Relating Fluctuating Asymmetries and Mean Values and Discordances of Asymmetries in a Set of Morphological Traits
Round 1
Reviewer 1 Report
In this paper, the author contends that the relationship between fluctuating asymmetry and the mean-value of a trait can vary, intrinsically so, and not just because of the mixture of multiplicative and additive errors due to active growth and measurement error. It is an interesting idea, though I am not sure that the author absolutely proves that the differences in the relationship between fluctuating asymmetry and mean-value exist even when measurement error is eliminated.
The author uses a variance components analysis to partition out measurement error. I don’t think, however, that you can conclude that “the Pearson-Davin-Lande-Soulé effect is not an artefact caused by measurement error” based on this approach. Active growth (new tissue to old tissue) generates log-normal distributions. Growth of inert structures generates normal distributions. The variance of a lognormal distribution is , whereas that of a normal distribution is σ2. A safer (and foolproof) approach would be to eliminate all measurement error by averaging replicate measurements, and only then comparing the asymmetry variances. There are assumptions implicit in the mixed-model analysis of variance that are not addressed. This requires a mathematical statistician to unravel the behavior of a mixed distribution. One could create an artificial data set that is a mixture of multiplicative and additive error and see how that behaves in the face of a mixed-model ANOVA. But having an actual mathematical proof is always more powerful.
Measurement error is likely to vary among traits and could account for some of the discordance among traits, but not among populations. This is why averaging replicates is such a powerful approach. Measurement error can be reduced to nothing with enough replicates. If the slopes of the fluctuating asymmetry and mean-value regressions are still negative, it suggests other sources of additive error. That would be very interesting in itself.
The log transformations are unclear—log r – log l is not the same as log(r – l) and log (r + l) is not the same as log r + log l. It is also unclear what is meant by fluctuating asymmetry and mean value. Is fluctuating asymmetry Var(r – l), or is it something else? Is mean-value E[(r + l)/2], or something else? How are you standardizing by size? Equations will make it clear.
While d’ = log r – log l makes the most sense for traits that exhibit purely active growth (multiplicative error), any mixture of additive and multiplicative error can be dealt with by using a Box-Cox Power transform, where y(λ) = (y λ - 1)/λ for λ ≠ 0 and y(λ) = log y for λ = 0. A power transformation can accommodate a linear transform (λ = 1), a log transform (λ = 0), and everything in between. A square-root transform, for example, is possible when λ = 0.5. A transformed asymmetry value, regardless of the transformation, could be d’ = [(r λ - 1)/λ] - [(l λ - 1)/λ]. Power transformations like this have been used in the literature when measurement error could not be reduced because replicate measurements were not made.
The author needs to address the traits used in the study. According to Mosimann and Campbell (1988), exoskeletons should exhibit additive error, not multiplicative error. If that is so, then no correction for size scaling should be required. It is also possible that a mixture of additive and multiplicative error might occur when there are several molts of the exoskeleton. And different populations may have different numbers of individuals at different molts and hence having different mixtures of additive and multiplicative error.
I think the main issue is with the traits used in this study. There is probably no need to standardize the asymmetry. Doing so can easily generate negative slopes. You need to address why the crustacean traits (involving an exoskeleton) and meristic traits in fish standardized? It is likely that these traits don’t involve active tissue growth and don’t require standardization. This could easily explain the negative slopes in the data.
Why does fluctuating asymmetry and mean-value represent a two-dimensional complex? This needs to be explained more thoroughly, and perhaps in the language of mathematics. I would think that it might actually be a three-dimensional (or more) complex. Fluctuating asymmetry is a function of measurements on right and left sides and so is the mean-value. One is a variance and the other a mean, based on the same measurements—Var(r – l) and E(r + l). I would think that r and l represent two dimensions, so if something else is influencing the relationship, it suggests at least three dimensions. A plot of r against l encapsulates both difference and sum in the same two-dimensional plot. What it doesn’t show is how the error ε might vary (increasing or decreasing at different rates) from population to population. Of course, one can boil r versus l down to one-dimension by using principal components analysis to factor out size. One would need to do an eigen-analysis on the variance-covariance matrix, not the correlations between r and l, and then examine the residuals. What I think you are saying is that the expansion of the residuals around PC1 vary from population to population. That could be viewed as two-dimensions, with the error ε increasing (or decreasing) with PC1. For each population, the rate of change is different. This all needs to be mulled over.
