Robust metabolic syndrome risk score based on triangular areal similarity

Introduction

Metabolic syndrome (MetS) is a constellation of risk factors that directly affect atherosclerotic cardiovascular disease (CVD) (Grundy et al., 2005). These constellations are factors of interrelated metabolic origin that are strongly associated with insulin resistance and type 2 diabetes (Grundy et al., 2005; Kahn et al., 2005).

The criteria for the diagnosis of MetS were established by several specialized groups, such as the World Health Organization (WHO), the European Group for the Study of Insulin Resistance (EGIR), the National Cholesterol Education Program-Third Adult Treatment Panel (NCEP-ATP III), the American Association of Clinical Endocrinologists; American Heart Association-National Heart, Lung, and Blood Institute (AHA/NHLBI), and the International Diabetes Federation (IDF) (Jeong et al., 2014). Although the diagnostic criteria presented by each group vary slightly, MetS is commonly diagnosed on the basis of five risk factors: elevated waist circumference (WC), elevated fasting glucose (GL), elevated blood pressure (BP), elevated triglycerides (TG), and reduced high-density lipoprotein cholesterol (HDL). For each factor, abnormalities are determined using a specific threshold, and a diagnosis of MetS ensues when there are three or more abnormal factors present (Stone, Bilek & Rosenbaum, 2005).

The MetS diagnosis based on thresholds has been widely used in clinical practice and epidemiological studies (Grundy et al., 2005). However, over the past 20 years, the limitations of categorical diagnosis have steadily increased (Kahn et al., 2005; Eisenmann, 2008; Okosun et al., 2010; Jeong et al., 2014; Soldatovic et al., 2016; Wiley & Carrington, 2016). MetS is a chronic disease that should be considered a progressive condition. However, because the current diagnostic criteria are dichotomous, information loss is inevitable. The current counting method does not sense nor reflect changes in risk factors. To overcome this limitation, various MetS risk scores with continuous values have been proposed (Eisenmann, 2008; Okosun et al., 2010; Jeong et al., 2014; Soldatovic et al., 2016; Wiley & Carrington, 2016).

The most common approach for deriving continuous risk scores is to use statistical techniques such as z-score, standardized residuals of linear regression, or principal component analysis (Eisenmann, 2008; Khazdouz et al., 2021). Khazdouz et al. (2021) collected and analyzed 1,113 studies related to continuous MetS (cMets) scores from 1980 to 2020 and ultimately selected 10 studies. However, most of these studies primarily employed the z-score approach, indicating that recent research on cMetS score calculation methods has not significantly diverged from established practices.

Among the key methodologies, Okosun et al. (2010) derived the cMets by: (1) converting the values of individual MetS factors into z-scores; (2) performing a linear regression depending on age, sex, and race; and (3) summing the standardized residuals obtained for each MetS factor. The diagnostic performance of the area under the receiver operating characteristic curve (AUC), recall, and specificity were 0.885, 0.831, and 0.833, respectively, with the population made up of American adults. The diagnostic threshold was set as the point at which the sum of recall and specificity was maximized, according to Youden’s (1950) index. However, cMetS is limited in that it is difficult to compare different populations because of dependency on the sample in the calculation process. In addition, it is difficult to interpret the meaning of cMetS intuitively because its range is not limited.

Another statistic-based score is the MetS severity score (MetSSS) proposed by Wiley & Carrington (2016). MetSSS is derived by: (1) subtracting thresholds from individual MetS factor values, (2) zeroing values below 0, (3) converting these values into z-scores, (4) calculating the factor loading value of the MetS factors, (5) multiplying the factor loading value by the z-score, and (6) calculating the standardized distance. Although the diagnostic performance was not excellent among Europeans (with an AUC of 0.77, accuracy of 0.68, recall of 0.82, and specificity of 0.57), the sample-dependent properties were notably reduced by the adjustment method using the diagnostic threshold. However, MetSSS has similar limitations as cMetS, such as sample-dependent properties inherent in the calculation process and a lack of a clear range of values.

Another statistical approach is the siMS method, a simple technique for quantifying MetS introduced by Soldatovic et al. (2016). The siMS score consists of a single sum of a line of formulas. This score is obtained by summing up each MetS component in the form of a linear regression. This simple method exhibited high diagnostic performance, with an AUC of 0.926. The sample-independent properties are an additional advantage of using MetS factor values without transformation. However, the population was relatively small with 528 Serbians, and the correlation between the number of MetS factors and siMS was 0.745, indicating the need for additional verification and improvement.

