Navigating morphometric minefields: Modelling heteroscedasticity in length‐conversion models

Abstract

A common problem when combining data on species' traits from multiple sources is that researchers often measure the same trait in different ways. For example, the length of a shark can be measured in a straight‐line or over‐the‐body, and for lengths that include the tail (e.g., fork and total lengths), with the tail in ‘stretched’ or ‘natural’ position. Statistical models comparing one variant to another can be used to standardise length measurements, thus allowing data to be combined. Often, when fitting such models, little attention is paid to the patterns of residuals, trusting that log‐transforming the data adequately accounts for heteroscedasticity, or whether conversion models built with data from defrosted specimens can be used on data from fresh (or live) individuals. Using a Bayesian modelling approach, we compared the out‐of‐sample predictive performance of linear and log‐linear models, with and without a model term that explicitly modelled heteroscedasticity as a function of the predictor variable, for converting between several length variants for school shark (Galeorhinus galeus). We found that point predictions were effectively identical across all four model forms. However, models including a term for heteroscedasticity produced superior predictive performance and prediction interval coverage. Measurements recorded from fresh individuals also fell within the prediction intervals of values estimated by models built using length variants measured from defrosted animals, suggesting that such models can be used to convert between length variants measured from fresh animals. Although the simpler models may be adequate for producing point estimates of the mean, models used to convert between length variants should include a term that explicitly models heteroscedasticity wherever prediction intervals or propagated uncertainty matter.

1. INTRODUCTION

Sustainable management of elasmobranch populations requires accurate knowledge of biological information, such as life history characteristics (Heithaus et al., 2022; Simpfendorfer & Dulvy, 2017; Stevens et al., 2000). For widely distributed species, it can be valuable to compare life‐history parameters across geographically dispersed populations. However, compiling data collected by different research groups can be problematic, as different research groups often have different measurement protocols for the same trait (De Wysiecki & Braccini, 2017; Francis, 2006; Natanson et al., 2022). For example, lengths may be recorded as fork length or total length, total length may be recorded with the tail in a stretched or a natural position, and sexual maturity may be determined according to the ability to produce gametes or when the copulation organs mature (Francis, 2006; Walker, 2005).

Having different variants of measured traits can hamper efforts to combine datasets and conduct comparisons across populations because the different variants are often confounded with time and/or geographic location. When this is the case, it is useful to have a dataset in which multiple trait variants are measured on the same individuals. With such a dataset, it is possible to build statistical models that can convert one variant to another, thus ameliorating the problem of having different variants and enabling cross‐population analyses of combined datasets (Bartes & Braccini, 2023; Francis, 2006).

Allometry is the theory of how different morphological traits relate to one another and, as such, allometric regression models are well suited to the task of converting among different measurement variants (Downs & Cheng, 2013; Manaster & Manaster, 1975; Siers, 2021; Xiao et al., 2011).

The basic allometric equation can be written as

$$\begin{array}{ll}Y& =\alpha X^{\beta }\\ & \end{array}$$

where Y is the target trait measurement and X is the predictor trait measurement. When the trait measurements are log‐transformed, the allometric equation can be expressed as follows:

$$\begin{array}{l}\log \left(Y\right)=\log \left(\alpha \right)+\beta \log \left(X\right)\end{array}$$

where $\log \left(\alpha \right)$ and $\beta$ are intercept and slope parameters, respectively, of a simple linear regression.

The approach of using simple linear regression models on log‐transformed trait variables (hereafter, the ‘log‐linear model’) dates back to the early 20th century (Clark, 1928; Green & Green, 1932), though some authors have warned against uncritical application of this approach for relating biological traits to one another (Nijhout & German, 2012; Packard, 2023; Smith, 1980; Thompson, 1945). Relationships between biological traits can be non‐linear, and deviations from the relationship (i.e., the errors) are almost always heteroscedastic (Francis, 2006; Gingerich, 2000). That is, the deviations are generally greater on average for larger fitted values. The utility of the log‐linear model is that it can fit linear and curvi‐linear relationships while allowing for some degree of heteroscedasticity. Yet, little attention is given to whether the log‐linear model adequately captures the patterns of heteroscedasticity in real biological data. As a result, parameter estimates may be biased by a small number of high‐leverage data points (Carroll & Ruppert, 1988a).

