Lorenzo Lolli

A comprehensive allometric analysis of 2nd digit length to 4th digit length in humans

It has been widely reported that men have a lower ratio of the 2nd and 4th human finger lengths (2D : 4D). Size-scaling ratios, however, have the seldom-appreciated potential for providing biased estimates. Using an information-theoretic approach, we compared 12 candidate models, with different assumptions and error structures, for scaling untransformed 2D to 4D lengths from 154 men and 262 women. In each hand, the two-parameter power function and the straight line with intercept models, both with normal, homoscedastic error, were superior to the other models and essentially equivalent to each other for normalizing 2D to 4D lengths. The conventional 2D : 4D ratio biased relative 2D length low for the generally bigger hands of men, and vice versa for women, thereby leading to an artefactual indication that mean relative 2D length is lower in men than women. Conversely, use of the more appropriate allometric or linear regression models revealed that mean relative 2D length was, in fact, greater in men than women. We conclude that 2D does not vary in direct proportion to 4D for both men and women, rendering the use of the simple 2D : 4D ratio inappropriate for size-scaling purposes and intergroup comparisons.

The acute-to-chronic workload ratio: An inaccurate scaling index for an unnecessary normalisation process?

An important question for researchers and practitioners is whether an individual’s risk of injury increases if they make prior changes to their training load.1 In this field of research, ‘load’ typically refers to in-training distances covered, speed and accelerations.1 Attention has generally focused on whether a person’s acute (eg, 7 day) increase in load, normalised to that person’s prior ‘baseline’ of chronic (eg, 28 day) load, predicts injury.1 To obtain this normalised predictor, acute load is typically divided by chronic load to provide the acute-to-chronic workload ratio (ACWR).1 Fundamentally, simple ratios (Y/X) are formulated to ‘ control for ’ a denominator variable (eg, preceding chronic load) that is perceived to have an important biological influence on the numerator variable (eg, acute load).2 Within this notion of ‘ control for ’,3 it is generally posited that the denominator is a ‘nuisance’ variable that is associated with the numerator of interest.2 Logically, a simple ratio index provides meaningful relative measures for clinical and prognostic purposes only if (1) there is a ‘true’ and ‘proportional’ association between numerator and denominator in the first place, and (2) the ratio normalises for the denominator in a consistent manner for all individuals in the measurement range.2 We have demonstrated recently that the typical practice in the current literature1 of including, …

Correct allometric analysis is always helpful for scaling flow-mediated dilation in research and individual patient contexts

McLay et al. (Clin Physiol Funct Imaging (2017); DOI: 10.1111/cpf.12465) recently examined whether the allometric scaling of flow‐mediated dilation influenced the mean difference between samples of young and older adults compared with the traditional percentage change approach. They also explored whether a new scaling calculation improved the ability to obtain individually scaled flow‐mediated dilation. In our response to their study, we can demonstrate that McLay et al. (Clin Physiol Funct Imaging, 2017) have (i) managed to formulate a new scaling index which does nothing to remove the dependency of that index on baseline diameter and (ii) suggested, incorrectly, that the original allometric approach cannot be used to derive individually‐adjusted values of flow‐mediated dilation, which can be interpreted in a similar way to a percentage change.

Ejection fraction as a statistical index of left ventricular systolic function: The first full allometric scrutiny of its appropriateness and accuracy

