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*Corresponding Author: Arne Torbjørn Høstmark, Institute of Health and Society, Faculty of Medicine, University of Oslo, Norway, Box 1130 Blindern, 0318 Oslo, Norway.
Citation: Arne T. Høstmark, (2022). Intended Ranges and Correlations between Relative Amounts: a Review, J. Nutrition and Food Processing; 5(1); DOI:10.31579/2637-8914/077
Copyright: © 2022, Arne Torbjørn Høstmark. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: 05 November 2021 | Accepted: 07 December 2021 | Published: 01 January 2022
Keywords: correlation rules; relative amounts; ranges; biological regulation; fatty acids; white blood cells
The present article is a review of our previously suggested concepts of “Distribution Dependent Correlations” (DDC), and “Intended Ranges”. DDC concern associations between relative amounts of positive scale variables, in unit systems where sum of the percentages is 100%. Such correlations arise mathematically on the condition that the variables in question have particular (“intended”) ranges. For example, with three variables, two of which (A, B) having very low variability relative to a third one (C), we should expect a positive association between percent A and percent B, the slope being estimated by the B/A ratio. In addition, we should anticipate a negative relationship between %C and %A (%B). On the other hand, if A and B have high numbers and variability relative to C, then %A should relate inversely to %B. Furthermore, alterations in the ranges may have appreciable effects to change the associations. We present examples from physiology, where ranges seem to give strong DDC (positive and negative). The examples relate to body fatty acids, and white blood cell counts. Possibly, Intended Ranges could represent a case of evolutionary selection, to ensure proper balance between particular metabolites.
Definitions and Abbreviations:
Variability: the width or spread of a distribution, measured e.g. by the range and standard deviation.
Distribution: graph showing the frequency distribution of a variable within a particular range. In this article, we also use distribution when referring to a particular range, a – b, on the scale.
Uniform distribution: every value within the range is equally likely. In this article, we may write, “Distribution was from a to b”, or “Distributions of A, B, and C were a - b, c - d, and e - f, respectively”.
OA = Oleic Acid (18:1 c9); LA = Linoleic Acid (18:2 n6); ALA = Alpha Linolenic Acid (18:3 n3); AA = Arachidonic Acid (20:4 n6); EPA = Eicosapentaenoic Acid (20:5 n3); DPA = Docosapentaenoic Acid (22:5 n3); DHA = Docosahexaenoic Acid (22:6 n3); DGLA= dihomo-gammalinolenic acid (20:3 n6)
“Low–number variables” have very low numbers relative to “high-number variables”.
This article is a review and extension of our previously suggested concepts of “Distribution Dependent Correlations” (DDC), and “Intended Ranges”. Below, we first present some theoretical considerations to explain mathematically the phenomenon of DDC. Next, we show results of computer experiments with random numbers, to test the hypotheses. Finally, we show some examples from physiology, where “intended ranges” seem to govern DDC.
Particular background
The idea of “intended ranges” and “distribution dependent correlations” originated from a diet trial in chickens, carried out for a specific purpose, without any bearing on the present subject [1]. During post-trial analyses of the data, we observed remarkably strong positive and negative correlations between relative amounts of the measured fatty acids (in breast muscle). The present work relates to some of these correlations, and our efforts to explain them, as well as our attempts to find some general rules concerning associations between relative amounts.
A major part of the current article relates to body fatty acids. In brief, dietary intake is a major factor to regulate the concentration of fatty acids in blood and tissues, and these lipids are important in health and disease [2-4]. For example, poly-unsaturated fatty acids with 20 or 22 carbon atoms are precursors of eicosanoids and docosanoids, which are important regulatory molecules formed in most organs and cell types, through the actions of cyclooxygenases, lipoxygenases, and epoxygenases [5]. EPA (20:5 n3) - derived eicosanoids may decrease inflammatory diseases [6,7], decrease the risk of coronary heart diseases [8,9], and cancer [10], but the beneficial effects of long-chain n3 fatty acids on all-cause and cardiovascular mortality have been questioned [11].
