**4. What is the justification for making our arguments?**

#### **4.1 Lessons from anthropometry, mathematics, geometry and epidemiology**

Arithmetic value and true risk measured from each anthropometric depends on formulae, unit of measure and body measurements derived from different structural components. Mathematical understanding of some concepts turns out to be key to detecting unhealthy BC and anthropometrically-measured risk. From this perspective, weight, height, height/2, WC and HC represent absolute values without expressing equality for risk as a mathematical object. Consequently, in assessing anthropometricsassociated risk, mathematical relation of equivalence between simple measurements and indicators or ratios to be compared should be recognised by the researchers

#### **Figure 1.**

*The standard human body and simple anthropometric measurements. Geometrical lines drawn from anthropometry for understanding metrics and rays of risk for WC and WHR >1. Mathematical principles and anthropometric arguments that hold true in an anthropometrically healthy population. Anthropometrics at baseline would represent mean values per standard deviation for height, height/2, WC, HC, WHR, WHtR and "X" distance being actually valid for any anthropometrically healthy population and ethnicity. On the respective rays of risk for WC (in red colour) and WHR >1 would lie points of increased abdominal obesity representing mean values (SD) for thousands of cases of MI/CVD as well as biological changes pointing towards greater excess risk as WC increases and while height may no condition the whole-risk measured by WC alone. On the ipsilateral segment, which length value is "X" positive (cm)/2 would lie all the points for WHR <1 (including WHR risk cutoff) from the lowest value up to 0.999 (X = 0.1: X/2 = 0.05) just before the outer limit of HC where X = 0. HC indicates hip circumference; WC, waist circumference; WHR, waist-tohip ratio; WHtR, waist-to-height ratio; X, subtracting HC by WC; X/2, ipsilateral segment as horizontal distance between any point of WC in the mid-axillary line and the vertical top line for HC in their outer limit. Footnote: Original drawings built and designed by the author. Dimensions are not to scale. Anthropometric evidence supports the referred mathematical inequalities between the simple measurements in the standard human body.*

(**Figure 1**, **Table 1**). Thus, when comparing with anthropometrically healthy subjects and with the evidence of CVD epidemiology, the rationale is as follows.

Muscle, bone, fat and residual mass as different biological components present no differentiation by body weight (unit of mass), and therefore, a higher BMI does not always involve greater body fat excess, at least in normal or overweight people [2, 24, 27]. Weight and height differences between sexes are not recognised by the BMI formula. Thereby, an equal BMI does not mean the same degree of fatness or unhealthy BC. In this sense, the error of estimation for high risk BC or risk may occur in comparing BMI with others, and either by age or by sex.

Height length depends on the bone structure of the adult. In this sense, height never correlates with adiposity [10, 21, 23, 27, 31, 48], and, therefore, it does not account for the true-risk per se. However, height as a volume factor would exert a modulating effect for conditioning the storage and distribution of the body fat as well as the relative volume that it occupies in the three-dimensional abdominal space [24, 27]. Thereby, a significant difference in height between groups and sexes conditions the risk estimated by each concerned anthropometric, and therefore, height as longitudinal dimension also has important implications.

Mathematically, WC and WHtR would be equivalent for the same estimated risk if, and only if, mean WC = height/2. Therefore, WHtR risk cut-off =0.5 is the entity of risk conditioned on WC, but height/2 taking the same value as WC (e.g., 80/160, 84/168, 85/170, 88/176 etc., all =0.5). If not, the error of estimation for both the true high risk BC and risk may occur in comparing WC alone with WHtR, and either by age or by sex. Thus, if the mean WC is >height/2 (WHtR risk cut-off >0.5) (e.g., 80.5/158, 82.6/162, 82.8/162.4, 95.4/187 etc., all =0.51) protective underestimation occurs for height with respect to WC, whether WC alone assigns the risk from a defined risk cut-off.

