A Bayesian Alternative to the Two One-Sided Tests (TOST) Procedure: State of the Art and a Proposal for a Generalized Bayesian Equivalence Model

Technical Summary: A Bayesian Alternative to the Two One-Sided Tests (TOST) Procedure

Problem Statement
Equivalence testing is a critical statistical task in bioequivalence, biosimilar development, and reproducibility research, aiming to demonstrate that an effect lies within a pre-specified negligible interval $(-\delta, \delta)$. The standard frequentist solution, the Two One-Sided Tests (TOST) procedure, faces three primary limitations:

1. Lack of Probabilistic Interpretation: It yields a binary rejection/non-rejection decision rather than a direct probability statement regarding the parameter's location within the equivalence region.
2. Inability to Incorporate Historical Evidence: It lacks a native mechanism to formally integrate prior or historical data (e.g., from previous bioequivalence studies or historical control frequencies), despite regulatory encouragement to do so.
3. Asymmetry: Like standard null hypothesis significance testing, failing to reject the TOST null does not constitute positive evidence for equivalence; the procedure remains silent in such cases.

While Bayesian alternatives exist (e.g., Region of Practical Equivalence [ROPE] approaches and interval-null Bayes factors) and methods for historical borrowing exist (e.g., power priors), the literature lacks a unified, practically implementable framework that simultaneously handles both continuous and frequency-based (count) historical data with an explicit, tunable discounting mechanism. Furthermore, closed-form theoretical properties for such integrated models have been largely absent.

Methodology: Generalized Bayesian TOST with Power-Discounted Priors (GBT-PP)
The paper proposes the GBT-PP, a model that unifies conjugate updating with power priors and a dual decision rule.

• Prior Elicitation and Discounting: The model accepts historical data in two forms:

  • Continuous: Summarized as mean ($\bar{x}_0$), standard deviation ($s_0$), and sample size ($n_0$).
  • Frequency: Summarized as successes ($s_0$) and failures ($f_0$) in $n_0$ trials.
  • Power-Discounted Priors: Following Ibrahim and Chen (2000), historical likelihoods are raised to a power $w \in [0, 1]$. This weight $w$ acts as a pre-registered, fixed discount factor.
    • For continuous data, this results in a Normal prior with variance scaled by $1/w$.
    • For frequency data, this results in a Beta prior where the historical counts are scaled by $w$.
  • The model deliberately fixes $w$ rather than estimating it hierarchically to avoid technical pathologies associated with unnormalized power priors (Neuenschwander et al., 2009).

• Posterior Update:

  • The continuous case utilizes Normal-Normal conjugacy.
  • The frequency case utilizes Beta-Binomial conjugacy.
  • The posterior distribution combines the power-discounted historical prior with the current study's likelihood.

• Dual Decision Rule: Equivalence is declared only if both of the following conditions are met:

  1. Posterior Probability ($P_{eq}$): The posterior probability mass within the equivalence region $E = (-\delta, \delta)$ exceeds a threshold $\gamma$ (e.g., 0.95).
  2. Interval-Null Bayes Factor ($BF_{01}$): The Bayes factor comparing the interval null $H_0: \theta \in E$ against $H_1: \theta \notin E$ exceeds a threshold $BF^*$ (e.g., 3).
  • Crucial Design Feature: The $BF_{01}$ is computed using a hybrid approach. The posterior $P_{eq}$ is derived from the power-discounted prior (dependent on $w$), while the prior probability $\pi_{eq}$ in the Bayes factor denominator is computed under a separate, fixed, pre-registered reference prior (independent of $w$). This decoupling prevents the Jeffreys–Lindley paradox from trivially driving the Bayes factor to infinity as the prior becomes diffuse.

• Approximation for Frequency Data: For the two-proportion case, where the posterior equivalence probability lacks a closed form, the paper proposes a deterministic Edgeworth expansion (Proposition 6) to approximate $P_{eq}$, offering a skewness-corrected alternative to Monte Carlo simulation.

Key Theoretical Contributions
The paper establishes five original theoretical results for the GBT-PP, proved in full in Appendix C:

1. Monotonicity of the Posterior Mean (Proposition 1): The posterior mean $\theta_n(w)$ is monotonic in the discount weight $w$. The direction of the shift depends strictly on the discrepancy between the historical mean and the current estimate, governed by the ratio of their precisions.
2. Asymptotic Consistency (Theorem 2 & Corollary 3): Regardless of the fixed weight $w$, as the current sample size $n \to \infty$, the GBT-PP decision rule converges in probability to the classical TOST decision. This formally demonstrates that historical borrowing is asymptotically "harmless" and that current data eventually dominate the inference.
3. Jeffreys–Lindley-Type Paradox (Theorem 4): The paper characterizes a paradox where, if the reference prior used for the Bayes factor component becomes increasingly diffuse, the Bayes factor diverges to $+\infty$ regardless of the data. This result necessitates the design recommendation that the reference prior variance must be fixed and pre-registered, not allowed to become arbitrarily large.
4. Closed-Form Sample Size Formula (Corollary 5): A formula is derived to calculate the required current sample size to achieve a target posterior equivalence probability. It explicitly quantifies the number of observations "saved" by borrowing historical information ($\Delta n^*$), expressed in units of the current study's variance.
5. Edgeworth Approximation (Proposition 6): A first-order Edgeworth expansion is provided for the posterior equivalence probability in the two-proportion case, offering a deterministic, auditable alternative to Monte Carlo methods with explicit skewness correction.

Results and Illustrations
The methodology is illustrated through two worked examples:

• Continuous Bioequivalence: Demonstrates that increasing $w$ pulls the posterior mean toward the historical mean and increases both $P_{eq}$ and $BF_{01}$ monotonically when the historical data supports equivalence.
• Frequency-Based Cure Rates: Demonstrates that if historical data is less favorable (further from the comparator) than current data, increasing $w$ can actually decrease the probability of equivalence. This highlights that borrowing is not inherently beneficial; it depends on the alignment of historical and current estimates. The Edgeworth approximation is shown to track Monte Carlo estimates closely across the range of $w$.

Significance and Claims
The paper claims to fill a specific methodological gap by providing a unified framework that treats continuous and frequency-based historical data symmetrically within a single decision rule. Its significance lies in:

• Practical Implementability: Offering a reproducible, closed-form (or deterministic approximation) approach that avoids reliance on complex hierarchical modeling or pure simulation for sensitivity analysis.
• Theoretical Rigor: Providing the first closed-form characterizations of monotonicity, asymptotic consistency, paradox behavior, and sample-size savings specifically for the power-discounted interval-null setting.
• Regulatory Relevance: Addressing the need for tools that can formally combine diverse historical evidence types (analytical/pharmacokinetic vs. clinical event rates) in line with evolving regulatory guidance (e.g., FDA 2026 draft guidance) toward justification-first, evidence-integrating frameworks.

The author positions GBT-PP not as a replacement for all hierarchical borrowing methods (like commensurate priors) but as a minimal, transparent, and theoretically grounded baseline that explicitly manages the trade-off between historical influence and current data through a fixed, auditable weight $w$.