Approaches other than statistics-based methods have also been proposed. Notably, a method exists for evaluating risk using the area of radar charts (Jeong et al., 2014), which are graphical representations designed to display multiple performance indicators concurrently in a circular format (Jeong et al., 2014; Stafoggia et al., 2011). Each indicator, standardized between 0 and 1, is positioned along the axes radiating from the center of the circle and connected to form a closed polygon (Jeong et al., 2014; Saary, 2008; Stafoggia et al., 2011). As a result, radar charts are valuable for visualizing multivariate data and provide an intuitive means of comparing overall performance (Jeong et al., 2014; Saary, 2008; Peng et al., 2019). Radar charts are widely utilized in business management, social science, and general engineering for performance measurement and risk assessment, and are being increasingly applied in healthcare, medicine, and biomedical engineering (Jeong et al., 2014; Saary, 2008; Peng et al., 2019).

Jeong et al. (2014) pioneered the use of radar charts for risk assessment in their groundbreaking study on MetS and introduced the areal similarity degree (ASD) method tailored for this purpose. ASD was derived by calculating the intersection area of the radar chart. Each MetS factor is represented by an axis on the radar chart. For each of the two adjacent axes, ASD quantifies the degree of overlap between a chart consisting of thresholds and the chart consisting of MetS factor values. The final score is derived by adding each resulting value to reflect the weight of each MetS factor. After evaluating 5,335 Koreans in various subgroups by sex and age, a strong correlation was observed between the number of risk factors and ASD in all groups. One advantage of ASD is that it ranges between 0 and 1, which allows for an intuitive interpretation. The closer it is to one, the closer the chart areas of MetS factors and thresholds. Another advantage of ASD is that weight reflects the importance of MetS factor.

However, ASD has three limitations. First, ASD is affected by the arrangement of the axes. The ASD method constructs a weighted radar chart with different central angles in proportion to the importance of MetS factors. In this design, the shape of the polygon depends on the arrangement of the axes and is asymmetric along the axis because of the different center angles. Second, ASD depends on the maximum and minimum values of the sample because it uses min–max scaling, which is vulnerable to outliers. Third, ASD exhibits sample-dependent properties. The weight of each MetS factor is determined by counting the number of specific cases in which the factor occurs within the sample. This count is then used to assign a measure of relative importance to each factor within the sample.

The cMetS risk score introduced thus far has one or more of the following limitations: (1) sample-dependent attributes, (2) uncertainty in the range or interpretation of values, (3) insufficient performance, and (4) variability under detailed conditions.

In this study, we introduce a robust risk score, the Robust MetS Risk Score (RMRS), which was designed to overcome these limitations. The ‘Methods’ section provides a detailed explanation for calculating RMRS with a specific focus on the innovative triangular areal similarity (TAS) method developed in this study. In the ‘Results’ section, we objectively compare the robustness of the RMRS across sex, age, and race, using extensive datasets from Korean and American populations, alongside previously introduced risk scores and using objective measures. The ‘Discussion’ section critically examines the reasons for the superior performance of the proposed method and reflects on the broader significance of our study. The ‘Conclusion’ section summarizes the key findings and outlines potential directions for future studies.

Methods

This section presents the proposed TAS method, which creates an RMRS by integrating two triangular areas on the radar charts. This process is schematically illustrated in Fig. 1. The details of each block are presented in the following subsections.

Calculating TAS score per radar chart

The TAS is a scoring method that uses four triangles on a three-axis radar chart, as shown in Fig. 2.

Four triangles

The four triangles are divided into two groups: criteria and measurement triangles.

The two criteria triangles refer to the threshold triangle ( $\Delta DEF$), to which each threshold (=0.5) is connected, and the maximum triangle $(\mathrm{\Delta }ABC),$ to which each maximum value (=1) is connected (Fig. 2).

