Relative growth of the semi-terrestrial crab Pachygrapsus marmoratus: an information-theory approach

Growth in animals is often accompanied by changes in proportion as well as in size, i.e. some body parts grow at a different rate than others. This phenomenon is called relative or allometric growth. One of the consequences of crustaceans having an exoskeleton is that growth proceeds in steps by a series of moults (or ecdyses), which separate the stages (or instars). Usually the change in shape in SCIENTIA MARINA 71(2) June 2007, 383-394, Barcelona (Spain) ISSN: 0214-8358


INTRODUCTION
Growth in animals is often accompanied by changes in proportion as well as in size, i.e. some body parts grow at a different rate than others.This phenomenon is called relative or allometric growth.One of the consequences of crustaceans having an exoskeleton is that growth proceeds in steps by a series of moults (or ecdyses), which separate the stages (or instars).Usually the change in shape in crustaceans is progressive, but sometimes particularly marked changes occur at a single moult, which in extreme cases are regarded as a metamorphosis (Hartnoll, 1985).Using the allometric equation (Huxley, 1932) is the most extensively used method for analysing relative growth during ontogeny.The relationship between the size of a body part Y relative to another body part X, usually carapace width or length, has the form Y = aX b , where the exponent b is a measure of the difference in the growth rates of the two body parts.To estimate the coefficients of the allometric equation, the data are usually logtransformed and a linear regression is fitted to the equation logY = loga + blogX (L model), where the allometric exponent b is the slope of the resulting linear equation.Logarithmic transformation is generally appropriate because morphological data tend to have log-normal structure, as they are non-negative, with positively-skewed distributions and variances that increase with the mean (Jolicoeur, 1990;Ebert and Russell, 1994).
However, the classic allometric equation frequently fails to adequately fit the data and more complex models of the form logY = f(logX) should be used.The reason might be due either to the existence of non-linearity (i.e.f is non-linear) or the existence of breakpoints (i.e.f and/or its first derivative f´ are not continuous functions).The existence of breakpoints in allometric data has been recognized since the allometric equation was first proposed (Huxley, 1932).These breakpoints are usually identified visually, when data are obviously separated, and then linear regression analysis is done on both sides of the selected breakpoint.Estimating breakpoints visually is not, however, an accurate method and using segmented regression models has been proposed to be a better alternative (Somerton, 1983;Shea and Vecchione, 2002).
The patterns of relative growth vary greatly among different crab families (Hartnoll, 1983).For example, in logarithmic plots of the chelae of male crabs against carapace length or width, three distinct segments were present in majid crabs (Hartnoll, 1963), a change in slope but no discontinuity was found for Portunus pelagicus (Hall et al., 2006), while gradual changes in b that appeared to follow a smooth, curvilinear trend were demonstrated for Chaceon bicolor (Hall et al., 2006).
One approach to study relative growth, followed quite often by many researchers, is to fit more than one model to the data and then use a criterion, like minimizing the residual sum of squares or maximizing the adjusted R 2 , to select the 'best' model.When the models are nested, a statistical hypothesis testing approach is often used with an F test or with a likelihood ratio test when the probability distribution of the error structure is specified.However, there are several practical pitfalls and theoretical problems with such an approach (Burnham and Anderson, 2002).During the past twenty years, modern statistical science has been moving away from traditional formal methodologies based on statistical hypothesis testing.In particular, hypothesis testing for model selection using traditional approaches (such as forward, backward, and stepwise selection) is often poor and of limited value (Akaike, 1981) and it has been suggested that it will be used less and less in the years ahead (Burnham and Anderson, 2002).Hypotheses testing schemes are based on arbitrary a levels (commonly 0.05 or 0.01), multiple testing is challenging, and tests between models that are not nested are problematic.The adjusted coefficient of multiple determination (R 2 ) that is often used in model selection was found to be a very poor approach (McQuarrie and Tsai, 1998).
