Is the Linear Threshold Good Enough? A Scale-Free Parameter and Adequacy Test for Curvature-Induced Threshold Displacement

Technical Summary: Is the Linear Threshold Good Enough?

Problem Statement
Applied statistical work frequently requires locating a threshold—a break-even point, benchmark dose, or critical exposure—by solving for the crossing of a smooth function $m(x)$ and a target value $c$. A common practice is to linearize the function about a reference point $x_0$ and solve for the root of the first-order expansion, yielding a displacement $h^*_1 = -a/b$ (where $a = m(x_0)-c$ and $b = m'(x_0)$). This approach is computationally simple and propagates uncertainty via the delta method. However, when the underlying function possesses curvature ($q = m''(x_0) \neq 0$), the linear crossing is displaced from the true crossing. This approximation error can alter substantive conclusions (e.g., declaring a dose safe or unsafe) even when standard errors for the linear estimate are reported as valid.

Existing literature addresses this through direct inversion of nonlinear models or by testing for the presence of curvature ($q=0$). However, a significant gap remains: a test for $q=0$ answers whether the function is linear, not whether the resulting linear threshold is substantively adequate for a specific application. Strong curvature with a steep slope may displace the threshold negligibly, while weak curvature with a shallow slope can cause massive displacement. Furthermore, failing to reject $q=0$ is not evidence of adequacy, particularly in small samples. There is no existing scale-free parameter or formal adequacy test to determine if a first-order threshold is accurate within a prespecified practical tolerance.

Methodology and Framework
The paper introduces a framework to quantify the displacement caused by curvature relative to the linear approximation. Let $h^*_1$ be the linear threshold displacement and $h^*_2$ be the principal root of the second-order expansion $F_2(h) = a + bh + \frac{1}{2}qh^2 = 0$. The principal root is defined as the one continuous with the linear solution as $q \to 0$.

The core contribution is the Curvature-Overstatement Parameter ($\Theta_{COT}$), defined as the log-ratio of the second-order to the first-order displacement:
$$\Theta_{COT} = \log \left| \frac{h^*_2}{h^*_1} \right|$$
The paper establishes that $\Theta_{COT}$ is not a function of the three local parameters ($a, b, q$) independently, but depends entirely on a single dimensionless index $u = 2qa/b^2$. This leads to an exact reduction:
$$\Theta_{COT} = \phi(u) = \log 2 - \log(1 + \sqrt{1-u})$$
This structural result implies that $\Theta_{COT}$ is invariant to affine reparameterization of $x$ and rescaling of the response $m-c$. The parameter range in the regular domain ($D = b^2 - 2qa > 0$) is $(-\infty, \log 2)$, with $\log 2$ approached only at the tangency boundary ($u=1$).

Key Theoretical Results

1. Structural Properties: The paper proves that the principal branch is the globally nearest crossing, resolving root-selection ambiguity without arbitrary convention. It demonstrates that the two standard regularity conditions (existence of a real root and non-vanishing derivative) are equivalent ($D > 0$).
2. Asymmetry: The parameter exhibits an asymmetric range. Linearization can understate the required displacement by a factor approaching 2 (as $u \to 1$), but can overstate it without bound (as $u \to -\infty$).
3. Approximation Error: The paper distinguishes $\Theta_{COT}$ (displacement between first- and second-order approximations) from the error against the true root. A remainder theorem bounds the distance between the second-order root and the true root, showing that tangency ($u \to 1$) degrades the second-order approximation itself, not just the inference.
4. Sampling Theory:
   • Regular Case: Under standard conditions ($a \neq 0, b \neq 0, u < 1$), the estimator $\hat{\Theta}_{COT}$ is $\sqrt{n}$-normal with an explicit variance derived via the delta method.
   • Tangency Regime: As the system approaches tangency ($u \to 1$), the estimator exhibits non-standard asymptotics. Along local-to-tangency sequences, the estimator exists with probability $\Phi(\kappa/\omega)$, converges at rate $n^{1/4}$, and has a non-normal limit.
   • Weak Slope Regime: As the slope $b \to 0$, the linear threshold diverges, and $\Theta_{COT}$ inherits the Gleser–Hwang pathology (unbounded confidence sets). The parameter is not locally identified at $b=0$.

Inference and Adequacy Testing
The paper proposes two inferential procedures based on $\Theta_{COT}$:

• Relevance Testing: Testing whether the linear threshold is materially distorted (i.e., $|\Theta_{COT}| > \delta$).
• Adequacy Testing (Equivalence): Testing whether the linear threshold is accurate within a practical tolerance (i.e., $|\Theta_{COT}| < \delta$). This is formulated as a Two One-Sided Test (TOST). The paper emphasizes that failure to reject the null of inadequacy does not confirm adequacy; only rejection of the inadequacy null licenses the affirmative claim that the linear threshold is sufficient.

Monte Carlo Evidence
A simulation study validates the theoretical results across four conditions:

1. Exact Linearity ($q=0$): The adequacy test correctly affirms adequacy in 76.3% of samples at $n=500$ and nearly 100% at $n=2000$.
2. Regular Distortion: The test correctly identifies material distortion when present.
3. Near Tangency: Regular delta-method intervals fail to achieve nominal coverage, confirming the need for diagnostics (e.g., studentized discriminant) to detect non-regularity.
4. Weak Slope: The method fails as predicted by Gleser–Hwang theory, with coverage dropping to ~53% and estimates diverging, illustrating that short regular intervals near $b=0$ cannot be trusted.

Significance and Claims
The paper claims to provide a "well-behaved effect-size scale" for a question applied work frequently asks but answers poorly: Is a first-order threshold accurate enough to report?

• Not a New Threshold Estimator: The author explicitly states they do not claim thresholds lack inferential treatment; modern practice often inverts nonlinear models directly.
• Not a Curvature Test: Testing $\Theta_{COT} = 0$ is equivalent to testing $q=0$ but suffers from Wald non-invariance; the paper recommends testing $q=0$ directly for the sharp null.
• Primary Contribution: The synthesis of a scale-free estimand ($\Theta_{COT}$) that measures the consequence of curvature on the threshold location, coupled with a formal adequacy test. This allows researchers to make affirmative statements about the sufficiency of a linear approximation within a practical margin, rather than treating non-significant curvature as evidence of adequacy.
• Limitations: The parameter measures first- vs. second-order displacement, not total error against the truth (though bounds are provided). It does not supersede direct inversion of nonlinear models when available. The multivariate extension is flagged as a separate research problem due to metric dependencies.

In summary, the paper argues that while linearization is a convenient tool, its substantive adequacy is not guaranteed by the absence of significant curvature. The proposed $\Theta_{COT}$ and associated adequacy test provide a rigorous, scale-free method to determine when a linear threshold is "good enough" to report.