Using Observed Residual Error Structure Yields the Best Estimates of Individual Growth Parameters
, E. Alberto Aragón-Noriega 2, Miguel Á. Cisneros-Mata 1,*, Laura Sánchez-Velasco 3, S. Patricia A. Jiménez-Rosenberg 3 and Alejandro Parés-Sierra 4Round 1
Reviewer 1 Report
This is a good work and I would recommend publication after revisions.
1, First of all I would like to see a Monte Carlo simulation study to investigate the best error structure for the growth model, because in such study the true error structure is known. I understand that this may be a new paper, therefore I hope the authors to consider it in their further research.
2, In Table 2, units are not given for the growth parameters. In Table 3 the values of L∞ are in cm.
3, Please make sure that the MM section should be in the end.
4, P8L242, “length-at-age” should be “mean length-at-age”, because there can be multiple observed length-at-age.
5, P8L246,247, “?∞” is not explained. Please confirm that it is t0 or t1.
6, P8L249, please confirm that there is no bracket for “?σ̂?”. Please write this clearly for better understanding.
7, P8L253, there should be a mistake in “0.5ln(σ̂)+0.5ln(σ̂)”. Again please confirm that there is no bracket for “2σ̂2”. Please write this clearly for better understanding.
8, In Table 1, n and j are not explained, or maybe j can be removed. Wj can not be found in section 4.1, therefore we do not know how it is used.
9, It may be a good idea to also include subtitles in section 3 of DISCUSSION.
Author Response
Reviewer 1
This is a good work and I would recommend publication after revisions.
R: We thank the reviewer for her/his encouraging comment and recommendation.
1, First of all I would like to see a Monte Carlo simulation study to investigate the best error structure for the growth model, because in such study the true error structure is known. I understand that this may be a new paper, therefore I hope the authors to consider it in their further research.
R: The goal of the manuscript was precisely using the known error structure and compare model fits (and phi prime) using commonly assumed structures. We thank the reviewer for this interesting idea of a new publication where we explore an alternative error structure using Monte Carlo simulations. We will give serious consideration to her/his suggestion.
2, In Table 2, units are not given for the growth parameters. In Table 3 the values of L∞ are in cm.
R: Thank you for flagging these important details. Done. We added the units in tables 2 and 3, and transformed L∞ in table 3 to mm.
3, Please make sure that the MM section should be in the end.
R: We concur with the reviewer’s point; however, we followed the journal´s instructions to authors.
4, P8L242, “length-at-age” should be “mean length-at-age”, because there can be multiple observed length-at-age.
R: Done. Thank you for this clarification.
5, P8L246,247, “?∞” is not explained. Please confirm that it is t0 or t1.
R: We thank the reviewer for noting this omission and confusion. Done: we added the following text: “ is the asymptotic variance (for older organisms)”. Additionally, throughout we now use K to refer to either k or kg, and T to refer to t0 or t1 to differentiate parameters of either the von Bertalanffy or Gompertz models, respectively.
6, P8L249, please confirm that there is no bracket for “?σ̂?”. Please write this clearly for better understanding.
R: Thank you for noting this potentially confusing notation. Done: we added brackets for better comprehension of the formula.
7, P8L253, there should be a mistake in “0.5ln(σ̂)+0.5ln(σ̂)”. Again please confirm that there is no bracket for “2σ̂2”. Please write this clearly for better understanding.
R: Thank you very much for the observation. Done: the LL equation was corrected, and we added brackets as suggested, which makes the equation clearer. The line number changed because text was added above. It is highlighted in yellow for ease of identification.
8, In Table 1, n and j are not explained, or maybe j can be removed. Wj can not be found in section 4.1, therefore we do not know how it is used.
R: We thank the reviewer for asking for some clarification in Table 1. Done: we now include what n and j are.
9, It may be a good idea to also include subtitles in section 3 of DISCUSSION.
R: We thank the reviewer for her/his suggestion of including subtitles in the Discussion section. We, however, respectfully believe that they are not absolutely needed because this section is rather short and sectioned in distinct paragraphs.