Model selection based on information theory is a relatively new paradigm in the biological sciences and is quite different from the usual methods based on null hypothesis testing.Information theory has been increasingly proposed to be a better and advantageous alternative for model selection (Burnham and Anderson, 2002), e.g. in studies of fish growth (Katsanevakis, 2006) or aquatic respiration (Katsanevakis et al., 2007b).Katsanevakis et al. (2007a) recommended the information theory approach as a more accurate, robust and enlightening way to study allometric growth of marine organisms.They demonstrated that using the classical allometric model when it is not supported by the data, might lead to characteristic pitfalls, data misinterpretation, and loss of valuable biological information.
In the present study, the relative growth of the grapsid crab Pachygrapsus marmoratus (Fabricius, 1787) was studied, following the information-theory approach proposed by Katsanevakis et al. (2007a), accounting possible breakpoints and non-linearity in the equations relating log-transformed morphometric data.The pattern of relative growth was investigated for various somatic parts of this grapsid crab and the potential pitfalls of using the classic allometric equation were emphasized.

Test species -Morphometric measurements
The family Grapsidae is one of the richest among Decapoda, in terms of intertidal species.Pachygrapsus marmoratus is the most common grapsid crab in the intertidal belt of rocky shores throughout the Mediterranean Sea, Black Sea and northeastern Atlantic from Brittany to Morocco including the Canary Islands, the Azores and Madeira (Ingle, 1980;Cannicci et al., 1999;Flores andPaula, 2001, 2002).P. marmoratus has a semiterrestrial life-style and is an omnivorous species that actively searches for food, relying on the intertidal community throughout its post-larval life (Cannicci et al., 2002).
A total of 85 P. marmoratus individuals was collected by hand from rocky shores of the Saronikos Gulf (37°30'N-37°55'N; 23°E-24°E) from January to November 2005.The following somatic parts of each crab were measured: maximum carapace length (CL), maximum carapace width (CW; excluding the lateral spines), maximum length (LPL) and width (LPW) of the propodus of the left chela, maximum length (RPL) and width (RPW) of the propodus of the right chela, and maximum width of the six abdominal somites and the telson (S 1 to S 7 ; S 7 corresponding to the telson).A vernier caliper with 0.1 mm accuracy was used for lengths >14 mm and an Image Analysis system, comprising of a stereoscope (Wild M8), a Sony camera (Hyper HAD) and the software ImagePro plus v3.0.1, for lengths <14 mm.

Data analysis
The allometric growth of CL, LPL, LPW, RPL, RPW, S i (i = 1 to 7) in relation to CW was investigated separately for males and females.In larger crabs, sex was identified from the abdominal shape, while in smaller crabs, pleopods were examined under a dissecting microscope.
Five candidate models for the relationship logY = f(logX) were fitted to the log-transformed data (logarithms with base 10), as proposed by Katsanevakis et al. (2007a), with non-linear least squares with iterations: linear (L), quadratic (Q), cubic (C), broken-stick (BS), and two-segment (TS) (Table 1).In the current context, the allometric exponent b is generalized to mean the first derivative of f, i.e. b = f´(X).Hereafter, X = CW, while Y is any of the other measured body parts.
The L model is the classical allometric equation, assuming that allometry does not change as body size increases (b = a 2 ).The Q and C models assume that a non-linearity exists in the relationship of log Y and log X and that b changes continuously with increasing body size (b = a 2 + 2a 3 log CW and b = a 2 + 2a 3 log CW + 3a 4 (log CW) 2 respectively).The BS and TS models assume a marked morphological change at a specific size of CW = a 4 ; the BS represents two straight line segments with different slopes that intersect at CW = a 4 , while the TS represents two straight line segments that do not intersect; thus, there is a point of discontinuity at CW = a 4 , and their slope (i.e.b) may or may not be equal.

