Thomas J. Boardman

The Misuse of Ratios, Indices, and Percentages in Ecophysiological Research

Ratios are used by many ecological physiologists to adjust (or scale) data that vary allometrically with body size. We use two sets of real data from our laboratory to illustrate in detail how investigators may be misled by statistical analyses performed on such ratios. The first example concerns the use of ratios to increase the precision of data gathered in planned experiments where body size varies within experimental groups but not among them. The second example concerns the use of ratios to remove confounding effects of body size from studies where animals in one group are larger than those in other groups, as a result either of the experimental manipulation itself or of the procedure for assigning animals to treatment groups. In both of these examples, statistical analyses of ratios lead to conclusions that are inconsistent with impressions gained from visual examinations of data displayed in bivariate plots. In comparison, analyses of covariance lead to conclusions that agree with impressions gained from these same plots. We therefore recommend that ecological physiologists discontinue using ratios to scale data and that they use the ANCOVA instead.

THE USE OF PERCENTAGES AND SIZE-SPECIFIC INDICES TO NORMALIZE PHYSIOLOGICAL DATA FOR VARIATION IN BODY SIZE : WASTED TIME, WASTED EFFORT?

Abstract Researchers commonly compute percentages or size-specific indices in an attempt to remove effects of body size from physiological data. Unfortunately, such ratios seldom eliminate the influence of body size on a physiological response and the ratios introduce major (but often unrecognized) problems with respect to statistical analysis and interpretation of the data. Indeed, these shortcomings of ratios frequently lead investigators to arrive at incorrect conclusions in otherwise flawless experiments. A superior alternative to using ratios combines graphical analysis and the analysis of covariance, which is a widely available statistical routine that uses least-squares regression to remove effects of body size from physiological data. Accordingly, we counsel researchers to discontinue forming ratios in an attempt to normalize physiological data for variation in body size and to adopt a reliable alternative. We also advise readers of scientific research not to place great confidence in results of studies that use ratios for scaling.

Influence of Moisture, Temperature, and Substrate on Snapping Turtle Eggs and Embryos

Flexible-shelled eggs of common snapping turtles (Chelydra serpentina) were incubated on two substrates (sand and vermiculite) at each of three temperatures (26.0°, 28.5°, 31.0°C) and three moisture regimes (water potentials initially -150 kPa, -550 kPa, -950 kPa) in a factorial experiment assessing the influence of these variables on the water relations of eggs and the development of embryos. Hatching success was high on wet substrates at 26.0° and 28.5°, but declined at the highest temperature and on drier media. Net absorption of water by viable eggs, duration of incubation by embryos, and size of hatchlings were positively correlated with wetness of substrates and negatively correlated with temperature. Turtles hatching from eggs at 26.0° were males regardless of the wetness of the medium, whereas those emerging from eggs at 28.5° and 31.0° were females. These patterns of response characterized eggs incubated on sand as well as those on vermiculite. Findings from this study indicate that temporal and spatial variations in moisture and temperature within and among natural nests probably elicit ecologically important variation in size and sex of hatchling snapping turtles.

Influence of Water Exchanges by Flexible-Shelled Eggs of Painted Turtles Chrysemys picta on Metabolism and Growth of Embryos

Flexible-shelled eggs of painted turtles (Chrysemys picta) were incubated under controlled conditions eliciting different patterns of net water exchange between eggs and the environment. The temporal patterns of decline in dry mass of yolks and of increase in dry mass of embryos did not vary among eggs incubated in different hydric environments, indicating that rates of metabolism and growth of embryos were largely unaffected by variation in the amount of water available inside eggs to support embryogenesis. Nevertheless, embryos in wet environments assimilated more water during incubation than did embryos in dry conditions, and this extra water apparently enabled them to develop longer before hatching than was possible for embryos in the drier settings. Because of their longer incubation, hatchlings emerging from eggs in wet environments were larger (both in mass and in carapace length) and contained less residual yolk than turtles coming from eggs incubated in drier surroundings. Embryos accumulated three times more excretory nitrogen in the form of urea than in the form of ammonia, but the patterns of nitrogen accumulation did not vary among embryos exposed to different hydric conditions. Water potential of the yolk increased during the first 10 days of incubation, as water flowed from the albumen into the vitelline sac, and decreased linearly thereafter, as water was transferred from the yolk to the developing embryo. The predicted water potential of the yolk at the time of hatching was the same for eggs incubated in wet and dry environments, thereby raising the possibility that water potential of some compartment inside eggs provides the cue for hatching. Water exchanges by eggs of painted turtles incubating in natural nests probably affect survival of embryos to hatching as well as body size and level of tissue hydration in young turtles.

