Chapter 2.6 Examining the utility of ratios to describe leaf shape. (original) (raw)

Ratios as a Size Adjustment in Morphometrics

Simple ratios in which a measurement variable is divided by a size variable are commonly used but known to be inadequate for eliminating size correlations from morphometric data. Deficiencies in the simple ratio can be alleviated by incorporating regression coefficients describing the bivariate relationship between the measurement and size variables. Recommendations have included: 1) subtracting the regression intercept to force the bivariate relationship through the origin (intercept-adjusted ratios); 2) exponentiating either the measurement or the size variable using an allometry coefficient to achieve linearity (allometrically adjusted ratios); or 3) both subtracting the intercept and exponentiating (fully adjusted ratios). These three strategies for deriving size-adjusted ratios imply different data models for describing the bivariate relationship between the measurement and size variables (i.e., the linear, simple allometric, and full allometric models, respectively). Algebraic rearrangement of the equation associated with each data model leads to a correctly formulated adjusted ratio whose expected value is constant (i.e., size correlation is eliminated). Alternatively, simple algebra can be used to derive an expected value function for assessing whether any proposed ratio formula is effective in eliminating size correlations. Some published ratio adjustments were incorrectly formulated as indicated by expected values that remain a function of size after ratio transformation. Regression coefficients incorporated into adjusted ratios must be estimated using least-squares regression of the measurement variable on the size variable. Use of parameters estimated by any other regression technique (e.g., major axis or reduced major axis) results in residual correlations between size and the adjusted measurement variable. Correctly formulated adjusted ratios, whose parameters are estimated by least-squares methods, do control for size correlations. The sizeadjusted results are similar to those based on analysis of least-squares residuals from the regression of the measurement on the size variable. However, adjusted ratios introduce size-related changes in distributional characteristics (variances) that differentially alter relationships among animals in different size Classes. 0 1993 Wiley-Liss, Inc.

On the effect of using ratios in the analysis of variance

Biobehavioral Reviews, 1977

On the effects of using ratios in the analysis of variance. BIOBEHAV. REV. 1(4) [225][226][227][228][229] 1977. --In experimental designs used by a number of disciplines, raw data are transformed into ratios and statistical analyses are performed on the ratio data. Although the designs themselves may be appropriate, the mathematical properties induced by ratio transformations can produce a loss of sensitivity in statistical tests. Since there are times when such designs must be used, it would be desirable to characterize the circumstances under which the statistical analysis of ratio data is or is not appropriate. To provide such information, a computer simulation approach was used to generate bivariate normal observations X and Y which were analysed using: (1) an analysis of variance ignoring the covariate; (2) an analysis of covarianee; (3) an analysis of variance on the ratio Y/X. Comparisons were made between the three models by accumulating the number of rejections (1-/3) on critical F-values for each model. Results taken from over a million analyses are discussed for a wide range of specific treatment effects and known correlations between the independent variable X and the dependent variable Y.

NON-PROPORTIONALITY IN RATIOS: AN ALTERNATIVE APPROACH

The British Accounting Review, 1997

This study identifies general postulates underlying the validity of the financial ratio measurement. Then, new relationships are suggested obeying the same postulates, which may replace the ratio form in the case of non-proportionality. Where proportionality holds, these relationships revert to the traditional ratio. The paper also reviews the reasons for expecting non-proportional components in ratios and presents application examples of the new relationships.

"Proportion"

LexArt. Words for Painting (France, Germany, England, The Netherlands, 1600-1750), Edited by Michèle-Caroline Heck, PULM, 2018

The term proportion was widely used in artistic treatises in Europe. All the authors were in agreement regarding the need for artists to fully master this science, the aim of which was to succeed in representing the human gure with credibility. This is why theorists and artists gave practical advice, and de ned rules based on the calculation of measurements so as to correctly reproduce the proportions of a person, an animal or an object. But this matter of proportion was not only a question of artistic practice and pedagogy, it was also part of theoretical re!ection composed of a quest for ideal beauty. It was addressed as much to painters as to art lovers.