Model selection -Multi-model inference (MMI)
According to the information theory approach, data analysis is assumed to be the integrated process of a priori specification of a set of candidate models (based on the science of the problem), model selection based on the principle of parsimony according to Akaike Information Criterion AIC (Akaike, 1973), and the estimation of parameters and their precision.The principle of parcimony implies selecting a model with the smallest possible number of parameters to adequately represent the data, a bias versus variance trade off.Furthermore, rather than estimating parameters from only the 'best' model, parameters can be estimated from several or even all the models considered.This procedure is termed multi-model inference (MMI) and has several theoretical and practical advantages (Burnham and Anderson, 2002).Further details on using information theory to study allometric growth may be found in Katsanevakis et al. (2007a), while a thorough presentation of information theory, the philosophical principles behind it and many related issues are given in Burnham and Anderson (2002).
In the current context, the small-sample, biascorrected form AIC c (Hurvich and Tsai, 1989) of the AIC (Akaike, 1973;Burnham and Anderson, 2002) was used for model selection.The model with the smallest AIC c value (AIC c,min ) was selected as the 'best' one out of the models tested.The AIC c differences Δ i = AIC c,i -AIC c,min were computed over all candidate models.According to Burnham and Anderson (2002), models with Δ i >10 have essentially no support and could be omitted from further consideration, models with Δ i <2 have substantial support, while there is considerably less support for models with 4 <Δ i <7.The 'Akaike weight' w i of each model was calculated to quantify the plausibility of each model given the data and the set of five models.This is considered to be the weight of evidence in favour of model i which is actually the best model of the available set of models (e.g.Akaike, 1983;Buckland et al., 1997;Burnham and Anderson, 2002).'Average' models were estimated by averaging the predicted response variable across models, using the corresponding w i 's as weights (Burnham and Anderson, 2002).Absolute residuals were plotted against log CW for the linear and average model, as a diagnostic tool to check model assumptions and especially to check for curvature in the pattern of residuals; the display was enhanced by a smoothing spline curve (with 4 degrees of freedom) fitted to the residuals.
Confidence intervals (95%) of the model parameters (CI boot ), were estimated with non-parametric bootstrap (Efron and Tibshirani, 1993), with bootstrap sample size B = 2000, by resampling biometric data for the individual crabs in the original data set and fitting the respective model by non-linear least squares to each sample.

RESULTS
The CW of male and female P. marmoratus respectively ranged between 3.5 to 39.4 mm and 4.3 to 34.9 mm.No significant differences were found in comparison tests between left and right chelar propodus lengths and widths in males (paired t-tests; p = 0.15 for lengths and p = 0.18 for widths) or females (p = 0.18 for lengths and p = 0.23 for widths).Therefore, there was no asymmetry between left and right chelar propodus in either males or females.Consequently, data for left and right chelar propods were combined.For chelar propodus length (PL) and width (PW), the models were fitted to the pooled data.The regression parameters describing the allometry of the various somatic parts in relation to CW are given in Table 2.The AIC c differences and the 'Akaike weights of evidence' for the five models and for each biometric variable are given in Table 3.
In females, L was the best model for the growth of PL, while for PW there was almost equal support for the L and TS models (Table 3).There was also some support for other models for both PL and PW.
Based on the L model, the CI boot for the allometric exponent b = a 2 was (1.005, 1.031) and (1.002, 1.041) for PL and PW respectively.This indicates that the growth of female chelar propods may be considered slightly positive allometric.
In males, the best model for PL growth was the Q model.There was also substantial support for the C model and some support for other models (Table 3).For the growth of PW in males, the best model was the TS model, with some support for the BS (Table 3); total w i for the two models was 97%, i.e. with strong evidence of the existence of a breakpoint in the allometric growth of PW; the breakpoint was quite marked (Fig. 1) at CW ≈ 16 mm.It is worth mentioning, that there was no support for the linear model (which was best for females), which indicates sub-stantial sexual differentiation in the relative growth of chelar propods.The generalized allometric exponent for the growth of PL in males was an increasing function of the crab size (Fig. 1).The generalized allometric exponent for the growth of PW in males, based on the TS model, was initially b PW = 0.985, with CI boot = (0.893, 1.070).Then, after the breakpoint, it increased to b PW = 1.246, with CI boot = (1.153,1.322).From the smoothing spline fit of the residuals, it is clear that in males, both for PW and PL, the assumption of linearity is more or less violated as the residuals of the L model do not have a random distribution around zero but show evident curvature (Fig. 1).The situation clearly improved with the average model,  with the smoothing spline fit of the residuals much closer to the horizontal axis.
