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Efficient estimation of relative risk, odds ratio and their logarithms for rare events

This paper proposes sequential estimators for relative risk, odds ratio, and their logarithms that guarantee a target mean-square error while demonstrating high efficiency, particularly for rare or moderately rare events.

Original authors: Luis Mendo

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luis Mendo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery involving two different neighborhoods, Neighborhood A and Neighborhood B. In both neighborhoods, a very rare event happens—let's say, finding a specific, elusive type of blue butterfly.

Your goal is to figure out two things:

Relative Risk (RR): How much more likely is it to find a butterfly in Neighborhood A compared to Neighborhood B? (e.g., "Is it 10 times more likely?")
Odds Ratio (OR): A slightly more complex mathematical way of comparing the chances, often used by statisticians.

The Problem: The "Rare Event" Trap

The trouble is, these butterflies are extremely rare. If you just send a team to go out and count butterflies for a fixed amount of time (say, 1 hour), you might find zero in both neighborhoods, or just one in one of them. Your data would be useless, and your estimate would be a wild guess.

Usually, to get a good answer, you'd need to spend a massive amount of time searching, hoping to catch enough butterflies to make a pattern. But that's expensive and slow.

The Solution: The "Smart Pairing" Strategy

This paper proposes a clever, efficient way to hunt for these butterflies (or any rare event) without wasting time. Instead of sending teams out blindly, the author suggests a "Smart Pairing" method.

Think of it like a game of "Red Light, Green Light" played by two players, one in Neighborhood A and one in Neighborhood B.

Step 1: The "Magic Coin Flip" (The Inner Loop)

Every time you want to check for a butterfly, you don't just look at one neighborhood. You flip a coin:

Heads: You check Neighborhood A.
Tails: You check Neighborhood B.

You keep flipping and checking until you finally spot a butterfly in the neighborhood you chose.

If you picked A and found a butterfly, great!
If you picked A and found nothing, you flip again.
If you picked B and found nothing, you flip again.

The Magic Trick: By doing this specific dance of flipping and checking, the author proves that the ratio of butterflies you find in A versus B perfectly mimics the true "Relative Risk" you are trying to measure. It's like a magic filter that turns two messy, rare searches into one clean, reliable signal.

Step 2: The "Stop When You're Sure" Rule (The Outer Loop)

Now that you have this "magic signal," you don't stop after just one butterfly. You keep collecting these "magic signals" until you have enough to be statistically confident.

The paper introduces a rule: "Stop exactly when you have found $X$ butterflies in the 'Success' pile and $Y$ butterflies in the 'Failure' pile."

If you are looking for the Relative Risk, you stop when you have a specific number of "Successes" and "Failures."
If you are looking for the Logarithm (a math trick to make the numbers easier to handle), the stopping rule changes slightly.

Once you hit that number, you stop immediately. You don't waste a single extra second searching. This is called Sequential Sampling.

Why is this so efficient? (The "No Waste" Principle)

Imagine you are trying to fill two buckets with water, but the hose is very weak (because the butterflies are rare).

Old Method: You fill Bucket A until it's full, then fill Bucket B until it's full. You might end up with Bucket A overflowing while Bucket B is still half empty, wasting water.
This Paper's Method: You fill both buckets simultaneously in pairs. If you need one drop for Bucket A, you take one drop for Bucket B and save it for later. You only stop when both buckets have exactly the right amount of water needed for the calculation.

The paper proves that for rare events, this method is incredibly efficient. It gets you the answer with almost the same amount of work as the theoretical "perfect" method (the Cramér–Rao bound), which is the best anyone could possibly hope to do.

The "Rare Event" Superpower

The most exciting part of this paper is that the rarer the event, the better this method works.

If the butterflies are common, the method is good.
If the butterflies are extremely rare (like finding a needle in a haystack), this method becomes almost perfect. It wastes almost no time.