Reviewer 2 Report
Major comments
This MS aims to provide an improved method in fish growth model fitting by considering the observed errors instead of convention method of constant variance. The idea is good and fits the scope of this journal. However, there are several major flaws in the current version.
The major problem of this MS is that the length at age data used in the analysis were from different sources. The length at age data of the major source Román-Rodríguez and Hammann (1997) were obtained from digitizing of the data from growth curve. I am not sure whether this method violates the academic ethics without the permission of the authors. Other published data were shared by other authors. There are uncertainties of these data. The size range, the criteria of age determination (estimation), and sampling periods among studies were quite different which may bias the estimation of growth parameters. For example, the observations of 80-120 cm were lacking in Román-Rodríguez and Hammann (1997). In addition, the authors claimed that this approach – considering the observed residual error is a novel method in growth equation estimation. However, this approach increased the number of parameters from 4 to 29 which increased the fitting without surprise even the selection criterion of BIC has the penalty of number of parameters.
I suggest the authors make a major revision by taking account the comments raised in the report.
General comments
L 50, “Zu [16]”
L 88, The assumption for this study is that the age estimation is correct. This should be mentioned and discussed in Discussion section.
L 100, Please specify “256” is the sample size.
L 101, The larger the parameters of the model, the better the model fitting. So, the authors should compare with the original model with constant variance and point out the reason for using variant variances.
L 103, The sentence is not complete.
L 115, The meaning of k is also different for the two growth models. They can not be compared directly. I suggest the authors use different notations such as k and kG instead of k for the two models.
Table 2, The to of VB with observed error was estimated as 0.0 which seems not reasonable. Please clarify. Please specify the length is total length or fork length.
The L∞ (149.2 cm) estimated from most plausible model (vB with observed errors) was much smaller than the reported maximum size of 200 cm in Line 78.
I suggest the authors discuss possible reasons for this underestimation.
Fig. 2, Large sample size of age 0.5 yr may affect the estimation of growth parameters which lead to a poor fit for old ages. I suggest the authors address this issue in Discussion section.
L 134, “wrong”. When compared with previous studies, the size range and sample size used in different studies may affect the results. The different ageing methods (otolith with/without sectioning or scale) and criteria may also affect the results.
L 137, “less”.
Table 3, Please make sure "1916-1994" is correct or not. The smaller L∞ and large k obtained in this study implied that there is a density dependent growth after ban fishing was canceled in recent years?
L 161, “If younger fish”.
L 168, Different birth date is one of the factors may result in the different size at age. However, this MS did not take this into account. For example, the sizes at ages 0.5 and 1.0 varies in wide range (Figure 2) which I do not think is realistic. For age 1, the size ranging from 30-70 cm is probably not due to individual difference in growth but due to different birth dates or age estimation bias. I suggest the authors should explain possible reasons in Discussion section.
L 203, As Román-Rodríguez and Hammann (1997) did not present their raw length at age data in the article, I am not sure whether this method violates the academic ethics without the permission of the authors.
L 233, Before conducting any analysis, you must assume that the age estimations from the previous studies (the length at age data used in this study) were correct. Therefore, no observation error existed. In addition, the data were from different sources, similar or same age determination criteria should be assumed. However, different age determination criteria may be used in different studies.
L 238, The definitions of k of the two growth models are different. The authors should specify this and use different notations for these two models.
L 240, 241, 249, Since these are estimators of δ, should “n” be “n-1”?
L 267, In the model fitting, the more the parameters, the better the fitting is. So, it is not surprised that the observed variance case had the best fit. However, this model should be compared with the constant variance one to see the difference and make discussion.
L 273, Will MCMC be a better approach to calculate the 95% C.I.?
Author Response
Reviewer 2
Major comments
This MS aims to provide an improved method in fish growth model fitting by considering the observed errors instead of convention method of constant variance. The idea is good and fits the scope of this journal. However, there are several major flaws in the current version.
R: We thank the reviewer for her/his major comments. We will struggle to demonstrate that such potential flaws can be solved with better explanations.
The major problem of this MS is that the length at age data used in the analysis were from different sources.
R: We respectfully consider that this might not constitute a major problem because there is no indication or publications showing that for totoaba, mean length-at-age has changed over time.