The sexual differentiation in growth patterns of chelar propods is demonstrated in Figure 1, in which the 'average' models and the corresponding generalized allometric exponents are plotted.Initially there were no differences in the size of the chelar propods between males and females, but as the crabs grew larger, males attained significantly larger propods.The allometric exponent was initially near unity for both sexes, but for males it soon increased at a greater rate compared to females (Fig. 1B).
Female abdominal somites showed a variety of allometric growth patterns (Table 3).For S 1 the L model was the best, and the Q, TS and BS models also had substantial support.For abdominal somites S 2 , S 3 , S 4 , and S 5, the TS was the best model; for S 3 and S 4 in particular, no other competing model had substantial support.The breakpoint for somite S 3 and S 4 was at CW = a 4 ≈ 16.5 mm.For somites S 6 and S 7 , the C model was the best with much less support for TS and essentially no support for any other model.The variation of the growth pattern among the seven abdominal somites of the females is demonstrated in Figure 2, where log S i and the allometric exponents are plotted against log CW, based on the 'average' models.The residual plots of the average models were much improved in relation to those of the linear models in most cases; two examples (for S 4 and S 6 ) are given in Figure 2 (E-H).
In females, the corresponding allometric exponent and its CI boot were estimated using the L model for the growth of S 1 (Table 4).The two b values for each line segment of the TS model were estimated for the growth of each of S 2 , S 3 , S 4 , and S 5 , (Table 4).The change of the allometric exponent b at the breakpoint was not significant in any of the cases (Table 4, Fig. 2).The generalized allometric exponent for the growth of somites S 6 and S 7 has a maximum at an intermediate size (Fig. 2).The maximum value of b and its value for the smallest and greatest measured CW of our dataset were estimated (Table 4).The allometric growth of S 6 and S 7 was initially negative allometric, then it gradually became positive allometric and finally isometric.The differences in the growth pattern of the abdominal somites cause a morphological change in the abdominal shape during the ontogeny of P. marmoratus females (Fig. 3).
The best model for the growth of male abdominal somites was the TS in most cases, except for S 1 where it was the L model (as in females) and for S 4 where it was the BS model.In most cases, models  other than the best ones were also substantially supported by the data (Table 3).The b values and the corresponding confidence intervals were estimated based on the best model (Table 4).The corresponding plots based on the 'average' models are given in Figura 4. Growth of the abdominal somites of the male P. marmoratus was either isometric or negative allometric (Table 4, Fig. 4).
The overall outcome of the differences in the relative growth of abdominal somites between males and females is that at small sizes there was no morphological sexual differentiation of the abdomen, but at larger sizes there was substantial differentiation with the abdomen of the females becoming relatively larger and more rounded, while that of the males remained lance-shaped (Fig. 3).

DISCUSSION
The information theory approach frees the analyst from the limiting concept that the proper approximating model is somehow 'given'.When a model is 'picked' in some way, independent of the data, and used to approximate the data as a basis for inference, both the uncertainty associated with model selection and the benefits of selecting a parsimonious model are ignored.This strategy incurs substantial costs in terms of reliable inferences because model selection uncertainty is assumed to be zero.When the data support evidence for more than one model, model-averaging the predicted response variable across models is advantageous for reaching a robust inference that is not dependant on a single model (Burnham and Anderson, 2002;Katsanevakis, 2006;Katsanevakis et al., 2007a).
