Bayes-Estimation

Conceived, supervised design and testing: Dr G Narenthiran MB ChB BSc(MedSci)(Hons) MRCS(Ed.) FEBNS FRCS(SN), g_narenthiran@hotmail.com

Interactive Clinical Bayesian Estimation, A/B Testing, Two-Group Comparison & Decision Suite

Green Panels: Estimation Statistics (Observed sample data, empirical counts, event rates, sample mean differences, and sample effect sizes like Cohen's h & d)
Light Sky Blue Panels: Bayesian Analysis (Prior belief parameters, full posterior distributions, 95% Highest Density Intervals [HDI], ROPE [Region of Practical Equivalence] decisions, and Bayes Factors)
🟒 Study Presets (Clinical Trials)
🟒 Group A (Control / Standard of Care)
🟒 Group B (Treatment / Experimental)
🟒 Clinical Equivalence Margin (ROPE)
ℹ️ What is ROPE? (Region of Practical Equivalence)

ROPE defines an interval around zero difference that represents clinical indifference or practical equivalence. In classical testing, any microscopic difference can achieve p < 0.05 simply with large sample sizes. ROPE guarantees that we only declare superiority if the credible difference exceeds this threshold.

Clinical Example: If reducing 30-day complication rates by less than ±3% (-0.03 to +0.03) does not change patient management, setting ROPE to [-0.03, +0.03] tests whether the treatment is clinically superior, practically equivalent, or inconclusive.
P(Treatment > Control) -- Probability of Superiority
Risk Difference (ARR) -- 95% CI: [--, --]
95% Highest Density (HDI) -- Shortest 95% Credible Band
Standardized Effect Size (Cohen's h) -- 95% CI: [--, --] Effect Size
Relative Risk (RR) -- Median (95% CrI)
Odds Ratio (OR) -- Median (95% CrI)
Number Needed to Treat -- NNT = 1 / |ARR|
Bayes Factor (BF₁₀) -- Savage-Dickey Density Ratio
ROPE Clinical Equivalence Verdict:
Evaluating clinical equivalence...
Calculating...
BF₁₀ Calculating...
πŸ”΅ Posterior Distribution of Rate Difference (95% HDI, ROPE & Null Marker)
πŸ”΅ Medical Journal / APA Bayesian Summary Report
Generating Bayesian clinical report...
Methodological Interpretation Guide: What Results Mean & What They Do NOT Mean
🟒 1. Estimation Statistics Method (Bootstrap CIs & Effect Sizes)
βœ“ What Estimation Statistics Means
  • Risk Difference (ARR): The direct empirical difference in observed success rates is --.
  • 95% Bootstrap Confidence Interval: Over 5,000 resamples with replacement, 95% of generated intervals contain the true effect size. Its width reflects sampling precision without assuming Gaussian normality.
  • Standardized Effect Size (Cohen's h): Standardized arcsine difference of -- allows unbiased comparison across trials regardless of baseline event rates.
  • Number Needed to Treat (NNT): On average, treating -- patients with Intervention B rather than Control A produces one additional successful outcome.
βœ— What Estimation Statistics Does NOT Mean
  • NOT a Bayesian Parameter Probability: A 95% Bootstrap CI does NOT mean there is a 95% probability the true parameter is inside it. That is the Bayesian HDI! Frequentist CIs describe long-run procedure coverage.
  • Crossing Zero is NOT Proof of Equality: If the 95% CI includes zero, it does NOT prove the treatments are equal. It simply means the study had insufficient sample size to rule out zero.
  • Cohen's h Does NOT Dictate Clinical Value: A "small" effect size (h = 0.2) in a fatal outcome is lifesaving, whereas a "large" effect size in a minor side effect is clinically unimportant.
  • Odds Ratio is NOT Relative Risk: If event rates are high (>10%), the Odds Ratio (--) noticeably overstates the Relative Risk (--).
πŸ”΅ 2. Bayesian Analysis Method (Posterior Credibility & Decisions)
βœ“ What Bayesian Analysis Means
  • Probability of Superiority: There is a direct -- posterior probability that Treatment B is superior to Control A, conditional on prior clinical belief and data.
  • 95% Highest Density Interval (HDI): The narrowest credible range: there is an actual 95% probability that the true difference lies between --.
  • ROPE Clinical Decision: Explicitly evaluates whether the credible difference is practically superior, practically equivalent, or inconclusive relative to the minimal important margin.
  • Bayes Factor (BF₁₀): The data are -- times more probable under the hypothesis of a difference than under strict equality.
βœ— What Bayesian Analysis Does NOT Mean
  • NOT a Frequentist P-Value: P(Superiority) = 99% does NOT equal (1 - p). It is a direct posterior belief update.
  • Superiority Does NOT Mean 100% Patient Benefit: Even when P(B > A) is 99%, individual patient responses vary due to biological dispersion.
  • ROPE Overlap is NOT Proof of Equivalence: Equivalence is confirmed ONLY if the full 95% HDI lies strictly inside the ROPE. Overlap indicates an inconclusive/underpowered study.
  • BF is NOT the Posterior Probability of H₁: The Bayes Factor is an evidence multiplier (likelihood ratio); it only equals posterior odds if prior odds were 50:50.
🟒 Clinical Presets (Continuous Endpoints)
🟒 Group A (Control)
🟒 Group B (Treatment)
🟒 Equivalence Margin (ROPE)
ℹ️ What is ROPE? (Region of Practical Equivalence)