The areas of the criteria triangles can be obtained using the sine law as follows:

(3) $${\mathrm{S}}_{\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}}={\displaystyle \frac{1}{2}\ast sin\left({\displaystyle \frac{\pi }{3}}\right)\ast \left(\overline{\mathrm{O}\mathrm{A}}\ast \overline{\mathrm{O}\mathrm{B}}+\overline{\mathrm{O}\mathrm{B}}\ast \overline{\mathrm{O}\mathrm{C}}+\overline{\mathrm{O}\mathrm{C}}\ast \overline{\mathrm{O}\mathrm{A}}\right),\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\overline{\mathrm{O}\mathrm{A}}=\overline{\mathrm{O}\mathrm{B}}=\overline{\mathrm{O}\mathrm{C}}=1}$$

(4) $${\mathrm{S}}_{\mathrm{\Delta }\mathrm{D}\mathrm{E}\mathrm{F}}={\displaystyle \frac{1}{2}\ast sin\left({\displaystyle \frac{\pi }{3}}\right)\ast \left(\overline{\mathrm{O}\mathrm{D}}\ast \overline{\mathrm{O}\mathrm{E}}+\overline{\mathrm{O}\mathrm{E}}\ast \overline{\mathrm{O}\mathrm{F}}+\overline{\mathrm{O}\mathrm{F}}\ast \overline{\mathrm{O}\mathrm{D}}\right),\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\overline{\mathrm{O}\mathrm{D}}=\overline{\mathrm{O}\mathrm{E}}=\overline{\mathrm{O}\mathrm{F}}=0.5}$$

The two measurement triangles refer to the internal triangle ( $\mathrm{\Delta }DQR$) connecting each value, with the maximum value limited to the threshold value, and the external triangle ( $\mathrm{\Delta }PQR$) connecting the original values (Fig. 2). The original input values $X=\left[x_1,x_2,x_3\right]$ of the radar chart are converted into $X^{\mathrm{\prime }}=\left[x_1^{\prime },x_2^{\prime },x_3^{\prime }\right]$ to construct an internal triangle using Eq. (5).

(5) $$X^{\mathrm{\prime }}=\left[min\left(x_1,0.5\right),min\left(x_2,0.5\right),\mathrm{m}\mathrm{i}\mathrm{n}\left(x_3,0.5\right)\right].$$

The area of the measurement triangles can be obtained using the sine law as follows:

(6) $${\mathrm{S}}_{\mathrm{x}}:={\mathrm{S}}_{\mathrm{\Delta }PQR}={\displaystyle \frac{1}{2}\ast sin\left({\displaystyle \frac{\pi }{3}}\right)\ast \left(x_1{x}_2+x_2{x}_3+x_3{x}_1\right)}$$

(7) $${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}:={\mathrm{S}}_{\mathrm{\Delta }DQR}={\displaystyle \frac{1}{2}\ast sin\left({\displaystyle \frac{\pi }{3}}\right)\ast \left(x_1^{\prime }{x}_2^{\prime }+x_2^{\prime }{x}_3^{\prime }+x_3^{\prime }{x}_1^{\prime }\right)}.$$

TAS score

The TAS score is calculated by recombining the four previously obtained triangles and focusing on the following attributes:

• 1)

  ${\mathrm{S}}_{\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}}$ and ${\mathrm{S}}_{\mathrm{\Delta }\mathrm{D}\mathrm{E}\mathrm{F}}$ are constants $.$

• 2)

  Let ${\mathrm{C}}_{\alpha }:={\mathrm{S}}_{\mathrm{\Delta }\mathrm{D}\mathrm{E}\mathrm{F}}$ and ${\mathrm{C}}_{\beta }:={\mathrm{S}}_{\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}}-{\mathrm{S}}_{\mathrm{\Delta }\mathrm{D}\mathrm{E}\mathrm{F}}$, then ${\mathrm{C}}_{\beta }=3\ast {\mathrm{C}}_{\alpha }$; ${\mathrm{C}}_{\alpha }$ is an area-based diagnostic criterion for these three risk factors; ${\mathrm{C}}_{\beta }$ is the maximum value of ‘the area-based severity,’ which is the maximum excess allowance from ${\mathrm{C}}_{\alpha }$ (Fig. 2A).

• 3)

  Let ${\mathrm{S}}_{\beta }:={\mathrm{S}}_{\mathrm{x}}-{\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}$, then ${\mathrm{S}}_{\mathrm{x}}$ can be decomposed into ${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}$ and ${\mathrm{S}}_{\beta }$ (Fig. 2B). The range of ${\mathrm{S}}_{\beta }$ is between 0 and ${\mathrm{C}}_{\beta }$.

The idea of TAS is as follows:

1. ${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}$ and ${\mathrm{S}}_{\beta }$ contain different types of information. ${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}$ contains a degree close to the diagnostic threshold of MetS, and ${\mathrm{S}}_{\beta }$ contains a serious degree that is not reflected in the diagnosis.