In another conceptual consideration, evidence supports that there is a higher excess risk of MI/CVD when abdominal obesity increases [13, 14]. However, when comparing between-groups abdominal obesity may be expressed either in cm2 (twodimensional area determined from WC length) or in cm3 (three-dimensional volume of a solid abdominal disk determined from WC and height of the disk = WHtR cm), (**Figures 2** and **3**) [24, 37]. From this new insight, WC and WHtR do not express the same risk when comparing healthy people and MI/CVD cases. This is because WC < height/2 (WHtR <0.5) is a natural inequality. In this way, WC and WHtR refer to the same risk only if the mean WC = height/2 (WHtR risk cut-off =0.5). However, when the mean WC increases above height/2 (WHtR risk cut-off >0.5), the distribution curves of WC and height/2 appear unbalanced between healthy and cases, and only WHtR as an entity of volume may described the risk that is conditional on both WC and height. Otherwise, if we accept WC alone as an anthropometrically-measured causal risk factor, this will lead to an overestimation of risk for WC concerning height, or a protective underestimation of height with respect to WC. It is clear that, if WHtR risk cut-off is >0.5 (the mean WC > height/2), height appears to be inversely associated with the group of cases, and WHtR is the indicator of risk when comparing by ethnicity and sex, but not WC alone. This is because risk is conditional on both WC and height as independent volume factors.

HC length depends on the breadth between both trochanters, the gluteal mass and the gluteal–femoral fat to determine a two-dimensional geometric area on a transverse plane of defined bodily components, but HC neither discriminates between them nor describes cardiometabolic risk. Therefore, it does not account for the true high risk BC or risk [10, 24, 27, 31, 37, 48]. Thus, either the high risk BC or raised %BF is not affected by

*Anthropometrics in Predicting Cardiovascular Disease Risk: Our Research Work Mathematically… DOI: http://dx.doi.org/10.5772/intechopen.105098*

#### **Figure 2.**

*Anthropometric length measurements in the standard body human and considerations for differencing between volume of a three-dimensional abdominal disk and WC as two-dimensional area. Measurements at baseline would represent mean values per standard deviation for WC, HC, height, height/2 and WHtR being actually valid for any study population and ethnicity. The model of disk for representing volume of an abdominal segment may be applied for both case–control and cohort studies from the respective mean values (SD) and risk cut-offs for WHtR. Anthropometric considerations are explained for understanding volume and excess risk of MI/CVD as WHtR increases. CVD denotes cardiovascular disease; H, body height; height/2, dividing height by 2; h, height of the disk; HC, hip circumference; MI, myocardial infarction, r, radius of the base; V, volume of the disk; WC, waist circumference; WHtR, waist-to-height ratio. Footnote: Original graphical abstract was built and designed by the author.*

HC, but vice versa. HC can be modified by physical activity or the ageing process, etc., in both sexes, but this does not justify a direct impact on MI/CVD risk. With modifications in HC, neither WC nor high risk BC and %BF are necessarily affected. In this sense, WC and WHR would be mathematically equivalent for the same estimation of risk if, and only if, the mean HC = WC, and therefore, WHR risk cut-off = 1 being the entity of risk conditional on WC, but HC taking the same value as WC. In this case, subtracting HC by WC we obtain an X value of zero (**Figures 1** and **3**) [36, 37]. If not, the error of estimation for both the true high risk BC and risk may occur in comparing WHR with WC alone, and either by age or by sex. Thus, the mean HC > WC protective overestimation occurs for HC with respect to WC, and WHR <1 may present a risk overestimation by selecting false-positive points as compared to those true-negatives conditional on WC values as the numerator. It is clear that, if WHR risk cut-off is <1 (mean HC > WC: similar to natural inequality), not all subjects in that stratum may present risk because HC as risk factor appears not to be associated with any group when compared. Similarly, if HC > WC (WHR <1: X > 0) is a true premise applicable to a healthy population, the question arises as to how it may be applied to cases of CVD without being a false premise? From an epidemiological viewpoint, effectively only WHR <1 may represent a risk associated to cases when conditioning WC as numerator. This value lies above their

#### **Figure 3.**

*Number lines in a Cartesian plane for representing values in healthy population and cases of MI/CVD: Metrics-associated risk increases as each anthropometric ray of risk move to the right (site of cases). Subtitled curves of distribution, overlapping area, risk ray and bias zone as appropriate. It is transferable to any study population and ethnicity. All reference values may be represented lying on the respective number lines drawn. We may find the points with the lowest baseline values for WHtR, WC and WHR (healthy/ controls or cases) lying on a respective line in the origin. Similarly, risk cut-offs and cutting lines lying where appropriate. The highest baseline values (generally in unhealthy cases) would lie on the arrowhead of the anthropometric rays of risk moving further outwards (right site). Other points would represent mean values per standard deviation for WC, HC, height, height/2, WHR and WHtR in healthy and cases as appropriate. In the respective lines and risk rays drawn in magenta colour would lie points of increased abdominal obesity representing values for thousands of cases of MI/CVD as well as biological changes pointing towards greater excess risk as WC increases and HC and height condition the true risk from WHR and WHtR, respectively. Values for X (between the maximum positive in their origin and zero (WC = HC) would be represented lying on the corresponding partial ray of risk (in blue colour). We have also pointed the theoretical cutting lines for WHtR and WHR there where would occur a balanced distribution of WC-height/2, WC-HC and WC-height mean values (SD) when pooling healthy and unhealthy cases. The model plotted may be applied for both case–control and cohort studies. CVD denotes cardiovascular disease; H, height; HC, hip circumference; MI, myocardial infarction; WC, waist circumference; WHR, waist-to-hip ratio; WHtR, waist-to-height ratio; X, subtracting HC by WC. Footnote: Original graphical abstract was built and designed by the author. Dimensions are not to scale.*