ROPE defines the boundary below which continuous differences are considered clinically trivial (e.g., within measurement noise or below the Minimal Clinically Important Difference [MCID]).

Clinical Example: On a 0–10 visual analog pain scale, a score change of ±0.5 to ±1.0 point is often imperceptible to patients. Setting ROPE to [-1.0, +1.0] prevents declaring an undetectable pain reduction as meaningful superiority.
P(Treatment > Control) -- Probability of Superiority
Mean Difference (Δμ) -- 95% CI: [--, --]
95% HDI Difference -- Highest Density Interval
Standardized Effect Size (Cohen's d) -- 95% CI: [--, --] Effect Size
Common Language Effect (CLES) -- P(Random B > Random A)
Variance Ratio (Οƒ_b / σₐ) -- SD Ratio (Heteroscedasticity)
JZS Bayes Factor (BF₁₀) -- Cauchy Prior scale r=0.707
ROPE Clinical Equivalence Verdict:
Evaluating clinical equivalence...
Calculating...
πŸ”΅ Posterior Distribution of Mean Difference (95% HDI, ROPE & Zero Line)
πŸ”΅ Posterior Distribution of Standardized Effect Size (Bayesian Cohen's d)
πŸ”΅ Medical Journal / APA Summary (BEST)
Generating report...
Methodological Interpretation Guide: What Results Mean & What They Do NOT Mean
🟒 1. Estimation Statistics Method (Bootstrap CIs & Effect Sizes)
βœ“ What Estimation Statistics Means
  • Unstandardized Mean Difference: Point estimate in native clinical units: --.
  • 95% Bootstrap Confidence Interval: 5,000 resamples with replacement produce an empirical, distribution-free interval for the difference and standardized effect size.
  • Standardized Effect Size (Cohen's d & Hedges' g): Standardized effect of -- measures separation in pooled standard deviation units.
  • Common Language Effect Size (CLES): There is a -- probability that a randomly chosen patient from Group B will score higher than a randomly chosen patient from Group A.
βœ— What Estimation Statistics Does NOT Mean
  • NOT a Parameter Probability: A frequentist Bootstrap CI describes the long-run coverage of the resampling procedure, NOT the probability that the true parameter is within this specific interval.
  • Small Effect is NOT Clinically Useless: A "small" Cohen's d (0.2) in a critical endpoint (e.g. stroke deficit or mortality) can be practice-changing.
  • Wide CI Spanning Zero is NOT Equivalence: Crossing zero merely indicates insufficient power to demonstrate an effect (absence of evidence != evidence of absence).
  • Does NOT Guarantee Every Individual Improves: Group-level mean separation does not eliminate individual response variation.
πŸ”΅ 2. Bayesian Analysis Method (Posterior Credibility & Decisions)
βœ“ What Bayesian Analysis Means
  • Posterior Probability Distribution: Complete Bayesian probability updating for group means, unequal variances, and difference parameters.
  • 95% Highest Density Interval (HDI): There is an actual 95% probability that the true population difference lies within this exact range.
  • ROPE Clinical Verdict: Directly determines if the posterior probability distribution lies completely outside (superiority) or completely inside (practical equivalence) the clinical threshold.
  • Bayes Factor (BF₁₀): The data are -- times more likely under H₁ than under Hβ‚€.
βœ— What Bayesian Analysis Does NOT Mean
  • NOT a P-Value: Bayesian posterior probability does not reference hypothetical unobserved data under a null hypothesis.