The Takeaway

In plain English, this paper gives statisticians a new, super-efficient tool for comparing two groups when the thing they are looking for is very rare.

Instead of blindly counting for hours, they use a smart, paired, stop-when-ready strategy that guarantees:

Accuracy: You will hit your target level of certainty.
Efficiency: You won't waste time or resources, especially when the event is rare.

It's like upgrading from a rusty shovel to a laser-guided metal detector when looking for a lost coin in a vast, empty field. You find it faster, with less effort, and you know exactly when to stop digging.

1. Problem Statement

The paper addresses the statistical challenge of estimating four key parameters describing the association between a dichotomous attribute in two populations with probabilities $p_1$ and $p_2$ :

Relative Risk (RR): $p_1/p_2$
Odds Ratio (OR): $\frac{p_1(1-p_2)}{p_2(1-p_1)}$
Log Relative Risk (LRR): $\log(p_1/p_2)$
Log Odds Ratio (LOR): $\log\left(\frac{p_1(1-p_2)}{p_2(1-p_1)}\right)$

Key Constraints and Goals:

Sampling Scheme: Observations are taken in pairs (one sample from each population simultaneously). This is a common scenario in epidemiology and social sciences but poses challenges for sequential estimation.
Accuracy Guarantee: The estimator must guarantee a specific target accuracy, defined as the Mean-Square Error (MSE) for logarithmic versions (LRR, LOR) and the Relative MSE for the raw ratios (RR, OR).
Rare Events: The method is specifically optimized for scenarios where $p_1$ and $p_2$ are small (rare events), a regime where traditional fixed-sample estimators often suffer from high variance or bias.
Efficiency: The goal is to minimize the number of samples required to achieve the target accuracy, measured against the Cramér–Rao lower bound.

2. Methodology

The proposed approach combines sequential sampling with Bernoulli factory techniques to transform the two-population problem into a single-population estimation problem.

A. Probability Transformation (Bernoulli Factories)

The core innovation is generating a sequence of independent Bernoulli variables $Y$ with a specific parameter $p$ derived from $p_1$ and $p_2$ , such that estimating the odds $p/(1-p)$ (or log odds) is equivalent to estimating the target parameter.

For RR and LRR:
The target parameter is $p = \frac{p_1}{p_1 + p_2}$ $p = \frac{p _{1}}{p _{1} + p _{2}}$ .
- Algorithm 1: Randomly select population 1 or 2 with equal probability. Sample from the chosen population. If the result is a failure (0), repeat. If success (1), output $Y=1$ if population 1 was chosen, or $Y=0$ if population 2 was chosen.
For OR and LOR:
The target parameter is $p = \frac{p_1(1-p_2)}{p_1(1-p_2) + p_2(1-p_1)}$ $p = \frac{p _{1} ( 1 - p _{2} )}{p _{1} ( 1 - p _{2} ) + p _{2} ( 1 - p _{1} )}$ .
- Algorithm 2: Sample from both populations simultaneously. If $X_1 = X_2$ , discard and repeat. If $X_1 \neq X_2$ , set $Y=1$ if $X_1=1$ (and $X_2=0$ ), else $Y=0$ . This is an extension of von Neumann's rejection sampling.

B. Sequential Estimation (Inverse Binomial Sampling)

Once the sequence $Y$ is generated, the paper applies the sequential estimation methods from Mendo (2025) for odds and log odds:

Phase 1 (Successes): Collect samples of $Y$ until $r + \alpha$ successes are observed (where $r$ is a design parameter and $\alpha=1$ for RR/OR, $\alpha=0$ for LRR/LOR). Let $V'$ be the number of samples.
Phase 2 (Failures): Collect samples of $Y$ until $r - \alpha$ failures are observed. Let $V''$ be the number of samples.
Estimator Calculation:
- For RR/OR: $\hat{\zeta} = \frac{r V''}{(r-1)(V'-1)}$
- For LRR/LOR: $\hat{\zeta} = -H_{V'-1} + H_{V''-1}$ (where $H_k$ is the $k$ -th harmonic number).