In allometric growth studies, it is a common practice to 'pick' the simple linear model (for logtransformed data) or sometimes to investigate for breakpoints using a version of the broken-stick or the two-segment model; examples of model selection and multi-model inference are rare (e.g.Hall et al., 2006;Katsanevakis et al., 2007a).Had we not followed an information-theoretic approach in this study, a large part of information would have been lost.We found that the relative growth pattern in P. marmoratus varies substantially among somatic parts and between sexes; some parts grow according to the classic allometric equation in relation to a reference dimension, others exhibit a non-linear change of their generalized allometric exponent during ontogeny, while in others there is a discontinuity in f and/or b at a breakpoint.In many cases, the simple linear model had no support (e.g. in the chelar propod of males or in the abdominal somites S 3 , S 4 , S 5 , S 6 , and S 7 in females).Using the linear model when there is strong non-linearity would cause 'smoothing' of the true picture; e.g. the abdominal somites S 6 and S 7 initially grow isometrically, then positive allometrically and then again isometrically, but with the linear model, an 'average' positive allometry would be concluded throughout ontogeny, which is only a rough approximation.Thus, a set of candidate models including the simple linear model, models that assume a continuous change in allometry, and models that assume discontinuity at breakpoints should be considered in allometric growth studies.An information-theoretic approach is recommended to account for model uncertainty, while null hypothesis testing should be avoided.Three kinds of maturity can be distinguished in crustaceans: physiological maturity (mature gonads), functional maturity (ability to mate) and morphometric maturity (possession of morphometric characteristics that are distinct from those of immature individuals) (Hall et al., 2006).The attainment of morphometric maturity is often identified by a breakpoint in the allometric data, i.e. f and/or its first derivative f´ change markedly at a certain body size corresponding to the pubertal moult (i.e. the moult to maturity).Physiological, functional and morphometric maturity are not necessarily simultaneous.Functional and physiological maturity are prerequisites for a crab to reproduce, while mating may also depend on morphometric maturity (mostly in males), although this is not always necessary.In non-competitive environments, physiologically mature but morphometrically immature male Chionoecetes opilio mate successfully (Sainte-Marie and Lovrich, 1994).
In P. marmoratus, a sexual dimorphism in the chelae was evident in larger specimens, with males having significantly larger chelar propods than females, although these differences are not of the same magnitude as in many other crab species.Similar results were found for other grapsid species, e.g.Hemigrapsus oregonensis (Olmsted and Baumberger, 1923), Plagusia dentipes (Tsuchida and Watanabe, 1997), Pachygrapsus transversus (Flores and Negreiros-Fransozo, 1999).The breakpoint in the allometric growth of PW in males was marked (Fig. 1) at a CW ~16 mm, and thus the breakpoint of the PW may be used as an indicator of morphometric maturity in males, which represents the pubertal moult.It is worth mentioning that Flores and Negreiros-Fransozo (1999) found no inflection points in the chelar dimensions of the congeneric P. transversus males, using Somerton's techniques (Somerton, 1983), and consequently, chelar relative growth could not offer any clues for male morphological maturity in that species.It can be suggested that in P. marmoratus, large chelae seem to be advantageous for larger males, since, in the Mediterranean, these individuals have been observed to have territories that do not overlap and which they share with small males and females (Cannici et al., 1999).
The data of this study supported the existence of a breakpoint in the allometry of the third and fourth abdominal somites in females (S 3 and S 4 ) at a CW ~16.5 mm, according to the TS model.For S 3 and S 4 only the TS model, out of the set of candidate models, was substantially supported by the data (Table 3).The TS with the same breakpoint was the best model for S 5 as well, but there was also support for the C model.The generalized allometric exponent for the relative growth of S 3 and S 4 did not change at the breakpoint but there was a marked discontinuity in f.It should be noted that both in males and females the breakpoints were found to occur at similar sizes (CW of ~ 16.0 and 16.5 mm respectively).This breakpoint in females may also be an indicator of morphometric maturity, which represents the pubertal moult.However, Flores and Negreiros-Fransozo (1999) observed that young P. transversus females may, after breeding, moult from a morphotype of rounded abdomen to a morphotype with ogival abdomen.Therefore, for this species two separate regression lines differentiate young and resting individuals from potentially reproductive females and not immature from mature females.This phenomenon of abdomen degradation has also been reported for P. marmoratus (Vernet-Cornubert, 1958).So far, the existence and/or importance of the above mentioned abdomen degradation is not known for different P. marmoratus populations.Further studies are needed in order to check this hypothesis and also to combine morphological analysis with physiological and functional maturity data.