If you still have the raw data from all of these experiments it may be worthwhile looking at measurement error more closely. Moreover, each trait should be evaluated for whether the main variation is additive or multiplicative. If it is additive, you probably don’t need to standardize any of the variables.
The references need to be in the style used by mdpi, which uses numbers in order of citation.
The writing requires attention, especially the first several paragraphs and the abstract. The writing improves once you get into the meat of the paper.
I have made many comments and suggested changes in the manuscript itself.
Suggested References
Mosimann JE, Campbell G. 1988. Applications in biology: simple growth models. In: Crow EL, Shimizu K, eds. Lognormal distributions: theory and applications. New York: Marcel Dekker, 287–302.
Comments for author File: 
Comments.pdf
Author Response
Reviewer 1
Reviewer comments in italic font
Comments and Suggestions for Authors
In this paper, the author contends that the relationship between fluctuating asymmetry and the mean-value of a trait can vary, intrinsically so, and not just because of the mixture of multiplicative and additive errors due to active growth and measurement error. It is an interesting idea, though I am not sure that the author absolutely proves that the differences in the relationship between fluctuating asymmetry and mean-value exist even when measurement error is eliminated.
I understand that this statement of the Respected Reviewer is described in more detailed below and I’ll mostly respond below. Here, I would like to respond that I do not use some “intrinsic” factors for explanation of empirically observed patterns of relationship between means and FAs, but try to explain them with a combination of additive and multiplicative errors generated by growth. And differences between different populations are explained by different ratio of additive and multiplicative errors. The measurement error may bring additional additive error and thus confound such analyses, and thus it has been removed as much as it possible. If the negative slope of FAs on means occurs, I believe, it cannot be absolutely the same for different sampes/popuations, and MAY differ between them, and thus may cause discordance between the FAs of various traits across the samples.
The author uses a variance components analysis to partition out measurement error. I don’t think, however, that you can conclude that “the Pearson-Davin-Lande-Soulé effect is not an artefact caused by measurement error” based on this approach. Active growth (new tissue to old tissue) generates log-normal distributions. Growth of inert structures generates normal distributions. The variance of a lognormal distribution is, whereas that of a normal distribution is σ2. A safer (and foolproof) approach would be to eliminate all measurement error by averaging replicate measurements, and only then comparing the asymmetry variances. There are assumptions implicit in the mixed-model analysis of variance that are not addressed. This requires a mathematical statistician to unravel the behavior of a mixed distribution. One could create an artificial data set that is a mixture of multiplicative and additive error and see how that behaves in the face of a mixed-model ANOVA. But having an actual mathematical proof is always more powerful.
I agree that the measurement error causes some issues in such analyses, which is difficult to avoid. Here I can respond that (i) I am not sure that suggested by the Reviewer approach to eliminate all measurement error by averaging replicate measurement is practical because, in theory, to remove the measurement error completely, one needs to perform the endless number of replicates; (ii) I did not addressed the parameters of distributions I worked with in this manuscript in the first version because I paid quite a lot of attention to these issues in papers I took the empiric material from (Lajus and Alekseev 2000; Lajus 2001; Lajus et al, 2003b,c), and trying to avoid overloading of the manuscript. I added, however, a substantial information about methodology of my analysis in the updated version of the manuscript; (iii) in one paper, Lajus et al., 2003c, on isopod Saduria entomon, the measurement error accounts for only about 1% of observed fluctuating asymmetry, and thus it can be technically neglected. The Pearson-Davin-Lande-Soulé effect there is about the same magnitude as in other cases (see Table in the ms), thus it provides additional evidence that measurement error is not a reason of the Pearson-Davin-Lande-Soule effect. In the updated version, I have added quantitative data on contribution of measurement error in source papers; (iv) I agree that it is possible to create an artificial data set to demonstrate a mixture of multiplicative and additive error, and my preliminary modelling with that showed the agreement of the model with the hypothesis, but I do not have sufficient mathematical expertise to present this data in the paper format here.
