Formulae for Slopes and Curvatures of Two Curves in Quadratic Polynomial Regression Equations with Response Surface Analysis

Abstract

Response Surface Methodology (RSM) constitutes a suite of statistical and mathematical techniques employed for the development, enhancement, and optimization of processes with the aim of achieving the maximum (or minimum) value of a response variable. In the realm of organizational research, scholars have embraced RSM, utilizing quadratic regression equations illustrated on a three-dimensional surface to investigate congruence phenomena such as fit, match, similarity, or agreement. To date, RSM has found extensive application in exploring nuanced relationships among combinations of two predictor variables and an outcome variable within organizational studies. This paper takes a novel approach by incorporating directional derivatives and the rotation of axes from multivariable calculus. It formulates a set of equations to calculate the slopes and curvatures of two curves in quadratic polynomial regression equations through response surface analysis.

1. Introduction

Response surface methodology (RSM) serves as an expansive toolkit, encompassing statistical and mathematical techniques with the overarching goal of developing, improving, and optimizing processes to achieve the maximum or minimum value of a response [1]. This empirical model employs mathematical and statistical approaches to establish relationships between a response and various input variables that may influence it. In the field of statistics, RSM has proven invaluable for scrutinizing the intricate relationships between predictor and response variables.

Originally conceptualized by Box and Wilson [2], RSM was initially applied to explore the correlations between the yield of a chemical process and a set of input variables presumed to impact the yield [3]. The fundamental principle of RSM involves utilizing a second-degree polynomial model to achieve an optimal outcome through a series of experiments. Its simplicity in estimation and application makes this statistical technique particularly valuable for maximizing the results of a specific substance through the optimization of operational factors via well-designed and analyzed experiments [3].

In the context of organizational research, significant credit is attributed to Edwards and Parry [4], Edwards [5], and other researchers who integrated RSM into their methodologies. They employed quadratic regression equations depicted on a three-dimensional surface to investigate congruence phenomena within organizational research, encompassing concepts such as fit, match, similarity, or agreement. Edwards and Parry [4], for instance, contributed formulae for testing and interpreting quadratic regression equations in the study of congruence within organizational research. These equations enable researchers to assess conceptual models that difference scores aim to represent. Due to the substantial contributions and expansions of this technique by these scholars, RSM has become widely employed in studying intricate relationships between pairs of predictor variables and an outcome variable in organizational studies [6, 7, 8, 9] and beyond.

Barranti et al. [10] argue for several conceptual advantages offered by RSM. Firstly, it examines (mis)matches by modeling the association of input factors with a response using three-dimensional space. Secondly, RSM can address more nuanced questions than traditional approaches. Thirdly, it can be utilized to test whether one type of mismatch (e.g., an overestimate) is worse than another (e.g., an underestimate). Recognizing these strengths [4, 10, 11], RSM has been recommended for examining two-way interactions in moderated regression [12, 13]. This chapter aims to present a diverse array of alternative formulae for the slopes and curvatures of two curves in quadratic polynomial regression equations using response surface analysis, further advancing the applicability and understanding of this powerful methodology.

2. Existing formulas

The fundamental quadratic regression equation utilized in response surface methodology (RSM) is expressed as follows:

$$\mathrm{Z}={\mathrm{b}}_0+{\mathrm{b}}_1\mathrm{X}+{\mathrm{b}}_2\mathrm{Y}+{\mathrm{b}}_3{\mathrm{X}}^2+{\mathrm{b}}_4\mathrm{XY}+{\mathrm{b}}_5{\mathrm{Y}}^2+\mathrm{e},$$E1

where Z denotes the outcome variable (response), while X and Y are two predictors. Also, in this equation, b₀ is a constant, while b₁, b₂, b₃, b₄, and b₅ are linear, quadratic, and interaction terms. The outcome variable Z is regressed on two predictors X and Y, their respective square terms X² and Y², and their interaction XY.

