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On the suitability of an allometric proxy for nondestructive estimation of average leaf dry weight in eelgrass shoots I: sensitivity analysis and examination of the influences of data quality, analysis method, and sample size on precision
- Héctor Echavarría-Heras^{1}Email author,
- Cecilia Leal-Ramírez^{1},
- Enrique Villa-Diharce^{2} and
- Nohe Cazarez-Castro^{3}
https://doi.org/10.1186/s12976-018-0076-y
© The Author(s). 2018
- Received: 11 September 2017
- Accepted: 22 January 2018
- Published: 6 March 2018
Abstract
Background
The effects of current anthropogenic influences on eelgrass (Zostera marina) meadows are noticeable. Eelgrass ecological services grant important benefits for mankind. Preservation of eelgrass meadows include several transplantation methods. Evaluation of establishing success relies on the estimation of standing stock and productivity. Average leaf biomass in shoots is a fundamental component of standing stock. Existing methods of leaf biomass measurement are destructive and time consuming. These assessments could alter shoot density in developing transplants. Allometric methods offer convenient indirect assessments of individual leaf biomass. Aggregation of single leaf projections produce surrogates for average leaf biomass in shoots. Involved parameters are time invariant, then derived proxies yield simplified nondestructive approximations. On spite of time invariance local factors induce relative variability of parameter estimates. This influences accuracy of surrogates. And factors like analysis method, sample size and data quality also impact precision. Besides, scaling projections are sensitive to parameter fluctuation. Thus the suitability of the addressed allometric approximations requires clarification.
Results
The considered proxies produced accurate indirect assessments of observed values. Only parameter estimates fitted from raw data using nonlinear regression, produced robust approximations. Data quality influenced sensitivity and sample size for an optimal precision.
Conclusions
Allometric surrogates of average leaf biomass in eelgrass shoots offer convenient nondestructive assessments. But analysis method and sample size can influence accuracy in a direct manner. Standardized routines for data quality are crucial on granting cost-effectiveness of the method.
Keywords
- Eelgrass conservation
- Nondestructive estimation
- Allometric projection suitability, methodological influences
- Sensitivity analysis
Background
Based on published data for 17 ecosystem services in 16 biomes, Costanza et al. [1] estimated the value of ecosystem services at the level of the whole biosphere. They found a lower bound in the range of US$16–54 trillion (1012) per year, with an average of US$33 trillion per year. Marine systems produced near 63% of the annual value. Almost half derived from coastal ecosystems. Approximately 25% of this share related to algal beds and seagrasses. This contribution to human welfare is deem relevant. Thus maintaining the health of marine ecosystems is subject of scientific concerns. Currently, increasing anthropogenic pressures pose important threats. And global climate change could also threaten future viability of seagrass meadows [2, 3]. For instance, water quality and other local stressors promote unprecedented meadow loss [4–6]. This reduces mitigation of wave action [7] and filtration [8]. Diminishes food and shelter for a myriad of organisms [9–11]. Weakens nutrient cycling [12, 13], erosion abatement and shoreline stabilization [14–16]. Moderates support for detrital food web foundation [17]. And inhibits carbon sequestration [18, 19].
Eelgrass (Zostera marina L.) is a dominant along the coasts of both the North Pacific and North Atlantic [20]. This species supports communities known as among the richest and most diverse in sea life [21]. Contribution of organic materials for food webs in shallow environments [22] is noticeable. Indeed, eelgrass produced up to 64% of the whole primary production of an estuarine system [23]. Current deleterious effects of anthropogenic influences on eelgrass prompted special restoration strategies. Among remediation efforts replanting plays an important role [24–27]. Transplant success amounts to reinstatement of ecological functions of natural populations. Evaluation relies on monitoring standing stock and productivity of transplanted plants. Then comparing with assessments of a reference population, which usually settle nearby [28].