Measurement error is likely to vary among traits and could account for some of the discordance among traits, but not among populations. This is why averaging replicates is such a powerful approach. Measurement error can be reduced to nothing with enough replicates. If the slopes of the fluctuating asymmetry and mean-value regressions are still negative, it suggests other sources of additive error. That would be very interesting in itself.
I already responded partly to this comment above. Here I would like to address the statement that the “Measurement error is likely to vary among traits and could account for some of the discordance among traits, but not among populations”. It seems to be true in most cases, but not in all. For instance, in our analysis of phenotypic variation of muscle prints on blue mussels (Lajus et al., Symmetry 2015, 7, 488–514 https://doi.org/10.3390/ sym7020488) we found that measurement error can differ between the populations.
The log transformations are unclear—log r – log l is not the same as log (r – l) and log (r + l) is not the same as log r + log l. It is also unclear what is meant by fluctuating asymmetry and mean value. Is fluctuating asymmetry Var(r – l), or is it something else? Is mean-value E[(r + l)/2], or something else? How are you standardizing by size? Equations will make it clear.
I understand differences between Log r – log L and Log (L-r). I have added the clarifications about the used methodology in the updated version of the manuscript (Section Relationship between FAs and trait means in a set of multiple traits).
While d’ = log r – log l makes the most sense for traits that exhibit purely active growth (multiplicative error), any mixture of additive and multiplicative error can be dealt with by using a Box-Cox Power transform, where y(λ) = (y λ - 1)/λ for λ ≠ 0 and y(λ) = log y for λ = 0. A power transformation can accommodate a linear transform (λ = 1), a log transform (λ = 0), and everything in between. A square-root transform, for example, is possible when λ = 0.5. A transformed asymmetry value, regardless of the transformation, could be d’ = [(r λ - 1)/λ] - [(l λ - 1)/λ]. Power transformations like this have been used in the literature when measurement error could not be reduced because replicate measurements were not made.
Thank you. I did log (r-l), as most of researchers of FA. The methods are provided in the manuscript.
The author needs to address the traits used in the study. According to Mosimann and Campbell (1988), exoskeletons should exhibit additive error, not multiplicative error. If that is so, then no correction for size scaling should be required. It is also possible that a mixture of additive and multiplicative error might occur when there are several molts of the exoskeleton. And different populations may have different numbers of individuals at different molts and hence having different mixtures of additive and multiplicative error.
The traits are briefly described in the caption of the Table 1. More detailed description is available from the papers used in these analyses. I do not think it is needed here to provide more information to avoid the overloading of the manuscript with unnecessary details. In fact, the traits are quite different (number of fat lacunae in fish bones, fish bone dimensions, various dimensions of crustaceans, number of chaetae in crustaceans). I tried to put additional accent to this in the paper. Thanks for pointing out reference to Mosimann and Campbell (1988). The paper by Graham et al 2003, which I cited while writing about multiplicative and additive error, uses Mosimann and Campbell paper, thus it implicitly presented in my work also. I cited it explicitly in the updated version.
I think the main issue is with the traits used in this study. There is probably no need to standardize the asymmetry. Doing so can easily generate negative slopes. You need to address why the crustacean traits (involving an exoskeleton) and meristic traits in fish standardized? It is likely that these traits don’t involve active tissue growth and don’t require standardization. This could easily explain the negative slopes in the data.
