Statistical Analysis Tool

Bayesian Posterior
Probability Calculator

Converts frequentist study results into Bayesian posterior probabilities that an effect exceeds a specified threshold, assuming a non-informative prior.

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Study Results — Mean Difference

Observed Mean Difference e.g. 2.5

Standard Error of Mean Diff or enter CI below

Lower 95% CI optional — derives SE

Upper 95% CI optional — derives SE

Observed Mean Difference the point estimate

One-tailed p-value in the direction of the observed effect

Derived SE computed from p-value — used in the calculation

Threshold (minimum meaningful effect) posterior prob that effect ≥ this value will be computed

Use informative prior

Show non-informative prior in distribution plot

Posterior Probability

— %

0%50%100%

Posterior distribution

Dashed line marks the threshold τ.

Note on equivalence: With a non-informative prior and normally distributed data, the posterior probability that the effect exceeds zero equals 1 − (one-tailed p-value). For other thresholds the calculation generalises straightforwardly using the normal posterior.

Study Results — Odds Ratio

Observed Odds Ratio on natural scale, e.g. 1.54

SE of log(OR) or enter CI below

Lower 95% CI (OR scale) optional — derives SE

Upper 95% CI (OR scale) optional — derives SE

Observed Odds Ratio on natural scale, e.g. 1.54

One-tailed p-value in the direction of the observed OR

Derived SE of log(OR) computed from p-value — used in the calculation

Threshold OR (minimum meaningful effect) e.g. 1.0 = any benefit; 1.2 = at least 20% increase in odds

Use informative prior (on log-OR scale)

Show non-informative prior in distribution plot

Posterior Probability

— %

0%50%100%

Posterior distribution

Dashed line marks the threshold on the analysis scale.

Analysis on log scale: The calculation is performed on the log(OR) scale where normality holds. The threshold OR is log-transformed before comparison. The SE of log(OR) is derived from the CI using: SE = [ln(upper CI) − ln(lower CI)] / 3.92.

Study Results — Risk Ratio

Observed Risk Ratio on natural scale, e.g. 0.75

SE of log(RR) or enter CI below

Lower 95% CI (RR scale) optional — derives SE

Upper 95% CI (RR scale) optional — derives SE

Observed Risk Ratio on natural scale, e.g. 0.75

One-tailed p-value in the direction of the observed RR

Derived SE of log(RR) computed from p-value — used in the calculation

Threshold RR (minimum meaningful effect) e.g. 1.0 = any increase; 0.8 = at least 20% risk reduction

Use informative prior (on log-RR scale)

Show non-informative prior in distribution plot

Posterior Probability

— %

0%50%100%

Posterior distribution

Dashed line marks the threshold on the analysis scale.

Analysis on log scale: As with odds ratios, the analysis proceeds on the log(RR) scale. SE is derived from the CI using: SE = [ln(upper CI) − ln(lower CI)] / 3.92.

Study Results — Correlation

Observed Correlation (r) Pearson r, e.g. 0.32

Sample Size (n) derives SE = 1/√(n−3); or enter CI below

Lower 95% CI (r scale) optional — derives SE

Upper 95% CI (r scale) optional — derives SE

Derived SE of Fisher z computed from CI if given, otherwise from n — used in the calculation

Observed Correlation (r) Pearson r, e.g. 0.32

One-tailed p-value in the direction of the observed correlation

Derived SE of Fisher z computed from p-value — used in the calculation

Threshold correlation (minimum meaningful effect) e.g. 0 = any positive correlation; 0.1 = at least a small positive correlation

Use informative prior (on correlation scale)

Show non-informative prior in distribution plot

Posterior Probability

— %

0%50%100%

Posterior distribution

Dashed line marks the threshold on the analysis scale.

Analysis on the Fisher-z scale: The correlation is transformed with z = arctanh(r), where the sampling distribution is approximately normal with SE = 1/√(n−3). When a confidence interval is supplied, SE is taken instead from SE = [arctanh(upper) − arctanh(lower)] / 3.92. The threshold is transformed the same way, and posterior summaries are back-transformed to the r scale via r = tanh(z). An informative prior is entered in correlation units: its mean is converted exactly (z = arctanh(r)), and its SD is converted with the delta method, σ_z ≈ σ_r / (1 − r²), evaluated at the prior mean.

Methodology

This calculator implements Bayesian inference under a normal likelihood model. With a non-informative (flat) prior, the posterior distribution of the parameter equals the likelihood, and the posterior probability that the effect exceeds any threshold τ is:

P(θ ≥ τ | data) = 1 − Φ( (τ − θ̂) / SE )

where Φ is the standard normal CDF, θ̂ is the observed estimate, and SE is its standard error. For ratio measures (OR, RR), the analysis is conducted on the log scale. For correlations, it is conducted on the Fisher z scale (z = arctanh(r)), where z is approximately normal with SE = 1/√(n−3); the threshold and posterior are transformed back with r = tanh(z).

With an informative prior N(μ₀, σ₀²), the posterior is also normal with:

Posterior mean = (μ₀/σ₀² + θ̂/SE²) / (1/σ₀² + 1/SE²)
Posterior SD = 1 / √(1/σ₀² + 1/SE²)

The key equivalence with frequentist statistics: when the threshold is zero (or 1 for ratios) and a flat prior is used, the posterior probability of a positive effect equals 1 − (one-tailed p-value). The Bayesian framing generalises this to arbitrary thresholds and admits prior knowledge.