ELIR ESS: Bayesian Information Metric

• ELIR ESS is a Bayesian metric that quantifies the effective sample size by equating prior information with data observations via the Fisher information scale.
• It is defined using local information ratios, with unique predictive consistency ensuring that new data contribute additively to the prior ESS.
• The metric facilitates external information borrowing in clinical trials by guiding design decisions such as control arm sizing and subgroup analyses through rigorous computational methods.

The Expected Local-Information-Ratio (ELIR) Effective Sample Size (ESS) quantifies the information contained in a Bayesian prior distribution, or more generally a distribution encoding external evidence, in terms of the number of observations from the current data-generating process that would carry equivalent Fisher information. Unlike earlier ESS definitions, the ELIR ESS is uniquely characterized by the property of predictive consistency—meaning, after observing $N$ new data points, the expected posterior ESS equals the prior ESS plus $N$. Modern applications are widespread in Bayesian clinical trial designs that borrow external information on the treatment effect scale, requiring rigorous methods for calibrating the prior’s informativeness (Zhang et al., 2024, Neuenschwander et al., 2019).

1. Foundations of ELIR ESS

The ELIR ESS is rooted in a Bayesian updating context, where the parameter of interest, typically denoted $\delta$, represents a summary treatment effect (e.g., mean difference, log-odds ratio, risk ratio). External information is encoded via a prior $p(\delta)$, and current trial data produce a likelihood $L(\delta; y)$. The ELIR framework asks: How many new-subject data points, under the current likelihood, are informationally equivalent to the prior $p(\delta)$? This equivalence is formalized on the scale of Fisher information.

For a scalar parameter $\theta$ with prior $\pi(\theta)$ and likelihood $f(y|\theta)$, the ELIR is defined as:

$$r(\theta) = \frac{-\frac{d^2}{d\theta^2}\log \pi(\theta)}{-E_{Y|\theta} \left[ \frac{d^2}{d\theta^2} \log f(Y; \theta)\right ]}$$

with overall ELIR ESS:

$$\operatorname{ESS}_{\mathrm{ELIR}} = E_{\theta \sim \pi}[r(\theta)]$$

(Neuenschwander et al., 2019).

For frequentist–Bayesian calibration, the denominator defines the Fisher information per “information unit” (commonly, one subject or a minimal (a:b) randomized block).

2. ELIR ESS for Continuous and Binary Endpoints

ELIR ESS extensions encompass a broad class of endpoint models and effect measures.

• Continuous (Normal) Endpoints: For $y_{Ci} \sim N(\mu_C, \sigma^2)$ and $y_{Tj} \sim N(\mu_T, \sigma^2)$, with $\delta = \mu_T - \mu_C$ and $\delta \sim N(\delta_0, \sigma_0^2)$, the ELIR simplifies due to constant variances:

$$\operatorname{ESS}_{\mathrm{ELIR}} = \frac{\sigma^2 (1/a + 1/b )}{\sigma_0^2}$$

where $(a, b)$ are treatment:control ratio in the information unit (Zhang et al., 2024).

• Binary Endpoints: Let $(p_C, p_T)$ denote control and treatment event probabilities. For risk difference, log-odds ratio, or log-risk ratio, joint bivariate normal priors are placed on $(\theta_1, \theta_2)$, with $\theta_1 = \operatorname{logit}(p_C)$ and $\theta_2 = \delta$ (the effect measure).

The information-unit variance and corresponding Fisher information ($I_{\text{like}}$) depend on the parameterization: - Risk difference: $\delta = p_T - p_C$, variance $= p_T(1 - p_T)/a + p_C(1 - p_C)/b$ - Odds ratio: variance $= 1/[a p_T(1-p_T)] + 1/[b p_C(1-p_C)]$ - Risk ratio: variance $= 1/[a p_T] + 1/[b p_C]$

The ESS requires evaluation of two-dimensional integrals over the prior-predictive distribution for $(p_C, p_T)$, typically performed via Gaussian quadrature or Monte Carlo (Zhang et al., 2024).

3. Predictive Consistency Property

The key rationale for ELIR ESS is predictive consistency. Let $N$ denote the number of additional independent data points to be collected. The ELIR ESS satisfies:

$$E_{Y_N}\left[ \operatorname{ESS}_{\mathrm{ELIR}}(\pi(\cdot | Y_N)) \right ] = \operatorname{ESS}_{\mathrm{ELIR}}(\pi) + N$$

This property fails for older variance-ratio, precision-ratio, or plug-in information-based ESS definitions, except in textbook exponential-family conjugate prior settings. Predictive consistency ensures that new data incrementally and additively contribute to the information accounting, matching the intuition behind “effective sample size” (Neuenschwander et al., 2019).

