0.3649A higher vague weight makes the prior more diffuse and reduces its influence on the posterior:
par(mfrow = c(1, 1))
weights <- c(0.10, 0.20, 0.30, 0.50)
cols <- c("#185FA5", "#1D9E75", "#D85A30", "#888780")
x <- seq(0, 0.8, length.out = 300)
plot(x, bayprior:::.eval_density_vec(informative, x),
type = "l", lwd = 2, col = "#185FA5",
xlab = "Response rate", ylab = "Density",
main = "Effect of vague weight on robust prior",
ylim = c(0, 6))
for (i in seq_along(weights)) {
r <- robust_prior(informative, vague_weight = weights[i])
lines(x, bayprior:::.eval_density_vec(r, x),
col = cols[i], lwd = 1.5, lty = i + 1)
}
legend("topright",
legend = c("Informative",
paste0("w = ", weights)),
col = c("#185FA5", cols),
lwd = 2,
lty = c(1, 2, 3, 4, 5),
bty = "n", cex = 0.8)
The vague component’s SD defaults to 10× the informative prior’s SD. Adjust with vague_sd:
rob_narrow <- robust_prior(informative, vague_weight = 0.20,
vague_sd = 2 * informative$fit_summary$sd)
rob_wide <- robust_prior(informative, vague_weight = 0.20,
vague_sd = 20 * informative$fit_summary$sd)
cat("Narrow vague SD: ", round(rob_narrow$components$vague$fit_summary$sd, 3), "\n")
#> Narrow vague SD: 0.16
cat("Default vague SD:", round(rob$components$vague$fit_summary$sd, 3), "\n")
#> Default vague SD: 0.8
cat("Wide vague SD: ", round(rob_wide$components$vague$fit_summary$sd, 3), "\n")
#> Wide vague SD: 1.6The sceptical prior (Spiegelhalter & Freedman, 1994) represents the view of a conservative regulator who is sceptical of a treatment effect. It is centred at the null value of the treatment effect with width calibrated to a chosen strength of scepticism.
This is the FDA’s recommended sensitivity prior for trials using informative priors: the trial conclusions should hold even under a prior that places most mass at “no effect.”
For continuous or log-scale quantities where the null is typically 0:
sc_weak <- sceptical_prior(
null_value = 0, family = "normal", strength = "weak",
label = "Log OR (weak sceptic)"
)
sc_moderate <- sceptical_prior(
null_value = 0, family = "normal", strength = "moderate",
label = "Log OR (moderate sceptic)"
)
sc_strong <- sceptical_prior(
null_value = 0, family = "normal", strength = "strong",
label = "Log OR (strong sceptic)"
)
cat("Weak SD: ", sc_weak$fit_summary$sd, "\n")
#> Weak SD: 1
cat("Moderate SD:", sc_moderate$fit_summary$sd, "\n")
#> Moderate SD: 0.5
cat("Strong SD: ", sc_strong$fit_summary$sd, "\n")
#> Strong SD: 0.25plot(sc_moderate)
The SD mapping by strength:
| Strength | SD | Interpretation |
|---|---|---|
weak |
1.0 | Vague scepticism — wide prior around null |
moderate |
0.5 | 2-SD departure from null has ~5% prior probability |
strong |
0.25 | Very concentrated at null — very sceptical |
For binary endpoints, null_value must be in \((0, 1)\) — it represents the null response rate, not a difference:
# Null response rate of 20%: sceptic believes treatment is no better than 20%
sc_beta <- sceptical_prior(
null_value = 0.20,
family = "beta",
strength = "moderate",
label = "Response rate (sceptical)"
)
plot(sc_beta)
For hazard ratios, the null is HR = 1, which corresponds to null_value = 0 on the log scale:
sc_hr <- sceptical_prior(
null_value = 0, # log(1) = 0, i.e. HR = 1
family = "lognormal",
strength = "moderate",
label = "Hazard ratio (sceptical)"
)
plot(sc_hr)
The FDA recommends presenting conclusions under both an enthusiastic prior (favouring treatment benefit) and a sceptical prior:
enthusiastic <- elicit_beta(
mean = 0.45, sd = 0.08,
method = "moments", label = "Response rate (enthusiastic)"
)
sceptical <- sceptical_prior(
null_value = 0.20, family = "beta", strength = "moderate",
label = "Response rate (sceptical)"
)
data_obs <- list(type = "binary", x = 18, n = 40)
post_enth <- bayprior:::.conjugate_update(enthusiastic, data_obs)
post_scep <- bayprior:::.conjugate_update(sceptical, data_obs)
cat("Posterior mean (enthusiastic):", round(post_enth$fit_summary$mean, 3), "\n")
#> Posterior mean (enthusiastic): 0.45
cat("Posterior mean (sceptical): ", round(post_scep$fit_summary$mean, 3), "\n")
#> Posterior mean (sceptical): 0.356
cat("Posterior SD (enthusiastic): ", round(post_enth$fit_summary$sd, 3), "\n")
#> Posterior SD (enthusiastic): 0.056
cat("Posterior SD (sceptical): ", round(post_scep$fit_summary$sd, 3), "\n")
#> Posterior SD (sceptical): 0.059The power prior (Ibrahim & Chen, 2000) provides a principled method for incorporating historical data by down-weighting it by a factor \(\delta \in (0, 1]\):
\[\pi(\theta | D_0, \delta) \propto L(\theta | D_0)^\delta \cdot \pi_0(\theta)\]
where \(D_0\) is the historical data and \(\delta\) controls how much weight it receives. \(\delta = 1\) fully incorporates the historical data (standard Bayesian updating); \(\delta \to 0\) ignores it entirely.