The necessity of FA standardization (as well as other types of phenotypic variance) come from heterogeneity of individuals in a sample and because trait mean increases with body size. If all individuals would be the same size, or if variances would not increase with size and average size of individuals would be the same, it would be no necessity for standardization. But unfortunately, this is not the case. Also, standardization is often used when one wants to compare and/or combine different traits with different means, for instance, while using composite FA indices. This requires the partition out phenotypic into factorial component, stochastic component (true fluctuating asymmetry), and measurement error. It is why researches so often use standardization. Standardization would not result in negative FAs vs means slope if only multiplicative errors would present. It only occurs because of additive errors. Thus standardization allows to detect presence of additive error. And from the presented data we see that the additive error is very common – at least it occurs in all structure which were studied.
Why does fluctuating asymmetry and mean-value represent a two-dimensional complex? This needs to be explained more thoroughly, and perhaps in the language of mathematics. I would think that it might actually be a three-dimensional (or more) complex. Fluctuating asymmetry is a function of measurements on right and left sides and so is the mean-value. One is a variance and the other a mean, based on the same measurements—Var(r – l) and E(r + l). I would think that r and l represent two dimensions, so if something else is influencing the relationship, it suggests at least three dimensions. A plot of r against l encapsulates both difference and sum in the same two-dimensional plot. What it doesn’t show is how the error ε might vary (increasing or decreasing at different rates) from population to population. Of course, one can boil r versus l down to one-dimension by using principal components analysis to factor out size. One would need to do an eigen-analysis on the variance-covariance matrix, not the correlations between r and l, and then examine the residuals. What I think you are saying is that the expansion of the residuals around PC1 vary from population to population. That could be viewed as two-dimensions, with the error ε increasing (or decreasing) with PC1. For each population, the rate of change is different. This all needs to be mulled over.
When I say about two-dimensionality of FA, I try to explain that FA cannot be described just with one number, i.e. in one-dimensional space as it is usually done, because of losing the essential information. Description of FA in two dimensional space, i.e. FA vs mean is more adequate. Definitely, there are other important relevant variables, not only these two, and thus the complex has more dimensions, but in the framework of this model and this manuscript I would be happy to prove that FA is not one-dimensional variable, but at least two-dimensional.
It is difficult for me to use language of mathematics because of my biological background, but also, I am not sure that it is necessary, because I am trying to approach biologists, not mathematicians.
Trying to describe the whole picture only using l and r values looks impossible for me because here I do not see how I can incorporate in the model different traits, which, due to variation in their means, create the second dimension of the complex. Also, l and r are normally made in at least two replicates to address measurement error, which is also needed to be reflected in some way.
If you still have the raw data from all of these experiments it may be worthwhile looking at measurement error more closely. Moreover, each trait should be evaluated for whether the main variation is additive or multiplicative. If it is additive, you probably don’t need to standardize any of the variables.
I have most of raw data available, but I understand that the study you suggest requires special design, in particular, a wide range of sizes of individuals in the same sample, and a large number of individuals per samples – only this will allow one to quantify ratio of additive and multiplicative variation in each trait (I am quite convinced that most of traits do generate both types of error during their growth, but not either additive or multiplicative error). The samples I have are rather small and usually are not very diverse in terms of size heterogeneity of individuals in order to minimize additional variation caused, for instance, by allometric and nonlinear growth patterns. Thus, I believe, the suggested needs to be designed specially.
The references need to be in the style used by mdpi, which uses numbers in order of citation.
Thanks, from the Instructions for authors I understood that Symmetry allows to use any format of references during the preliminary stages of submission, thus I used what is more convenient for me to work with and for readers (according to my assumption). I’ll follow the Symmetry format on the later steps of the publication process.
The writing requires attention, especially the first several paragraphs and the abstract. The writing improves once you get into the meat of the paper.
Thanks, I try to improve it as much as I can paying a special attention to sections you indicated, also, I asked a native speaker to work with the manuscript.
I have made many comments and suggested changes in the manuscript itself.