Figure 1 visually represents a hypothetical response surface Z (depicted in blue) based on the quadratic regression equation mentioned above. The curve M (in red) is the intersected curve of the surface Z and plane ACC′A′ (Y = X). Plane BDD′B′ (Y = −X) intersects the surface Z at the curve M′ (dotted yellow curve). Line L (in black) is the tangent line to the intersected curve M at point P (0, 0, b₀) (in yellow). The origin point is O (0,0,0) and is marked in red.

Figure 1.

A hypothetical response surface. The two predictor variables X and Y are located on the two axes at the bottom plane ABCD. The vertical axis is the Z-axis. Plane ACC′A′ (Y = X) intersects the surface Z at the curve M. Plane BDD′B′ (Y = −X) intersects the surface Z at the curve M′. The origin point is O (0,0,0).

According to Edwards and Parry [4] and Edwards (2002), the slopes and curvatures of the intersected curves M and M′ of the surface Z and two planes are important in exploring the nuanced relationships between two predictors and the outcome. One plane is the plane ACC′A′ (Y = X) (see Figure 1), while another is the plane BDD′B′ (Y = −X), which is perpendicular to the plane ACC′A′ at Z-axis. To estimate the slope and curvature of intersection curve M, X is substituted for Y in the quadratic regression Eq. (1), which yields Eq. (2):

$$\mathrm{Z}={\mathrm{b}}_0+\left({\mathrm{b}}_1+{\mathrm{b}}_2\right)\mathrm{X}+\left({\mathrm{b}}_3+{\mathrm{b}}_4+{\mathrm{b}}_5\right)X^2+\mathrm{e},$$E2

Likewise, they derive quadratic regression Eq. (3) to estimate the slope and curvature of the curve M′,

$$\mathrm{Z}={\mathrm{b}}_0+\left({\mathrm{b}}_1–{\mathrm{b}}_2\right)\mathrm{X}+\left({\mathrm{b}}_3–{\mathrm{b}}_4+{\mathrm{b}}_5\right)X^2+\mathrm{e},$$E3

Then, they have

$${\mathrm{a}}_1={\mathrm{b}}_1+{\mathrm{b}}_2$$E4

$${\mathrm{a}}_2={\mathrm{b}}_3+{\mathrm{b}}_4+{\mathrm{b}}_5$$E5

$${\mathrm{a}}_3={\mathrm{b}}_1–{\mathrm{b}}_2$$E6

$${\mathrm{a}}_4={\mathrm{b}}_3–{\mathrm{b}}_4+{\mathrm{b}}_5$$E7

Where the term (b₃ + b₄ + b₅) is the curvature of the curve M and (b₁ + b₂) represents its slope at the point P (0, 0, b₀). Likewise, the term (b₃ – b₄ + b₅) represents the curvature of the curve M′, and (b₁ – b₂) is its slope at the point P (0, 0, b₀).

The formulas for calculating the significance tests of the curve values (a₁ through a₄) are also given as follows [13]:

$$\text{For variable}{\mathrm{a}}_1\mathrm{t}=\frac{\left(b1+b2\right)}{\sqrt{\left(\mathit{SEb}{1}^2+\mathit{SEb}{2}^2\right)+2\mathit{COV}b1b2}}$$E8

$$\text{For variable}{\mathrm{a}}_2\mathrm{t}=\frac{\left(b3+b4+b5\right)}{\sqrt{\left(\mathit{SEb}{3}^2+\mathit{SEb}{4}^2+\mathit{SEb}{5}^2\right)+2\mathit{\text{COVb}}3b4+2\mathit{\text{COVb}}3b5+2\mathit{\text{COVb}}4b5}}$$E9

$$\text{For variable}{\mathrm{a}}_3\mathrm{t}=\frac{\left(b1-b2\right)}{\sqrt{\left(\mathit{SEb}{1}^2+\mathit{SEb}{2}^2\right)-2\mathit{COV}b1b2}}$$E10

$$\text{For variable}{\mathrm{a}}_4\mathrm{t}=\frac{\left(b3-b4+b5\right)}{\sqrt{\left(\mathit{SEb}{3}^2+\mathit{SEb}{4}^2+\mathit{SEb}{5}^2\right)-2\mathit{\text{COVb}}3b4+2\mathit{\text{COVb}}3b5-2\mathit{\text{COVb}}4b5}}$$E11

Where SE = standard error, COV = covariance.