On the other hand, eicosanoids derived from AA (20:4 n6), such as thromboxane A2 (TXA2) and leukotriene B4 (LTB4), have strong proinflammatory and prothrombotic properties, and are involved in allergic reactions and bronchoconstriction [2, 3, 5]. Furthermore, AA- derived endocannabinoids may have a role in adiposity and inflammation [12]. Additionally, low serum EPA/AA ratio may be a risk factor for cancer death [10]. Thus, the EPA and AA antagonism could explain many of the alleged positive health effects of EPA.
Also docosanoids, originating from C22 fatty acids (DPA, DHA), have strong metabolic effects. Among these latter compounds are protectins, resolvins, and maresins, which may strongly counteract immune- and inflammatory reactions [5]. Also eicosatrienoic acid, i.e. 20:3 n6 (dihomo-gammalinolenic acid, DGLA) may give eicosanoids [5].
To obtain a proper balance between the metabolic influences of the many eicosanoids and docosanoids, we should anticipate a coordinated regulation of their precursor fatty acid percentages, e.g. of % EPA, percentage AA, and %DGLA. Indeed, we might expect in general that these particular percentages of the total sum of fatty acids were positively associated, so that an increase (decrease) in e.g. percentage AA would be accompanied by a concomitant increase (decrease) in other fatty acid precursor percentages as well, in order to obtain the required balance. We previously reported that percentageAA, %EPA, %DHA, as well as other eicosanoid (docosanoid) percentages were positively associated in breast muscle lipids of chickens [12 - 15], as discussed below. We also showed that the correlation outcomes related to the particular concentration distributions of the fatty acids. This finding seemed to be in line with the remarkably similar outcomes with true values and with surrogate random numbers, found on the condition that we sampled the numbers within the true ranges of the fatty acids [14 - 18]. Furthermore, computer experiments showed that altering the ranges strongly influenced the correlation outcomes, in support of our suggested name: Distribution Dependent Correlations, DDC [14, 18 - 27].
In addition to the situation with body fatty acids, we observed that also the ranges of white blood cell (WBC) counts influenced the inverse relationship between e.g. the relative amounts of blood neutrophil granulocytes and lymphocytes [27]. We subsequently suggested the name “intended ranges” [28] to indicate ranges that might possibly serve to make strong correlations (positive and negative) between relative amounts of biological variables in “unit systems”, as exemplified by WBC, and by particular fatty acids
Since DDC rules are general, they should apply to any unit system in nature. However, investigations specifically focusing upon this issue seem hard to find, in a literature search. The apparent lack of interest might possibly relate to a methodological concern encountered when correlating percentages of the same sum, since the associations arise mathematically. On the other hand, it may not be obvious whether we should reject strong positive (negative) associations between percentages of the same sum as correlation bias. Rather, we previously suggested that intended ranges could be a case of evolutionary selection to obtain strong DDC [20 - 22]. The first part of this review article concerns mathematical explanations of correlations arising between percentages of the same sum. The presentation does not necessarily reflect our opinions in chronological order. Rather, we try to give a systematized presentation, based upon our present knowledge. However, in the second part, a synopsis of papers related to this topic, appear chronologically. We present some examples from physiology where evolution seems to have selected particular ranges to make relative amounts of some variables to be positively or negatively associated, mathematically, i.e. Distribution Dependent Correlations.
The present review concerns associations between relative amounts of positive scale variables in “unit systems”, where sum of relative amounts is 100%. We define A, B, C… to be positive scale variables and S their sum, i.e. S = A + B + C +… All of the variables should have the same unit, for example, g/kg, g/L, moles/L, or counts/L, i.e. the absolute amounts of muscle fatty acids may appear as g/kg wet weight. Furthermore, each of the variables should have particular ranges.
Previously [21], we investigated the association between relative amount of e.g. arachidonic acid (AA, 20:4 n6) and percentage of eicosapentaenoic acid (EPA, 20:5 n3), in chicken lipids. From histograms, the physiological concentration distributions (g/kg wet weight) for the fatty acids were determined. Next the sum (S, g/kg wet weight) of all fatty acids was computed, as well as and the remaining sum (R) when omitting the couple of fatty acids under investigation, thereby apparently obtaining 3 positive scale variables. With these variables, and with surrogate random number variables, generated with the true concentration distributions, computer analyses as described in detail below, were carried out. Our previous analyses [20, 21] demonstrated that correlations between e.g. percentage A and percentage B depended upon the particular range of each of the variables involved, and we obtained qualitatively similar correlations using the true (measured) values, or random numbers, if ranges were like the measured ones.
