Robust, Sceptical, and Power Priors

Author

Ndoh Penn

Published

September 24, 2026

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)

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

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")
#> 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

Deriving the Informative Prior from Historical Trials (MAP)

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.

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)
#> 
#> -- bayprior: Historical control (log-odds) --
#> • Distribution : NORMAL
#> • Parameters   : mu = 0.1115, sigma = 0.0944
#> • Method       : map elicitation
#> • Expert       : map_historical
#> • Mean (SD)    : 0.1115 (0.0944)
#> • 95% CrI      : [-0.0736, 0.2966]

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)

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)
#> 
#> -- bayprior: Historical control (from raw counts) --
#> • Distribution : NORMAL
#> • Parameters   : mu = -1.1311, sigma = 0.203
#> • Method       : map elicitation
#> • Expert       : map_historical
#> • Mean (SD)    : -1.1311 (0.203)
#> • 95% CrI      : [-1.529, -0.7332]

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.

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().

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")

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")
#> 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

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")
#> 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

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)
#> 
#> -- Power Prior Calibration --
#> • Method          : bayes_factor
#> • Target BF       : 3
#> • Optimal delta   : 0.05
#> • Power prior mean: 0.4448
#> • Power prior SD  : 0.173

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

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)
#> 
#> -- Power Prior Calibration --
#> • Method          : bayes_factor
#> • Target BF       : 3
#> • Optimal delta   : 0.95
#> • Power prior mean: 0.3478
#> • Power prior SD  : 0.0396

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

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.

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.

Overview

Robust Mixture Prior

Concept

Usage

Varying the Vague Weight

Vague SD Multiplier

Deriving the Informative Prior from Historical Trials (MAP)

Concept

Usage

Getting y and se from Raw Trial Data

Choosing the Heterogeneity Prior Scale

Inspecting the Heterogeneity Posterior

Sceptical Prior

Concept

Normal Family (Mean Differences, Log Odds Ratios)

Beta Family (Response Rates)

Log-Normal Family (Hazard Ratios)

Enthusiastic vs Sceptical Pair

Power Prior

Concept

Calibrating \(\delta\)

Compatibility Method

Power Prior for Other Families

Choosing the Right Alternative Prior

Including Robust Priors in the Regulatory Report

References

Getting `y` and `se` from Raw Trial Data