Thanks a lot, I have accepted most of them. Below there are some which need my special comments:
Line 29. The first papers on environmental stress and FA are a great deal earlier than this. The set of papers by DW Valentine, ME Soule, P Samollow come to mind, in 1973.
Yes, it is true. But here I wanted to indicate not the first pioneering studies, but the period when substantial information on FA was already accumulated, which allowed to propose FA for a wide use as stress indicator. This happened approximately in late 1980s-early 1990s, it is why I did not cite the first references.
35: I don't see the number of new papers starting to slow until 2005.
I provide here a picture from paper Lajus et al 2019 for you I guess that it allows to see the pattern I mentioned quite clearly (black rows).
47: Waddington also addressed different buffering of traits. Some traits are highly stable, others are not. If symmetry is important for fitness it is likely to be buffered. Selection can act on different parts of the genome.
I agree. But here I am dealing with somewhat different aspect, discordance of FAs across traits in different samples/populations, not with different buffering of the traits. They can be buffered differently, and thus manifest different levels of FA., but it does not mean that they vary disconcordantly in different samples/populations.
52: Hasn't this been realized from the beginning?
Here I do not consider temporal aspects, just logical.
167: please define what factorial and stochastic components are
Factorial component is defined at line 115: “…among-individual is non-stochastic, or the factorial component which results from genetic and environmental heterogeneity (Kozhara 1994; Lajus et al 2003 a)”.
187: Measurement error is additive as well. It is the main source of the mixture of additive and multiplicative error in many traits. This is easily tested by averaging replicate measurements.
Here, I was citing paper Graham et al., 2003a, which was mentioned in the previous sentence. I repeated this citation here to avoid misunderstanding.
195: If you are estimating variance components, you are making certain assumptions regarding their behavior in the face of mixed distributions.
For the data, which were used for partitioning total variance to stochastic and factorial components, we tested l-r distribution with analysis of skewness and kurtosis and removed traits which showed significant departures from normality after application of Bonferroni correction. Also, we studied these parameters for L,R distributions.
207: Why? Wouldn't this depend on the nature of the error distributions?
I do not know why, I am just stating the fact in this case. Here it is important that not square-root transformation, but log-transformation results in better linearization of the relationship between FAs and means.
209 shouldn't the slope decrease from the effect of additive error?
Yes, I have mistaken, it decreases in terms of signed values
Fig. 1: log-transform of an already standardized FA is like applying standardization twice. If these traits all involve exoskeletal elements, they may not require any standardization.
Standardization involves two usually interconnected variables, one divided to another. Log-transformation is a change a shape of distribution of the same variable.
232: meristic traits are also likely to exhibit additive variation. See Mosimann and Campbell.
Yes, they, as all other traits, show negative slope meaning presence of additive error.
256: One might use the methods of meta-analysis here. For each trait, one estimates an effect size. This is legitimate because all traits are potentially measuring developmental instability.
The aim of this study was to demonstrate the very existence of negative slope and FA discordances. I was thinking about using literature data to expand material for such research, but this might be a topic for further studies because requires a special design. I think it is quite difficult to find papers with trait set containing a sufficient number of traits considerably differing in means, partitioning out total phenotypic variance on components and measurement error. Because of that combining of data from different studies may result in high heterogeneity of data and confound the relationships which are analyzed.
In the end of my response, I have to say that I agree that “it all needs to be mulled over” more, even despite I am dealing with this topic quite long time. I very much appreciate your detailed and substantial discussion of this work, and that you would like to sign your review report. However, I have to say that according to the journal rules, a name of the reviewer cannot be disclosed.
Author Response File: 
Author Response.docx
Reviewer 2 Report
I found this manuscript an interesting addition to the research on fluctuating asymmetry and have only very few comments for improvement:
-Please check the language quality throughout (e.g. in the Acknowledgements, it must be "kind" instead of "kindly")
-I would recommend also discussing "Fluctuating asymmetry: A tool for impact assessment on fish populations in a tropical polluted bay, Brazil" by Seixas et al. (2016) as it provides some additional aspects on the topic.
