What is $\frac{a}{b}$ called in a $y=a x+b$ regression in the context of a physical detection test?

Question

Let's say we have 10 calibrated reference samples of chemical products with a known concentration $x_i$ of a certain chemical component A.
$x_i$ is different for each sample.

We are building a chemical test workflow to detect the concentration of A in any chemical product, and we get a measurement $y_i$ for each of the 10 samples.

Note: if there is no "component A" in the chemical sample ($x_i=0$), then $y_i$ should be as close as zero as well, but it's not really true, due to real-world conditions.

Now here is the question:

We do know that there is a linear relation between the real concentration $x_i$ and our measurement $y_i$. Therefore we do a least-square regression $y = a x + b$, let's say:

$$y = 1.2 x + 100$$

Now if we change some workflow settings in our chemical test (modification in our chemistry protocol), our raw measurements data values change, and we get another regression $y = c x + d$, let's say

$$y = 2.3 x + 150$$

Question: how to evaluate the "quality" of these 2 different detection tests? (except using the $R^2$ parameter which is always close to 0.98 or 0.99)

Intuitively it seems that the highest $$Q = \frac{a}{b},$$ the more our test is sensitive to the presence of the chemical component A. But does this quantity $Q$ have a name in the context of a statistical regression?

Example: if we had $$y=0.001x + 50,$$ empirically, we would have very similar measurement values $y_i$ even if $x_i$ changes a lot from $1$ to $10$. Here, the ratio $Q = \frac{a}{b}$ is very low, indicating a poor sensitivity in the detection of the chemical component A.

Are there other "quality factors" for comparing multiple workflows (each of them giving different $a$ and $b$ in the $y=ax+b$ regression) that show a relationship between a known real-world quantity $x_i$ and a measurement $y_i$?

---

TL;DR: we have 3 different chemistry protocols for detecting the amount $x_i$ of a component A in chemical products. Our measurements are noted $y_i$. For each protocol we have a different relationship between $x_i$ and $y_i$:

$$\begin{align}y & = 1.2 x + 100\\ y & = 2.3 x + 150\\ y & = 0.001 x + 50\end{align}$$

with similar $R^2$ coefficients. How to find which protocol is the most sensitive?

Answer 1

$-b/a$ is equivalent to the intercept with the x-axis and may sometimes have specific meaning.

For example in a Lineweaver Burk plot the value is related to the Michaelis-Menten kinetics model as $a/b = K_m$. That is, when we plot $1/v$ versus $1/c$ then the Michaelis-Menten equation

$$v = \frac{Vc}{K_m +c}$$ becomes equivalent to your linear regression equation

$$\underbrace{\vphantom{\frac{Km}{V}}(1/v)}_{y} = \underbrace{\frac{K_m}{V}}_{a} \cdot \underbrace{\vphantom{\frac{Km}{V}}(1/c)}_{x} + \underbrace{\frac{1}{V}}_{b}$$

and the quotient of regression coefficients is $$\frac{a}{b} = \frac{K_m/V}{1/V}= K_m$$

---

In linear regression it doesn't have any specific name that I am aware of (except for possibly something like 'the negative of the x-axis intercept').

---

In your specific example it may play some role, but it is not very clear. It depends on how the error of measurements are made by your device.

The error might increase with the magnitude of $y$, but this may not need to be linear. For example reading of some voltage on a digital meter will have a round of error, and that has the same error whether is a large or small value. The same is true for many other instruments which have relatively similar error for smaller and larger values.

It might potentially be better to quantify the error of the observed/estimated concentration as function of the concentration $x$ and express something like the difference in two concentrations that is noticeable at some given power. Certainly a larger intercept $b$, and a smaller slope $a$ will make this relative error worse, but it is better to express it more directly. The quotient of $a/b$ doesn't cover all of the nuances.