3. Alternative formulas

Presently, the formulas derived from Eqs. (2) and (3) play a pivotal role in various studies that leverage the response surface methodology (RSM) across a diverse array of fields. However, it is crucial to highlight a limitation in their applicability when transitioning to the context of three-dimensional geometry. Specifically, when substituting X (or −X) for Y in these equations, which are designed to represent curves M and M′ formed by the intersection of surface Z with planes ACC′A′ (Y = X) and BDD′B′ (Y = −X) a discrepancy arises.

Contrary to expectations, Eqs. (2) and (3) do not precisely depict curves M and M′ but rather represent two-dimensional projections onto the XOZ plane by the intersection curves M and M′. This distinction is particularly relevant when interpreting the slopes and curvatures of these projected curves in relation to the intersected curves within the three-dimensional space. Caution is warranted in utilizing these formulae as it may not always be appropriate or accurate to extrapolate the slopes and curvatures of projections to represent those of intersected curves, especially under specific circumstances.

This cautionary note emphasizes the need for a nuanced and context-specific approach when applying these formulas in three-dimensional geometric scenarios. Researchers and practitioners should be aware of the limitations associated with the projection of curves onto two-dimensional planes and exercise discretion in interpreting and applying the derived slopes and curvatures, ensuring that the chosen methodology aligns with the geometric intricacies of the study at hand. By acknowledging and addressing this caveat, the broader scientific community can enhance the precision and reliability of results when utilizing these formulas within the context of three-dimensional geometry.

The highlighted distinction holds paramount importance for researchers engaged in the intricate exploration of lines and curves within the realm of three-dimensional geometry. To achieve an accurate determination of slopes and curvatures concerning the intersected curves between surface Z and the two planes (Y = X and Y = −X), a more optimal approach is recommended. This involves leveraging the principles of directional derivatives and the rotation of axes, as commonly employed in multivariable calculus.

The application of directional derivatives facilitates a more nuanced examination of how the surface Z intersects with the planes, allowing researchers to discern the changes in the surface’s behavior along specific directions. Additionally, the rotation of axes provides a transformative framework that aligns with the geometric intricacies of the study. By appropriately rotating the coordinate system, researchers can simplify the analysis and enhance the precision of calculations related to slopes and curvatures.

This advanced method not only ensures a more robust analysis but also aligns with the inherent complexities of three-dimensional geometry. It enables researchers to overcome the limitations associated with the two-dimensional projections mentioned earlier, providing a more accurate representation of the relationships between the curves and the intersecting planes. Embracing such sophisticated mathematical tools from multivariable calculus contributes to a higher level of precision and reliability in the study of intersected curves within three-dimensional spaces, ultimately enhancing the quality of research outcomes in this domain.

$$\text{For}\mathrm{Z}={\mathrm{b}}_0+{\mathrm{b}}_1\mathrm{X}+{\mathrm{b}}_2\mathrm{Y}+{\mathrm{b}}_3{\mathrm{X}}^2+{\mathrm{b}}_4\mathrm{XY}+{\mathrm{b}}_5{\mathrm{Y}}^2+\mathrm{e},$$E12