2. The effects of ${\mathrm{S}}_{\beta }$ differ before and after MetS onset.

3. These differences are adjusted by reflecting ${\mathrm{C}}_{\alpha }$ and ${\mathrm{C}}_{\beta }$.

Based on these properties and ideas, the formula for the TAS score is defined as

$$\begin{array}{r}TAS\left(x_1,x_2,x_3\right)={\displaystyle \frac{1}{2}\left\{closeness\left({\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}},{\mathrm{S}}_{\beta }\right)+severity\left({\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}},{\mathrm{S}}_{\beta }\right)\right\}}\\ ={\displaystyle \frac{X}{0.75+X-X^{\mathrm{\prime }}}+I\left({\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}\right)\ast {\displaystyle \frac{4}{9}\left(X-X^{\mathrm{\prime }}\right)}}\end{array}$$where:

$$closeness\left({\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}},{\mathrm{S}}_{\beta }\right)={\displaystyle \frac{{\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}+{\mathrm{S}}_{\beta }}{{\mathrm{C}}_{\alpha }+{\mathrm{S}}_{\beta }},severity\left({\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}},{\mathrm{S}}_{\beta }\right)=I\left({\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}\right)\ast {\displaystyle \frac{{\mathrm{S}}_{\beta }}{{\mathrm{C}}_{\beta }},}}$$

$$I\left({\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}\right):=if{\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}={\mathrm{C}}_{\alpha }then1else0,$$

(8) $$X=x_1{x}_2+x_2{x}_3+x_3{x}_1,X^{\mathrm{\prime }}=x_1^{\prime }{x}_2^{\prime }+x_2^{\prime }{x}_3^{\prime }+x_3^{\prime }{x}_1^{\prime }$$

Properties:

• 1)

  The TAS score is the average of the closeness and severity functions and has a value between 0 and 1. Closeness is defined as the degree of proximity to MetS.

• 2)

  Before the onset of MetS ( ${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}<{\mathrm{C}}_{\alpha })$, ${\mathrm{S}}_{\beta }$ only affects the closeness function and not the severity function. Based on the ratio of ${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}$ to the maximum value ( $={\mathrm{C}}_{\alpha })$ of ${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}$, the closeness function is modified to reflect severity ${\mathrm{S}}_{\beta }$ and is in the range of 0 and 1.

• 3)

  After the onset of MetS ( ${\mathrm{S}}_{{\mathrm{x}}^{\mathrm{\prime }}}={\mathrm{C}}_{\alpha })$, ${\mathrm{S}}_{\beta }$ affects only the severity function and not the closeness function. The severity function is the ratio of ${\mathrm{S}}_{\beta }$ to the maximum value ( $={\mathrm{C}}_{\beta })$ of ${\mathrm{S}}_{\beta }$, which is in the range of 0 to 1.

Integrating TAS score into RMRS

RMRS

RMRS is defined as Eq. (9). The RMRS is between 0 and 1 because it is the square root of the mean of the TAS scores which range from 0 to 1.

(9) $$RMRS=\sqrt{{\displaystyle \frac{1}{10}{\sum }_{i=1}^{10}TASscor{e}_i}}.$$

The RMRS has a structural characteristic that adds to the impact of the risk factors beyond the threshold. The example in Fig. 3 delineates the area (blue) where ${\mathrm{S}}_{\beta }$ occurs in each radar chart because $x_1$ and $x_3$ exceed the threshold among the five risk factors; $x_1$ and $x_3$ add to this effect by generating ${\mathrm{S}}_{\beta }$ in a total of six radar charts. In addition, the three radar charts (second and third graphs in the upper row and second graph in the lower row) further reflect the interaction effects of $x_1$ and $x_3$.

Diagnostic threshold

A diagnostic threshold was established so that RMRS could also be used to diagnose MetS. Although the current diagnostic criteria for MetS require the presence of at least three risk factors, the RMRS is a continuous value that can vary within a certain range, even when the three factors are equal. To address this, we calculated the distribution ranges of MetS risk factors for both two- and three-risk factor scenarios and set the threshold for MetS diagnosis at the center of the interval where the two ranges overlapped. The overlapping range was defined as the range from the minimum RMRS score (three risk factors) to the maximum RMRS score (two risk factors). The calculation was performed using two inputs for the risk factors, constructed regardless of order, namely, [0.5, 0.5, 0.5, 0, 0] and [1, 1, 0.5- $ϵ$, 0.5- $ϵ$, 0.5- $ϵ$, 0.5], where $ϵ$ is an arbitrarily small value close to zero. Based on the calculation results of 0.387 and 0.707, a threshold value of 0.547 was determined as the center of the overlapping range. The threshold is based only on the structural properties of the RMRS and is consistently applicable to all samples.