Author Response
Reviewer 2
Reviewer comments in italic font
Comments and Suggestions for Authors
I found this manuscript an interesting addition to the research on fluctuating asymmetry and have only very few comments for improvement:
Thanks for reading this.
-Please check the language quality throughout (e.g. in the Acknowledgements, it must be "kind" instead of "kindly")
Thank you, I’ve sent the ms for proofreading by native speaker and now, I hope the writing is much better.
-I would recommend also discussing "Fluctuating asymmetry: A tool for impact assessment on fish populations in a tropical polluted bay, Brazil" by Seixas et al. (2016) as it provides some additional aspects on the topic.
Thanks for pointing this paper. I read it attentively, but I did not find its direct relevance for the Discussion in my manuscript. Seixas et al (2016) paper’s main conclusion (in the part related to difference in response of different traits) reads that “These differences appeared to be mostly related to the synergetic effects of measurement errors and the interference of other types of asymmetry, since none of these potential biases were detected for the number of gill rakers while the opposite trend (i.e. significant measuring errors and DA) was found for the diameter of the eye” (p. 529). This paper discusses difference between traits indeed, but the discussion is related to different aspects of trait FA heterogeneity than my paper – presence of measurement error and other types of asymmetries in addition to fluctuating asymmetry in different traits. In material I used, the measurement error has been partitioned out, and tests for other types of asymmetries were performed and proven that they are absent.
Author Response File: Author Response.docx
Reviewer 3 Report
As stated in the Ms this review aims to analyse the relationship between fluctuation asymmetry (FA) and mean values in a set of morphological traits to address a problem of FA concordance across traits.
Overall I consider that the presentation throughout the Ms is not clear (sometimes very confusing) and there is not a clear focussed text or description which does not allow a clear understanding of the explanation of the author's ideas. This also involves the Abstract which is insufficiently prepared as some basic aspects are not mentioned: objectives of the study, main methods and main conclusions.
Under my point of view, the analysis of FA is interesting and relevant if connected or related to biological aspects related the relative growth. In the Ms this relation fails to be convincingly (or even sufficiently) demonstrated and in the end it apparently resumes to a manipulation of FA data obtained in some biological models.
In sum, I consider that this Ms brings no relevant contribution to the use of morphological traits to analyse or describe relative growth and this it has no conditions to be accepted for publication in this Journal
Author Response
Reviewer 3
Reviewer comments in italic font
As stated in the Ms this review aims to analyse the relationship between fluctuation asymmetry (FA) and mean values in a set of morphological traits to address a problem of FA concordance across traits.
Overall I consider that the presentation throughout the Ms is not clear (sometimes very confusing) and there is not a clear focussed text or description which does not allow a clear understanding of the explanation of the author's ideas. This also involves the Abstract which is insufficiently prepared as some basic aspects are not mentioned: objectives of the study, main methods and main conclusions.
Thanks for the comments. I tried to address it them as much as I can by improving the updated version of the ms.
Under my point of view, the analysis of FA is interesting and relevant if connected or related to biological aspects related the relative growth. In the Ms this relation fails to be convincingly (or even sufficiently) demonstrated and in the end it apparently resumes to a manipulation of FA data obtained in some biological models.
This manuscript in indeed addresses the biological aspects related to growth. The whole manuscript is built up on multiplicative and additive errors generated during the growth. I do not know how it is possible to demonstrate relation of topic of my study to growth. Data manipulation is quite a serous accusation for a researcher. I strongly disagree with this and I would not like to leave it unanswered, but I cannot respond it right now because the accusation is very generic.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
You have addressed most of the critical comments I had.
Reviewer 3 Report
I had access to a revised and improved Ms and I think that the comments issued in my previous review (which had problems during the uploading to the platform) we duly considered and included in this revised outline.
I think that the Ms might be accepted as it is in this revised version.
