Docs/Bayesian Borrowing

Bayesian Historical Borrowing

Name: Zetyra
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Author: Zetyra

Technical documentation for incorporating historical control data into current trial design with appropriate discounting. This module implements Power Priors, Commensurate Priors, and Meta-Analytic Predictive (MAP) Priors for external data synthesis.

1. Overview & Motivation

Historical borrowing leverages data from prior studies to strengthen inference in the current trial. When historical and current populations are similar, borrowing can reduce sample size requirements while maintaining statistical rigor.

Key Benefits

Smaller Trials

Reduce required sample size by 20-40% when historical data is highly relevant

Ethical

Fewer patients randomized to control when effect is well-established

Efficiency

Faster trials with preserved statistical precision

The Exchangeability Assumption

Borrowing is valid only when historical and current populations areexchangeable—meaning they can be treated as samples from the same underlying distribution. Key similarity dimensions:

Patient Population: Same disease stage, demographics, prior treatments
Endpoints: Identical definitions and assessment methods
Standard of Care: Similar background therapies
Time Period: No temporal drift in outcomes

Critical Warning

Inappropriate borrowing (from dissimilar populations) can inflate Type I error or bias treatment effect estimates. Always use conflict diagnostics and consider discounting when similarity is uncertain.

2. Power Prior Method

The power prior (Ibrahim & Chen, 2000) discounts historical likelihood by raising it to a power $\delta \in [0, 1]$ :

\pi(\theta | D_0, \delta) \propto L(\theta | D_0)^\delta \cdot \pi_0(\theta)

For Beta-Binomial models with historical data $(k_0, n_0)$ and base prior $\text{Beta}(\alpha_0, \beta_0)$ :

\theta | D_0 \sim \text{Beta}(\alpha_0 + \delta k_0, \beta_0 + \delta(n_0 - k_0))

Effective Sample Size

The ESS from the power prior is:

ESS_{historical} = \delta \cdot n_0

Discount Factor	Interpretation	When to Use
$\delta = 1.0$	Full borrowing (100% weight)	Identical population, same sponsor's prior trial
$\delta = 0.5$	Skeptical borrowing (50% weight)	Similar population, minor protocol differences
$\delta = 0.2$	Conservative borrowing (20% weight)	Different indication, mechanism-based only
$\delta = 0.0$	No borrowing (ignore historical)	Populations clearly different

Choosing the Discount Factor

The discount factor should be pre-specified in the protocol based on clinical judgment about similarity. A common approach: start with $\delta = 0.5$ as a “skeptical default” and adjust based on formal similarity assessment.

3. Commensurate Prior Method

The commensurate prior (Hobbs et al., 2011) uses a hierarchical model where the commensurability parameter $\tau$ controls borrowing strength dynamically.

\theta_{current} | \theta_{historical}, \tau \sim N(\theta_{historical}, \tau^2)

Key insight: when $\tau \to 0$ , no borrowing (fully skeptical); when $\tau \to \infty$ , approaches full borrowing.

Simplified Implementation

Zetyra uses a computationally efficient approximation that maps the commensurability parameter to an effective discount factor:

\delta_{effective} = \frac{\tau}{1 + \tau}

$\tau = 0$

$\delta = 0$ (no borrowing)

$\tau = 1$

$\delta = 0.5$ (balanced)

$\tau = 9$

$\delta = 0.9$ (strong borrowing)

$\tau \to \infty$

$\delta \to 1$ (full borrowing)

4. Meta-Analytic Predictive (MAP) Prior

When multiple historical studies are available, the MAP prior (Schmidli et al., 2014) synthesizes them using random-effects meta-analysis:

\theta_i \sim N(\mu, \tau^2)

Where $\mu$ is the pooled effect and $\tau^2$ captures between-study heterogeneity.

Heterogeneity Assessment (I²)

The calculator reports the I² statistic to quantify heterogeneity:

I^2 = \max\left(0, \frac{Q - (k-1)}{Q}\right) \times 100\%

I² Range	Interpretation	Recommendation
0-25%	Low heterogeneity	Full borrowing appropriate
25-75%	Moderate heterogeneity	Use robust MAP
>75%	High heterogeneity	Borrow cautiously

Robust MAP Component

To protect against prior-data conflict, the robust MAP mixes the informative MAP prior with a vague component:

\pi_{robust}(\theta) = (1 - w) \cdot \pi_{MAP}(\theta) + w \cdot \pi_{vague}(\theta)

Where $w$ is typically 0.1–0.2 (10–20% vague component).

5. Prior-Data Conflict Diagnostics

The calculator assesses whether current trial data conflicts with the historical prior using a prior predictive check.