Results

The number of risk factors and the risk score

Since current MetS diagnostic criteria are based on the number of factors exceeding the threshold, i.e., the number of risk factors, RMRS analyzed the relevance of the number of these risk factors. The distribution of the number of risk factors is shown in Fig. 4A; the ratio decreased as the number increased. This trend appears in both datasets, with similar ratios (%) of each number to the whole: 36.9, 31.0, 19.2, 9.4, 3.1, and 0.6 for KoGES_HEXA, and 30.7, 29.8, 21.3, 12.1, 4.8, and 1.2 for NHANES (Table 3).

Table 3:

Distribution of risk score by number of MetS risk factors.

Risk factors (N)	Range of risk score	KoGES_HEXA		NHANES
		Mean ± sd (Range)	%	Mean ± sd (Range)	%
0	0–0.707	0.248 ± 0.057 (0.127–0.560)	36.9	0.240 ± 0.055 (0.123–0.525)	30.7
1	0–0.707	0.375 ± 0.062 (0.210–0.624)	31.0	0.359 ± 0.067 (0.203–0.661)	29.8
2	0.224–0.707	0.494 ± 0.055 (0.322–0.681)	19.2	0.476 ± 0.058 (0.330–0.681)	21.3
3	0.387–0.742	0.606 ± 0.044 (0.451–0.729)	9.4	0.595 ± 0.046 (0.445–0.716)	12.1
4	0.548–0.837	0.720 ± 0.033 (0.613–0.807)	3.1	0.723 ± 0.033 (0.632–0.797)	4.8
5	0.707–1	0.850 ± 0.033 (0.753–0.945)	0.6	0.866 ± 0.040 (0.760–0.953)	1.2

DOI: 10.7717/peerj-cs.2015/table-3

Figure 4B shows the distribution of RMRS, with each bar in Fig. 4A converted into a histogram. Figure 4B shows how the existing discrete space (number of risk factors) expands into a continuous space. As the measurement space expands with the continuous risk score, an overlapping interval occurs between each histogram corresponding to the number of risk factors.

The range of the risk score column in Table 3 shows the theoretically possible range for the number of risk factors. For the two empirical datasets, the ranges were narrower than the theoretical range. In addition, as the number of risk factors increased, the minimum, maximum, and average values of the range increased proportionally. Pearson’s correlation analysis showed a statistically significant positive correlation (p < 0.05) between the number of risk factors and the average, maximum, and minimum values of the risk score: 0.9997, 0.9946, and 0.9806 for KoGES_HEXA and 0.9993, 0.9937, and 0.9611 for NHANES, respectively.

Figure 4C is a display of Fig. 4B in boxplot format where the distribution characteristics of the RMRS according to the number of risk factors are clearly observed. For each number of risk factors, the RMRS scores corresponding to the 2nd and 3rd quartiles did not overlap. However, the RMRS corresponding to the fourth quartile overlapped the RMRS with more risk factors. Therefore, even with fewer than three risk factors, some cases fell into the range of MetS diagnoses based on RMRS (refer to the example provided in Table S1).

Discussion

The RMRS was proposed based on the TAS method to overcome the limitations of existing methods. Our method shares the underlying idea of ASD in calculating the overlay area of radar charts, but reflects robust and sample-independent properties. ASD constructs a radar chart using the center angle as the weight, resulting in asymmetric polygons along the axis. Although this asymmetry reflects the importance of the MetS factors, it also leads to deviations in the risk score. We addressed this problem by reconstructing the five-axis radar chart of ASD in the form of 10 three-axis radar charts. Consequently, this difference also reflects the weights. The ASD reflects global weights as the ratio of each anomaly to the overall anomaly of the sample, whereas RMRS reflects instance-level weights using the method shown in Fig. 3.