defined risk cut-off. Obviously, WHR ≥1 (WC ≥ HC: X ≤ 0) will always represent risk associated to group of cases irrespective of HC value (**Figure 1**). Therefore, the true risk assignment for WHR only depends on WC receiving risk as numerator, and besides, WC as the entity of risk compared according to ethnicity and sex, but never WHR alone as an abstract fraction.

WC length depends on specific biological components that determine a twodimensional geometric area (cm2 ) on a transverse plane. Evidence supports WC as the strongest simple metric linked to visceral adiposity that provides a solid estimation of risk [13, 14, 17, 46, 47, 49]. On the other hand, in the standard human body, WC can

#### *Anthropometrics in Predicting Cardiovascular Disease Risk: Our Research Work Mathematically… DOI: http://dx.doi.org/10.5772/intechopen.105098*

be lower than height/2 (WHtR <0.5) without posing any putative risk or protective effect (**Figures 1** and **2**). Only when WC and height/2 are mathematically equivalent (WC = height/2: WHtR = 0.5) is there a notion of equality and balance for the same estimation of risk from WC and WHtR. However, evidence also supports the notion that WHtR >0.5 is strongly associated to cases of MI [15, 18, 21, 23, 27, 37]. When the WHtR risk cut-off is >0.5, equality does not exist between WC and height/2, and only WHtR may be used to draw a valid conclusion for estimating the risk (**Figure 3**, **Table 1**). Thus, if the mean WC > height/2 risk overestimation occurs for WC with respect to height, WC alone will present an overestimation of risk in the tallest people and an underestimation in the shortest. Mathematically, WHtR >0.5 and < 1 is a proper abstract fraction (part/whole) whose decimal value up to 1 (theoretical) tells us the equal parts of WC that we have in height (whole), but never WC (part) referring to the entity of whole-risk as a mathematical object. Quite the opposite is the case; the higher the WHtR (whereas being <1), the higher the risk overestimation for WC as compared to WHtR. Similarly, the higher the WHtR between 0.51 and 0.999, the higher the probability of bias for WC. If WHtR cannot record true risk, WC might capture false risk beyond the true risk of WHtR. Hence, WC might present an error of estimation in women compared to men due to differences in WC and height between both sexes and, therefore, different risks to be compared. Only when the mean WC is lower than height/2 (WHtR risk cut-off <0.5), WC and its risk cut-off would represent the entity of risk without accounting for bias, but only up to WHtR = 0.5 (**Figure 3**). That way, only in unrepresentative, small samples where the mean WC is lower than height/2 or in women where differences between mean WC and height/2 are less important, WC and WHtR would capture similar risk as being close to WHtR = 0.5. However, if WHtR risk cut-off is >0.5 (mean WC > height/2) not all subjects in that stratum will present risk from WC alone because it may not capture true risk, at least without accounting for height. In this regard, if the mean WC > height/2 (WHtR risk cut-off >0.5) is a true premise applicable to MI/ CVD cases, how can it be applied to a healthy population without being a false premise? Epidemiologically, those values for WHtR from 0.51 up to any other defined risk cut-off of >0.51, while lying on the overlapping zone of the distribution curves between groups, they may be true-negatives for healthy subjects when conditional on WHtR >0.5 as the true predictive variable, effectively being the mean WC higher than height/2. In this situation, those true-negative points for WHtR always lie before the line of their defined risk cut-off, which is much further on from 0.5 (bias zone for WC, **Figure 3**). Indisputably, if in any study population's WHtR risk cut-off is of >0.5 (mean WC > height/2), the concrete value of this metric while measuring the relative volume and being conditional on both WC and height predicts the received risk, but never WC alone.