Combined biomass of leaves in shoots is an important component of standing stock. Assessments rely on the estimation of the biomass of individual leaves. This requires shoot removal followed by dry weight measurement procedures in the laboratory. Elimination of shoots could infringe damage to natural eelgrass populations [29]. And reduced shoot density makes these effects even more perceptible for transplanted plots. Allometric methods make it possible simplified-indirect estimations of eelgrass productivity and standing stock. Echavarría-Heras et al. [30] considered an allometric representation for eelgrass leaf biomass and related length. Agreeing with Solana et al. [31], the involved parameters are invariant within a given geographical region. Estimates and leaf length measurements grant nondestructive approximations of observed leaf biomass values. This way, leaf length measurements grant nondestructive approximations for observed leaf biomass values. Leaf growth rates estimation relies on successive measurements of leaf biomass. Then the allometric model in [30] entails nondestructive assessments of eelgrass productivity. But, invariance does not impede local factors to imply variability of parameter estimates. Besides, local influences other factors could explain numerical differences in parameter estimates. There are methodological influences that may render biased parameter estimates. Analysis method, sample size, and data quality can influence scaling results (e.g. [32, 33]). And, since scaling relationships are particularly sensitive to parametric uncertainties, Echavarría-Heras et al. [30] concluded that the actual precision of derived allometric surrogates requires clarification.
Here we deal with allometric surrogates for average leaf biomass in eelgrass shoots. These derive from the model w(t) = βa(t)^{ α } for leaf biomass w(t) and area a(t) measured at time t, and α and β parameters. Leaf area is more informative of eelgrass leaf biomass than corresponding length. Thus, the present scaling endures a boost in precision of parameter estimates by the model in [30]. This could increase the accuracy of derived surrogates for leaf biomass in shoots. Besides, eelgrass leaf area and length admit an isometric representation [34, 35]. Then, the time invariance found by Solana-Arellano et al. [31] also holds for parameter estimates of the present scaling. This by the way imbeds a nondestructive advantage to the present shoot-biomass substitutes. But, agreeing with Echavarría-Heras et al. [30], we must examine influences on precision of estimates for suitability of projections. Since, such an analysis was not produced before, we took here the try of filling that gap. Achieving the related goals, required the assemblage of an extensive data set. It comprises coupled measurements of eelgrass leaf biomasses and related areas. This is further called “raw data set”. A data cleaning procedure adapted from Echavarría-Heras et al. [30] removed inconsistent leaf biomass replicates from the raw data. Thereby forming what we call a “processed data set”. Differences in reproducibility strength allowed to assess data quality effects in precision. A similar procedure evaluated sample size effects. And a sensitivity analysis evaluated robustness of the projection method. This supports consistent, cost-effective allometric projections of observed values from raw data. But, this depends on nonlinear regression as an analysis method. Besides, sample size must be optimal. Data quality as expected improved reproducibility strength of the allometric projection method. But, this factor was more relevant in optimizing sample size. A detailed explanation of used procedures appears in the methods section. The results section is not only devoted to the presentation of our findings. It also examines the relative strengths of factors influencing the precision of proxies. A Discussion section emphasizes on the gains and the limitations of the method. Appendix 1 deals with the model selection problem. Appendix 2 is about data processing methodology. Appendix 3 presents the procedure for sensitivity assessment.
Methods
The present raw data come from a coastal lagoon located in San Quintin Bay, México [30]. This comprises 10,412 leaves and measured lengths [mm], widths [mm] and dry weights (g). The product of length times width provided estimations of leaf area [mm^{2}] [36]. In what follows the symbol n_{ ra } stands for number of leaves in raw data. Processed data results by applying direct and statistical data cleaning techniques. The direct hinges on the consistency of allometric models for eelgrass leaf biomass. Leaf length or area are allometric descriptors of eelgrass leaf biomass [30, 31, 34]. A model selection exploration corroborated a power function like trend assumption for the present data. Details appear in Appendix 1. Severe deviations, from the mean response curve, are inconsistent and must be removed. This took care of sets containing less than ten leaf dry weight replicates. The statistical procedure worked on sets with a larger number of replicates. It centers on properties of the median of a group of data. This is immune to sample size and also a robust estimator of scale. The adapted Median Absolute Deviation (MAD) data cleaning procedure [37] appears in Appendix 2. Processing data resulted in a number of n_{ qa } = 6094 pairs of leaf dry weights and areas.