The directional derivative D_L$f$(X₀, Y₀) equals the slope of the tangent line (L) to the intersected curve M of surface Z and the plane ACC′A′ (Y = X) at the point (X₀, Y₀, $f$(X₀, Y₀)). The formula to calculate the directional derivative of a curve at this point is:

$${\mathrm{D}}_{\mathrm{L}}=\frac{\partial Z}{\partial l}=\frac{\partial Z}{\partial x}\mathit{cos}\theta +\frac{\partial Z}{\partial y}\mathit{sin}\theta$$E13

Since angle θ = 45°,

$$\begin{array}{lll}{\mathrm{D}}_{\mathrm{L}}=\frac{\partial Z}{\partial l}& =& \frac{\partial Z}{\partial x}\mathit{cos}{45}^{°}+\frac{\partial Z}{\partial y}\mathit{sin}{45}^{°}\\ & =& \left({\mathrm{b}}_1+{\text{2b}}_3\mathrm{X}+{\mathrm{b}}_4\mathrm{Y}\right)\mathit{cos}{45}^{°}+\left({\mathrm{b}}_2+{\mathrm{b}}_4\mathrm{X}+{\text{2b}}_5\mathrm{Y}\right)\mathit{sin}{45}^{°}\\ & =& \frac{\sqrt{2}}{2}\left[\left({\mathrm{b}}_1+{\mathrm{b}}_2\right)+\left({\text{2b}}_3+{\mathrm{b}}_4\right)\mathrm{X}+\left({\mathrm{b}}_4+{\text{2b}}_5\right)\mathrm{Y}\right]\end{array}$$E14

When X = 0 and Y = 0, the directional derivative D_L$f$(0,0) equals $\frac{\sqrt{2}}{2}$ (b₁ + b₂) at the point P (0, 0, b₀). Obviously, the slope of the intersected curve M of surface Z and plane ACC′A′ (Y = X) is not (b₁ + b₂) at the point P (0, 0, b₀) as calculated by Edwards and Parry [4] and Edwards [5]. Similarly, the slope of the intersected curve M′ is $\frac{\sqrt{2}}{2}$ (b₁ – b₂) at the point P (0, 0, b₀), not (b₁ – b₂).

To capture both slope and curvature of the intersected curves, the rotation of axis is used to derive new rotated equations of intersected curves M and M′. The formula is provided as follows:

$$\left\{\begin{array}{l}\mathrm{Z}={\mathrm{Z}}^{´}\\ \mathrm{X}={\mathrm{X}}^{´}cos\theta -{\mathrm{Y}}^{´}sin\theta \\ \mathrm{Y}={\mathrm{X}}^{´}sin\theta +{\mathrm{Y}}^{’}cos\theta \end{array}\right.$$E15

For the intersected curve M, it is obtained by rotating ${45}^{°}$ counterclockwise from the plane XOZ (X = 0); therefore, the new equation of curve M is

$$\left\{\begin{array}{l}Z=Z^{´}\\ X=X^{´}\mathit{cos}{45}^{°}-Y^{´}\mathit{sin}{45}^{°}\\ Y=X^{´}\mathit{sin}{45}^{°}+Y^{´}\mathit{cos}{45}^{°}\end{array}\right.$$E16

Since $\mathit{sin}{45}^{°}=\mathit{cos}{45}^{°}$ = $\frac{\sqrt{2}}{2}$, we have

$$\left\{\begin{array}{l}\mathrm{Z}={\mathrm{Z}}^{´}\\ \mathrm{X}=\frac{\sqrt{2}}{2}\left(X^{´}-Y^{´}\right)\\ \mathrm{Y}=\frac{\sqrt{2}}{2}\left(X^{´}+Y^{´}\right)\end{array}\right.$$E17

