Prior-Data Conflict
Diagnostics
Before updating the prior with trial data it is good practice to
check compatibility between the prior and the observed data.
prior_conflict() computes four complementary
diagnostics:
| Box p-value |
Prior predictive p-value; < 0.05 flags conflict |
| Surprise index |
Standardised distance; > 3 indicates high surprise |
| KL divergence |
Information distance from prior to likelihood |
| Bhattacharyya overlap |
Distributional overlap; < 0.3 is concerning |
prior <- elicit_beta(
mean = 0.30,
sd = 0.10,
method = "moments",
label = "Response rate"
)
# Observed: 18 events in 40 patients
cd <- prior_conflict(
prior = prior,
data_summary = list(type = "binary", x = 18, n = 40), # also: "continuous", "poisson", "survival"
alpha = 0.05
)
print(cd)
Visualise the prior-likelihood-posterior overlay to assess
compatibility graphically:
plot_prior_likelihood(
prior,
data_summary = list(type = "binary", x = 18, n = 40),
show_posterior = TRUE
)
When the prior (blue) and likelihood (orange) overlap substantially
there is no meaningful conflict, and the posterior (green) represents a
credible update.
Sensitivity
Analysis
Regulatory guidelines require demonstrating that trial conclusions
are robust to plausible prior variations.
sensitivity_grid() evaluates how posterior inferences
change across a grid of hyperparameter values.
Note: the grids below are intentionally coarse for vignette build
speed. In practice, use finer grids (e.g. seq(1, 8, 0.5))
to obtain smooth surfaces.
sa <- sensitivity_grid(
prior = prior,
data_summary = list(type = "binary", x = 14, n = 40), # also: "continuous", "poisson", "survival"
param_grid = list(alpha = seq(1, 6, 1), beta = seq(2, 14, 2)),
target = c("posterior_mean", "prob_efficacy"),
threshold = 0.30
)
Tornado Plot
The tornado plot ranks posterior quantities by their range across the
hyperparameter grid. Wider bars indicate higher prior sensitivity:
Influence
Heatmap
The heatmap shows how the posterior mean changes as both
hyperparameters vary. The orange diamond marks the reference prior:
plot_sensitivity(sa, target = "posterior_mean")
Credible Interval
Sensitivity
sensitivity_cri() focuses specifically on the posterior
credible interval width — a key regulatory quantity:
cri_sa <- sensitivity_cri(
prior = prior,
data_summary = list(type = "binary", x = 14, n = 40), # also: "continuous", "poisson", "survival"
param_grid = list(alpha = seq(1, 6, 1), beta = seq(2, 14, 2)),
cri_level = 0.95,
threshold = 0.30
)
plot_sensitivity(cri_sa, target = "cri_width")
Robust and Sceptical
Priors
When prior-data conflict is detected, or when a conservative
regulatory stance is required, bayprior provides ready-made alternative
prior constructions.
Robust Mixture
Prior
A robust prior mixes an informative component with a vague (diffuse)
component (Schmidli et al., 2014). The vague component ensures the
posterior is never dominated by a conflicting informative prior. The
default vague_weight = 0.20 gives an 80/20
informative/vague split:
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)
Sceptical Prior
A sceptical prior is centred at the null value of the treatment
effect and represents a conservative regulatory stance (Spiegelhalter et
al., 1994).
For a Normal family (e.g. log odds ratio, mean
difference), null_value is the null treatment effect
(typically 0):
sc_norm <- sceptical_prior(
null_value = 0,
family = "normal",
strength = "moderate",
label = "Log odds ratio (sceptical)"
)
plot(sc_norm)
For a Beta family (binary response rate),
null_value must be in (0, 1):
sc_beta <- sceptical_prior(
null_value = 0.20, # null response rate of 20%
family = "beta",
strength = "moderate",
label = "Response rate (sceptical)"
)
plot(sc_beta)
Power Prior
Calibration
When relevant historical data exist, a power prior down-weights the
historical evidence by \(\delta \in (0,
1]\) before incorporating it.
calibrate_power_prior() selects \(\delta\) to achieve a target Bayes Factor
(Ibrahim & Chen, 2000; Gravestock & Held, 2017).
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(0.10, 1.0, by = 0.10), # coarse grid for vignette speed
method = "bayes_factor"
)
print(calib)
plot(calib)
The calibration curves show the Bayes Factor (top) and Box p-value
(bottom) across the \(\delta\) grid.