Parameter estimates \( \widehat{\alpha} \) and \( \widehat{\beta} \) and leaf area values yield allometric proxies \( {w}_m\left(\widehat{\alpha},\widehat{\beta},t\right) \) for w_{ m }(t). (cf. Eq. (19)). The symbol \( \left\langle {w}_m\left(\widehat{\alpha},\widehat{\beta},t\right)\right\rangle \) (cf. Eq. (20)) stands for the pertinent average over sampling dates. We use Lin’s Concordance Correlation Coefficient (CCC) [38] as an evaluation of reproducibility. This meant as the extent to which two connected variables fall on a line through the origin and with a slope of one. We represent this statistic by means of the symbol \( \widehat{\rho} \). Agreement defined as poor whenever \( \widehat{\rho}<0.90 \), moderate for \( 0.90\le \hat{\rho}<0.95 \), good for \( 0.95\le \hat{\rho}\le 0.99 \) or excellent for \( \widehat{\rho} \)>0.99 [39]. Values of \( \widehat{\rho} \) gave an evaluation of the strength of the \( {w}_m\left(\widehat{\alpha},\widehat{\beta},t\right) \) devise to reproduce observed values.
In getting parameter estimates \( \widehat{\alpha} \) and \( \widehat{\beta} \) we relied on two procedures. The traditional analysis method of allometry and nonlinear regression. Assessing analysis method effects on reproducibility strength of \( {w}_m\left(\widehat{\alpha},\widehat{\beta},t\right) \) depended on testing differences in \( \widehat{\rho} \). The traditional approach involves a linear regression equation (cf. Eq. (4)). This obtained through logarithmic transformation of response and descriptor in Eq. (1). The nonlinear regression analysis method relied on maximum likelihood [40, 41]. This approach fitted the model of Eq. (1) in a direct way in the original arithmetical scale. The nonlinear fit allowed the consideration of homoscedasticity or heteroscedasticity (cf. Eqs. (5) and (6)). All the required fittings for both raw and processed data depended on the use of the R software.
We also fitted the model of Eq. (1) to samples of different sizes taken out from primary and processed data sets. Each sample of size k; with 100 ≤ k ≤ n_{ ra } produced estimates \( \widehat{\alpha}(k) \) for α and \( \widehat{\beta}(k) \) for β, and resulting \( {w}_m\left(\widehat{\alpha}(k),\widehat{\beta}(k),t\right) \) projections. The symbol \( \widehat{\rho}(k) \) denotes the value of \( \widehat{\rho} \) for a sample of size k. Differences in \( \widehat{\rho}(k) \) allow exploring sample size influences in reproducibility.
Deviations ∆α_{ q } and ∆β_{ r } convey fluctuating values α_{ q } = α + ∆α_{ q } and β_{ r } = β + ∆β_{ r } for the parameters α and β one to one. The modulus of the vector of parametric changes (∆α_{ q }, ∆β_{ r }) defines a tolerance range θ(q, r). And the value of θ(q, r) determined by the standard errors of parameter estimates denoted by mean of θ_{ ste }. A fixed value of θ(q, r) leads to four possible characterizations of the pair (∆α_{ q }, ∆β_{ r }). Each one associates to a trajectory w_{ m }(α_{ q }, β_{ r }, t) shifting from a reference one w_{ m }(α, β, t). The symbol δw_{ mθ }(α_{ q }, β_{ r }, t) (cf. Eq. (42)) denotes deviations between reference and average of shifting trajectories at sampling dates. And the average of δw_{ mθ }(α_{ q }, β_{ r }, t) values taken over all sampling dates denoted through 〈δw_{ mθ }(α_{ q }, β_{ r }, t)〉 (cf. Eq. (43)). The absolute value of the ratio of 〈δw_{ mθ }(α_{ q }, β_{ r }, t)〉 to 〈w_{ m }(α, β, t)〉 defines a relative deviation index ϑ(θ). It measures sensitivity of 〈w_{ m }(α, β, t)〉 to fluctuations of tolerance θ(q, r) on α and β. Appendix 3 presents detailed formulae.