The length at age data of the major source Román-Rodríguez and Hammann (1997) were obtained from digitizing of the data from growth curve. I am not sure whether this method violates the academic ethics without the permission of the authors.
R: We recognize that this could constitute an ethical issue and appreciate the reviewer’s concern. This is, however, not our case as we have written permission from the first author (Román-Rodríguez), who most kindly agreed that we use her published material. In the revised version of the manuscript, we now acknowledge this. We thank the reviewer for flagging her/his concern about this important matter.
Other published data were shared by other authors. There are uncertainties of these data. The size range, the criteria of age determination (estimation), and sampling periods among studies were quite different which may bias the estimation of growth parameters. For example, the observations of 80-120 cm were lacking in Román-Rodríguez and Hammann (1997). In addition, the authors claimed that this approach – considering the observed residual error is a novel method in growth equation estimation. However, this approach increased the number of parameters from 4 to 29 which increased the fitting without surprise even the selection criterion of BIC has the penalty of number of parameters.
R: We thank the reviewer for such challenging comments, which we address below.
I suggest the authors make a major revision by taking account the comments raised in the report.
R: Thank you for suggesting the above.
General comments
L 50, “Zu [16]”
R: To keep up with the instructions to authors, we changed the wording so that reference [16] is no longer at the beginning of the sentence. Thank you for your suggestion.
L 88, The assumption for this study is that the age estimation is correct. This should be mentioned and discussed in Discussion section.
R: Although this is beyond the scope of the present work, we acknowledge that errors due to incorrect age determination could be an issue. At the end of the Introduction, we mention that our present work does not account for such potential errors and include some wording in the Discussion section. We thank the reviewer for flagging this.
L 100, Please specify “256” is the sample size.
R: Thank you for noting this omission. Done in the new version
L 101, The larger the parameters of the model, the better the model fitting. So, the authors should compare with the original model with constant variance and point out the reason for using variant variances.
R: The reviewer is right, the larger the set of parameters in the model, the better the model fit. In our case, variance was considered as a parameter to be included in the information criterion analysis. The original model was contrasted with constant variance and observed variance. In our case, the BIC (stricter and more consistent than AIC) penalized the model with more parameters and yet the von Bertalanffy model with observed variance and many parameters was the most plausible. We believe that this is precisely a good reason to use observed variance at the risk of augmenting the number of parameters. On the other hand, the Gompertz model with observed variance was not plausible at all, even when it contained the same number of parameters as the von Bertalanffy. Respectfully, we believe that herein lies the importance of our procedure and the manuscript itself.
L 103, The sentence is not complete.
R: Thank you for the observation. Done: the new text was written as: In this particular case, the vB was selected as the most plausible model.
L 115, The meaning of k is also different for the two growth models. They can not be compared directly. I suggest the authors use different notations such as k and kG instead of k for the two models.
R: We truly appreciate this observation. We have now clarified that “k” in the Gompertz model refers to the intrinsic growth rate at age t1. Accordingly, we included this correction throughout the text and equations. We took the advice of the reviewer and differentiated between “k” for the von Bertalanffy and “kg” for the Gompertz models. Generically, we use K. Thank you for allowing improvement of the new version of the manuscript.
Table 2, The to of VB with observed error was estimated as 0.0 which seems not reasonable. Please clarify. Please specify the length is total length or fork length.
R: We welcome this set of comments by the reviewer. The total length was used, and this is now explained in the text and the heading of Table 1. Regarding the t0 value, all we can say is that the iterative methods search freely for the parameter value and establishes the best choice when it meets the criteria of objective function. In this case, a 0 value was selected as best value in two cases. Respectfully, we see no reason to change these 0 values.
The L∞ (149.2 cm) estimated from most plausible model (vB with observed errors) was much smaller than the reported maximum size of 200 cm in Line 78.
R: Thank you for this non-trivial observation. We added text in Materials and Methods (M&M) explaining that L∞ is the average length that individuals of a population would reach if they were to grow indefinitely, and often one can sample individuals with lengths larger than this asymptotic length. We supported this with several citations.