It has recently become more straightforward to fit more sophisticated models with components that explicitly model heteroscedastic errors as a relationship between the error standard deviation and the predictor (Packard, 2017, 2020, 2023; Xu et al., 2017), for example, using the R packages glmmTMB (Brooks et al., 2017) or brms (Bürkner, 2017). Moreover, there have been significant advances in methods of comparing out‐of‐sample predictive performance in the context of Bayesian models (Magnusson et al., 2020; Vehtari et al., 2017). Yet, few studies have investigated whether the log‐linear model is a better fit than the linear model with heteroscedasticity, or whether the log‐linear model adequately accounts for heteroscedasticity in the context of trait‐variant conversion models.

For sharks, length conversion models are typically used to convert between measurements from fresh (or live) individuals (e.g., Braccini et al., 2022; Bradley et al., 2017; Drew et al., 2015), as these measurements best represent the size of animals observed in situ. However, collecting sufficient data from live individuals to build length conversion models can be difficult due to various constraints in the field such as depredation and limiting air exposure (Francis, 2006; Natanson et al., 2022). Although multiple measurements of length can be more easily obtained from dead individuals, specimens may need to be frozen prior to processing, leading to shrinkage of measurements by up to 5% (Francis, 1997; Francis & Francis, 1992). Theoretically, the relationship among different length variants should be conserved between fresh and defrosted individuals, meaning models derived from measurements of defrosted specimens could be used to convert between fresh length variants. Yet, few studies have tested this assumption directly, leaving uncertainty about the reliability of models built from defrosted specimens to convert between fresh length measurements.

Here, we utilise a modern Bayesian modelling approach to investigate the out‐of‐sample predictive performance of four different models for converting between different variants of length measurements taken from school sharks (Galeorhinus galeus, Linnaeus, 1758). School sharks were chosen due to the urgent need to improve our understanding of their biology, following the collapse of all but the New Zealand population and the species' subsequent global reclassification to Critically Endangered in 2020 (Walker et al., 2020). However, comparing data across several regions requires standardisation of trait measurements, such as length, as multiple variants of the same trait are present in data collected both within and among school shark populations. The four models we considered were simple linear regressions on the untransformed (linear) and on the natural log‐transformed (log‐linear) variables and, in each case, with and without an explicit term for heteroscedasticity (where the log of the error standard deviation was modelled as a linear function of the log of the predictor variable). Models with and without different parameter values for males vs. females were also compared. Summarising the results of 17 different pairs of length variants, we aimed to investigate whether (1) the relationships were linear or curvi‐linear, (2) models required a component that explicitly models heteroscedasticity, even when the log‐linear form of the model was fit, and (3) sex‐specific parameters were required. Finally, we (4) examined whether models fit to length variants of defrosted animals generalised to converting length variants of fresh animals when data was lacking. Robust length conversion models that account for data structure and are validated across fresh and defrosted specimens would allow more reliable integration of length data from multiple sources and regions. For a Critically Endangered species like school sharks, where population assessments depend on pooling historical and contemporary data, this directly improves the quality of biological metrics underpinning management decisions.

2. METHODS

2.1. Ethics statement

The capture, euthanasia and use of sharks used in this study were approved under Massey University Animal Ethics Committee (MUAEC) Protocols 19/98 and 22/44 and Ministry for Primary Industries Special Permits 632‐2 and 814‐2.

2.2. Morphological data

The morphological data used in this study were collected from school sharks captured in New Zealand's exclusive economic zone between February 2020 and October 2023. Data from 286 animals were analysed. These were captured opportunistically by commercial and recreational fishers and during research surveys using various methods, including angling, longlining, set netting and trawling. When a school shark was captured, variants of the total, fork and precaudal lengths were measured using measure boards or measuring tapes (Figure 1). In instances where the specimens were collected rather than returned alive, multiple measurements were taken during post‐mortems on fresh or defrosted specimens, including full and partial body lengths (Figures 1, S1.2 and S1.3; Tables S1.1–1.3). In some cases, not all the length variants could be taken due to the condition of the animal. Measurements at capture and post‐mortem were taken with an individual lying flat on the ventral surface of its abdomen (except for preoral length, which was measured with the shark on the lateral surface of its abdomen) and recorded to the nearest mm, if possible, otherwise to the nearest 5 mm.

