Canonical analysis measures nonlinear selection on latent axes from a rotation of the gamma matrix (γ) of quadratic and correlation selection gradients. Here, we document that the conventional method of testing eigenvalues (double regression) under the null hypothesis of no nonlinear selection is incorrect. Through simulation we demonstrate that under the null the expectation of some eigenvalues from canonical analysis will be nonzero, which leads to unacceptably high type 1 error rates. Using a two-trait example, we prove that the expectations for both eigenvalues depend on the sampling variability of the estimates in γ. An appropriate test is to slightly modify the double regression method by calculating permutation P-values for the ordered eigenvalues, which maintains correct type 1 error rates. Using simulated data of nonlinear selection on male guppy ornamentation, we show that the statistical power to detect curvature with canonical analysis is higher compared to relying on the estimates from γ alone. We provide a simple R script for permutation testing of the eigenvalues to distinguish curvature in the selection surface induced by nonlinear selection from curvature induced by random processes. K E Y W O R D S :Fitness surface, nonlinear selection, phenotypic selection, selection surface, stabilizing selection.Empirical estimation of nonlinear selection (i.e., stabilizing, disruptive, or correlational) on phenotypes, or curvature in the selection surface, bears on such important topics as the topography of the adaptive landscape (Lande and Arnold 1983;Phillips and Arnold 1989;Arnold et al. 2008), and the genetic architecture of complex traits (Blows and Hoffmann 2005;Hunt et al. 2007). Kingsolver et al.'s (2001) review of estimates of nonlinear selection from the literature revealed that disruptive selection and stabilizing selection were generally weak (16% were declared statistically significant), of similar magnitudes, and that correlational selection was rarely estimated at all. It is possible that lack of natural phenotypic variation in traits has limited the power to detect nonlinear selection, such that clear demonstration of stabilizing selection may require experimentally manipulated phenotypes (Cresswell 2000;Conner et al. 2003). Although a simple analytical error in not doubling the quadratic selection gradients of the selection model may be partially to blame (Stinchcombe et al. 2008), it is curious that a more pervasive signal of stabilizing selection has not been found in natural populations using phenotypic selection approaches.Although nonlinear selection estimates appear to be weak, the observed magnitude of stabilizing selection gradients on traits measured in natural populations are too strong to explain the observed level of genetic variation for those traits (Johnson and Barton 2005). Ample additive genetic variation is found for almost all traits measured in nature (Lynch and Walsh 1998), (Lande and Arnold 1983;Phillips and Arnold 1989;Hunt et al. 2007;Reynolds et al. 2009). Trait c...

ObjectiveCurrent clinical guidelines and public health statements generically prescribe body mass index (BMI; kgm2) categories regardless of the individual’s situation (age, risk for diseases, etc.). However, regarding BMI and mortality rate (MR), two well-established observations are (1) there is a U-shaped (i.e., concave) association - people with intermediate BMIs tend to outlive people with higher or lower BMIs; and (2) the nadirs of these curves tend to increase monotonically with age. Multiple hypotheses have been advanced to explain either of these two observations. Here we introduce a new hypothesis that may explain both phenomena, by drawing on the so-called obesity paradox: the unexpected finding that obesity is often associated with increased survival time among people who have some serious injury or illness despite being associated with reduced survival time among the general population.ResultsWe establish that the obesity paradox offers one potential explanation for two curious but consistently observed phenomena in the obesity field.ConclusionFurther research is needed to determine the extent to which the obesity paradox is actually an explanation for these phenomena, but if our hypothesis proves true the common practice of prescribing overweight patients to lower their BMI should currently be applied with caution. In addition, the statistical modeling technique employed here could be applied in such other areas involving survival analysis of disjoint subgroups, in order to explain possible interacting causal associations and to determine clinical practice.

Two observations favor the presence of a lower mass-specific resting energy expenditure (REE/weight) in taller adult humans: an earlier report of height (H)-related differences in relative body composition; and a combined model based on Quetelet and Kleiber's classic equations suggesting that REE/weight proportional, variantH(-0.5). This study tested the hypothesis stating that mass-specific REE scales negatively to height with a secondary aim exploration of related associations between height, weight (W), surface area (SA), and REE. Two independent data sets (n = 344 and 884) were evaluated, both with REE measured by indirect calorimetry and the smaller of the two including fat estimates by dual-energy X-ray absorptiometry. Results support Quetelet's equation (W proportional, variantH(2)), but Kleiber's equation approached the interspecific mammal form (REE proportional, variantW(0.75)) only after adding adiposity measures to weight and age as REE predictors. REE/weight scaled as H( approximately (-0.5)) in support of the hypothesis with P values ranging from 0.17 to <0.001. REE and SA both scaled as H( approximately 1.5), and REE/SA was nonsignificantly correlated with height in all groups. These observations suggest that adiposity needs to be considered when evaluating the intraspecific scaling of REE to weight; that relative to their weight, taller subjects require a lower energy intake for replacing resting heat losses than shorter subjects; that fasting endurance, approximated as fat mass/REE, increases as H(0.5); and that thermal balance is maintained independent of stature by evident stable associations between resting heat production and capacity of external heat release. These observations have implications for the modeling of adult human energy requirements and associate with anthropological concepts founded on body size.

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