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
The SD mapping by strength:
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)
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)
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.
