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int3ract: Johnson-Neyman Technique and its Three-Way Extension for Frequentist and Bayesian Models in R

The `int3ract` R package implements the Johnson-Neyman and its three-way extension (JNK) techniques to facilitate the interpretation of interaction effects by identifying significant moderator regions across both frequentist and Bayesian modeling frameworks.

Original authors: Robert W. Krause

Published 2026-04-27

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CC BY 4.0

Original authors: Robert W. Krause

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

The "It Depends" Problem: A Guide to the int3ract Tool

Imagine you are a chef testing a new recipe for spicy pasta. You find that, on average, adding chili flakes makes the dish more popular. But then you realize something: the effect of the chili depends on whether the customer is eating it with a cold soda or a hot tea.

If they have a cold soda, the spice is perfect. If they have hot tea, the spice is overwhelming.

In statistics, this is called an interaction effect. It’s the scientific way of saying, "The answer isn't just 'yes' or 'no'; the answer depends on something else."

The Problem: The "Spotlight" Mistake

Most researchers make a common mistake when studying these "it depends" scenarios. Instead of looking at the whole picture, they use a "Spotlight Approach."

Imagine you are in a dark room with a tiny flashlight. You only shine the light on two or three specific spots (e.g., "What happens if the temperature is exactly 70 degrees? What about 90?"). You conclude that the spice works at 70 and fails at 90, but you have no idea what happens at 75, 80, or 85. You’ve missed the "gray areas" where the effect might be fading in or out.

The Solution: The Johnson–Neyman Technique

The Johnson–Neyman (JN) technique is like turning on the overhead lights. Instead of checking a few random spots, it maps out the entire room. It tells you exactly where the "magic zone" is—the precise range where the effect is actually significant—and where it disappears into the noise.

What is int3ract?

The paper introduces a new R software package called int3ract. Think of it as a high-tech, automated mapping drone for researchers.

Here is why this "drone" is special:

1. It handles "Triple Threats" (Three-Way Interactions)
Most tools can only handle two variables (Spice $\times$ Temperature). But life is more complex. What if it’s Spice $\times$ Temperature $\times$ The type of drink? This is a "three-way interaction."
Mapping this is like trying to draw a map of a mountain range on a flat piece of paper. int3ract uses heatmaps (colorful, glowing maps) to show researchers exactly where the "sweet spot" exists in a three-dimensional world.

2. It works for both "Traditionalists" and "Modernists"
In the world of statistics, there are two main schools of thought:

Frequentists (The Traditionalists): They look for "yes/no" answers based on strict rules.
Bayesians (The Modernists): They look at "probabilities" and "likelihoods," updating their beliefs as they get more data.
Usually, you need different tools for each. int3ract is a universal translator; it works perfectly for both, providing clear visual maps regardless of which mathematical philosophy the researcher uses.

3. It’s a "Plug-and-Play" Engine
Whether a researcher is studying how social networks evolve (SAOMs), how people behave in groups (Mixed-effects models), or simple linear trends, int3ract can "auto-detect" their work. It’s like a universal charger that fits almost any scientific device.

The Bottom Line

Instead of researchers saying, "The chili works... mostly," int3ract allows them to say, "The chili is statistically effective only when the temperature is between 65 and 82 degrees and the customer is drinking a cold beverage."

It moves science away from "guessing in the dark" and toward precise, colorful, and honest mapping of how the world actually works.

Technical Summary: int3ract: Johnson–Neyman Technique and its Three-Way Extension for Frequentist and Bayesian Models in R

1. The Problem: Limitations of Traditional Interaction Analysis

In applied statistical modeling, multiplicative interaction terms are used to determine if the relationship between a predictor ( $X$ ) and an outcome ( $Y$ ) is moderated by one or more variables ( $M, W$ ). However, the paper identifies two major pitfalls in current reporting practices:

The "Spotlight" Approach: Researchers often rely on "simple-slope" analysis, evaluating the effect of $X$ only at a few arbitrary values of the moderator (e.g., mean, $+1$ SD, $-1$ SD). This provides a narrow, potentially misleading snapshot of the interaction.
Complexity of Higher-Order Interactions: While two-way interactions are well-studied, interpreting three-way interactions is mathematically and visually complex. Existing R packages lack the capability to visualize the significance regions of three-way interactions effectively.

2. Methodology: The Johnson–Neyman (JN) and JNK Techniques

The paper introduces the int3ract R package, which implements two primary methodologies:

The Johnson–Neyman (JN) Technique (Two-Way): Instead of discrete points, this technique identifies the continuous range of a moderator over which the focal effect is statistically significant. For a linear model, it calculates the simple slope $\hat{\theta}_X(m) = \hat{\beta}_1 + \hat{\beta}_3 m$ and uses the delta method to derive its sampling variance, identifying regions where the $z$ -score exceeds the critical threshold ( $|z(m)| > z_{\alpha/2}$ ).
The Johnson–Neyman–Krause (JNK) Technique (Three-Way): This is the paper's core methodological extension. For a three-way interaction, the conditional effect of $X$ depends on two moderators ( $M$ and $W$ ). The package uses the multivariate delta method to calculate the variance of the conditional effect across a two-dimensional grid, allowing for the identification of significance regions in a plane rather than a line.

3. Key Contributions and Implementation

The int3ract package provides a unified framework for both frequentist and Bayesian paradigms:

Frequentist Engine (JNK_freq()):
- Auto-detection: It automatically extracts coefficients and covariance matrices from lm, glm, lme4 (mixed-effects), and RSiena (SAOMs) objects.
- Model Agnosticism: It can be applied to any model (survival, ordinal, beta regression, etc.) if the user provides a coefficient vector and covariance matrix.
- Mixed-Effects Support: For lme4 models with random interaction terms, it can generate per-group JN plots.
Bayesian Engine (JNK_bayes()):
- Instead of point estimates, it uses conditional posterior distributions.
- For two-way interactions, it produces density plots of the posterior.
- For three-way interactions, it produces heatmaps of the posterior mean and Bayesian $p$ -values, using crosshatch overlays to denote non-significant regions.
Advanced Visualization:
- Two-way: Shaded ribbon plots (green for significant, chocolate for non-significant).
- Three-way: Color-gradient heatmaps that map the magnitude of the effect across a 2D moderator grid.

4. Results and Application

The paper demonstrates the package's utility through several simulated scenarios:

Two-way interactions: Visualizing how the significance of a predictor fluctuates across the range of a single moderator.
Three-way interactions: Generating 2D heatmaps that reveal complex "islands" of significance within a moderator space.
Stochastic Actor-Oriented Models (SAOMs): Demonstrating how the package handles the specialized requirements of network evolution models (RSiena), including the ability to assess heterogeneity across different groups/networks.

5. Significance and Impact

The significance of this work lies in its push for statistical transparency. By moving away from "spotlight" testing and toward continuous region-mapping, int3ract allows researchers to:

Avoid False Conclusions: Prevent the misinterpretation of interactions that may only be significant in narrow, unexamined ranges.
Enhance Reporting: Provide richer, more granular data in supplementary materials, aligning with modern calls for reproducible and complete statistical reporting.
Bridge Paradigms: Offer a consistent workflow for both frequentist and Bayesian researchers, making sophisticated moderation analysis accessible across different modeling philosophies.