Results
Parameter estimates \( \widehat{\alpha} \) and \( \widehat{\beta} \) associated standard errors (\( ste\left(\widehat{\alpha}\right), ste\Big(\widehat{\beta} \))) found by fitting the model of Eq. (1). Nonlinear regression estimates associate to the homoscedastic case of the model of Eqs. (5) and (6) (see Appendix 1). Values of \( \widehat{\rho} \) give an evaluation of reproducibility strength of the proxy of Eq. (1)
Analysis method Data | \( \widehat{\beta} \) | \( ste\left(\widehat{\beta}\right) \) | \( \widehat{\alpha} \) | \( ste\left(\widehat{\alpha}\right) \) | \( \widehat{\rho} \) |
---|---|---|---|---|---|
Log-linear Transformation Raw | 1.3674x10^{−5} | 2.9355 × 10^{− 7} | 1.023 | 3.662 × 10^{− 3} | 0.8910 |
Nonlinear Regression Raw | 8.718x10^{−6} | 3.530 × 10^{−7} | 1.104 | 5.101 × 10^{−3} | 0.9307 |
Log-linear Transformation Processed | 1.142 x10^{−5} | 2.0831 × 10^{−7} | 1.046 | 3.035 × 10^{−3} | 0.9455 |
Nonlinear Regression Processed | 6.956 x10^{−6} | 2.200 × 10^{−7} | 1.132 | 3.954 × 10^{−3} | 0.9777 |
For raw data the log-linear transformation method produced \( \widehat{\rho}=0.8910 \), entailing poor reproducibility. This explains a biased distribution of replicates around the mean response curve (Fig. 3a). Meanwhile, estimates acquired by nonlinear regression from raw data conveyed adequate reproducibility (\( \widehat{\rho}=0.9307\Big) \). This explains a displayed fair distribution of projected leaf biomass values (Fig. 3b).
Estimates via log-linear transformation for processed data seemed enhance reproducibility (\( \widehat{\rho}=0.9777 \) ̂=0.9455). But, Fig. 4a, reveals a bulk of inconsistent replicates for leaves areas under 5000 mm^{2}. Notice that this subset of replicates distributes almost evenly around the mean response curve. Yet replicate spread for areas beyond 5000 mm^{2} shows significant bias (Fig. 4a). Meanwhile, nonlinear regression and processed data associate to \( \widehat{\rho}=0.9777 \). This corresponded to good reproducibility strength. Indeed, spread of replicates around the mean response is fair (Fig. 4b).
Reproducibility results for w_{ m }(α, β, t). Entries include, Lin’s concordance correlation coefficients \( \left(\hat{\rho}\right) \), root mean square deviations (rms) and ratios of 〈w_{ m }(α, β, t)〉 to 〈w_{ m }(t)〉 averages
Analysis method Data | 〈w_{ m }(α, β, t)〉/〈w_{ m }(t) 〉 | \( \widehat{\rho} \) | rms |
---|---|---|---|
Log-linear Transformation Raw | 0.8436 | 0.9285 | 0.01265 |
Nonlinear Regression Raw | 0.9773 | 0.9915 | 0.00460 |
Log-linear Transformation Processed | 0.8588 | 0.9489 | 0.01264 |
Nonlinear Regression Processed | 0.9975 | 0.9976 | 0.00293 |
Instead, nonlinear regression and raw data produced a value of \( \widehat{\rho}=0.9915. \) And root mean squared deviation attained a value of rms = 0.00460 (Table 2). This suggest a remarkable reproducibility strength for \( {w}_m\left(\hat{\alpha},\hat{\beta},t\right) \) projections (Table 2). Correspondence between projected and observed values, shown in Fig. 5b. corroborates high agreement. Moreover, the ratio of projected to observed leaf dry weight averages attained an outstanding value of 0.9773 (Table 2).
In turn, w_{ m }(α, β, t) projections made by nonlinear regression and processed data yield the highest value of \( \hat{\rho}=0.9976 \). (Table 2). And also the smallest root mean squared deviation among analysis method–data set combinations (Table 2). As shown by Fig. 6b this corresponds to a fairly good reproducibility strength. Additionally, data quality and nonlinear regression led to an outstanding value of 0.9975 for the ratio of projected 〈w_{ m }(α, β, t)〉 to observed 〈w_{ m }(t)〉 averages.
The simulation code of Eqs. (39) through (44) explored the sensitivity of the w_{ m }(α, β, t) projection method, to numerical variation of parameters α and β. Available parameter estimates yield reference values for α and β (Table 2). Again, since nonlinear regression associates to highest reproducibility strength, for easier presentation, we only explain results using this analysis method.