The vertical dotted green line marks the optimal weight and the
horizontal dashed lines mark the target BF and \(\alpha = 0.05\).
Multivariate Conflict
Diagnostics
For trials with co-primary endpoints,
conflict_mahalanobis() extends the conflict check to the
multivariate setting. The Mahalanobis distance has a known \(\chi^2\) reference distribution under
compatibility:
mv <- conflict_mahalanobis(
prior_means = c(0.35, 0.60),
prior_cov = matrix(c(0.01, 0.003, 0.003, 0.015), 2, 2),
obs_means = c(0.55, 0.58),
obs_cov = matrix(c(0.008, 0.002, 0.002, 0.010), 2, 2) / 50,
labels = c("Response rate", "OS rate")
)
print(mv)
#>
#> Marginal z-scores per parameter:
#> Response rate OS rate
#> 1.984 -0.162
The marginal_z scores identify which endpoint
contributes most to any detected conflict.
Generating a Prior
Justification Report
Once the analysis is complete, prior_report() renders a
self-contained regulatory-ready document in HTML, PDF, or Word format.
The report includes the prior specification, conflict diagnostics,
sensitivity visualisations, and an FDA/EMA compliance checklist.
# prior_report() calls rmarkdown::render() internally and cannot be run
# inside a vignette build. Run interactively after loading bayprior.
prior_report(
prior = prior,
conflict = cd,
sensitivity = sa,
output_format = "html", # or "pdf" or "docx"
trial_name = "TRIAL-001",
sponsor = "BioPharma Ltd",
author = "J. Smith",
notes = paste0(
"Prior based on Phase 2 internal data and two external expert ",
"elicitations. Sensitivity analysis confirms robustness across a ",
"wide range of prior hyperparameter values."
)
)
Complete Worked
Example
The following end-to-end example mirrors a realistic binary-endpoint
trial workflow, combining three expert opinions and assessing
sensitivity.
# 1. Elicit priors from three experts using different methods
e1 <- elicit_beta(mean = 0.28, sd = 0.08, method = "moments",
expert_id = "E1", label = "ORR")
e2 <- elicit_beta(mean = 0.35, sd = 0.10, method = "moments",
expert_id = "E2", label = "ORR")
e3 <- elicit_beta(
quantiles = c("0.10" = 0.15, "0.50" = 0.30, "0.90" = 0.52),
expert_id = "E3", label = "ORR"
)
# 2. Pool into a consensus prior
consensus <- aggregate_experts(
priors = list(E1 = e1, E2 = e2, E3 = e3),
weights = c(0.40, 0.35, 0.25),
method = "linear"
)
# 3. Check conflict with interim data (20 responses in 55 patients)
data_obs <- list(type = "binary", x = 20, n = 55)
cd_full <- prior_conflict(consensus, data_obs)
print(cd_full)
# 4. Visualise prior-likelihood-posterior
plot_prior_likelihood(consensus, data_obs, show_posterior = TRUE)
# 5. Sensitivity analysis (coarse grid for vignette speed)
sa_full <- sensitivity_grid(
prior = consensus,
data_summary = data_obs,
param_grid = list(alpha = seq(1, 6, 1), beta = seq(2, 12, 2)),
target = c("posterior_mean", "prob_efficacy"),
threshold = 0.25
)
plot_tornado(sa_full)
plot_sensitivity(sa_full, target = "prob_efficacy")
# 6. Robust alternative in case conflict worsens at final analysis
rob_full <- robust_prior(consensus, vague_weight = 0.20)
plot(rob_full)
References
Box, G. E. P. (1980). Sampling and Bayes’ inference in scientific
modelling and robustness. Journal of the Royal Statistical Society
A, 143, 383–430.
Gravestock, I. & Held, L. (2017). Adaptive power priors with
empirical Bayes for clinical trials. Pharmaceutical Statistics,
16, 349–360.
Ibrahim, J. G. & Chen, M.-H. (2000). Power prior distributions
for regression models. Statistical Science, 15, 46–60.
O’Hagan, A., Buck, C. E., Daneshkhah, A., Eiser, J. R., Garthwaite,
P. H., Jenkinson, D. J., Oakley, J. E., & Rakow, T. (2006).
Uncertain Judgements: Eliciting Experts’ Probabilities.
Wiley.
Oakley, J. E. & O’Hagan, A. (2010). SHELF: the Sheffield
Elicitation Framework. University of Sheffield.
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.