FIGURE 1.

Most common body morphometrics that can be used to derive total body length for sharks. The grey outlined tail to illustrate when the tail is in a stretched position. Definitions of the length variants are available in Table S1.1.

In addition to the main dataset collected in New Zealand, we used supplementary data from school sharks caught in Australia (Braccini unpublished data). The Australian data were not used for fitting any models; they were used as an independent test set. Specifically, they were used to assess whether conversion models fit to data measured from defrosted sharks could be used to convert fresh variants (aim 4 in the introduction). These data were collected from school sharks captured by commercial fisheries using gillnets in Australian waters between 1993 and 1995 (Braccini unpublished data), with all measurements taken from fresh specimens.

2.3. Statistical models

All analyses were undertaken using R and R Studio (Posit team, 2024; R Core Team, 2024). Models were constructed and compared using the brms and rstan packages (Bürkner, 2017; Stan Development Team, 2024).

The four models described in Table 1 were tested and compared on a total of 17 conversions. These conversions consisted of converting nine length variants (six full and three partial body lengths) to the two most used length variants for school sharks; that is, total length measured in a straight line with the tail in the natural vs. stretched (tail in line with body axis) position (Figures 1 and 2; Table S2.1).

TABLE 1.

Different forms of the regression model with and without explicit terms for heteroscedasticity. $y_i$ is the desired length variant, and $x_i$ is the length variant being converted.

Model name	Model description	Mathematical notation
Linear	Linear model with homoscedastic errors	$\begin{array}{c}{y}_i\sim \text{Normal}\left({\mu }_i,\sigma \right)\\ {\mu }_i={\beta }_0+{\beta }_1{x}_i\end{array}$
Linear sigma	Linear model with explicit heteroscedasticity	$\begin{array}{l}{y}_i\sim \text{Normal}\left({\mu }_i,{\sigma }_i\right)\\ {\mu }_i={\beta }_0+{\beta }_1{x}_i\\ \log \left({\sigma }_i\right)={\zeta }_0+{\zeta }_1\log \left(x_i\right)\end{array}$
Log‐linear	Log‐linear (natural log‐normal) model without explicit heteroscedasticity	$\begin{array}{l}{y}_i\sim \log \text{−}\text{Normal}\left({\mu }_i,\sigma \right)\\ \log \left({\mu }_i\right)={\beta }_0+{\beta }_1\log \left(x_i\right)\end{array}$
Log‐linear sigma	Log‐linear (natural log‐normal) model with explicit heteroscedasticity	$\begin{array}{l}{y}_i\sim \log \text{−}\text{Normal}\left({\mu }_i,{\sigma }_i\right)\\ \log \left({\mu }_i\right)={\beta }_0+{\beta }_1\log \left(x_i\right)\\ \log \left({\sigma }_i\right)={\zeta }_0+{\zeta }_1\log \left(x_i\right)\end{array}$

Note: In all four models, ${\mu }_i$ and ${\sigma }_i$ denote the mean and standard deviation of $y_i$ on the arithmetic scale and the second argument of each distribution is a standard deviation rather than a variance. For models without a term for explicit heteroscedasticity, ${\sigma }_i$ is constant across all observations and may be replaced with the scalar $\sigma$; for models with such a term, ${\sigma }_i$ varies as a function of $x_i$. The linear models relate ${\mu }_i$ to $x_i$ through an identity link with a normal likelihood, whereas the log‐linear models relate ${\mu }_i$ to $\mathit{log}\left(x_i\right)$ through a log link with a log‐normal likelihood parameterised directly in terms of the arithmetic‐scale mean and standard deviation. ${\beta }_0$ and ${\beta }_1$ denote the intercept and slope parameters for the mean, and ${\zeta }_0$ and ${\zeta }_1$ the intercept and slope parameters for the standard deviation; these are defined on the scale of the corresponding link function (identity for ${\mu }_i$ in the linear models, log for ${\mu }_i$ in the log‐linear models and log for ${\sigma }_i$ throughout). The probability density functions of all four models are given in Supplementary Material S3.

FIGURE 2.