Sensitivity of the w_{ m }(α, β, t) projections to changes in estimates of the parameters α and β. Included are calculated θ_{ ste } values. This gives θ(q, r) as determined by the standard errors of estimates. We also present corresponding values of the relative deviation index ϑ(θ_{ ste })
Analysis method Data | θ _{ ste } | ϑ(θ_{ ste }). |
---|---|---|
Log-Linear transformation Raw | 3.662×10^{−3} | 0.1598 |
Nonlinear regression Raw | 5.101 × 10^{−3} | 0.0205 |
Log-Linear transformation Processed | 3.035×10^{−3} | 0.1419 |
Nonlinear regression Processed | 3.954 × 10^{−3} | 0.003 |
Discussion
Results of Solana-Arellano et al. [31] explain invariance of the allometric parameters α and β in Eq. (1). This suggest w_{ m }(α, β, t) proxies as possible nondestructive estimations of the average leaf dry weight in eelgrass shoots. These assessments are essential for monitoring the efficiency of transplanted eelgrass plots, fundamental in remediation aims. The present examination shows that the w_{ m }(α, β, t) proxies could in fact offer reliable and cost-effective assessments. This on condition that practitioners take in to account our guidelines. For instance, our results typify the extent on which accuracy of estimates of the parameters α and β influences the predictive power of the w_{ m }(α, β, t) projections. And, our findings clarify that there are methodological factors affecting the accuracy of estimates. Related influences could spread significant bias in approximations supported by the w_{ m }(α, β, t) device. Indeed, analysis method turned into a main factor inducing bias in parameter estimates of the model of Eq. (1). Moreover, only parameter estimates acquired by nonlinear regression yield consistency of the model of Eq. (1) (Table 1 and lines in Fig. 3b and Fig. 4b). And, only these estimates upheld conclusive predictive power of the w_{ m }(α, β, t) proxies (Table 2, as well as, lines in Fig. 5b and Fig. 6b). Our results also show that data quality could not improve the performance of w_{ m }(α, β, t) projections acquired via log-linear transformations. Without doubt, parameter estimates acquired from processed data by this method still led to significant bias in w_{ m }(α, β, t) projections (Fig. 6a). Meanwhile, data processing improved reproducibility of projections built for raw data using nonlinear regression (Table 2 and lines in Fig. 5b and Fig. 6b). Besides, relevance of data quality was also evident for optimizing sample size. Indeed, while for raw data, a sample of approximately 2000 leaves shows reasonable reproducibility, for the quality controlled data this threshold drops to near 1000. However, samples sized beyond these thresholds would not induce a noteworthy gain in reproducibility. This result on its own, ties to efficiency of the w_{ m }(α, β, t) projection method. Undoubtedly routines for leaf dry weight assessment are tedious and time consuming. So, reducing size of data set for parameter estimation increases cost-effectiveness of the w_{ m }(α, β, t) projection method.
Nonlinear regression estimation also showed advantages in sensitivity over the log-linear analysis counterpart. Estimates from raw data led to a largest absolute deviation between w_{ m }(α, β, t) and w_{ m }(t) values amounting only 3% of the average of w_{ m }(t) over sampling dates. And, for processed data, the fluctuation range for equivalent sensitivity widened to 2.8 times the range for standard errors of estimates. But, on spite of data quality relevance, sensitivity results for raw data reveal that the w_{ m }(α, β, t) projection method is robust relative to expected fluctuations in parameter estimates.
Our results show that both the accuracy and cost-effectiveness of projections can be enhanced by the addition of data quality control procedures. However, including data processing can become a weakness for the w_{ m }(α, β, t) projection method. Indeed, data cleaning procedures convey niceties that relate in a fundamental way to detection and rejection of inconsistent replicates. Also, compromising about which particular rejection edge should work, is hard to determine. Thus, the use of any data processing will endure a doubt, that the examiner selects an arrangement producing the most probable results [37]. In that order of ideas, when attempting to enhance the reproducibility power of w_{ m }(α, β, t) projections it is desirable to avoid depending in any form of data processing. For that aim, prior to data assembly, we must bear in mind standardized routines yielding accurate measurements for w(t) and a(t). This will favor direct identification of the model of Eq. (1) in a consistent way. It is of a fundamental importance to be aware, that errors in leaf dry weight or area assessment differentiate in terms of leaf size. Certainly, leaves produced anew normally present a complete and undamaged span. But, they normally yield very reduced dry weight values. Therefore, we can expect estimation errors imputable to the precision of the analytical scale for individual leaf dry weight assessments. To this, we may add errors in the reading and/or recording of the scale output. These issues could explain a perceptible accumulation of inconsistent replicates for leaves with areas between 2 mm^{2} and 350 mm^{2} (Fig. 1). And, even after data cleaning procedures, leaf dry weight replicate spread for leaves bigger than 2000 mm^{2} shows significant residual variability (Fig. 2). Likewise, as far as, bigger and older leaves is concerned, there are issues on dry weight estimation errors. These seem to relate to damage caused by exposure to environmental factors. The fact that we estimated leaf area by means of the product of related length and width could explain these effects. For complete undamaged eelgrass leaves, the use of a leaf times width proxy grants simplified and accurate estimations of leaf area [36].