I suggest the authors discuss possible reasons for this underestimation.
R: We appreciate the suggestion by the reviewer. Done: in addition to the text added in the M&M section, the new version of the manuscript also includes text at the onset of the Discussion section.
Fig. 2, Large sample size of age 0.5 yr may affect the estimation of growth parameters which lead to a poor fit for old ages. I suggest the authors address this issue in Discussion section.
R: Again, thank you for your interesting suggestion. The new version (Discussion section), however, now points out that the effect of overrepresentation of young individuals yields a biased-high estimate of L∞. This result of ongoing investigations is counterintuitive, and we so caution in the text.
L 134, “wrong”. When compared with previous studies, the size range and sample size used in different studies may affect the results. The different ageing methods (otolith with/without sectioning or scale) and criteria may also affect the results.
R: Thank you for pointing out the error in grammar. Done. With respect to the comments on potential errors having to do with samples, etc., we used some of your wording and expanded it to include some text in the Discussion section.
L 137, “less”.
R: Done. We corrected the word, thank you for the observation
Table 3, Please make sure "1916-1994" is correct or not. The smaller L∞ and large k obtained in this study implied that there is a density dependent growth after ban fishing was canceled in recent years?
R: The range of years spanning from 1916 to 1994 is correct. The explanation for this is that authors referenced as [39] pooled data from five published studies to calculate von Bertalanffy growth parameters. Regarding your second question, the answer is that density-dependence might explain a rather low L∞ and high k. However, there is also a more parsimonious explanation related to the known fact that L∞ and k are inversely related, as shown in the following plot (please, see attached document as the platform does not allow pasting figures):
In the new version of the manuscript, we include this in the Results section right before Table 3. We thank the reviewer for bringing this up.
L 161, “If younger fish”.
R: Done. We corrected the word, thank for the observation.
L 168, Different birth date is one of the factors may result in the different size at age. However, this MS did not take this into account. For example, the sizes at ages 0.5 and 1.0 varies in wide range (Figure 2) which I do not think is realistic. For age 1, the size ranging from 30-70 cm is probably not due to individual difference in growth but due to different birth dates or age estimation bias. I suggest the authors should explain possible reasons in Discussion section.
R: Thank you for flagging the need to discuss the size-at-age in young totoaba, which is an open issue, beyond the scope of our work. The topic, however, does deserve to be addressed and we have done so in the new version. First, in the Introduction section we included the spawning period of totoaba (Feb to Apr) and mention the particularly high variability of this very small region. In the Discussion section we now include text recognizing that the high variability in length-at-age observed in the youngest individuals could be explained by the different birth dates, the variable environment, also because both sexes were used to estimate parameters (we added a citation), and recognize that such variability carries throughout the lifespan of totoaba.
L 203, As Román-Rodríguez and Hammann (1997) did not present their raw length at age data in the article, I am not sure whether this method violates the academic ethics without the permission of the authors.-
R: Again, we recognize that this could constitute an ethical issue and appreciate the reviewer’s concern. This is, however, not our case as we have written permission from the first author (Román-Rodríguez), who most kindly agreed that we use her published material. In the revised version of the manuscript, we now acknowledge this. We thank the reviewer for flagging her/his concern about this important matter.
L 233, Before conducting any analysis, you must assume that the age estimations from the previous studies (the length at age data used in this study) were correct. Therefore, no observation error existed. In addition, the data were from different sources, similar or same age determination criteria should be assumed. However, different age determination criteria may be used in different studies.
R: Again, we acknowledge that errors due to incorrect age determination could be an issue. At the end of the Introduction, we mention that our present work does not account for such potential errors and include text in the Discussion section to address such issue.
L 238, The definitions of k of the two growth models are different. The authors should specify this and use different notations for these two models.
R: We thank the reviewer for pointing out the different definitions of k in the von Bertalanffy and Gompertz models. We have now corrected this throughout the text, including the equations.
L 240, 241, 249, Since these are estimators of δ, should “n” be “n-1”?