The relationships between length variants being modelled. (a) Defrosted, natural, total length given defrosted, natural, over‐the‐body total length; (b) Defrosted, natural, total length given defrosted, stretched, total length; (c) Defrosted, natural, total length given defrosted, stretched, over‐the‐body total length; (d) Defrosted, natural, total length given defrosted fork length; (e) Defrosted, natural, total length given defrosted precaudal length; (f) Defrosted, natural, total length given defrosted, stretched, over‐the‐body dorsal origin to caudal tip length; (g) Fresh, natural, total length given defrosted, natural, total length; (h) Fresh, natural, total length given defrosted head length; (i) Fresh, natural, total length given defrosted preoral length; (j) Defrosted, stretched, total length given defrosted, natural, total length; (k) Defrosted, stretched, total length given defrosted, natural, over‐the‐body total length; (l) Defrosted, stretched, total length given defrosted, stretched, over‐the‐body total length; (m) Defrosted, stretched, total length given defrosted fork length; (n) Defrosted, stretched, total length given defrosted precaudal length; (o) Defrosted, stretched, total length given defrosted, stretched, over‐the‐body dorsal origin to caudal tip length; (p) Defrosted, stretched, total length given defrosted head length; (q) Defrosted, stretched, total length given defrosted preoral length. Defrosted variants are variants measured from defrosted animals, whereas fresh variants are variants measured from fresh/live animals. Definitions of each length variant are in Supplementary Material 1. The dashed line is when x=y.

In addition, variants of these models that included different parameters for each sex were also fit. An example of the log‐linear sigma model with sex‐specific parameters is:

$$\begin{array}{l}{y}_i\sim \log \text{−}\text{Normal}\left({\mu }_i,{\sigma }_i\right)\\ \log \left({\mu }_i\right)={\beta }_{0,\mathrm{Sex}\left[i\right]}+{\beta }_{1,\mathrm{Sex}\left[i\right]}\log \left(x_i\right)\\ \log \left({\sigma }_i\right)={\zeta }_0+{\zeta }_1\log \left(x_i\right)\end{array}$$

To provide conversion models for measurements from fresh individuals (rather than frozen and defrosted) when data is lacking, we tested whether the use of a given length–length relationship built from defrosted sharks could accurately convert the same length variants measured from fresh animals.

For each model, weakly informative priors were used (Table S3.1). The code and data used for the models are available in Supplementary Material S3 and S4, respectively.

2.4. Model evaluation and comparison

Model adequacy was assessed using Pareto shape k values, posterior predictive density overlays and leave‐one‐out posterior predictive interval checks. Together these diagnostics assessed the reliability of the Pareto‐Smoothed Importance Sampling approximation, agreement between the observed and posterior predictive response distributions, and the ability of the model to predict individual observations under leave‐one‐out cross‐validation. The model adequacy diagnostics for each model across all conversions are available in Supplementary Material S5.

Model performance was assessed using Expected Log Predictive Density (ELPD), Leave‐One‐Out Relative Root Mean Square Error (LOO‐RRMSE) and Leave‐One‐Out prediction interval coverage. Additional evaluation included testing the selected model on new data and visualising residuals and observed data against model predictions. All evaluation metrics were also used to establish whether models needed to account for sex when converting between length variants. ELPD is a measure of predictive accuracy for out‐of‐sample data, which is estimated using Pareto‐Smoothed Importance Sampling (Vehtari et al., 2017). When the difference in ELPD between two models is approximately equal to 4, the probability of the model with the higher ELPD having superior predictive performance is approximately 98% (McLatchie & Vehtari, 2024). Hence, differences in ELPD <4 and/or the standard error of the difference was greater than the difference in ELPD, indicated there was no evidence in the difference of the predictive ability of a model (McLatchie & Vehtari, 2024; Sivula et al., 2025).

The Leave‐One‐Out Relative Root Mean Square Error quantifies out‐of‐sample prediction error relative to the observed response value:

$$\begin{array}{l}\text{LOO}\text{−}\text{RRMSE}=100\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(\frac{y_i-{\widehat{y}}_i^{\left(-i\right)}}{y_i}\right)}^2}\end{array}$$

where $y_i$ is the observed value of the response (the target length variant), ${\widehat{y}}_i^{\left(-i\right)}$ is the leave‐one‐out posterior predictive mean for observation $i$ estimated using Pareto‐Smoothed Importance Sampling and $n$ is the sample size.

