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Robust, Sceptical, and Power Priors

Ndoh Penn

2026-05-29

1 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)

2 Robust Mixture Prior

2.1 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).

2.2 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$vague_weight, "\n")
#> Informative component weight: 0.8
cat("Vague component weight:      ", rob$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:   0.3649

2.3 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)

2.4 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")
#> 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:    1.6

3 Sceptical Prior

3.1 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 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.”

3.2 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")
#> 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:   0.25

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

3.3 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)

3.4 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)

3.5 Enthusiastic vs Sceptical Pair

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):      0.059

4 Power Prior

4.1 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.

4.2 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.

4.3 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$delta_opt, "\n")
#> Optimal delta (BF method):            0.05
cat("Optimal delta (compatibility method):", calib_compat$delta_opt, "\n")
#> Optimal delta (compatibility method): 1

4.4 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)

5 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",
    "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

6 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.

7 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.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.