Patterns and Possible Significance of Water Exchange by Flexible-Shelled Eggs of Painted Turtles (Chrysemys picta)

Samples of eggs laid by nine painted turtles (Chrysemys picta) were half buried in substrates differing in water potential and incubated at 29 C. Other eggs from these clutches were incubated on wire platforms above the same substrates. Eggs on the wire platforms declined in mass between the outset of study and hatching owing to a net outward diffusion of water vapor, with eggs held above relatively dry substrates losing more vapor than eggs held above relatively wet substrates. Eggs half buried in the substrates increased in mass during the first half of incubation by amounts that were directly related to substrate water potential, but decreased in mass during the second half of incubation as the net flux of water shifted from inward to dutward. The conductance of painted turtle eggs to water vapor was 70 times higher than expected for avian eggs of comparable mass, yet transpirational loss of water from incubating turtle eggs nonetheless was small owing to the very small gradient in vapor pressure between eggs and air trapped inside the containers. Eggs exposed to wet substrates had longer incubation periods and higher hatching success than eggs exposed to drier substrates. Moreover, turtles hatching from eggs exposed to relatively wet conditions were larger than hatchlings emerging from eggs incubated in slightly drier settings.

Model Selection and Logarithmic Transformation in Allometric Analysis

The standard approach to most allometric research is to gather data on a biological function and a measure of body size, convert the data to logarithms, display the new values in a bivariate plot, and then fit a straight line to the transformations by the method of least squares. The slope of the fitted line provides an estimate for the allometric (or scaling) exponent, which often is interpreted in the context of underlying principles of structural and functional design. However, interpretations of this sort are based on the implicit assumption that the original data conform with a power function having an intercept of 0 on a plot with arithmetic coordinates. Whenever this assumption is not satisfied, the resulting estimate for the allometric exponent may be seriously biased and misleading. The problem of identifying an appropriate function is compounded by the logarithmic transformations, which alter the relationship between the original variables and frequently conceal the presence of outliers having an undue influence on properties of the fitted equation, including the estimate for the allometric exponent. Much of the current controversy in allometric research probably can be traced to substantive biases introduced by investigators who followed standard practice. We illustrate such biases with examples taken from the literature and outline a general methodology by which the biases can be minimized in future research.

Effects of temperature and moisture during incubation on carcass composition of hatchling snapping turtles (Chelydra serpentina)

SummaryFlexible-shelled eggs of common snapping turtles (Chelydra serpentina) were incubated on each of two substrates (vermiculite, sand) at each of three temperatures (26.0°C, 28.5°C, 31.0°C) and three moisture regimes (wet, intermediate, dry). Embryos developing in cool, wet environments mobilized the largest amounts of protein from their yolk and attained the largest size before hatching, whereas turtles developing in warm, dry environments mobilized the smallest quantities of protein and were the smallest in body size at hatching. Embryos on wet substrates mobilized more lipid from their yolk than did embryos on dry media, but ambient temperature had no demonstrable influence on patterns of lipid mobilization. The total reserve of neutral lipid available in residual yolk plus carcass to sustain neonates in the interval prior to the beginning of feeding was largest in hatchlings from dry environments and smallest in animals from wet environments, but was unaffected by temperature during incubation. Hydration of tissues in hatchlings was higher when incubation was in cool, moist conditions than when incubation was in warm, dry settings, thereby indicating that some of the effects of moisture and temperature on mobilization of nutrients by embryos may be mediated by differences in intracellular water. Patterns of response to temperature and moisture recorded for turtles emerging from eggs on sand were similar to those recorded for hatchlings on vermiculite, so no important conclusion would have been affected by incubating eggs on one medium instead of the other.

Fitting statistical models in bivariate allometry.

Several attempts have been made in recent years to formulate a general explanation for what appear to be recurring patterns of allometric variation in morphology, physiology, and ecology of both plants and animals (e.g. the Metabolic Theory of Ecology, the Allometric Cascade, the Metabolic‐Level Boundaries hypothesis). However, published estimates for parameters in allometric equations often are inaccurate, owing to undetected bias introduced by the traditional method for fitting lines to empirical data. The traditional method entails fitting a straight line to logarithmic transformations of the original data and then back‐transforming the resulting equation to the arithmetic scale. Because of fundamental changes in distributions attending transformation of predictor and response variables, the traditional practice may cause influential outliers to go undetected, and it may result in an underparameterized model being fitted to the data. Also, substantial bias may be introduced by the insidious rotational distortion that accompanies regression analyses performed on logarithms. Consequently, the aforementioned patterns of allometric variation may be illusions, and the theoretical explanations may be wide of the mark. Problems attending the traditional procedure can be largely avoided in future research simply by performing preliminary analyses on arithmetic values and by validating fitted equations in the arithmetic domain. The goal of most allometric research is to characterize relationships between biological variables and body size, and this is done most effectively with data expressed in the units of measurement. Back‐transforming from a straight line fitted to logarithms is not a generally reliable way to estimate an allometric equation in the original scale.

A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.

Estimation In Compound Exponential Failure Models

When items on life-test are subject to two independent single parameter exponential causes of failure, the resulting failure times can be used to estimate these parameters. The maximum likelihood estimates of the parameters are derived for the case when exact failure times are recorded and when the data is censored at a fixed time T. Properties of the estimators are considered conditioned upon the event of obtaining a failure by either cause. The expressions for the means, variances, and covariance of the estimator are found to be functions of expected values of ratios and reciprocals of multinomial random variables. Two examples are included to show possible applications of estimation from this type of compound model.