Unanticipated consequences of logarithmic transformation in bivariate allometry

Z-score regression model for coronary artery diameter in healthy Chinese Han children

BACKGROUND

Coronary artery distension and aneurysm are complications of Kawasaki disease in children.

PURPOSE

To develop a Z-score regression model for coronary artery diameter in children that could be used as reference.

MATERIAL AND METHODS

This retrospective analysis included children with normal heart structure between March 2013 and April 2017. Body surface area (BSA) was calculated. The diameters of the right coronary, left main coronary, left anterior descending, and circumflex arteries were measured by echocardiography. Pearson correlation analysis was used to establish linear, exponential, logarithmic, power, and square root regression models.

RESULTS

The analysis included 509 children (280 boys) aged 1 day to 15.2 years. Coronary artery diameters were significantly correlated with age, height, body mass, BSA, and BSA (r = 0.663-0.826; P < 0.05), with a stronger correlation for BSA than BSA (P < 0.05). The adjusted determination coefficients (Ra2) were higher for the exponential and square root models than for the other models (P < 0.05). The random error term variance was constant for the exponential model (P > 0.05), and processing with the weighted least-square methods eliminated heteroscedasticity in the other models. The Z-scores were normally distributed for the exponential and square root models (P > 0.05).

CONCLUSION

Overall, the square root model was the optimal equation for the calculation of coronary artery Z-score in Chinese Han children. This model could be used to facilitate the diagnosis of coronary artery distension in children with suspected Kawasaki disease.

Modeling allometric variation: lessons from the metabolic allometry of black carp (Mylopharyngodon piceus)

I used linear and nonlinear regression to re-examine published data on the scaling of metabolic rate vs. body mass in an ontogenetic series of black carp (Mylopharyngodon piceus (Richardson, 1846)). My objective was to expose shortcomings of the conventional procedure for fitting statistical models to bivariate observations (i.e., the procedure that is widely attributed to J.S. Huxley) and simultaneously to outline a more general and utilitarian protocol for analyzing bivariate data in studies of allometry. Authors of the original study on carp reported exponents of 0.83 and 0.78 for two-parameter power functions fitted to observations for resting metabolism and maximum metabolism, respectively. However, metabolic scaling in these fishes actually is described best by straight lines having positive intercepts with the Y axis. The allometric exponent is 1 for a straight line, so interpretations from the current analyses differ substantially from those reached in the original investigation. Contemporary theories for the evolution of optimal body size (e.g., the Metabolic Theory of Ecology) are based on patterns of metabolic allometry that have been estimated by the conventional analytical method. Thus, the current investigation raises questions about generally accepted patterns of metabolic allometry and theoretical models based upon them.

A model for conditional male trimorphisms

Conditional dimorphisms are widespread in color, morphology, behavior, and life history. Such traits have been successfully modeled in game theory as conditional strategies, and in quantitative genetics as threshold traits. Conditional trimorphisms have recently been unveiled, and here we combine the rock-paper-scissors (RPS) model of game theory and the environmental threshold (ET) model of quantitative genetics to model trimorphisms that are environmentally induced and result from the expression of two thresholds. We investigated the tactic fitness structure for maintenance of alternative reproductive tactics in scarab dung beetles that constitute the first known examples of conditional male trimorphism. We parameterized a novel ternary fitness landscape that explains how conditional male trimorphism in these beetles can be maintained. We tracked changes in tactic frequencies in a wild population of Phanaeus triangularis and detected fitness intransitivity consistent with RPS dynamics. Quantitative predictions of our model compare favorably with corresponding observed parameters. The ternary landscape further reveals how geographic populations of these beetles can evolve between conditional trimorphism and dimorphism. The ternary model also suggests that polyphenic systems could potentially evolve between conditional and purely genetic mediation.

A framework for identifying plant species to be used as 'ecological engineers' for fixing soil on unstable slopes

Major reforestation programs have been initiated on hillsides prone to erosion and landslides in China, but no framework exists to guide managers in the choice of plant species. We developed such a framework based on the suitability of given plant traits for fixing soil on steep slopes in western Yunnan, China. We examined the utility of 55 native and exotic species with regard to the services they provided. We then chose nine species differing in life form. Plant root system architecture, root mechanical and physiological traits were then measured at two adjacent field sites. One site was highly unstable, with severe soil slippage and erosion. The second site had been replanted 8 years previously and appeared to be physically stable. How root traits differed between sites, season, depth in soil and distance from the plant stem were determined. Root system morphology was analysed by considering architectural traits (root angle, depth, diameter and volume) both up- and downslope. Significant differences between all factors were found, depending on species. We estimated the most useful architectural and mechanical traits for physically fixing soil in place. We then combined these results with those concerning root physiological traits, which were used as a proxy for root metabolic activity. Scores were assigned to each species based on traits. No one species possessed a suite of highly desirable traits, therefore mixtures of species should be used on vulnerable slopes. We also propose a conceptual model describing how to position plants on an unstable site, based on root system traits.

Exaggerated trait allometry, compensation and trade-offs in the New Zealand giraffe weevil (Lasiorhynchus barbicornis)

Sexual selection has driven the evolution of exaggerated traits among diverse animal taxa. The production of exaggerated traits can come at a cost to other traits through trade-offs when resources allocated to trait development are limited. Alternatively some traits can be selected for in parallel to support or compensate for the cost of bearing the exaggerated trait. Male giraffe weevils (Lasiorhynchus barbicornis) display an extremely elongated rostrum used as a weapon during contests for mates. Here we characterise the scaling relationship between rostrum and body size and show that males have a steep positive allometry, but that the slope is non-linear due to a relative reduction in rostrum length for the largest males, suggesting a limitation in resource allocation or a diminishing requirement for large males to invest increasingly into larger rostra. We also measured testes, wings, antennae, fore- and hind-tibia size and found no evidence of a trade-off between these traits and rostrum length when comparing phenotypic correlations. However, the relative length of wings, antennae, fore- and hind-tibia all increased with relative rostrum length suggesting these traits may be under correlational selection. Increased investment in wing and leg length is therefore likely to compensate for the costs of flying with, and wielding the exaggerated rostrum of larger male giraffe weevils. These results provide a first step in identifying the potential for trait compensation and trades-offs, but are phenotypic correlations only and should be interpreted with care in the absence of breeding experiments.

Fitting statistical models in bivariate allometry: scaling metabolic rate to body mass in mustelid carnivores

The ongoing debate about methods for fitting the two-parameter allometric equation y=ax(b) to bivariate data seemed to be resolved recently when three groups of investigators independently reported that statistical models fitted by the traditional allometric method (i.e., by back-transforming a linear model fitted to log-log transformations) typically are superior to models fitted by standard nonlinear regression. However, the narrow focus for the statistical analyses in these investigations compromised the most important of the ensuing conclusions. All the investigations focused on two-parameter power functions and excluded from consideration other simple functions that might better describe pattern in the data; and all relied on Akaike's Information Criterion instead of graphical validation to identify the better statistical model. My re-analysis of data from one of the studies (BMR vs. body mass in mustelid carnivores) revealed (1) that the best descriptor for pattern in the dataset is a straight line and not a two-parameter power function; (2) that a model with additive, normal, heteroscedastic error is superior to one with multiplicative, lognormal, heteroscedastic error; and (3) that Akaike's Information Criterion is not a generally reliable metric for discriminating between models fitted to different distributions. These findings have apparent implications for interpreting the outcomes of all three of the aforementioned studies. Future investigations of allometric variation should adopt a more holistic approach to analysis and not be wedded to the traditional allometric method.