Thus,

$$\begin{array}{lll}{\mathrm{Z}}^{\prime }& =& {\mathrm{b}}_0+{\mathrm{b}}_1\ast \frac{\sqrt{2}}{2}\left(X^{´}-Y^{´}\right)+{\mathrm{b}}_2\ast \frac{\sqrt{2}}{2}\left(X^{´}-Y^{´}\right)+{\mathrm{b}}_3{\left[\frac{\sqrt{2}}{2}\left(X^{´}-Y^{´}\right)\right]}^2\\ & & +{\mathrm{b}}_4\left[\frac{\sqrt{2}}{2}\ast \frac{\sqrt{2}}{2}\left(X^{\prime }-Y^{\prime }\right)\left(X^{\prime }+Y^{\prime }\right)\right]+{\mathrm{b}}_5{\left[\frac{\sqrt{2}}{2}\left(X^{\prime }+Y^{\prime }\right)\right]}^2\end{array}$$E18

Since in the intersected curve M, Y′ = 0

$${\mathrm{Z}}^{\prime }={\mathrm{b}}_0+\frac{\sqrt{2}}{2}\left({\mathrm{b}}_1+{\mathrm{b}}_2\right){\mathrm{X}}^{\prime }+\frac{1}{2}\left({\mathrm{b}}_3+{\mathrm{b}}_4+{\mathrm{b}}_5\right)X{´}^2+\mathrm{e}$$E19

Similarly, the new equation of intersected curve M′ is

$${\mathrm{Z}}^{\prime }={\mathrm{b}}_0+\frac{\sqrt{2}}{2}\left({\mathrm{b}}_1–{\mathrm{b}}_2\right){\mathrm{X}}^{\prime }+\frac{1}{2}\left({\mathrm{b}}_3–{\mathrm{b}}_4+{\mathrm{b}}_5\right)X{\prime }^2+\mathrm{e}$$E20

Therefore,

$${\mathrm{a}}_1=\frac{\sqrt{2}}{2}({\mathrm{b}}_1+{\mathrm{b}}_{2)}$$E21

$${\mathrm{a}}_2=\frac{1}{2}({\mathrm{b}}_3+{\mathrm{b}}_4+{\mathrm{b}}_{5)}$$E22

$${\mathrm{a}}_3=\frac{\sqrt{2}}{2}({\mathrm{b}}_1–{\mathrm{b}}_{2)}$$E23

$${\mathrm{a}}_4=\frac{1}{2}({\mathrm{b}}_3–{\mathrm{b}}_4+{\mathrm{b}}_{5)}$$E24

The significance test t is calculated by using the following formula: t = $\frac{\underset{\_}{X}-\mu }{\mathit{SE}}$.

In the case of the significance test of coefficient of X´, a₁

$$\begin{array}{lll}\mathrm{t}& =& \frac{a1}{\mathit{SEa}1}=\frac{\sqrt{2}}{2}\left({\mathrm{b}}_1+{\mathrm{b}}_2\right)/\frac{\sqrt{2}}{2}\sqrt{\left(\mathit{SEb}{1}^2+\mathit{SEb}{2}^2\right)+2\mathit{COV}b1b2}\\ & =& \left({\mathrm{b}}_1+{\mathrm{b}}_2\right)/\sqrt{\left(\mathit{SEb}{1}^2+\mathit{SEb}{2}^2\right)+2\mathit{COV}b1b2}\end{array}$$E25

Similarly,

$$\text{for variable}{\mathrm{a}}_2\mathrm{t}=\frac{\left(b3+b4+b5\right)}{\sqrt{\left(\mathit{SEb}{3}^2+\mathit{SEb}{4}^2+\mathit{SEb}{5}^2\right)+2\mathit{\text{COVb}}3b4+2\mathit{\text{COVb}}3b5+2\mathit{\text{COVb}}4b5}}$$E26

$$\text{for variable a3}\mathrm{t}=\frac{\left(b1-b2\right)}{\sqrt{\left(\mathit{SEb}{1}^2+\mathit{SEb}{2}^2\right)-2\mathit{COV}b1b2}}$$E27

$$\text{for variable}{\mathrm{a}}_4\mathrm{t}=\frac{\left(b3-b4+b5\right)}{\sqrt{\left(\mathit{SEb}{3}^2+\mathit{SEb}{4}^2+\mathit{SEb}{5}^2\right)-2\mathit{\text{COVb}}3b4+2\mathit{\text{COVb}}3b5-2\mathit{\text{COVb}}4b5}}$$E28