calibrate_power_prior() selects \(\delta\) to achieve a target Bayes Factor between the historical-data-informed prior and the current likelihood — ensuring the historical data is incorporated only to the extent it is compatible with current data:
base <- elicit_beta(
mean = 0.50,
sd = 0.20,
method = "moments",
label = "Response rate"
)
calib <- calibrate_power_prior(
historical_data = list(type = "binary", x = 12, n = 40),
current_data = list(type = "binary", x = 18, n = 50),
base_prior = base,
target_bf = 3,
delta_grid = seq(0.05, 1.0, by = 0.05),
method = "bayes_factor"
)
print(calib)plot(calib)
The calibration curves show:
Alternatively, select \(\delta\) to be the largest value for which the historical-data-informed prior shows no conflict with the current data:
calib_compat <- calibrate_power_prior(
historical_data = list(type = "binary", x = 12, n = 40),
current_data = list(type = "binary", x = 18, n = 50),
base_prior = base,
method = "compatibility",
delta_grid = seq(0.05, 1.0, by = 0.05)
)
cat("Optimal delta (BF method): ", calib$delta_opt, "\n")
#> Optimal delta (BF method): 0.05
cat("Optimal delta (compatibility method):", calib_compat$delta_opt, "\n")
#> Optimal delta (compatibility method): 1Power prior updating is supported for Beta, Normal, Gamma, Log-Normal, and Mixture priors. For a Normal prior with continuous historical data:
base_norm <- elicit_normal(
mean = 0.0, sd = 0.5,
method = "moments", label = "Mean difference"
)
calib_norm <- calibrate_power_prior(
historical_data = list(type = "continuous", x = 0.35, sd = 0.3, n = 60),
current_data = list(type = "continuous", x = 0.42, sd = 0.3, n = 80),
base_prior = base_norm,
target_bf = 3,
delta_grid = seq(0.05, 1.0, by = 0.10),
method = "bayes_factor"
)
print(calib_norm)library(knitr)
kable(data.frame(
Situation = c(
"No conflict, regulatory requirement",
"Mild conflict detected",
"Severe conflict detected",
"Historical data available",
"FDA enthusiastic/sceptical pair required"
),
`Recommended prior` = c(
"Robust mixture (w = 0.20)",
"Robust mixture (w = 0.30-0.40)",
"Sceptical prior (moderate-strong)",
"Power prior (calibrated)",
"Sceptical prior as second arm"
),
check.names = FALSE
), align = "ll")| Situation | Recommended prior |
|---|---|
| No conflict, regulatory requirement | Robust mixture (w = 0.20) |
| Mild conflict detected | Robust mixture (w = 0.30-0.40) |
| Severe conflict detected | Sceptical prior (moderate-strong) |
| Historical data available | Power prior (calibrated) |
| FDA enthusiastic/sceptical pair required | Sceptical prior as second arm |
All three robust prior types — robust mixture, sceptical, and power prior — are automatically included in the downloaded prior justification report when they have been computed in the session. Pass them directly to prior_report():
# After running the analyses above...
prior_report(
prior = prior,
conflict = cd,
sensitivity = sa,
robust_prior = rob, # adds "Robust Mixture" section to report
sceptical_prior = scep, # adds "Sceptical Prior" section to report
power_prior = calib, # adds "Power Prior" section with calibration table
output_format = "html",
output_file = "prior_justification_report",
trial_name = "TRIAL-001",
sponsor = "BioPharma Ltd",
author = "J. Smith, Biostatistician"
)Each section in the report includes a parameter summary table and the corresponding density or calibration plot. The compliance checklist in the report automatically marks “Robust / sceptical prior computed” as Complete when any of the three types is supplied.
When using the Shiny app, the robust priors flow into the report automatically — simply run the analyses in the Robust Priors panel before clicking Download Report.
Note:
prior_report()requiresdevtools::install(), not justdevtools::load_all(). Quarto spawns a fresh R session that requires the package to be properly installed.
Ibrahim, J. G. & Chen, M.-H. (2000). Power prior distributions for regression models. Statistical Science, 15, 46–60.
Schmidli, H., Gsteiger, S., Roychoudhury, S., O’Hagan, A., Spiegelhalter, D., & Neuenschwander, B. (2014). Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics, 70, 1023–1032.
Spiegelhalter, D. J., Freedman, L. S., & Parmar, M. K. B. (1994). Bayesian approaches to randomized trials. Journal of the Royal Statistical Society A, 157, 357–416.
Gravestock, I. & Held, L. (2017). Adaptive power priors with empirical Bayes for clinical trials. Pharmaceutical Statistics, 16, 349–360.