A gradual increase is observed (Fig. 2) in the generalized allometric exponents of the relative growth equations of the abdominal somites in females, with b S1 < b S2 < b S3 < b S4 < b S5 during ontogeny (where b Si is the generalized allometric exponent of the growth of somite S i ), while b S6 and b S7 are also >b S5 in the middle size range.Furthermore, the allometric exponent of S 5 has an intermediate form between the approximately constant b type of abdomen somites S 1 -S 5 and the parabolic type of b through ontogeny of S 6 and S 7 .Such a smooth change of b from each abdominal somite to the next is anticipated, as consecutive somites are bonded together with joints, and thus the growth of each somite restricts the growth rate of the adjacent ones.
In conclusion, relative growth patterns, as demonstrated in the present study, can diverge in different ways from the classical linear model, the main outcome of which is the definition of the allometric exponent and the type of allometry.Marked discontinuities of the allometric exponent can not be detected using the linear model, and thus using it can give misleading results.Such changes in the growth trajectories of morphological characters during ontogeny are a potentially useful source of information as they may be caused by marked events in the life history of the species or fast ecological change; therefore, they should not be overlooked.The attainment of morphometric maturity in crabs is identified by finding breakpoints in the allometry of certain body parts at the puberty moult, usually chelae in male crabs and abdominal segments in females (Somerton 1981;Hall et al. 2006).The present study revealed the possibility of gradual changes in the generalized allometric exponent, which may follow a smooth, curvilinear trend (e.g. in female abdominal somites S 6 and S 7 ).This growth pattern could not be detected by the classic allometric model or models incorporating discontinuities.The information-theoretic approach proved to be quite effective in deciding on the most appropriate model from among a set of biologically meaningful candidate models and provided more information than the classic approach.
FIG. 1. -(A) The relative growth patterns of chelar propods (PL and PW in relation to CW) in male and female P. marmoratus; the datapoints and the 'average' models are given in absolute scales.(B) The generalized allometric exponent b for the relative growth of PL and PW, calculated as the slope of the corresponding 'average' models.(C-F) Residual plots against log CW for the linear (C, D) and the average (E, F) models for log PW (C, E) and log PL (D, F) in males, with a smoothing spline fitted to the residuals.
FIG. 3. -The morphological sexual differentiation of the abdomen of P. marmoratus; The S i 's are given for seven different sizes of CW (from 5 to 35 mm) for each sex.
the relative growth patterns of the width of abdominal somites (S i ) of male P. marmoratus according to the corresponding 'average' models; the Akaike weights w i were used for the weighted averages; somites S 2 , S 3 and S 4 were separated for clarity.(Bottom) the allometric exponent b for the relative growth of each S i of male P. marmoratus, calculated as the slope of the corresponding 'average' models between log S i and log CW.The scales of the axes were kept the same with the corresponding plots in Figure2for comparison.

TABLE 2 .
-The parameters a i and the residual sum of squares RSS of the regression equations between the logarithms of the measured biometric variables Y in relation to CW. Model abbreviations as in Table1.

TABLE 3 .
-Values of the AIC c differences (Δ i ) and of the 'Akaike weights' w i for the five models of the measured biometric variables.

TABLE 4 .
-The allometric exponents b i and their 95% bootstrap confidence intervals for the growth of the abdominal somites S i in females and males, according to the corresponding 'best' models.+: positive allometry, -: negative allometry, =: isometry.