Upon close examination of the aforementioned analysis, it becomes apparent that the values of coefficients (a1, a2, a3, and a4) and their corresponding standard errors exhibit notable discrepancies compared to the values reported by Edwards and Parry [4] and Edwards [5]. Interestingly, the t values, however, remain consistent, retaining their original significance due to a canceling-out effect.

The observed differences in coefficient values and standard errors suggest a departure from the previously established models proposed by Edwards and Parry [4] and Edwards [5]. It is essential to scrutinize these distinctions meticulously to discern the potential implications for the interpretation and generalization of the quadratic polynomial regression equations under consideration. Despite the dissimilarities in coefficient values and standard errors, the unchanged t values indicate that the significance levels associated with the coefficients remain unaltered. This underscores the robustness of the statistical significance of the individual coefficients, emphasizing their reliability even in the face of differing parameter estimates.

4. An empirical example

In order to elucidate the differentiation between the two previously mentioned formulae, we conducted an empirical analysis using the initial dataset derived from polynomial regression studies carried out by Qiu, Dooley, and Xie [12]. The referenced paper encompasses a comprehensive exploration involving two distinct studies, each contributing valuable insights to the field.

The dataset for the first study was meticulously gathered from front-line employees within a prominent restaurant chain. The primary aim of this study was to delve into the intricate dynamics of the interaction between servant leadership and self-efficacy, examining its profound impact on service quality within the context of the hospitality industry. This research undertaking reflects a targeted effort to understand the nuanced relationships and interplay between leadership styles and individual efficacy in a service-oriented setting, specifically within the dynamic environment of a restaurant chain.

By selecting this particular dataset, we sought to ground our exploration in a real-world scenario, one that encapsulates the complexities and intricacies of human interactions within the hospitality sector. The utilization of data from front-line employees adds a practical dimension to our analysis, enabling us to draw meaningful conclusions about the potential variations in slopes and curvatures of the two curves within quadratic polynomial regression equations. Through this empirical approach, we aim to not only highlight the distinctions between the formulae but also to provide valuable insights that contribute to the broader understanding of how these mathematical models can be applied and interpreted in the realm of organizational research, particularly within the hospitality industry.

The polynomial regression equation they derived from the response surface analysis was expressed as Z = 3.481 + 0.551X + 0.227Y – 0.234XY + 0.119Y² + e. Based on Eq. (1) described above, it can be easily seen that b₀ = 3.481, b₁ = 0.551, b₂ = 0.227, b₃ = 0, b₄ = −0.234, b₅ = 0.119. They then calculated the values of a₁, a₂, a₃, and a₄ as follows: a₁ = 0.778^**, a₂ = 0.051, a₃ = 0.324^**, and a₄ = 0.417^** (^**p < 0.01) following the formulae provided by Edwards and Parry [4]. However, if readers use the correct formulas presented previously, it can be derived that a₁ = 0.550^**, a₂ = 0.026, a₃ = 0.229^**, and a₄ = 0.209^** (^**p < 0.01). Here, t scores were not calculated because the significance levels of the coefficients remain unchanged.

For a more detailed examination of the statistical results, including t scores and additional relevant information, we recommend referring to Table 2 in the paper authored by Qiu, Dooley, and Xie [12]. This table likely provides a comprehensive summary of the regression analysis, offering insights into the specific values of coefficients, standard errors, t scores, and corresponding significance levels. By consulting this table, readers can gain a more nuanced understanding of the statistical nuances and intricacies involved in the analysis, thereby enhancing the overall comprehension of the research findings presented in the paper.