Conflict Detection Algorithm

Given current data $(k_{current}, n_{current})$ and effective prior $\text{Beta}(\alpha, \beta)$ :

1.Compute current rate: $\hat{\theta} = k/n$
2.Compute prior mean: $\mu_0 = \alpha/(\alpha + \beta)$
3.Compute predictive variance (includes sampling variability)
4.Calculate z-score and two-tailed p-value

P-value	Conflict Level	Action
> 0.10	None	Proceed with borrowing
0.01–0.10	Moderate	Consider reducing discount (δ × 0.5)
< 0.01	Severe	Minimal borrowing (δ ≤ 0.2) or none

Regulatory Requirement

The FDA guidance recommends pre-specifying how prior-data conflict will be handled in the Statistical Analysis Plan (SAP). Document the conflict detection criteria and fallback procedures.

6. Sample Size Impact

The calculator compares sample size requirements with and without historical borrowing to quantify the efficiency gain.

Comparison Framework

With Borrowing

Use effective prior $\text{Beta}(\alpha, \beta)$ derived from historical data

Without Borrowing

Use uninformative prior $\text{Beta}(1, 1)$

For each scenario, the calculator finds the minimum $n$ achieving target power (80%) and Type I error control (5%).

Sample Size Reduction

\text{Reduction \%} = \frac{n_{without} - n_{with}}{n_{without}} \times 100\%

Typical reductions range from 15–40% depending on historical data quality and discount factor.

7. Regulatory Considerations

FDA Bayesian Guidance Section V.D.4

“When utilizing external data, sponsors should describe methods for assessing the similarity of external data to trial data, including approaches for adjusting the degree of borrowing if inconsistencies are identified.”

Documentation Requirements

Historical Data Source: Study ID, publication reference, patient population, endpoints, and quality assessment
Similarity Justification: Explicit comparison of inclusion/exclusion criteria, endpoints, and standard of care
Borrowing Method: Power prior, commensurate, or MAP with parameter specifications
Conflict Handling: Pre-specified criteria and fallback procedures
Operating Characteristics: Type I error and power under various scenarios
Sensitivity Analysis: Results under alternative discount factors and prior specifications

8. API Quick Reference

POST /api/v1/calculators/bayesian-borrowing

Key Parameters

Parameter	Type	Description
method	string	"power_prior" \| "commensurate_prior" \| "map_prior"
historical_events, historical_n	int	Historical study data (power/commensurate)
discount_factor	float	Power prior δ ∈ [0, 1] (default: 0.5)
studies	array	List of studies for MAP prior (min 2)
robust_weight	float	MAP robust component weight (default: 0.1)

Key Response Fields

effective_prior — Resulting Beta(α, β) parameters
ess — Effective sample size breakdown
comparison — Sample size with/without borrowing
conflict_assessment — Prior-data conflict analysis

View full API documentation →

Bayesian Historical Borrowing

Contents

1. Overview & Motivation

Key Benefits

Smaller Trials

Ethical

Efficiency

The Exchangeability Assumption

2. Power Prior Method

Effective Sample Size

Choosing the Discount Factor

3. Commensurate Prior Method

Simplified Implementation

$\tau = 0$

$\tau = 1$

$\tau = 9$

$\tau \to \infty$

4. Meta-Analytic Predictive (MAP) Prior

Heterogeneity Assessment (I²)

Robust MAP Component

5. Prior-Data Conflict Diagnostics

Conflict Detection Algorithm

6. Sample Size Impact

Comparison Framework

With Borrowing

Without Borrowing

Sample Size Reduction

7. Regulatory Considerations

Documentation Requirements

8. API Quick Reference

Key Parameters

Key Response Fields

Related Documentation

Bayesian Historical Borrowing

Contents

1. Overview & Motivation

Key Benefits

Smaller Trials

Ethical

Efficiency

The Exchangeability Assumption

2. Power Prior Method

Effective Sample Size

Choosing the Discount Factor

3. Commensurate Prior Method

Simplified Implementation

τ=0\tau = 0τ=0

τ=1\tau = 1τ=1

τ=9\tau = 9τ=9

τ→∞\tau \to \inftyτ→∞

4. Meta-Analytic Predictive (MAP) Prior

Heterogeneity Assessment (I²)

Robust MAP Component

5. Prior-Data Conflict Diagnostics

Conflict Detection Algorithm

6. Sample Size Impact

Comparison Framework

With Borrowing

Without Borrowing

Sample Size Reduction

7. Regulatory Considerations

Documentation Requirements

8. API Quick Reference

Key Parameters

Key Response Fields

Related Documentation

$\tau = 0$

$\tau = 1$

$\tau = 9$

$\tau \to \infty$