C. Sample Pairing Strategy

Since the inner algorithms (1 and 2) may consume different numbers of samples from $X_1$ and $X_2$ , but the physical constraint requires taking samples in pairs, the paper adopts a "conservative" pairing strategy:

Samples are taken in pairs.
If one population's sample is needed but the other is not, the extra sample is stored.
The total number of pairs used is $U = \max(U_1, U_2)$ .
This introduces a potential efficiency loss factor $\sigma$ , which the paper analyzes.

3. Key Contributions

Unbiased Estimators with Guaranteed Accuracy: The proposed estimators are unbiased and mathematically guarantee that the MSE (or relative MSE) is below a user-defined target $A$ for any $p_1, p_2 \in (0,1)$ .
High Efficiency for Rare Events: The estimators achieve efficiency (ratio of variance to the Cramér–Rao bound) approaching 100% (i.e., $\eta \to 1$ ) as $p_1$ and $p_2$ become small. This is superior to previous group-sampling methods (e.g., Mendo 2026) in the rare-event regime.
Theoretical Bounds on Efficiency:
- The paper derives lower bounds for the estimation efficiency $\eta$ that are independent of the ratio $\theta = p_1/p_2$ .
- Theorem 3 & 4: Show that for small probabilities, $\eta > \frac{1}{1 + A(\mu + \alpha)}$ . For example, with a target relative RMSE of 10% ( $A=0.01$ ), the efficiency is guaranteed to be above ~91-95% regardless of the specific values of $p_1$ and $p_2$ , provided they are small.
Analysis of Pairing Loss: The paper quantifies the efficiency loss due to the "conservative" pairing of samples ( $\sigma$ ). It proves that for RR/LRR, this loss is negligible when events are rare ( $\sigma \to 1$ ), and for OR/LOR, there is no loss at all ( $\sigma = 1$ ) because the inner algorithm consumes inputs in perfect pairs.

4. Results

Simulation vs. Theory: Monte Carlo simulations (10 $^7$ realizations) confirm that the theoretical lower bounds on efficiency are tight and accurate.
Performance in Rare Event Regime:
- As the geometric mean of probabilities $\phi = \sqrt{p_1 p_2}$ decreases, the efficiency $\eta$ increases, approaching 1.
- For moderately rare events (e.g., $\phi \approx 0.01$ ), the proposed method significantly outperforms the reference group-sampling method from Mendo (2026).
Impact of Target Accuracy: Higher accuracy requirements (smaller $A$ ) lead to higher efficiency. This is counter-intuitive but explained by the fact that as $r$ increases (to meet stricter accuracy), the variance terms in the efficiency formula become more favorable.
Symmetry: For LRR and LOR, the efficiency is symmetric with respect to swapping $p_1$ and $p_2$ (i.e., $\theta$ vs $1/\theta$ ).

5. Significance

Practical Utility: The method is highly relevant for epidemiology and clinical trials where rare adverse events or rare diseases are studied. In these fields, obtaining large fixed sample sizes is expensive, and rare events lead to unstable estimates.
Resource Optimization: By guaranteeing a specific accuracy with minimal expected sample size, the method reduces the cost and time of data collection.
Theoretical Advancement: It extends the "Bernoulli factory" concept to multi-parameter estimation (RR/OR) and provides rigorous proofs for the efficiency of sequential estimators under the constraint of paired sampling.
Robustness: Unlike many estimators that rely on asymptotic normality or large-sample approximations, this approach provides finite-sample guarantees on accuracy and performs optimally precisely when data is scarce (rare events).

In conclusion, Luis Mendo proposes a mathematically rigorous, highly efficient sequential sampling framework that solves the problem of estimating risk ratios and odds ratios for rare events with guaranteed accuracy, outperforming existing methods in the most critical (rare event) scenarios.