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New Equivalence Tests for Approximate Independence in Contingency Tables

This paper introduces new asymptotic and bootstrap-based equivalence tests for approximate independence in two-way contingency tables, featuring a computationally efficient estimator for boundary points, and validates their performance through simulations and real-world applications with an accompanying R implementation.

Original authors: Vladimir Ostrovski

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Vladimir Ostrovski

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: Are two things in your data truly connected, or are they just pretending to be? Maybe you're looking at whether eye color has anything to do with hair color, or if a patient's gender changes how they react to a specific medicine. In the world of statistics, this is called testing for "independence."

Usually, detectives look for a perfect match. If the data isn't a perfect 100% match, they say, "Aha! They are connected!" But in real life, nothing is ever perfectly perfect. Sometimes, two things are almost independent, just close enough that the tiny differences don't really matter. This is where "approximate independence" comes in.

The problem? The old detective tools were a bit clumsy. They relied on a method that required solving a massive, complicated math puzzle (called "numerical optimization") every single time to find the answer. It was like trying to find a needle in a haystack by building a new robot for every single piece of hay. It was slow, and the robot sometimes got stuck.

The New Detective Kit
Vladimir Ostrovski, a statistician from Cologne, Germany, has built a brand new, sleeker detective kit. Instead of wrestling with that heavy math puzzle, he invented two new, super-fast ways to measure the "distance" between your data and the idea of total independence.

Think of it like measuring how far a wobbly table is from being perfectly flat.

The "Absolute" Ruler ( $d_a$ ): This measures the raw difference. It's like checking how many millimeters the table leg is off the floor.
The "Relative" Ruler ( $d_r$ ): This measures the difference compared to the size of the table. It's like asking, "Is that wobble huge compared to the table's size, or is it just a tiny speck?"

These new rulers are easy to use. You don't need a supercomputer to read them; you can calculate the answer instantly.

The "Bootstrap" Magic Trick
Here is the tricky part: To know if your measurement is a "real" difference or just a fluke caused by having a small sample of data, you need to know what the "critical value" is. It's like knowing how much wobble is too much before you call the table broken.

The old way calculated this using a theoretical formula that works great when you have millions of data points, but gets a bit wobbly when you only have a few. To fix this, the author uses a trick called the bootstrap.

Imagine you have a bag of marbles representing your data. The bootstrap method is like reaching into the bag, pulling out a handful, making a copy, putting them back, and repeating this thousands of times to see how much the results wiggle around. This gives a much more accurate picture of what's happening with smaller groups of data.

But there was a catch: To do this bootstrap trick for independence, you usually need to find a specific "boundary point"—a magical spot where the data is exactly on the edge of being independent. Finding this spot was like trying to find a specific grain of sand on a beach without a map.

The Solution: A Randomized Map
The author solved this by creating a "randomized estimator." Instead of trying to find that one perfect grain of sand, the method throws a net over the beach. It generates a bunch of random "exterior points" (places that are definitely not independent) and then draws a line between your data and those points to find the boundary.

It's like saying, "I don't know exactly where the edge of the forest is, but if I walk from my house toward a mountain I know is far away, I'll eventually hit the edge." This makes the whole process fast and computationally possible, even for complex tables.

What the Simulations Showed
The author didn't just guess; they ran a massive simulation lab. They created thousands of fake data tables of different sizes (from tiny $2 \times 4$ grids to huge $5 \times 5$ grids) and tested their new method against the old one.

The Good News: The new "bootstrap" tests are much better at handling small sample sizes. They stay true to the rules (the "nominal level" of 0.05) much better than the old "asymptotic" tests, which tended to get too cautious (conservative) as the tables got bigger.
The Catch: The tests aren't perfect everywhere. If the data has categories with almost zero people in them (like a category where the probability is close to zero), the tests can get a little too eager to find a connection (anti-conservative). The author warns that if your data has empty or nearly empty boxes, you need to be extra careful.
The "Shrinking" Trick: To make the tests even safer, the author showed that if you shrink the "tolerance parameter" (the amount of wobble you're willing to accept) to 0.18, the tests become very reliable and rarely make false alarms.

Real-World Tests
The author took this new kit out for a spin on three real-life mysteries:

Nitrendipine Therapy: Does gender affect how patients react to this blood pressure medicine? With a sample of 217 people, the new tests suggested that gender and the outcome are approximately independent (they aren't strongly linked).
Eye and Hair Color: Do brown eyes always go with brown hair? The tests confirmed what we already know: they are not independent. The data showed a strong link, and the tests only accepted independence if you allowed for a huge amount of "wobble" (a tolerance of about 0.58).
Children and Income: Does the number of children a family has depend on their income? This dataset was huge (25,263 people), but some categories were very empty. The tests suggested they might be approximately independent, but the approximation was "very inaccurate" because of those empty categories.

The Bottom Line
This paper doesn't claim to have solved the mystery of independence forever. Instead, it offers a new, faster, and more reliable set of tools for statisticians. It replaces a slow, heavy math puzzle with a quick, smart calculation and a clever "random walk" method to handle small data sets.

The author provides a free "detective kit" (software code) online for anyone to use. While the tests work great for most situations, the paper explicitly warns that they need to be used with caution when dealing with data categories that have very few entries. It's a powerful new way to look at the world, but like any good detective, you still have to know when to double-check your clues.