But, this approach could deliver inaccurate estimations for long and damaged leaves. Actually, bigger leaves remain exposed during significant periods of time to environmental influences such as drag forces or herbivory. This could remove large amounts of leaf tissue while leaving length unaffected. Thus, causing overestimation of true leaf area when using a width and length product estimation. At the same time, lost portions of a leaf will produce a smaller dry weight than expected for an overestimated area. These effects will bring dry weight replicates that deviate from the power function–like trend associated to the model of Eq. (1). Estimation bias for the dry weights of smaller and longer leaves could explain the anomalous proliferation of inconsistencies (around 41 % ) found while applying the proposed data cleaning procedure to the present raw data. These effects will propagate uncertainty of parameter estimates of the model of Eq. (1), influencing accuracy of the w_{ m }(α, β, t) projections. Hence, for the sake of consistent reproducibility of observed values via w_{ m }(α, β, t) projections, we need to be aware of these effects. And as elaborated, a good starting point for granting consistency, is appropriate gathering of primary data for the identification of the model of Eq. (1). This will make subsequent data cleaning procedures unnecessary.
Conclusion
This research show that precise estimates of allometric parameters in Eq. (1) grant accurate non-destructive projections of the average leaf dry weight in eelgrass shoots. These projections offer efficient appraisal of eelgrass restoration projects, thereby contributing to the conservation of this important seagrass species. Our findings support views in Hui and Jackson [32], Packard and Birchard [33] and Packard et al. [44], on the relevance of analysis method in scaling studies. Indeed, we found that for assuring suitability of the w_{ m }(α, β, t) proxies, the use of nonlinear regression is crucial. On spite of claims that the use of the traditional log-linear analysis method is a must in allometric examination [45], exploration of spread of present crude data reveals curvature [46]. This explaining failure of the traditional analysis method to produce unbiased results for the present data. Besides proxies supported by nonlinear regression and raw data, are robust.
Data cleaning could only marginally enhance the accuracy of proxies produced by nonlinear regression and raw data. But results underline a relevant influence of data quality in setting optimal sample size for a suitable precision of parameter estimates. Nevertheless, the use of data cleaning procedures leads to intricacies. They in a fundamental way relate to choosing thresholds for rejection of detected inconsistencies, which are often regarded as subjective. Thus, instead of using later data cleaning, data gathering should seek for suitability. Special care must be taken when processing bigger and older leaves. These are often damaged or even trimmed so that their dry weights do not conform to a true weight to area relationship. Irregularities in raw data may also associate to biased estimation of leaf length or width. Moreover, in a lesser way faulty equipment for leaf dry weight assessment, rounding off, or even incorrect data recording could as well contribute.
Taking advantage of a time invariance of the parameters in Eq. (1) the w_{ m }(α, β, t) device could offer to the eelgrass conservation practitioner highly consistent and truly nondestructive assessments of the average value of leaf dry weight in shoots. But the explained guidelines on analysis method, sample size and data appropriateness are mandatory for cost-effectiveness. Moreover, the present results suggest that the use of the w_{ m }(α, β, t) method could be extended to other seagrasses species with similar leaf architecture to eelgrass.
Declarations
Acknowledgements
We thank Francisco Javier Ponce Isguerra for art work.
Funding
All funding for the present research were obtained from institutional grants
Availability of data and materials
Data and simulation codes will be available from corresponding author upon request.
Authors’ contributions
HEH conceived, designed performed and supervised the whole research. CLR performed formal derivations simulation and numerical tasks. EVD performed the statistical analysis and interpretation of data. NCC revised the manuscript critically both at statistical and formal levels. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interest.
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