A major part of the present work consists of computer experiments using random numbers to explore further, how ranges of A, B, and C might influence correlations between relative amounts of the sum, S = A + B + C. This equation implies that sum of the A (B, C) percentages of S is 100; i.e. %A + %B + %C = 100, showing dependency between the percentages. We studied histograms, scatterplots, and correlations (Spearman’s rho). Computer experiments were performed, to study how alterations in ranges might change associations between %A, %B, and %C. Several repeats were carried out, with new sets of random numbers (for simplicity, n = 200 each time); the general outcome was always the same, but corresponding correlation coefficients and scatterplots varied slightly. We present the results mainly as scatterplots with correlation coefficients. In most of the computer experiments, the random numbers had uniform distribution, but we used random numbers with normal distribution as well, however obtaining qualitatively similar results. We used SPSS 27.0 for the analyses, and for making figures. The significance level was set at p less than 0.05. We present further details under Results and Discussions.
Unit systems with two variables only
We name the two variables A and B, respectively, and choose their ranges arbitrarily to be e.g. 11-37 for A, and 51 - 83 for B. Thus, A + B = S. Within these ranges, and for each of for example 200 cases, we generate random numbers of the variables. Thereby, we obtain 200 S-values: S1, S2, S3,. .. S200, each of which representing one of the 200 cases. We do not assume any relationship between the absolute amounts (Fig. 1, left panel). In contrast to this, the relative amounts of A and B must vary inversely, since sum of the relative amounts will always be 100%. A (B) percentages of S are %A = (A/S) x 100, and %B = (B/S) x 100, respectively. Thus, %A + %B = 100, or %B = -%A + 100, showing a perfect inverse linear relationship (Fig. 1, right panel). All %A values must correspond inversely to %B, since %B = - %A + 100, irrespective of which of the allowed values of A (B) appearing for a certain case. For example, the lowest (highest) %A must correspond to the highest (lowest) %B. Unlike this strict requirement put upon A and B percentages, for each case, the absolute A-value could have any values from 11 to 37, and B any value from 51 to 83, explaining the lack of correlation between A and B.
Accordingly, in a unit system with two variables only, each of which with a particular range, we should find a perfect inverse relationship between their relative amounts, irrespective of their ranges. We may raise the question of whether the inverse relationship between relative amounts, in a “two-variable unit system”, could be of any physiological interest.
The equation %B= -%A + 100 shows that the regression line for %A vs. %B should pass through 100% on both axes, irrespective of the sizes of the A and B values. However, the A and B ranges determine where we find the points on the %A vs. %B scatterplot. Thus, if A has low (high) numbers relative to B, then the points will appear in the lower (upper) part of the %A scale, implying that the %B to %A ratio is distributed among high (low) values, respectively (Fig. 1). The %B to %A ratio is equal to the B/A ratio, as seen from the calculation of the percentages. In medicine, we might encounter close to “two variable unit systems” where distribution of the B/A ratio among high or low values could be of clinical interest [29, 30].
Below, we shall consider “unit systems” involving more than two positive scale variables, each of which having particular ranges. In the theoretical reasoning, we first utilize the equation of a straight line (y = ax + b). Next, we consider the relationship between relative amounts of the variables and their sum. The theory and computer experiments suggest that relative amounts (percentages) should correlate positively or negatively, because of their particular ranges.
Unit systems with three variables [16]
Applying the equation of a straight line (y = ax + b)
In the current context, we have A + B + C = S, i.e. %A + %B + %C = 100, or %B = -%A + (100 - %C). This equation resembles that of a straight line (y = ax + b), however involving relative amounts of the three variables (A, B, C), each of which having a particular range. We will consider this equation in three particular situations: 1) if the [removed]100 - %C) is approaching zero, i.e.
These studies seem to suggest that the particular distributions (range/place on the scale/spread) of variables such as fatty acids will determine whether their relative amounts are positively or negatively associated, or not correlated at all.
None
I want to thank Professor emerita Anna Haug very much for allowing me to use data from her excellent diet trial in chickens.
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