The standard human body can have a HC higher than WC without posing any putative risk or protective effect (**Figure 1**). By deduction, HC > WC is an anthropometrically healthy natural inequality, which responds to a linear equation: HC = WC + X, where by subtracting HC from WC we calculate X (>zero) as a unit of length with one decimal digit-tenths; the standard value is higher in women and the middle-aged than in men and elderly subjects, respectively, but higher than zero in all cases. Mathematically, WHR <1 is a proper abstract fraction whose decimal value ranged from hundredths up to 1, which states that equal parts of WC in HC, but it shows no anthropometric consistency or true risk beyond that of WC or X distance. It is clear that WHR <1 is simply a way of representing size (part/whole) that is not a whole number or entity of whole-risk as a mathematical object, unlike WC or X. In this sense, WHR <1 might represent a higher risk than WC and X, when HC has the importance of being overestimated as a protective factor with respect to WC

and, therefore, creating bias for WHR. This is because fractions of equal value do not refer to the same risk and the sensitivity of WHR (hundredths) is different from X (tenths). It is clear that between two consecutive values of WHR <1 we have 10 of X (e.g., between 0.95 and 0.96. we have from 5 up to 4.1 for X, but not all referring to the same risk as it is 0.95, which misclassifies risk). Thus, the higher the positive value of X (e.g., in women, middle-aged people, athletes), the higher the probability of bias for WHR when compared to WC, and if values of WC (numerator) and X as true-negatives below their respective risk cut-offs receive no true risk, WHR may effectively capture false-positive points in the stratum of <1. "From a proper abstract fraction, if WHR risk cut-off is of <1, WC turns out to be the entity of risk to be compared, but never WHR performing better than WC, at least while understanding maths and biases" [37].

Anthropometrically, in any study population, from the lowest baseline up to the highest values there is a direct correlation between cardiometabolic risk for WC and WHtR indicating the corresponding risk cut-offs. As WC and WHtR increase, the respective risk cut-offs and points with greater excess risk move further outwards lying on their geometric rays. However, WC may only represent risk when WHtR = 0.5 and the mean WC and height/2 are balanced in their data distribution (**Figure 3**). Similarly, WC alone may represent risk with respect to WHR when the WC cut-off lies before the line where WHR = 1. When the WHR risk cut-off is ≥1 (improper fraction), WC and WHR express the same risk. On the contrary, while WHtR may demonstrate a risk cut-off between 0.51 and 0.999 (<1), neither WC nor WHR will represent risk due to overlapping and bias zones where false-positive points might be selected from both with respect to WHtR, which would receive no risk up to their risk cut-off lying on their ray of risk further outwards (site of cases), (**Figure 3**). Indisputably, the risk points from WHR and WC in bias zones before the WHtR risk cut-off will never capture the true risk while not being true positives lying on their respective rays of risk after the same WHtR risk cut-off. The risk captured by WHR and WC in the identified bias zones will always be false, at least partially.

Epidemiologically, neither height nor HC correlate to cardiometabolic risk. Hence, in predicting MI/CVD risk, HC and height may only be conditional risks for WHR and WHtR as area and volume factors, respectively. HC never appears to take the same cut-off value as height (the mean height is always higher than HC and HC > height/2). WC hardly reaches the same cut-off value as height or HC (mathematically it is always fulfilled as the mean height > HC > WC. The mean HC > WC > height/2, see **Table 1**). In addition, as WC increases, WHR >1 (whole/part) may also draw a similar correlation of risk up to the highest WC values because it directly depends on WC as a total area of risk, irrespective of HC (**Figure 1**). Nevertheless, WHR <1 (part/whole) draws neither ray nor greater excess risk, at least between their risk cut-offs and the 0.999 value where a higher or lesser bias occurs as HC increases or decreases and WC does not move in its respective ray of risk. On the other hand, only WHtR as a relative volume allows a clear indication of risk to be recognised up to value of 1, which theoretically would represent the unity of risk corresponding to the total volume where WC would take the same value as height (in a balanced distribution). In this approach, we will always find the point for WHtR = 0.5 before the line for WHR = 1, and the WHtR risk cut-off lies much more outwards (in the site of cases) than WC and WHR. Thereby, the curves of distribution and overlapping zones explain that, in capturing risk, WHtR presents much more sensitivity (true-positive fraction) than WC or WHR. This is because true-negative values conditioned on the WHtR risk

*Anthropometrics in Predicting Cardiovascular Disease Risk: Our Research Work Mathematically… DOI: http://dx.doi.org/10.5772/intechopen.105098*

cut-off are not selected as false-positive ones, unlike WC and WHR between their respective risk cut-offs and the end of the bias zones (**Figure 3**).

