Nothing
Robust, Sceptical, and Power Priors
In bayprior: Bayesian Prior Elicitation and Diagnostics for Clinical Trials
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.width = 6,
fig.height = 3.8,
out.width = "95%",
dpi = 120,
warning = FALSE,
message = FALSE
)
library(bayprior)
Overview
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) |
Robust Mixture Prior
Concept
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).
Usage
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$vague_weight, "\n")
cat("Vague component weight: ", rob$vague_weight, "\n")
cat("Mixture mean:", round(rob$fit_summary$mean, 4), "\n")
cat("Mixture SD: ", round(rob$fit_summary$sd, 4), "\n")
Varying the Vague Weight
A higher vague weight makes the prior more diffuse and reduces its influence
on the posterior:
oldpar <- 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)
par(oldpar)
Vague SD Multiplier
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")
cat("Default vague SD:", round(rob$components$vague$fit_summary$sd, 3), "\n")
cat("Wide vague SD: ", round(rob_wide$components$vague$fit_summary$sd, 3), "\n")
Sceptical Prior
Concept
The 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 sensitivity prior follows Spiegelhalter and Freedman's (1994) approach
for trials using informative priors: the trial conclusions should hold even
under a prior that places most mass at "no effect."
Normal Family (Mean Differences, Log Odds Ratios)
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")
cat("Moderate SD:", sc_moderate$fit_summary$sd, "\n")
cat("Strong SD: ", sc_strong$fit_summary$sd, "\n")
plot(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 |
Beta Family (Response Rates)
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)
Log-Normal Family (Hazard Ratios)
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)
Enthusiastic vs Sceptical Pair
A common regulatory sensitivity practice is to present conclusions under both
an enthusiastic prior (favouring treatment benefit) and a sceptical prior,
following Spiegelhalter and Freedman (1994):
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")
cat("Posterior mean (sceptical): ", round(post_scep$fit_summary$mean, 3), "\n")
cat("Posterior SD (enthusiastic): ", round(post_enth$fit_summary$sd, 3), "\n")
cat("Posterior SD (sceptical): ", round(post_scep$fit_summary$sd, 3), "\n")
Power Prior
Concept
The 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.
Calibrating $\delta$
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:
- Top panel: Bayes Factor vs $\delta$. The dashed red line is the target BF;
the dotted green line marks the optimal $\delta$.
- Bottom panel: Box p-value vs $\delta$. Values below 0.05 indicate
conflict at that weight.
Compatibility Method
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$delta_opt, "\n")
cat("Optimal delta (compatibility method):", calib_compat$delta_opt, "\n")
Power Prior for Other Families
Power 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)
Choosing the Right Alternative Prior
library(knitr)
kable(data.frame(
Situation = c(
"No conflict, regulatory requirement",
"Mild conflict detected",
"Severe conflict detected",
"Historical data available",
"Enthusiastic/sceptical pair for sensitivity analysis"
),
`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")
Including Robust Priors in the Regulatory Report
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() requires devtools::install(), not just
devtools::load_all(). Quarto spawns a fresh R session that requires the
package to be properly installed.
References
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.
Try the bayprior package in your browser
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bayprior documentation built on Aug. 27, 2026, 1:09 a.m.
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.width = 6, fig.height = 3.8, out.width = "95%", dpi = 120, warning = FALSE, message = FALSE ) library(bayprior)
Overview
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) |
Robust Mixture Prior
Concept
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).
Usage
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$vague_weight, "\n") cat("Vague component weight: ", rob$vague_weight, "\n") cat("Mixture mean:", round(rob$fit_summary$mean, 4), "\n") cat("Mixture SD: ", round(rob$fit_summary$sd, 4), "\n")
Varying the Vague Weight
A higher vague weight makes the prior more diffuse and reduces its influence on the posterior:
oldpar <- 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) par(oldpar)
Vague SD Multiplier
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") cat("Default vague SD:", round(rob$components$vague$fit_summary$sd, 3), "\n") cat("Wide vague SD: ", round(rob_wide$components$vague$fit_summary$sd, 3), "\n")
Sceptical Prior
Concept
The 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 sensitivity prior follows Spiegelhalter and Freedman's (1994) approach for trials using informative priors: the trial conclusions should hold even under a prior that places most mass at "no effect."
Normal Family (Mean Differences, Log Odds Ratios)
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") cat("Moderate SD:", sc_moderate$fit_summary$sd, "\n") cat("Strong SD: ", sc_strong$fit_summary$sd, "\n")
plot(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 |
Beta Family (Response Rates)
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)
Log-Normal Family (Hazard Ratios)
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)
Enthusiastic vs Sceptical Pair
A common regulatory sensitivity practice is to present conclusions under both an enthusiastic prior (favouring treatment benefit) and a sceptical prior, following Spiegelhalter and Freedman (1994):
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") cat("Posterior mean (sceptical): ", round(post_scep$fit_summary$mean, 3), "\n") cat("Posterior SD (enthusiastic): ", round(post_enth$fit_summary$sd, 3), "\n") cat("Posterior SD (sceptical): ", round(post_scep$fit_summary$sd, 3), "\n")
Power Prior
Concept
The 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.
Calibrating $\delta$
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:
- Top panel: Bayes Factor vs $\delta$. The dashed red line is the target BF; the dotted green line marks the optimal $\delta$.
- Bottom panel: Box p-value vs $\delta$. Values below 0.05 indicate conflict at that weight.
Compatibility Method
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$delta_opt, "\n") cat("Optimal delta (compatibility method):", calib_compat$delta_opt, "\n")
Power Prior for Other Families
Power 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)
Choosing the Right Alternative Prior
library(knitr) kable(data.frame( Situation = c( "No conflict, regulatory requirement", "Mild conflict detected", "Severe conflict detected", "Historical data available", "Enthusiastic/sceptical pair for sensitivity analysis" ), `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")
Including Robust Priors in the Regulatory Report
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.
References
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.
Try the bayprior package in your browser
Any scripts or data that you put into this service are public.
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