  • ROPE Overlap is NOT Proof of Equivalence: Overlapping the ROPE margin means the clinical trial is inconclusive, NOT that the treatments are identical.
  • BF₁₀ is NOT the Probability of H₁: The Bayes Factor is a likelihood multiplier, not the posterior probability of the hypothesis itself.
  • Prior Sensitivity: In small trials, posterior conclusions can be influenced by strong priors; testing robust/skeptical priors is essential.
🟒 Single Proportion Presets
🟒 Observed Sample Data
🟒 Prior Belief Specification
Posterior Mean -- Estimated Proportion
95% HDI -- Credible Interval
Effect Size vs Benchmark (h) -- Effect Size
P(Rate > Benchmark) -- Exceedance Probability
πŸ”΅ Bayesian Tri-Plot: Prior (amber dashed), Likelihood (green dotted), Posterior (blue solid)
Clinical Interpretation Guide
βœ“ What These Results Mean
  • Posterior Distribution: Combines past evidence (Prior Beta) with observed cohort data to produce the updated probability density for the true rate.
  • Benchmark Comparison: Quantifies the exact posterior probability that the complication or success rate exceeds the threshold pβ‚€.
βœ— What These Results Do NOT Mean
  • Not Purely Data-Driven: With small sample sizes (n < 30), the posterior is heavily shaped by the prior; always test sensitivity to prior choice.
  • Does Not Guarantee Future Permanence: Assumes future surgical/clinical conditions match the cohort population and procedural standard.
🟒 Observed Sample Data
🟒 Prior Belief Specification
Posterior Mean -- Precision-Weighted Mean
95% Credible Interval -- Highest Density Interval
Effect Size vs Benchmark (d) -- Effect Size
P(Mean > Benchmark) -- Exceedance Probability
πŸ”΅ Continuous Mean Tri-Plot (Prior vs Data Likelihood vs Posterior)
Clinical Interpretation Guide
βœ“ What These Results Mean
  • Precision-Weighted Updating: The posterior mean balances prior clinical experience against new trial data in proportion to their respective variances.
  • Credible Range: 95% certainty that the true population mean lies within the calculated boundaries.
βœ— What These Results Do NOT Mean
  • Does Not Imply Homogeneity: Individual patients will score above and below this mean; standard deviation must always be evaluated alongside the mean.
🟒 Diagnostic Presets
🟒 Diagnostic Parameters
Likelihood Ratio (LR+) -- Diagnostic Effect
Likelihood Ratio (LRβˆ’) -- Diagnostic Effect
Diagnostic Odds Ratio (DOR) -- Overall Discriminative Power
Post-Test Prob (+Test) -- After Positive Result
Post-Test Prob (βˆ’Test) -- After Negative Result
πŸ”΅ Bayesian Clinical Fagan Nomogram (Two-Step Likelihood Ratio Updating)
Clinical Diagnostic Interpretation Guide
βœ“ What These Results Mean
  • Post-Test Probability: The probability a patient truly has the pathology after taking the diagnostic test, combining pre-test clinical suspicion with test accuracy.
  • Likelihood Ratio Impact: An LR+ > 10 substantially increases disease probability; an LR- < 0.1 substantially decreases disease probability.
βœ— What These Results Do NOT Mean
  • Sensitivity is NOT Positive Predictive Value: A test with 95% sensitivity can still have a poor positive predictive value if the pre-test disease prevalence in that clinic is low.
  • Negative Test is NOT an Absolute Rule-Out: Unless LR- is extremely small and pre-test suspicion was low, a negative test does not guarantee disease absence.