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Hybrid Neutrosophic Regression ANOVA (HNRA): An Uncertainty-Aware Statistical Inference Framework Under Measurement Imprecision

This paper introduces Hybrid Neutrosophic Regression ANOVA (HNRA), a novel statistical framework that models measurement imprecision as neutrosophic intervals to provide a bounded, uncertainty-aware decomposition of sums of squares and a formally guaranteed F-statistic, thereby extending classical regression ANOVA to scenarios with instrument calibration errors and biological variability without requiring prior distributional assumptions.

Original authors: Jaida Najihah Jamidin, Zahari Md Ro, Sayang Mohd Deni, Siti Meriam Zahari

Published 2026-07-23

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CC BY 4.0

Original authors: Jaida Najihah Jamidin, Zahari Md Ro, Sayang Mohd Deni, Siti Meriam Zahari

Original paper licensed under CC BY 4.0 (https://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 using a magnifying glass. In the world of science, that magnifying glass is a statistical tool called Regression ANOVA. It helps researchers figure out if two things are connected—like whether getting older makes your muscles shrink. For decades, scientists have used this tool assuming their measurements are perfect, like reading a ruler that never wobbles. But in the real world, nothing is perfect. A scale might be off by a tiny bit, a sensor might glitch, or a person might forget exactly how old they are. These tiny errors are called "measurement imprecision."

When you ignore these errors, your detective work might look clear, but it could be misleading. It's like trying to guess the weight of a cat while wearing thick, foggy gloves; you might get a number, but you don't know how much that number could be wrong. Some scientists have tried to fix this by using "fuzzy" logic (where things are a little bit true and a little bit false) or "Bayesian" methods (which require guessing a starting belief before you begin). But these methods can be complicated, require extra assumptions, or don't give you the standard "yes/no" answers that doctors and regulators are used to. The big question is: Can we build a detective tool that admits the gloves are foggy, gives us a clear answer, and still follows the old, trusted rules of the game?

This paper introduces a new tool called Hybrid Neutrosophic Regression ANOVA (HNRA). Think of it as a super-powered magnifying glass that doesn't just give you one single number for the cat's weight. Instead, it gives you a range—a "safe zone" of possibilities. If the cat weighs between 4 and 6 pounds, the tool says, "We are sure it's in this range." The authors, Jaida Najihah Jamidin and her team from Universiti Teknologi MARA, developed a mathematical framework that takes these fuzzy ranges and runs them through the standard regression math.

Here is what they found and proved. First, they showed that this new method works like a perfect translation of the old method. If you take away all the fog (the measurement errors), HNRA turns back into the classic, trusted tool exactly as it was. This is crucial because it means the new method isn't a wild guess; it's a safe upgrade. Second, they proved that as the fog gets thicker (meaning the measurements get worse), the "safe zone" gets wider, but it never shrinks or behaves strangely. It grows in a predictable, steady way.

The team tested this on a real dataset involving 60 adults, looking at the link between muscle mass and age. In the real world, measuring muscle mass isn't exact; it can vary by about ±10% due to how the machine works, and age can be off by ±0.5 years because people round up or down. When they applied their new HNRA tool, the result wasn't a single number. Instead, they got a range for the strength of the connection: the F-statistic (a score that tells you how strong the evidence is) fell between 116.52 and 260.02.

Why does this matter? In the old way, a score of 174.06 (the average) was already considered a "win" because it was way higher than the critical threshold of 4.00. But with HNRA, the scientists can say something even stronger: "Even in the worst-case scenario, where the errors are at their absolute maximum, the lowest possible score is still 116.52." This means the conclusion—that age and muscle mass are definitely connected—is rock solid, no matter how much the measuring tools wobble.