The Leave‐One‐Out prediction interval coverage assesses how well the predictive uncertainty estimated by the model reflects the observed variability in the data by calculating the proportion of observations contained within the 95% out‐of‐sample prediction intervals. Coverage was summarised across four length classes to assess model performance across the observed size range. When model performance metrics indicated no evidence of a difference between the models, the simplest model was selected.

3. RESULTS

Most of the relationships between variants of length measurements appeared to be linear (Figure 2a–g,j–o). Relationships with non‐linearity/minor curvature appeared to be limited to partial length measurements; specifically, head measurements such as head length and preoral length (Figure 2h,i,p,q). There also seemed to be no difference in the relationship between sexes over the range of the length variants (Figure 2a–q).

The linear sigma and log‐linear sigma models had better predictive accuracy compared to the non‐sigma models in all but two conversions (Figure 3a; Table S2.1). Sigma models on average also had high prediction interval coverage across the range of observed lengths, whereas the coverage of non‐sigma models was too high for smaller animals and too low for larger animals (Figure 3c; Table S2.1). However, there was little evidence of a difference in the fit (Figure 4; Supplementary Material S6), predictive ability (Figures 3a and 5; Supplementary Material S6) and prediction error (Figure 3b; Table S2.1) between the linear and log‐linear models in most of the conversions. This resulted in the simpler, i.e., non‐sigma models being selected due to their simplicity. There were five conversions where the log‐linear sigma model had superior predictive ability (Figures 3a and 6; Table S2.1) and was subsequently selected. The cases where the log‐linear sigma model was selected included three relationships between partial and full body lengths that were non‐linear (Figure 2; Supplementary Material S6).

FIGURE 3.

Comparison of (a) differences in Expected Log Predictive Density (ELPD), (b) Leave‐One‐Out Relative Root Mean Square Error (LOO‐RRMSE) and (c) Average 95% Leave‐One‐Out Prediction Interval Coverage between the four models across the 17 conversions. The solid and dashed vertical lines in (a) represents a difference of 0, which is the model with the greatest ELPD, and a difference of 4, respectively. Any differences in model ELPD $\le$ 4 can be taken as no evidence of a difference in the predictive accuracy of those models. The vertical lines coloured by ‘Model class’ in (a) and (b) are the averages of the metric value for each Model class.

FIGURE 4.

Predicted fit of the four conversion models where there was no evidence of a difference in the ELPD between the models within the same error structure. PI corresponds to 80% and 95% posterior prediction intervals. Defrosted, stretched, total length is the total length measured from a defrosted shark, in a straight line, with the tail in a stretched position. Defrosted, natural, total length is the total length measured from a defrosted shark, in a straight line, with the tail in a natural position.

FIGURE 5.

Residual plots of the four conversion models where there was no evidence of a difference in the ELPD between the models within the same error structure. Points represent the point estimates of the residual posterior distributions. Vertical lines indicate the 66% and 95% credible intervals of each residual's posterior distribution, characterising the uncertainty in its location. The shaded band is the 95% residual spread implied by each model's error structure. Defrosted, stretched, total length is the total length measured from a defrosted shark, in a straight line, with the tail in a stretched position. Defrosted, natural, total length is the total length measured from a defrosted shark, in a straight line, with the tail in a natural position.

FIGURE 6.

Predicted fit of the four conversion models where there was evidence of a difference in the fit between all four models. PI corresponds to 80% and 95% posterior prediction intervals. Defrosted, stretched total length is the total length measured from a defrosted shark in a straight line with the tail in a stretched position. Defrosted preoral length is the preoral length measured from a defrosted shark in a straight line.

Plots of fits and residuals of the four models across all conversions are available in Supplementary Material S6. Point estimates (i.e., the mean of the posterior distribution of estimates) of the log‐linear sigma model parameters to convert between the tested length variants are available in Table S7.1.

Comparing the ELPD, LOO‐RRMSE and prediction interval coverage values of the log‐linear sigma model with and without the inclusion of a sex covariate showed no evidence of a difference between the models (Table S8.1; Figure S8.1).