R: We acknowledge the potential confusion that the use or misuse of n or n-1 could cause. Here, however, we respectfully provide the reviewer with some references for clarification:
Page 84 of Haddon (2011): The variance estimated from the data (hence the hat). It is the maximum likelihood estimate so we divide by n and not n – 1. By expanding Equation 3.13, using Equation 3.14, we can produce a simplification for easier calculations.
Page 91 of Haddon (2011): Note the maximum likelihood version of σ2 using n instead of n-1 (Example Box 3.5). The ΣLn(x) term is a constant and is usually omitted from calculations
Pages 205, 300 and 393 of Haddon (2011). (…maximum likelihood estimate of the variance and uses n rather than n – 1).
Page 332 of Haddon (2011): Note the division by n instead of n – 1 to give the maximum likelihood estimate (Neter et al., 1996, p. 34).
References:
Neter, J., M. H. Kutner, C. J. Nachtsheim, and W. Wasserman. 1996. Applied linear statistical models.
Haddon, M. 2011. Modelling and quantitative methods in fisheries, 2nd ed. Chapman and Hall/CRC, London, U.K.
L 267, In the model fitting, the more the parameters, the better the fitting is. So, it is not surprised that the observed variance case had the best fit. However, this model should be compared with the constant variance one to see the difference and make discussion.
R: As commented above, the reviewer is right: the larger the set of parameters in the model, the better the model fit. As explained, in our case variance was considered as a parameter included in the information criterion analysis. The original model was contrasted with constant variance and observed variance. In our case, the BIC (stricter and more consistent than AIC) penalized the model with more parameters and yet the von Bertalanffy model with observed variance and many parameters was the most plausible. We believe that this is precisely a good reason to use observed variance at the risk of augmenting the number of parameters. On the other hand, the Gompertz model with observed variance was not plausible at all, even when it contained the same number of parameters as the von Bertalanffy. Respectfully, we believe that herein lies the importance of our procedure and the manuscript itself.
L 273, Will MCMC be a better approach to calculate the 95% C.I.?
R: Although the reviewer might be right, the Markov Chain Monte Carlo method is more appropriate for a nonparametric problem which is not our case. The question is interesting and certainly deserves a specific investigation. However, we respectfully believe that for the purpose of our work, the commonly used Chi-square based method suffices.
Reviewer 3 Report
The author should be congratulated for conducting an important study. But the manuscript needs to be edited for grammar and syntax.
Please check and cite this paper 10.3750/AIEP/01977 (if it is possible).
Author Response
Reviewer 3
The author should be congratulated for conducting an important study. But the manuscript needs to be edited for grammar and syntax.
Please check and cite this paper 10.3750/AIEP/01977 (if it is possible).
R: We thank the reviewer for her/his encouraging comment and also for flagging some general improvements needed in the grammar and syntax. We note that the manuscript was revised and corrected by two US English language speaker researchers, which we now acknowledge in the new version. We also included the recommended paper in the Discussion section; thank you for your suggestion.
Round 2
Reviewer 2 Report
This revised MS has answered most questions I raised in the first round review. Some minor corrections have to be made before it can be fully accepted for publication as follows:
L 120, 121, As these two models also applied to the Gompertz model, K should be k or kG. Please specify that K represents k or kG.
Table 3, As size range and sample size of the specimens may affect the estimations of growth parameters, I suggest the authors add the information of all studies to this table.
Author Response
1) "L 120, 121, As these two models also applied to the Gompertz model, K should be k or kG. Please specify that K represents k or kG." R: This observation was already taken into account and, accordingly, the text and equations were modified to differentiate between the growth coefficient for both models (von Bertalanffy and Gompertz). This can be observed in lines 110 to 122, where K is used generically whereas k is used for the von Bertalanffy and kg for the Gompertz models, respectively. Such changes, as all others, are highlighted in yellow. 2) "Table 3, As size range and sample size of the specimens may affect the estimations of growth parameters, I suggest the authors add the information of all studies to this table." R: We thank Reviewer 2 for this suggestion, which brings even more clarity to our manuscript. The revised manuscript's Table 3 was modified as follows: addition of a) sample size (N) per author, and b) size range (R) per author (highlighted in yellow for quick reference). The table heading was amended according to these additions.