The paper also ran 500 computer simulations to see how the tool would handle different levels of messiness. They found that even when they made the measurements very sloppy (up to 25% uncertainty), the tool still held up, keeping the "safe zone" wide but the conclusion stable. However, the authors are careful to note that this tool is best for situations where you need to account for known measurement errors, like in a hospital lab or with wearable fitness trackers. It is not a magic wand that fixes everything; if the data is too messy or the signal is too weak, the "safe zone" can get so wide that it stops being helpful. But for the cases where we know our tools have limits, HNRA offers a way to be honest about those limits while still getting a trustworthy answer. It's a way to say, "We know our ruler isn't perfect, but we know exactly how imperfect it is, and we can still solve the mystery."

Technical Summary: Hybrid Neutrosophic Regression ANOVA (HNRA)

Problem Statement
Classical regression-based Analysis of Variance (ANOVA) underpins quantitative biomedical inference but relies on the assumption that predictor ( $X$ ) and response ( $Y$ ) variables are observed as exact, error-free quantities. In practice, this assumption is routinely violated by instrument calibration bounds, biological variability, sensor-reporting errors, and rounding in self-reported data. While existing methodologies—such as interval regression, fuzzy regression, and Bayesian regression—attempt to address measurement imprecision, they suffer from specific limitations:

Interval Regression: Often lacks a standard ANOVA decomposition or a conventional $F$ -statistic.
Fuzzy Regression: Requires subjective membership function parameterization and lacks standard ANOVA tables.
Bayesian Regression: Requires prior distribution specification and yields posterior distributions rather than frequentist $F$ -statistics.
Existing Neutrosophic Models: Prior works have established neutrosophic correlation and simple regression but have not derived a complete regression ANOVA decomposition ( $SST_N = SSR_N + SSE_N$ ) with interval-valued mean squares and a formally bounded $F$ -statistic.

Consequently, there is a gap in statistical frameworks that can propagate structural measurement uncertainty through the sums-of-squares decomposition of regression ANOVA while maintaining the interpretability and computational simplicity of classical Ordinary Least Squares (OLS).

Methodology: The HNRA Framework
The paper introduces Hybrid Neutrosophic Regression ANOVA (HNRA), a framework that represents observations as neutrosophic intervals and propagates them through regression ANOVA using closed interval arithmetic.

Neutrosophic Interval Representation:
Observations are transformed into closed intervals $A_N = [A_L, A_U]$ . For a dataset $(x_i, y_i)$ , the neutrosophic representations are defined as:
- $X_{i,N} = [x_i - \delta_x, x_i + \delta_x]$
- $Y_{i,N} = [(1-\delta_y)y_i, (1+\delta_y)y_i]$
  where $\delta_x$ and $\delta_y$ represent non-negative uncertainty parameters for predictors and responses, respectively.
Model Formulation:
The HNRA model retains classical OLS coefficient estimates ( $\hat{\beta}_0, \hat{\beta}_1$ ) calculated from the midpoints of the intervals. The model is expressed as $Y_{i,N} = \beta_0 + \beta_1 X_{i,N} + \epsilon_{i,N}$ .
- Fitted Values: Calculated using interval arithmetic, with bound swapping applied if $\beta_1 < 0$ to maintain $[L, U]$ ordering.
- Sums of Squares: Total ( $SST_N$ ), Regression ( $SSR_N$ ), and Error ( $SSE_N$ ) sums of squares are computed as intervals using closed-interval arithmetic rules for addition, subtraction, and squaring.
- $F$ -Statistic: The interval-valued $F$ -statistic is defined as $F_N = [F_L, F_U] = [MSR_L / MSE_U, MSR_U / MSE_L]$ , where Mean Squares are derived from the interval sums of squares.
Decision Classification:
The framework introduces a three-way significance classification based on the relationship between the $F$ -interval $[F_L, F_U]$ and the classical critical value $F^*$ :
- Uniformly Significant: $F_L > F^*$ (Significance guaranteed across all uncertainty realizations).
- Uniformly Non-Significant: $F_U < F^*$ .
- Uncertainty-Sensitive Region: $F_L \leq F^* \leq F_U$ (Conclusion depends on the specific realization within the uncertainty bounds).