Given the disparity between the sample sizes of fresh and defrosted measurement variants of some length–length relationships (e.g., for total length natural ~ total length stretched, n = 6 for fresh vs. n = 141 for defrosted), conversion models that used defrosted length variants performed well when used to convert between fresh variants of the same lengths. For most individuals, including data measured from a different population (Australia) by a different research group, the observed lengths were within the prediction error intervals of predicted variant of the same length (Figure 7a,b, Supplementary Material S9).

FIGURE 7.

Predictions for fresh measurement variants made from log‐linear sigma models trained on the same variants measured from defrosted individuals. Fresh, natural, total length: Total length, measured in a straight line, tail in natural position, measured from a fresh or live shark. Fresh, stretched, total length: Total length, measured in a straight line, tail in stretched position, measured from a fresh or live shark. Fresh fork length: Fork length, measured in a straight line, tail in natural position, measured from a fresh or live shark. The filled black circles are the point estimates of the predicted length variants. The black horizontal lines are the 80% and 95% prediction intervals. The grey distributions around the filled points and horizontal lines are the posterior distributions of the predicted length variants. The orange plus symbols are the observed values of the fresh variants. (a) Fresh, natural, total length given fresh, stretched, total length. (b) Fresh, stretched, total length given fresh fork length.

4. DISCUSSION

Relationships between partial and full‐body measurements for many elasmobranchs are typically modelled with linear models on the arithmetic or log‐scale, without a ‘sigma’ model component that explicitly fits the residual error standard deviation as a function of the predictors (e.g., Francis, 2006). Although log‐transformation is commonly used to address non‐linearity and heteroscedasticity, it does not always adequately account for heteroscedasticity (Carroll & Ruppert, 1988b; Manning, 1998; Packard, 2017). In our study, the little difference in the predictive error and fit of the models (Figures 3, 4, 5; Table S2.1; Supplementary Material S6) indicated that point predictions were statistically indistinguishable across all four model forms for many conversions. However, even when the log‐transformation was applied, including a model component for the error standard deviation resulted in superior out‐of‐sample predictive performance and prediction interval coverage in most conversions (i.e., the linear sigma or log‐linear sigma models; Figure 3; Table S2.1). In all cases, the estimated slope of the sigma term was positive (Table S7.2), suggesting that the log‐linear models underestimated the degree of heteroscedasticity. Overall, explicitly modelling the error standard deviation led to more robust predictions.

Across a range of shark species, few have sex‐specific relationships between length measurements (e.g., De Wysiecki & Braccini, 2017; Natanson et al., 2022), yet allometric relationships have been observed between head‐related measurements and full body length measurements (e.g., Ahnelt et al., 2020; Gayford, Godfrey, & Whitehead, 2023; Sternes & Higham, 2022). A similar pattern was observed in this study, with non‐linear relationships between head‐related (i.e., preoral and head lengths) and full body length measurements, but no evidence that sex‐specific length conversion parameters were required (Figures 2 and 6; Supplementary Material S6 and S8). Although not directly investigated here, one possible explanation for the observed non‐linearity is the allometric growth of head structures during ontogeny, potentially associated with shifts in the diet (Lucifora et al., 2006) and habitat use (Hurst et al., 1999; Thorburn et al., 2019), as suggested by Gayford, Whitehead, et al. (2023). However, further research examining the relationship between morphometrics and ecology in school sharks is required to test this potential explanation. Given the presence of non‐linearity in some conversions and the absence of an effect of sex, we recommend that the log‐linear sigma model and pooled sex data should be used for converting between body length variants of school sharks. This is due to the model being flexible to non‐linear relationships, where they arise, and able to appropriately account for heteroscedasticity.

An important issue with measuring the lengths of sharks is whether the measurements were taken when the animal was fresh, or after being frozen and defrosted (Francis, 1997; Francis & Francis, 1992; Olsen, 1954). Similar to patterns observed in other species (e.g., Francis, 1997; Francis & Francis, 1992), we estimated length variants from fresh school sharks to be around 5% larger on average than the defrosted equivalents (Figure S10.1). Depending on how the animals are captured and processed, one usually obtains measurements either fresh or defrosted, and rarely both. In this study, we had fewer measurements from fresh animals than defrosted ones and, as a result, conversion models for several variant pairs could not be built for fresh animals. However, when models trained on length data from defrosted animals were applied to convert between the same measurement variants from fresh animals, most of the observed (fresh) values were within the expected posterior distributions of predicted values, including for individuals from the Australian population (Figure 7; Supplementary Material S9). Although the number of fresh specimens for validation was limited for some conversions, this finding suggests that conversion models built with data from defrosted animals may be useful for converting within the same variant pairs taken from fresh animals. Furthermore, although the application of the conversion parameters to other school shark populations requires further validation, the successful prediction of measurements from the Australian population suggests that morphometric relationships are likely to be similar across populations, consistent with the conserved body morphology expected within a species.