Left ventricular ejection fraction (EF) is a ratio that is deemed to accurately normalize stroke volume (SV) to end‐diastolic volume (EDV). Ratios are now well‐recognized for not normalizing the numerator, in this case SV, consistently for the denominator, EDV. We aimed to provide the first allometric‐based scrutiny of the conventional assumptions that underpin the EF ratio. We allometrically modelled untransformed SV and EDV measurements from 112 preclinical heart failure patients in the Multi‐Ethnic Study of Atherosclerosis (MESA), and 864 chronic heart failure patients in the Treatment of Preserved Cardiac Function Heart Failure with an Aldosterone Antagonist (TOPCAT) study. An information‐theoretic approach was adopted to assess the relative quality of twelve candidate models for normalizing SV to EDV. None of the conventional underlying assumptions for accurate ratio normalization, for example an allometric exponent ≈1, were upheld for EF. A two‐parameter power function with normal, heteroscedastic error was the best model for scaling SV to EDV in both samples. The allometric exponent (95% confidence interval) was 0·776 (0·682 to 0·869) in MESA, and 0·860 (0·857 to 0·864) in TOPCAT. EF was inversely correlated with EDV in MESA (r = −0·67, 95% CI: −0·76 to −0·55) and TOPCAT (r = −0·41, 95% CI: −0·46 to −0·35). Consequently, for fundamental statistical reasons, EF was biased low for people with generally larger EDVs, and vice versa. For the first time, we have demonstrated that EF is an inaccurate statistic for scaling SV to EDV, leading to potential biased inferences for research and individual patients.

Presence of a high-flow-mediated constriction phenomenon prior to flow-mediated dilatation in normal weight, overweight, and obese children and adolescents

There are now many indices of vascular function that are based on imaging techniques to quantify the difference in arterial diameter between a resting baseline (Dbase) and after a certain blood flow intervention. For example, researchers have reported one or more of the following indices: flow-mediated dilatation (FMD%), nitroglycerin-mediated dilatation (NMD%), lowflow-mediated constriction (L-FMC%), high-flowmediated constriction (H-FMC%), and even a “composite” constriction and dilatation index, eg, L-FMC% 1 FMD%. Some researchers have also suggested dividing some of these indices by each other, eg, FMD%/FMC%. Importantly, all these proposed indices of vascular function are expressed in ratio terms as the percentage change from Dbase. This percentage-based approach was adopted in an attempt to normalize consistently for artery size and, therefore, compare the indices between certain populations or conditions. Nevertheless, it is now well-documented that the first-proposed index, FMD%, does not serve this size-scaling role sufficiently well, leading to biased estimates of mean differences in vascular function and the emergence of spurious correlations. Because L-FMC% and H-FMC% are also ratio indices, it is likely that they suffer from the same size-scaling drawbacks as FMD%. Crucially, the Dbase that is the denominator in the calculation of FMD% (equation 1) is also the denominator in the FMC% calculation (equation 2). This essentially paves the way for spurious (nonbiologic) correlations between the indices, and potentially other variables of interest, because they are already mathematically coupled by their common denominator of Dbase. This confounding would also be present in a composite index. These issues could be relevant to the data reported by Ostrem et al, as well as any other study in which FMD% and FMC% are examined together.

Letter to the Editor: Presence of a high-flow-mediated constriction phenomenon prior to flow-mediated dilation in normal weight, overweight, and obese children and adolescents

There are now many indices of vascular function that are based on imaging techniques to quantify the difference in arterial diameter between a resting baseline (Dbase) and after a certain blood flow intervention. For example, researchers have reported one or more of the following indices: flow-mediated dilatation (FMD%), nitroglycerin-mediated dilatation (NMD%), lowflow-mediated constriction (L-FMC%), high-flowmediated constriction (H-FMC%), and even a “composite” constriction and dilatation index, eg, L-FMC% 1 FMD%. Some researchers have also suggested dividing some of these indices by each other, eg, FMD%/FMC%. Importantly, all these proposed indices of vascular function are expressed in ratio terms as the percentage change from Dbase. This percentage-based approach was adopted in an attempt to normalize consistently for artery size and, therefore, compare the indices between certain populations or conditions. Nevertheless, it is now well-documented that the first-proposed index, FMD%, does not serve this size-scaling role sufficiently well, leading to biased estimates of mean differences in vascular function and the emergence of spurious correlations. Because L-FMC% and H-FMC% are also ratio indices, it is likely that they suffer from the same size-scaling drawbacks as FMD%. Crucially, the Dbase that is the denominator in the calculation of FMD% (equation 1) is also the denominator in the FMC% calculation (equation 2). This essentially paves the way for spurious (nonbiologic) correlations between the indices, and potentially other variables of interest, because they are already mathematically coupled by their common denominator of Dbase. This confounding would also be present in a composite index. These issues could be relevant to the data reported by Ostrem et al, as well as any other study in which FMD% and FMC% are examined together.