The RMRS structurally reflects the effect of the interaction between MetS factors. Although the TAS score calculation was derived from the triangular area, the equation can be interpreted as the sum of the interaction effects between each factor. The reconstruction of TAS Formula (8) from the interaction perspective is defined in Eq. (10): Input values $x_1,x_{2,}$ and $x_3$ are replaced with $0.5+{\alpha }_1,0.5+{\alpha }_{2,}$ and $0.5+{\alpha }_3$, respectively, based on the threshold of 0.5.

$$TAS\left(x_1,x_2,x_3\right)=TAS\left(0.5+{\alpha }_1,0.5+{\alpha }_2,0.5+{\alpha }_3\right)={\displaystyle \frac{I_T+I_P+I_N}{I_T+I_p}+I\ast \left({\displaystyle \frac{4}{9}}\right)\ast I_p,}$$

$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{I}_T=0.75,-0.5<{\alpha }_1,{\alpha }_2,{\alpha }_3<0.5,\mathrm{I}:=\mathrm{i}\mathrm{f}{\alpha }_1,{\alpha }_2,{\alpha }_3\ge 0\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}1\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}0$$

$$case1:{\alpha }_1,{\alpha }_2,{\alpha }_3<0,I_P=0,I_N={\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_1{\alpha }_2+{\alpha }_2{\alpha }_3+{\alpha }_3{\alpha }_1$$

$$case2:{\alpha }_1\ge 0,{\alpha }_2,{\alpha }_3<0,I_P={\alpha }_1+{\alpha }_1{\alpha }_2+{\alpha }_3{\alpha }_1,I_N={\alpha }_2+{\alpha }_3+{\alpha }_2{\alpha }_3$$

$$case3:{\alpha }_1,{\alpha }_2\ge 0,{\alpha }_3<0,I_P={\alpha }_1+{\alpha }_2+{\alpha }_1{\alpha }_2+{\alpha }_2{\alpha }_3+{\alpha }_3{\alpha }_1,I_N={\alpha }_3$$

(10) $$case4:{\alpha }_1,{\alpha }_2,{\alpha }_3\ge 0,I_P={\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_1{\alpha }_2+{\alpha }_2{\alpha }_3+{\alpha }_3{\alpha }_1,I_N=0$$where $I_P$ is the interaction caused by factors ${\alpha }_{\mathrm{i}}\ge 0$, and $I_N$ is the interaction caused by factors ${\alpha }_{\mathrm{i}}<0$. $I_T$ is an interaction that occurs based on the threshold value (0.5) of each factor and is a reference interaction. In Cases 1–3, MetS was not diagnosed based on these three factors. In this case, only the left-hand side of Eq. (10) remains. Because $I_T$ and $I_p$ are common factors in the numerator and denominator, respectively, $I_N$ is eventually reflected as a penalty. In other words, even if the effect exceeding the threshold is large, if the effect below the threshold is also large, the two effects of TAS are offset. In contrast, Case 4 exceeds all thresholds, reflecting the interaction effect of all factors without a penalty.

Finally, the contribution of this study can be summarized as follows:

Universal risk score without sample-dependence: By eschewing sample-dependent approaches such as the z-score, PCA loading values for MetSSS and cMetS, and sample-based weights for ASD, a universal risk score can be established by RMRS. This is achieved by relying solely on the threshold of the MetS factor, thereby ensuring independence from sample variation.

Fixed threshold stability: RMRS introduces stability through a fixed threshold derived from the structural characteristics that are unaffected by sample variations. This stability is demonstrated across diverse sample groups including different countries, age groups, sexes, and races.

Objective performance comparison with large datasets: The risk score, originally introduced in our previous study (Shin et al., 2021), was confined to Korean data. However, our current study expanded the dataset to include American data and conducted a comprehensive comparative analysis with four other methods, thereby establishing the superior performance of our approach. This study pioneers cross-country performance comparisons of existing risk scores by leveraging extensive Korean and American datasets collected by their respective governments.

Clinical implications: While recognizing the enduring value of dichotomous MetS scores in clinical practice, this study acknowledges the limitations in identifying early-stage abnormalities (Khazdouz et al., 2021). The proposed cMetS scores offer enhanced reliability in predicting MetS risk compared to traditional criteria. Given the global prevalence of MetS, our risk scores have substantial utility in preventive medicine worldwide.

Conclusion

This study addressed the limitations of the existing dichotomous diagnostic method for MetS and previously proposed calculations for a cMetS score. We introduced the robust MetS risk score RMRS based on the TAS method with a sample-independent three-axis radar chart, further verifying that RMRS exhibits consistently superior performance with universal properties.

Further research and verification of this scaling method are required. After setting 10% of the threshold to one unit using Eq. (1), the 10% interval around the threshold was nonlinearly expanded using Eq. (2). The threshold setting of 10% should be verified clinically to determine its appropriateness. Whether it is reasonable to assume equal weighting for all MetS factors at 10% and whether it would be more appropriate to adjust this interval individually also require further exploration.

Supplemental Information