Key Theoretical Contributions
The paper establishes four theoretical results proving HNRA is a conservative, well-behaved generalization of classical regression ANOVA:

Decomposition Theorem: Proves that $SST_N \approx SSR_N + SSE_N$ holds componentwise under interval arithmetic, with the discrepancy bounded by a function of the uncertainty parameters ( $O(\delta_x + \delta_y)$ ).
Convergence Theorem: Demonstrates that as uncertainty parameters ( $\delta_x, \delta_y$ ) approach zero, all HNRA interval quantities converge exactly to their classical crisp counterparts.
Monotonicity Theorem: Establishes that the width of all interval-valued statistics (including the $F$ -statistic) grows monotonically with increasing measurement uncertainty, ensuring that increasing the uncertainty budget never spuriously narrows inferential conclusions.
Boundedness and Positivity Theorem: Proves that if the lower bounds of the mean squares are positive, the interval $F$ -statistic is strictly positive and finite. A corollary guarantees uniform statistical significance if the lower bound $F_L$ exceeds the classical critical value.

Experimental Results
The framework was validated using a biomedical dataset ( $n=60$ ) relating muscle mass to age, with $\pm 10\%$ response uncertainty and $\pm 0.5$ -year predictor uncertainty.

Application: The classical $F$ -statistic was 174.06. The HNRA result yielded $F_N = [116.52, 260.02]$ . Since the lower bound (116.52) vastly exceeded the critical value ( $F^* \approx 4.00$ ), the relationship was classified as Uniformly Significant.
Sensitivity Analysis: Across uncertainty levels from 5% to 25%, the interval width of $F_N$ grew monotonically, but the lower bound remained significantly above the critical threshold, confirming the stability of the conclusion under high imprecision.
Simulation Study: A study of 500 replications confirmed that classical estimates consistently fell within the HNRA intervals. An expanded factorial simulation (18,000 datasets) showed 100% coverage of classical $F$ values by HNRA intervals when the error mean square remained positive.
Borderline Scenarios: Synthetic scenarios with weaker effect sizes demonstrated the "Uncertainty-Sensitive Region," where classical ANOVA might report significance while HNRA correctly identifies that the conclusion is fragile to measurement uncertainty.

Significance and Claims
The paper claims HNRA offers a distinct position in statistical inference by:

Preserving Interpretability: It uses classical OLS coefficients, making results familiar to practitioners trained in frequentist regression.
Eliminating Distributional Assumptions: Unlike Bayesian methods, it requires no priors; unlike fuzzy methods, it requires no membership function parameterization.
Providing Formal Guarantees: It offers a complete ANOVA decomposition and a bounded $F$ -statistic with proven convergence and monotonicity properties, which were absent in prior neutrosophic regression literature.
Addressing Regulatory Needs: The framework is directly applicable to clinical laboratory science and epidemiology where standards (e.g., ISO 5725, CLSI) require the reporting of measurement uncertainty alongside analytical results.

Limitations and Future Work
The authors explicitly note several limitations:

Type-I Error Control: Formal Type-I error control under neutrosophic intervals has not yet been established; thus, HNRA is currently positioned as a robust descriptive and sensitivity-analytic tool rather than a fully validated frequentist hypothesis test.
Operating Envelope: In low-signal regimes with high uncertainty, the $F$ -interval can become uninformatively wide (the "Uncertainty-Sensitive Region"), and the lower bound of the error mean square ( $MSE_L$ ) can approach zero.
Scope: The current formulation is limited to simple regression ( $k=1$ ). Extensions to multiple regression, asymmetric uncertainty bounds, and interval-valued model selection criteria (e.g., AIC) are identified as future work.

Technical Summary: Hybrid Neutrosophic Regression ANOVA (HNRA)

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