Given the importance of length measurements to biological analyses, accurate and unbiased conversions between length variants are essential. Using school shark as a case study, we found that although point predictions were effectively indistinguishable across all four model forms, only the models with explicit heteroscedastic error produced prediction intervals with reliable coverage across the full size range. We also found that models constructed using length variants measured from defrosted animals can be used to convert between length variants measured from fresh animals. The framework developed here provides improved, and in some cases new, conversions between full and partial body lengths of school sharks, facilitating more consistent data standardisation and comparisons of length‐based biological traits among populations. More broadly, this framework can be applied to other species to improve the reliability of length conversions for inter‐ and intra‐population analyses. However, to ensure accurate and unbiased conversions between length variants, we echo the appeals of Francis (2006) and Natanson et al. (2022) in calling for researchers to clearly and thoroughly define the length measurements they used in a study. Additionally, we recommend including a term in length conversion models that explicitly models the error standard deviation because, while log‐transforming the variables may go some way to account for heteroscedasticity, it did not do so sufficiently in the examples in this study. Robust standardisation of length data requires models that account for the structure of the data and are applied to clearly defined length variants, thereby improving the reliability of length‐based metrics used to inform population management.

AUTHOR CONTRIBUTIONS

Conceptualisation: Alex J. C. Burton, Adam N. H. Smith; Data curation: Alex J. C. Burton, J. Matías Braccini; Formal analysis: Alex J. C. Burton, Adam N. H. Smith; Funding acquisition: Alex J. C. Burton, Adam N. H. Smith; Investigation: Alex J. C. Burton, Adam N. H. Smith, J. Matías Braccini; Methodology: Alex J. C. Burton, Adam N. H. Smith; Supervision: Adam N. H. Smith; Writing – original draft: Alex J. C. Burton, Adam N. H. Smith; Writing – review & editing: Alex J. C. Burton, Adam N. H. Smith, J. Matías Braccini.

FUNDING INFORMATION

This research was funded by the Massey University Doctoral Scholarship, Massey University Research Fund (MURF), Ministry for Primary Industries Postgraduate Science Scholarship, Sea Through Science Postgraduate Scholarship. Transport of specimens was funded by the Ministry for Primary Industries and Talley's.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

Supporting information

Supplementary Material S1: Definitions of full and partial body length measurement variants of sharks mentioned in the main text and Supplementary Materials.

Supplementary Material S2: Expected Log Predictive Density (ELPD), Leave‐One‐Out Relative Root Mean Square Error (LOO‐RRMSE) and 95% Leave‐One‐Out prediction interval coverage values of the candidate models across the 17 length conversions and their associated length variants.

Supplementary Material S3: Priors and R code for each candidate model, and Probability Density Functions for the distribution families used for the response variable.

Supplementary Material S4: Data used to train the four candidate models for converting between length measurement variants.

Supplementary Material S5: Model adequacy checks for the candidate models across the 17 length conversions.

Supplementary Material S6: Fit and residual plots of the candidate models across the 17 length conversions.

Supplementary Material S7: Parameter point estimates from the log‐linear sigma model for converting between length variants for school sharks.

Supplementary Material S8: Comparison of Expected Log Predictive Density (ELPD), Leave‐One‐Out Relative Root Mean Square Error (LOO‐RRMSE) and 95% Leave‐One‐Out prediction interval coverage for the log‐linear sigma model, with and without sex‐specific parameters, across the 17 length conversions.

Supplementary Material S9: Comparison of predictions for converted fresh length variants, derived from models trained on defrosted length variants, with observed data.

Supplementary Material S10: Plot of the additive versus multiplicative effects between measurement variants.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available in Supplementary Material S4 of this article.