The (non)significance of a slope of 1.00
A common point of departure in discussions of static allometry is that a slope of 1.00 is assumed to characterize traits under natural selection. That is, the proportion of the body dedicated to wings, legs, eyes, etc. is expected to be the same in large individuals as in small individuals. The emphasis on 1.00 is such that allometry is sometimes reported simply in terms of positive (> 1.00) and negative (< 1.00), with a slope significantly > 1.00 being used as a litmus test for sexual selection (Green, 2000; Kelly, 2004; Tasikas et al., 2007). There are several reasons to think that this emphasis on the precise value 1.00 is probably misplaced.
The forms of cost and benefit curves for different designs of a structure can vary for different habitats and body sizes (Bonduriansky & Day, 2003; Dial et al., 2008). For example, two species of the water strider genus Aquarius have larger body sizes and proportionally longer middle and hind legs (involved in locomotion) than the corresponding legs of seven species of Gerris; the Aquarius species inhabit areas with more disturbed water surfaces, where proportionally longer legs are thought to be advantageous (Klingenberg & Zimmermann, 1992). Thus the value 1.00 is not necessarily always expected, even under natural selection.
Environmental variation can also influence sexual selection, and benefit curves for genitalia can also vary with habitat differences, as in greater relative gonopod length in the poeciliid fish at sites where dangerous predators are present (Kelly et al., 2000; Jennions & Kelly, 2002), or greater relative penis length in barnacles where wave action is less intense (Neufeld & Palmer, 2008). Population density could also affect benefits by altering the chances of males meeting and competing directly, or females meeting or mating with multiple males and exercising female choice (Oosthuizen & Miller, 2000); this could affect selection on genitalia (Wang et al., 2008). Smaller males may have proportionally less material reserves with which to construct sexually dimorphic structures, so the construction cost curves could vary across the range of body sizes (Petrie, 1988, 1992; Green, 1992), or across different environments with different resource availability.
Benefit curves could also be affected by other aspects of the same individual’s phenotype, such as behaviour. For instance, the advantage of relatively longer gonopodia in poeciliid fish would increase in smaller individuals if these individuals also had a greater propensity to execute successful sneak attack copulations (as opposed to mating attempts preceded by courtship), or if females were less likely to accede to mating attempts following courtship from smaller males; these correlations would favour lower allometric slopes for gonopod length. Behaviour can also affect biomechanical details of physical engagements, and thus the costs of structures that are able to function in these circumstances. In an example from animal weapons, the wrestling fights of antelopes produce less mechanical stress on their horns than the more forceful uses of horns in goats and sheep (Kitchener, 1985), and thus entail lower costs for effective weapons in antelopes.
Still another problem concerns interpretations based on whether or not male structures touch the female during copulation. A simple interpretation of the one-size-fits-all hypothesis would suppose that only those portions of the male genitalia that actually touch the female would be likely to be under post-copulatory sexual selection, and, conversely, that portions that do not contact the female would not be under post-copulatory sexual selection (Bertin & Fairbairn, 2007). This interpretation fails to recognize, however, that different portions of the male’s body are physically and functionally interconnected. For instance, the muscles that move more distal portions of a male’s genitalia often reside in more basal segments (Snodgrass, 1935; Chapman, 1998). The size of a portion of the genitalia that remains outside the female during copulation may influence both the kinds of movements made by structures that do contact her body, and the power that they exert. To a first approximation, shorter intromittent portions are likely to require smaller amounts of muscle, leading to reduced sizes of more basal, nonintromittent portions. Detailed understanding of internal functional morphology can be necessary to make confident interpretations of some allometric patterns.
There are several additional, perhaps more important, practical reasons to de-emphasize the value 1.00. The two regression techniques most frequently used to measure allometry are ordinary least squares (OLS) and reduced major axis (RMA), which give different slopes with the same data. OLS regressions generally give lower slopes than RMA regressions; this was true, for instance, in 41 of 42 pairs of regressions in Simmons & Tomkins (1996). There is controversy regarding which technique is more appropriate, and there may not be any single answer for all cases (Eberhard et al., 1999; Green, 1999; Palestrini et al., 2000; Cuervo & Møller, 2001; Bernstein & Bernstein, 2002; Kato & Miyashita, 2003; Ohno et al., 2003; Hosken et al., 2005; Warton et al., 2006; Warne & Charnov, 2008), and there are strong arguments against each. It is thus not clear whether the 1.00 given by OLS or by RMA should be the point of reference.
