Robust, Sceptical, and Power Priors

Ndoh Penn

2026-10-09

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
MAP (from historical trials) map_prior() Random-effects meta-analysis with explicit tau prior Roever et al. (2021)
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 Deriving the Informative Prior from Historical Trials (MAP)

3.1 Concept

The examples above assume the informative prior fed into robust_prior() already exists (from elicitation, or otherwise). In practice, that informative component is often itself derived from multiple historical trials via a random-effects meta-analysis. map_prior() performs this step directly: given trial-level effect estimates and standard errors, it returns a meta-analytic-predictive (MAP) prior suitable for passing straight into robust_prior().

Unlike treating the between-trial heterogeneity (the parameter \(\tau\)) as fixed, map_prior() always places an explicit prior on \(\tau\), following the weakly-informative recommendations of Roever et al. (2021). This matters because how much heterogeneity is assumed among historical trials directly affects how much borrowing is ultimately justified — an assumption that should be stated and justified explicitly, not left as an unstated default. Sensible outcome-specific defaults are built in; see ?resolve_tau_prior.

map_prior() fits this model itself, entirely in base R: the posterior of \(\mu\) given \(\tau\) has a closed form, so only the marginal posterior of \(\tau\) needs numerical integration (stats::integrate()). bayprior has no dependency, direct or optional, on any external meta-analysis package for this. See ?map_prior for the underlying model.

3.2 Usage

# Five historical control-arm trials -- i.e. a single-arm log-odds scale,
# not a two-arm log-odds RATIO. This distinction matters: see below.
y  <- c(0.10, -0.05, 0.22, 0.05, 0.15)
se <- c(0.12, 0.15, 0.10, 0.18, 0.14)

map <- map_prior(
  y, se,
  outcome_type = "single_arm_log_odds",
  label        = "Historical control (log-odds)"
)
print(map)

map_prior() returns an ordinary bayprior object (a Normal approximation to the marginal posterior of the meta-analytic mean, integrating over \(\tau\)), so it can be used exactly like any elicited prior:

rob_map <- robust_prior(map, vague_weight = 0.20)
plot(rob_map)

3.3 Getting y and se from Raw Trial Data

The example above assumes y/se are already computed. In practice they’re usually derived from each historical trial’s own reported summary statistics (event counts, arm means/SDs, …), and metafor::escalc() is the standard tool for that conversion. historical_effect_sizes() is a thin wrapper around it that maps map_prior()’s outcome_type vocabulary onto the matching metafor measure, so raw trial data can be turned into y/se in one step:

# The same five trials as above, but as originally reported -- event
# counts (xi out of ni) rather than pre-computed log-odds estimates.
trials <- data.frame(
  xi = c(11, 9, 14, 10, 12),
  ni = c(45, 40, 55, 48, 42)
)
es <- historical_effect_sizes("single_arm_log_odds", trials)
es[, c("xi", "ni", "y", "se")]
#>   xi ni          y        se
#> 1 11 45 -1.1284653 0.3468730
#> 2  9 40 -1.2367626 0.3786412
#> 3 14 55 -1.0745147 0.3095461
#> 4 10 48 -1.3350011 0.3554093
#> 5 12 42 -0.9162907 0.3415650

map_from_counts <- map_prior(
  y = es$y, se = es$se,
  outcome_type = "single_arm_log_odds",
  label        = "Historical control (from raw counts)"
)
print(map_from_counts)

This is not the only way to get y/se – they can equally be read by hand off a published point estimate and 95% CI (e.g. se = (log(upper) - log(lower)) / (2 * 1.96) on whatever scale the CI was reported) – and historical_effect_sizes() is only usable if metafor is installed (it is a Suggests dependency, not Imports, so it is not required just to use map_prior() itself). See ?historical_effect_sizes for the full column-name mapping across outcome types – two-arm odds ratios, standardised/raw mean differences, incidence-rate ratios, and correlations, in addition to the single-arm case shown here.

3.4 Choosing the Heterogeneity Prior Scale

outcome_type is not just a label: unless tau_prior is supplied explicitly, it selects the default \(\tau\) prior from Roever et al. (2021)’s outcome-specific recommendations, via resolve_tau_prior():

resolve_tau_prior("log_or")               # two-arm log-odds ratio
#> $family
#> [1] "half_normal"
#> 
#> $scale
#> [1] 0.5
resolve_tau_prior("single_arm_log_odds")  # e.g. a historical control rate
#> $family
#> [1] "half_normal"
#> 
#> $scale
#> [1] 1

Note that these give different scales (0.5 vs 1.0), despite both being “log-odds” in casual speech – they are different quantities with different heterogeneity properties, and conflating them silently applies the wrong default. "mean_difference" (raw, unstandardised units) has no preset at all – its scale is entirely endpoint-specific (mmHg, cm, days, …), so map_prior() requires tau_prior to be supplied explicitly for it:

# An explicit tau_prior always overrides the outcome_type preset, and is
# required (map_prior() errors otherwise) for "mean_difference".
map_tight <- map_prior(y, se, tau_prior = list(family = "half_normal", scale = 0.1))
map_wide  <- map_prior(y, se, tau_prior = list(family = "half_normal", scale = 1.0))

cat("Tight tau prior -- posterior SD:", round(map_tight$fit_summary$sd, 4), "\n")
#> Tight tau prior -- posterior SD: 0.0683
cat("Wide tau prior  -- posterior SD:", round(map_wide$fit_summary$sd, 4), "\n")
#> Wide tau prior  -- posterior SD: 0.0944

Allowing more heterogeneity (a wider \(\tau\) prior) generally widens the resulting informative prior, which in turn results in a more conservative (less confident) prior being carried forward into robust_prior().

3.5 Inspecting the Heterogeneity Posterior

The full posterior distribution of \(\tau\) (evaluated on a plotting grid whose range is chosen automatically from the fitted posterior, not a fixed guess) is available via $tau_posterior, useful for checking whether the data are actually informative about heterogeneity or whether the posterior for \(\tau\) still closely tracks the prior:

plot(map$tau_posterior$tau, map$tau_posterior$density, type = "l",
     xlab = expression(tau), ylab = "Posterior density",
     main = "Posterior distribution of the heterogeneity SD")

4 Sceptical Prior

4.1 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 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.”

4.2 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

4.3 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)

4.4 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)

4.5 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")
#> 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

5 Power Prior

5.1 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.

5.2 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:

5.3 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

5.4 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)

6 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")
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)
Enthusiastic/sceptical pair for sensitivity analysis Sceptical prior as second arm

7 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.

8 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.

Roever, C., Bender, R., Dias, S., Schmid, C. H., Schmidli, H., Sturtz, S., Weber, S., & Friede, T. (2021). On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects meta-analysis. Research Synthesis Methods, 12(4), 448–474.