In Gaussian models with known variance, predictive consistency holds exactly. For binomial endpoints, it holds approximately with bias diminishing as $n \rightarrow \infty$ (Zhang et al., 2024).

4. Comparison with Other ESS Definitions

Early ESS metrics include the variance-ratio ESS:

$$\operatorname{ESS}_{\mathrm{VR}} = \frac{E_{\theta}[I(\theta)^{-1}]}{\operatorname{Var}(\theta)}$$

the precision-ratio ESS, and variants based on prior “precision” at the mean or mode. While these agree with the textbook ESS in conjugate exponential family settings, they diverge for nonconjugate, mixture, or heavy-tailed priors. Only the ELIR metric produces additivity of ESS over future data, aligning the ESS with its natural decision-theoretic interpretation (Neuenschwander et al., 2019).

Method	Predictive Consistency	Closed Form in Exponential Family	Robust to Prior Misspecification
ELIR	Yes	Yes	Yes
VR, PR, MTM	No	Yes	No

For full column definitions, see [(Neuenschwander et al., 2019), Table 2–3].

5. Computation and Practical Implementation

The computation of ELIR ESS proceeds as follows:

1. Specify the information unit (usually tied to trial randomization).
2. Derive $I_{\mathrm{like}}$ from the variance formula appropriate for the endpoint.
3. Calculate $I_{\mathrm{prior}}$ (prior Fisher information).
4. For binary endpoints, compute the joint prior density over $(p_C, p_T)$, involving a Jacobian for variable transformation.
5. Evaluate

$$\operatorname{ESS}_{\mathrm{ELIR}} = \iint \frac {I_{\mathrm{prior}}(\delta)}{I_{\mathrm{like}}(p_C, p_T)} f(p_C, p_T) dp_C dp_T$$

using numerical integration or Monte Carlo sampling (Zhang et al., 2024).

For binomial models with effect-scale borrowing, the bivariate transformation and Monte Carlo averaging are implemented in publicly available R code (https://github.com/squallteo/TrtEffESS).

For mixture priors (e.g., mixture of Beta for event rates), one fits a parametric approximation, computes prior information for the mixture using analytical formulas, and integrates or averages the local information ratio (Neuenschwander et al., 2019).

6. Applications in External Borrowing and Hierarchical Modeling

Typical use cases include:

• External Data Borrowing on the Treatment Effect Scale: In modern Bayesian clinical trials, information from external sources is encoded as a prior on the treatment effect. ELIR ESS quantifies the amount of historical information (in subjects) being integrated, thereby guiding trial design and regulatory justification (Zhang et al., 2024).
• Control Arm Sizing: For example, using a meta-analytic-predictive prior fit as a Beta mixture to placebo rates from historical data, the ELIR ESS outputs the number of concurrent controls necessary for equivalence in information content (Neuenschwander et al., 2019).
• Subgroup Analysis in Hierarchical Models: ESS can be assigned to subgroups in partially or fully pooled hierarchical models, quantifying the borrowed strength from other subgroups, robustified against possible prior misspecification via mixture components (Neuenschwander et al., 2019).

A worked example from (Zhang et al., 2024): A 2:1 trial with a log-odds-ratio prior $\delta \sim N(0.5, 0.25)$, prior mean logit-control $-1$, prior s.d. $s_1=0.5$, correlation $\rho = -0.8$, and $s_2 = 0.5$ yields ELIR of 92.9 subjects; halving $s_2$ yields ESS ≃ 25.3 subjects, illustrating sensitivity to prior diffuseness.

7. Extensions and Algorithmic Notes

ELIR ESS accommodates continuous, binary (risk difference, odds ratio, risk ratio), and more general endpoint models. For normal endpoints, closed forms are often available; for more complex models, numerical or Monte Carlo integration is standard. Publicly available R implementations support both univariate and mixture prior settings.

Algorithmic steps often include: mixture model fitting, computation of prior information for component distributions, and either one- or two-dimensional integration via Gaussian quadrature or simulation. Provided R functions (e.g., ess_betamix, ess_normmix) operationalize these workflows for direct application in clinical trial planning (Neuenschwander et al., 2019, Zhang et al., 2024).