Gary C. Packard

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.

THE PHYSIOLOGICAL ECOLOGY OF REPTILIAN EGGS AND EMBRYOS. AND THE EVOLUTION OF VIVIPARITY WITHIN THE CLASS REPTILIA

1. Eggs of Crocodilia and Chelonia, like those of birds, have a pair of egg membranes separating a thick layer of albumen from the calcareous shell. In contrast, eggs of oviparous Lepidosauria have only a single shell membrane, upon which relatively small amounts of calcium carbonate are deposited; and the volume of albumen in eggs is extraordinarily small at the time of oviposition.

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 the Hydric and Thermal Environments on Eggs and Hatchlings of Bull Snakes Pituophis melanoleucus

Eggs of bull snakes (Pituophis melanoleucus) were incubated at temperatures of 22.0, 27.0, and 32.0 C on substrates with water potentials of -150, -300, and -1,100 kPa using a 3 X 3 factorial design. The thermal and hydric environments affect fluxes in water between eggs and their surroundings during the course of incubation. Eggs incubated at low temperature (22.0 C) and on wet substrate (-150 kPa) gain the greatest amount of mass during incubation, while eggs incubated at high temperature (32.0 C) on dry substrate (-1100 kPa) lose the greatest amount of mass. The proportion of eggs that hatch is lower at the low temperature, while incubation at the high temperature results in a greater proportion of hatchlings with abnormalities. In general, eggs incubated at the intermediate temperature produce larger hatchlings than do eggs held at either low or high temperatures. While the hydric environment does not affect hatching success, eggs exposed to wet or moist hydric conditions give rise to larger hatchlings than do eggs exposed to dry conditions. Both temperature and water availability affect the composition of hatchlings.

Water Relations of Chelonian Eggs

Eggs of painted turtles (Chrysemys picta) absorb water across that portion of the eggshell in contact with the substrate and simultaneously lose water by transpiration from that part of the eggshell exposed to air inside the nest chamber. Depending upon the rates of influx and efflux, eggs may experience increases or decreases in mass during incubation, or mass may remain essentially constant between oviposition and hatching. Water exchanges are especially sensitive to such factors as substrate water potential, relative surface exposed to the nest atmosphere, hydraulic conductance of the eggshell, and conductance of the eggshell to water vapor. Hatchlings emerging from eggs absorbing and storing large quantities of water are heavier than hatchlings emerging from eggs taking up smaller quantities of water from the substrate. Furthermore, water absorption equal to, or in excess of, water loss by transpiration assures that the original shape of the egg will be preserved, thereby guaranteeing that sufficient space is available within the egg for normal development of the embryo.

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.

Comparative aspects of calcium metabolism in embryonic reptiles and birds

Eggs of oviparous, amniotic vertebrates must be endowed at oviposition with all of the organic and most of the inorganic components required for embryonic growth. A major inorganic constituent of these eggs is calcium used for ossification of the skeleton. The two main sources of this element are the yolk and the eggshell, but the proportion of calcium supplied by these compartments varies among species. Some of the calcium absorbed from the eggshell by embryos of domestic fowl is stored in the yolk. causing the calcium content of this compartment to increase appreciably during incubation. In contrast, the calcium content of yolk declines throughout incubation in eggs of reptiles, and the yolk that is withdrawn into the abdominal cavity just prior to hatching contains only small quantities of this element. Thus, major differences in the pattern of calcium metabolism characterize avian and reptilian embryos, and studies of embryos of domestic fowl may not provide a broadly-based model with which to characterize calcium metabolism in embryos of other species. Control of calcium transport across the cellular epithelia (the yolk sac and chorioallantois) that separate embryos from their sources of calcium (yolk and eggshell) represents one aspect of control of calcium metabolism during embryogenesis, but this process has been examined only in eggs of domestic fowl and only in the chorioallantois. Calcium transport across the chorioallantois of embryonic chicks is influenced by a vitamin K-dependent calcium-binding protein, carbonic anhydrase, the level of calcium to which the chorioallantois is exposed, and vitamin D. However, a complete story concerning control of calcium transport across the chorioallantois and its relationship to calcium regulation in embryos of domestic fowl is not yet possible.