ALLOMETRIC SCALING OF VO2max: A SYSTEMATIC REVIEW AND META-ANALYSIS

We aimed to provide the first quantitative synthesis of derived static allometric coefficients (b) used for scaling maximal oxygen uptake (VO2max) to whole-body mass and fat-free mass in human samples. Eight electronic databases were searched for relevant peer-reviewed articles. Inclusion criteria comprised human cardiorespiratory fitness data; cross-sectional study designs; an empirical derivation of the exponent; reported precision statistics; and reported information regarding participant sex, age and sports background, VO2max protocol, body composition protocol and line-fitting methods. Thirty-seven studies, involving 7,851 participants, met the eligibility criteria and were dichotomized into two main domains relevant to whole-body mass (n=28) and fat-free mass (n=16), respectively. The pooled allometric exponent (95% CL) was found to be 0.71 (0.65 to 0.77) for body mass and 0.91 (0.83 to 0.98) for fat-free mass. The among-studies heterogeneity was substantial for both whole-body mass and fat-free mass (τ =±0.15). Participant sex explained 33% of the between-study variability in the whole body mass exponent, but only 5% of the variability in the fat-free mass exponent. While the body mass exponent was substantially lower in women (b=0.52; 95% CL: 0.41 to 0.64) than for men (b=0.77; 95% CL: 0.71 to 0.83), the fat-free mass exponent was similar for both sexes. None of the identified moderator variables was statistically significant. The body mass exponent estimate encompassed both ⅔- and ¾-power laws. Conversely, the fat-free mass exponent was substantially larger and more generalisable due to the heterogeneity of body composition in human samples. We conclude that the scaling of VO2max in humans is consistent with the allometric cascade model, with an estimated pooled exponent that precludes ⅔- and ¾-power scaling.

THE QUANTIFICATION OF CHANGES IN CARDIORESPIRATORY FITNESS INDEPENDENT FROM CHANGES IN BODY MASS: ILLUSTRATION OF AN ALLOMETRIC APPROACH

Both obesity and a low level of cardiorespiratory fitness (VO2max) are strong predictors of morbidity and mortality. Therefore, in any lifestyle intervention study, it is important to ascertain if any changes in VO2max are clinically important, while considering any parallel changes in body mass and/or composition. Most researchers adopt a simple ratiometric approach to normalising VO2max (ml·kg−1·min−1), even though this is associated with several important assumptions, which are rarely confirmed by researchers. Using data from the Activity Counselling Trial (ACT), we developed and applied a novel allometric approach for comparing normalised changes in VO2max between different treatment arms. Overall, 874 participants were randomly allocated into advice only (n=292), assistance (n=293) and counselling group (n=289), respectively. A repeated-measures allometric model was adopted to adjust the changes in VO2max for the concurrent changes in body mass and fat-free mass from baseline to 24-month follow-up. Magnitude thresholds for standardized differences in VO2max of 0.20, 0.60, 1.20, 2.0 and 4.0 were considered as a small, moderate, large, very large and extremely large effect, respectively. The within-subjects allometric exponent (90% CL) for body mass was 0.46 (0.40 to 0.53) and 0.37 (0.31 to 0.43) in men and women, respectively. The respective fat-free mass exponents were 0.79 (0.71 to 0.87) and 0.64 (0.56 to 0.73). Ratio-scaled VO2max increased 4.5% (2.1% to 7.0%) more in the assistance vs advice group. This increase was slightly smaller when allometrically scaled to body mass [3.9% (1.7% to 6.1%)] and fat-free mass [3.7% (1.5% to 6.0%)], although standardised mean differences were similar between approaches (≈ 0.2). Similar findings were obtained for the counselling vs advice groups. Changes in VO2max were small in all study arms for men. Contrary to the original study report and subsequent citations, scaled intervention effects on cardiorespiratory fitness in the ACT are unlikely to be clinically important. Given the non-linear relationship between changes in body size and VO2max, a within-subjects allometric modelling approach is recommended for accurately quantifying any intervention-induced effects in future studies.