Technical Summary: New Equivalence Tests for Approximate Independence in Contingency Tables

Problem Statement
The paper addresses the statistical challenge of testing for approximate row-column independence in two-way contingency tables. Unlike traditional independence testing, which seeks to reject the null hypothesis of perfect independence, this work focuses on equivalence testing. The objective is to test the null hypothesis $H_0: \{d(p, M) \geq \varepsilon\}$ against the alternative $H_1: \{d(p, M) < \varepsilon\}$ , where $p$ is the true probability matrix, $M$ is the independence model (the set of all product measures), $d$ is a distance metric, and $\varepsilon > 0$ is a pre-specified tolerance parameter.

The author notes that while previous methods exist (e.g., Hodges and Lehmann, 1954; Diaconis and Efron, 1985; Liu and Lindsay, 2009; Wellek, 2010), they often rely on semi-parametric tolerance regions or tests for consistency with fully specified multinomial distributions. A specific limitation identified in the author's prior work (Ostrovski, 2018) is the reliance on numerical optimization to calculate the minimum distance $d(p, M)$ , which makes bootstrap-based tests computationally intensive. Furthermore, the existence of a continuous minimizer for the independence model was assumed but not proven, creating a theoretical gap.

Methodology
The paper proposes a framework that avoids numerical optimization for the independence model by introducing two specific distance metrics derived from transformations of the probability matrix $p$ :

Absolute Deviation ( $d_a$ ): Defined via the transformation $h_a(p) = (p_{ij} - r_i c_j)$ , where $r$ and $c$ are marginal probability vectors. The distance is $d_a(p, M) = l(h_a(p), 0)$ .
Relative Deviation ( $d_r$ ): Defined via the transformation $h_r(p) = (p_{ij} / r_i c_j)$ . The distance is $d_r(p, M) = l(h_r(p), 1)$ .

Here, $l$ represents a differentiable distance (specifically scaled Euclidean distance in the simulations). These metrics allow for the direct calculation of the minimum distance to the independence model without iterative optimization.

The paper develops two types of tests:

Asymptotic Tests: The test statistics are defined as $T_*(p_n) = \sqrt{n}(d_*(p_n, M) - \varepsilon)$ . Using the Delta method, the author derives the asymptotic Gaussian distribution of these statistics. The critical values are calculated using the estimated asymptotic variance $\sigma_*(p_n)$ .
Bootstrap Tests: To improve finite sample performance, a parametric bootstrap approach is introduced. A key innovation is the development of a computationally feasible boundary point estimator ( $\tilde{p}_n$ $\tilde{p}_{n}$ ). Since the boundary $\partial H_0$ $\partial H_{0}$ is complex and difficult to parameterize, the author proposes a randomized estimator. This involves:
- Selecting a set of "exterior" points $Q$ where $d_*(q, M) > \varepsilon$ .
- Constructing linear combinations $c(p_n, q) = a_n p_n + (1-a_n)q$ such that the distance to the model equals $\varepsilon$ .
- Selecting the combination that minimizes the distance to the observed data $p_n$ as the estimator $\tilde{p}_n$ .
- Using Monte Carlo simulation based on $\tilde{p}_n$ to estimate the critical value $c(\alpha, \tilde{p}_n)$ .

Key Contributions

Analytical Simplification: The paper provides closed-form expressions for the minimum distance to the independence model using $d_a$ and $d_r$ , eliminating the need for the numerical optimization required in previous general methods.
Efficient Boundary Estimation: A randomized estimator for the boundary point is developed, making the bootstrap test computationally feasible for contingency tables.
Theoretical Guarantees: The author proves that the proposed boundary estimator is consistent (converges almost everywhere to the true boundary point) under specific conditions and establishes the asymptotic normality of the test statistics.
Implementation: The tests are implemented in R and made available online.

Results
The performance of the proposed tests was evaluated through simulation studies and applied to real-world datasets:

Finite Sample Performance:
- Power: The minimum tolerance parameter $\varepsilon$ required to achieve 90% power decreases with sample size ( $n^{-1/2}$ ) and increases with table size ( $k_1 + k_2$ ).
- Type I Error: Asymptotic tests were found to be non-conservative for small tables but highly conservative for larger tables. In contrast, bootstrap tests maintained rejection probabilities very close to the nominal level ( $\alpha = 0.05$ ) across various table sizes, though the $d_r$ -based bootstrap test showed a tendency to be anti-conservative in specific boundary regions (where marginal probabilities are close to zero).
- Type II Error: The power of $d_a$ -based tests was relatively stable across different points, whereas $d_r$ -based tests showed high variability.
Real Data Applications:
- Nitrendipine Therapy: With a small sample size ( $n=217$ ), both asymptotic and bootstrap tests suggested approximate independence between treatment outcome and gender.
- Eye and Hair Color: The tests correctly rejected independence for very large $\varepsilon$ , confirming the known lack of independence between these variables.
- Children and Income: Despite a large sample size ( $n=25,263$ ), the sparsity in the "4+ children" category led to low power for $d_r$ -based tests. The $d_a$ -based tests suggested approximate independence but noted the approximation was inaccurate.

Significance and Claims
The paper claims to offer a statistically efficient and computationally feasible method for testing approximate independence. By avoiding numerical optimization and unproven assumptions regarding the existence of continuous minimizers (which were necessary in Ostrovski, 2018), the proposed method provides a more robust theoretical foundation. The author emphasizes that the bootstrap tests, particularly when using the proposed boundary estimator, offer superior finite sample performance compared to asymptotic tests, especially for larger contingency tables. The work is presented as a practical improvement for statistical practice, with the tools made available to facilitate application.

Technical Summary: New Equivalence Tests for Approximate Independence in Contingency Tables

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