Anatomically, HC is also higher than height/2 and lower than height (height/ HC >1; HC/(height/2) >1) (**Figure 1**). Hence, there would be no equivalent relation between WHR and WHtR risk cut-offs to compare the same risk if the first is lower than the second × 2 (WHR/WHtR <2). According to this premise, WHtR ≥0.5 will always detect risk before WHR ≥1 (see **Figure 3**). Since the balanced distribution between WC and height/2 on the one hand, and between WC and HC on the other hand, may only be found on the risk cut-offs of WHtR =0.5 and WHR =1, respectively, both indices will never capture the same risk because it is anthropometrically impossible and epidemiologically false (**Table 1**). Therefore, bias will occur for WHR with respect to WHtR due to an unbalancing of HC and height/2 values between healthy and unhealthy cases (**Figures 1** and **3**). If WHR risk cut-off is lower than WHtR × 2 and WC does not move, WHR-associated risk above WHtR would be a

#### **Figure 4.**

*Lessons from geometry: Volume of solids. Geometric model representing the human body as a solid cylinder or two truncated cones joined together at their major bases. Geometry formulas and explanations for understanding the meaning of WHtR when comparing cardiometabolic risk between healthy population and cases of MI/CVD. Geometric values at baseline would represent the mean values per standard deviation for WC, radius, heights and WHtR being actually valid for any study population and ethnicity. The model may be applied for both case–control and cohort studies from the respective mean values (SD) and risk cut-offs for WHtR. "Volume" refers to the amount of three-dimensional space that bodily components occupy in relation to their mass and density. Volume is determined by geometry formulas. The base of the cylinder and the major base of the truncated cones have a length or perimeter equal to WC as appropriate. Dividing H by WHtR we get the total number of disks that fit into each three-dimensional shape. CVD denotes cardiovascular disease; H, total height corresponding to that of the cylinder or double truncated cone; h1, height or thickness of each disk or frustum; h2, height of a single truncated cone (H/2); MI, myocardial infarction, R or r, radius of each base as appropriate; V1, volume of the cylindrical disk; V2, volume of the conical frustum; WC, waist circumference; WHtR, waist-to-height ratio.*

false-positive due a protective overestimation for HC concerning height, either by age or by sex. "From a mathematical conception, …, if ratio of the risk cut-offs between WHR and WHtR is of <2 (WHR <WHtR x 2), WHtR turns out to be the entity of risk to be compared, but never WHR performing better than WHtR, at least while understanding maths and biases" [37].

From geometry, the concrete volume of a three-dimensional disk or frustum (e.g., at umbilicus level) may be quantified from the WHtR. Simulating a cylinder or truncated cone, the volume of this disk will depend on area of the base(s) (πr 2 , where WC =2πr: r = WC/2π) and their geometrical height (thickness of the disk = WHtR cm) [36, 37]. Geometrically, the human body as a solid from the head to the feet would have several disks, so that number of disks = body height (H)/WHtR, and the sum of the volume of all the disks would give us the total volume of the body. The total body volume would be the theoretical unity of risk where WC = height: WHtR =1: number of disks =1 (**Figure 4**). Obviously, only from this hypothetical situation WHtR ≥1 (improper fraction where the mean WC ≥ height) will always represent risk associated to group of cases irrespective of heiht value, and WC and WHtR ≥1 refering to the same risk. Thereby, an epidemiologically real WHtR gives us the corresponding relative volume (cm3 ) that we have by unit of height or disk in a directinverse relationship with WC-height. The higher the WHtR, the higher the volume of the disk. On the other hand, although WC values do not change, the disk volume may be modulated by body height towards a higher or lesser amount of three-dimensional space that risk components occupy and, therefore, modifying their cardiometabolic effect. Epidemiologically, WHtR is important because it captures risk above the WC area, at least when height may have significant differences between groups to be compared and with a WHtR risk cut-off >0.5 and < 1. In this approach, the area and volume from WC and WHtR, respectively, would not be comparable. "From a proper abstract fraction, if WHtR risk cut-off is of >0.5 and <1, the value of this metric is the entity of risk to be compared, but WC never performs better than WHtR, at least while understanding maths and biases" [37].