5. Conclusions

This paper utilized directional derivatives and rotations of axes, concepts from multivariable calculus, to derive an alternative set of formulae for calculating the slopes and curvatures of two curves within quadratic polynomial regression equations through response surface analysis. To emphasize, the accurate formulas for a₁, a₂, a₃, and a₄ are as follows:

$${\mathrm{a}}_1=\frac{\sqrt{2}}{2}({\mathrm{b}}_1+{\mathrm{b}}_{2)}$$E29

$${\mathrm{a}}_2=\frac{1}{2}({\mathrm{b}}_3+{\mathrm{b}}_4+{\mathrm{b}}_{5)}$$E30

$${\mathrm{a}}_3=\frac{\sqrt{2}}{2}({\mathrm{b}}_1–{\mathrm{b}}_{2)}$$E31

$${\mathrm{a}}_4=\frac{1}{2}({\mathrm{b}}_3–{\mathrm{b}}_4+{\mathrm{b}}_{5).}$$E32

The formulas for the significance of a₁, a₂, a₃, and a₄ are as follows:

$$\text{For variable}{\mathrm{a}}_1{\mathrm{t}}_{\text{a1}}=\frac{\left(b1+b2\right)}{\sqrt{\left(\mathit{SEb}{1}^2+\mathit{SEb}{2}^2\right)+2\mathit{COV}b1b2}}$$E33

$$\text{For variable}{\mathrm{a}}_2{\mathrm{t}}_{\text{a2}}=\frac{\left(b3+b4+b5\right)}{\sqrt{\left(\mathit{SEb}{3}^2+\mathit{SEb}{4}^2+\mathit{SEb}{5}^2\right)+2\mathit{\text{COVb}}3b4+2\mathit{\text{COVb}}3b5+2\mathit{\text{COVb}}4b5}}$$E34

$$\text{For variable}{\mathrm{a}}_3{\mathrm{t}}_{\text{a3}}=\frac{\left(b1-b2\right)}{\sqrt{\left(\mathit{SEb}{1}^2+\mathit{SEb}{2}^2\right)-2\mathit{COV}b1b2}}$$E35

$$\text{For variable}{\mathrm{a}}_4{\mathrm{t}}_{\text{a4}}=\frac{\left(b3-b4+b5\right)}{\sqrt{\left(\mathit{SEb}{3}^2+\mathit{SEb}{4}^2+\mathit{SEb}{5}^2\right)-2\mathit{\text{COVb}}3b4+2\mathit{\text{COVb}}3b5-2\mathit{\text{COVb}}4b5}}$$E36

Our primary goal is to systematically present a comprehensive assortment of mathematically rigorous formulae, poised to significantly enhance our comprehension of the slopes and curvatures inherent in quadratic polynomial regression equations through the prism of response surface analysis. The pursuit of this objective becomes imperative in light of the intricate mathematical intricacies entwined with such investigative endeavors. Our intent is not only to unveil theoretical frameworks but also to ground them in practicality, as evidenced by the empirical example we present. Through this illustration, we seek to shed light on the nuanced distinctions in slopes between the two previously outlined formulae, offering researchers a tangible and applicable perspective.

In elaborating on our empirical example, we emphasize its role as a pragmatic demonstration, providing a clear and tangible portrayal of how the theoretical constructs can be translated into real-world scenarios. By elucidating the disparities in slopes between the two curves within quadratic polynomial regression equations, we aim to demystify complex mathematical concepts and encourage a deeper understanding of the underlying principles.

Furthermore, we extend an invitation to the broader research community, urging scholars across diverse fields to adopt and integrate this newly proposed analytical tool into their investigations. The application of response surface analysis to explore the slopes and curvatures of two curves within quadratic polynomial regression equations holds promising potential for unlocking insights across various disciplines. By advocating for the widespread use of this innovative approach, we aspire to catalyze advancements in knowledge and contribute to the refinement of methodologies employed in quantitative analyses within respective research domains.