Another problem is that the choice of the variable used to estimate body size (total body length, prothorax length or width, elytrum length and femur length have been used in different insect studies) will influence the values obtained (Kratochvil et al., 2003). Perhaps the best body size indicator is some composite measure like a principal components variable that combines many different size measures (Uhl & Vollrath, 2000; Ohno et al., 2003; Pizzo et al., 2006), but the relative contributions of different body parts and shape will vary in different species, rendering inter-specific comparisons of absolute values difficult to interpret. Use of weight measurements is probably not a good idea, at least in some groups with large seasonal and life stage variations in weight (Miller & Burton, 2001).
Different body size indicators can give different allometric slopes. For instance, in the mosquito Aedes aegyptii the slopes for two different genital measures, when regressed on wing length instead of leg length, were similar but not identical: 0.31 vs. 0.34, and 0.38 vs. 0.32 (Wheeler et al., 1993). Differences can occur even with a body size indicator that combines several different dimensions. Regression slopes using the centroid for elytral measures vs. the centroid for prothoracic traits as an indicator of body size in two congeneric species of beetles were 1.03 vs. 0.93 for the head and 0.44 vs. 0.35 for the genitalia in one species, and 0.33 vs. 0.27 for the head and 0.28 vs. 0.23 for the genitalia in the other (Pizzo et al., 2006).
Measurement errors can also result in appreciable differences in allometric slopes. Means of allometric slopes calculated on the basis of repeated measurements of 14 different body parts in two species differed by a median of only 5.3%, but in three of the 14 they differed substantially (> 25%) (Eberhard et al., 1998).
Sharp geographical variations in allometric slopes for given structures also occur in some species, providing further reason to doubt the usefulness of absolute values of allometric slopes. For instance, slopes were measured in six different populations of the water strider A. remigis in California (Bertin & Fairbairn, 2007). When comparing slopes between different geographical populations of A. remigis, the largest slope was 214%, 178% and 198% of the smallest slope for three external genitalic traits; 167%, 180% and 193% for internal genitalic traits; and 163% and 129% for somatic traits. Two beetle species and a fish also showed substantial differences between different populations (Kelly et al., 2000; Bernstein & Bernstein, 2002; Kawano, 2002). The possible significance of these tantalizing differences remains to be determined.
Finally, there is the empirical fact that some nongenital structures that are not sexually dimorphic and with no apparent relation to sex, and which would thus theoretically be expected to show allometric slopes close to 1.00, nevertheless have values quite different from 1.00 (e.g. Schulte-Hostedde & Alarie, 2006). Even the same nonsexually selected trait sometimes has quite different allometric slopes in different species. For instance, elytron length scales more steeply on pronotum width in the firefly Photinus pyralis (1.62) than in P. macdermotti (0.80), presumably because individuals of P. pyralis fly longer distances (Vencl, 2004). A particularly dramatic example of intraspecific variation involves the internal epipharyngeal structures of the dung beetle O. taurus. Seven external body traits of this species had moderate slopes in both males (0.59–0.91) and females (0.79–1.29), but four epipharyngeal traits had extremely low slopes in both males (0.05–0.30) and females (0.04–0.39) (Palestrini et al., 2000). In addition, some nongenital structures that are probably not under sexual selection show positive allometry, such as the mandibular palps of some Scathophaga flies (Hosken et al., 2005) and the middle legs of male and female A. remigis water striders (Bertin & Fairbairn, 2007).
Several of these problems can be avoided or ameliorated by comparing the allometric slopes of different structures of the same individuals that are and are not thought to be under sexual selection with each other, instead of executing statistical tests of differences with 1.00, as has been common (Schmitz et al., 2000; Miller & Burton, 2001; Jennions & Kelly, 2002; Ohno et al., 2003; Mutanen & Kaitala, 2006; Mutanen et al., 2006; Bertin & Fairbairn, 2007; Kinahan et al., 2007). Using the same regression technique and the same body size indicator to obtain the slopes of all structures that are compared can probably largely correct for the possible peculiarities of the variable chosen as an indicator of body size, and of the regression technique (Eberhard et al., 1998, 1999; Cuervo & Møller, 2001; Bernstein & Bernstein, 2002). Comparisons of this sort should work better when many rather than only a few structures that are not thought to be under direct sexual selection are used (most studies to date are inadequate in this respect), and using median rather than mean values of their slopes in comparisons with genital slopes. This would reduce the chances of being misled by atypical values of any particular trait. When multiple slopes have been determined for both genital and nongenital traits, their means can be compared (Uhl & Vollrath, 2000). Inclusion of alternative slopes calculated using alternative variables such as body size indicators can also help avoid possibly atypical values. Similarly, uncertainty regarding the best regression technique can be avoided by reporting values for both. In practice, conclusions based on intraspecific comparisons between genital and nongenital structures have proved to be little affected by either the regression technique or the use of different body size indicators (Eberhard et al., 1999; Funke & Huber, 2005). Discussions in the literature have concentrated on absolute values of slopes, and are probably not very relevant to comparisons between slopes.