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Application of information theory-based decision support system for high precision modeling of the length-weight relationship (LWR) for five marine shrimps from the northwestern Bay of Bengal
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Accurate and unbiased modeling of length-weight relationship (LWR) is essential for precise estimation of biomass from length-based abundance data for aquatic ecosystem modeling and developing resource management strategies for the sustainable utilization of aquatic living resources. The present study was conducted to develop the best modeling method for establishing the relationship between the length and weight of the fish, giving adequate priority to the variance distribution structure, which is often overlooked. Three modeling methods viz. (1) NLM: nonlinear model with additive homogeneous variance structure, (2) wNLM: weighted nonlinear model with additive heterogeneous power variance structure and (3) LM: log-transformed linear model with multiplicative heterogeneous variance structure in untransformed scale were evaluated for their performance in accurately estimating the model parameters and their confidence intervals. As variance in fish weight increases with the increase in fish size, usually, a multiplicative heterogeneous variance structure is assumed while explaining the relationship between length and weight, which was evidenced from the residual plot of NLM. As the data did not follow the additive homogenous variance assumption of NLM, the worst performance (highest AIC and BIC) was obtained when NLM was used. The variance distribution structure could be homogenized and normalized in three out of six fishes (i.e., Plicofollis layardi, Cynoglossus arel, Otolithes ruber) investigated in the study using LM, resulting in superior performance (lowest AIC and BIC) among the three models. For two out of three remaining species (i.e., Nemipterus japonicus and Parastromateus niger) for which the variance distribution cannot be normalized by log transformation (LM), the residuals could be homogenized and normalized by the wNLM which was evidenced by the lowest AIC and BIC for the model. Even for the species (i.e., Pampus argenteus), where the residuals could not be normalized by any of the models, better performance was obtained from wNLM. A meta-analysis was performed to validate the hitherto available information on the LWRs of these six species by regressing the regression parameters [log(a) vs. b] of the available LWRs and by computing the form factors. The erroneous LWRs were identified as doubtful reports by carefully scrutinizing the outliers obtained from Cook’s distance method and subsequently validating them by observing their dispersion from modeled prediction intervals (PI) and interquartile range (IQR) based outlier detection methods using the form factor analysis. The study presents the decision support framework for an appropriate modeling approach while dealing with certain assumptions such as variance normality and homoscedasticity. The study also describes a combined approach for the validation of derived LWRs by a comprehensive meta-analysis.
The magnetospheric physics research community uses a broad array of quantitative data-model comparison methods (metrics) when conducting their research investigations. It is often the case, though, that any particular study will only use one or two metrics, with the two most common being Pearson correlation coefficient and root mean square error (RMSE). Because metrics are designed to test a specific aspect of the data-model relationship, limiting the comparison to only one or two metrics reduces the physical insights that can be gleaned from the analysis, restricting the possible findings from modeling studies. Additional physical insights can be obtained when many types of metrics are applied. We organize metrics into two primary groups: 1) fit performance metrics, often based on the data-model value difference; and 2) event detection metrics, which use a discrete event classification of data and model values determined by a specified threshold. In addition to these groups, there are several major categories of metrics based on the aspect of the data-model relationship that the metric assesses: 1) accuracy; 2) bias; 3) precision; 4) association; 5) and extremes. Another category is skill, which is a measure of any of these metrics against the performance of a reference model. These can be applied to a subset of either the data or the model values, known as reliability and discrimination assessments. In the context of magnetospheric physics examples, we discuss best practices for choosing metrics for particular studies.
Growth models estimate life history parameters (e.g., growth rates and asymptotic size) that are used in the management of fisheries stocks. Traditionally in fisheries science, it was common to fit one growth model—the von Bertalanffy growth model—to size-at-age data. However, in recent years, fisheries science has seen an increase in the number of growth models available and the evaluation of multiple growth models for a given species or study. We reviewed n = 196 peer-reviewed age and growth studies and n = 50 NOAA (National Oceanic and Atmospheric Administration) regional stock assessments to examine trends in the use of growth models and model selection in fisheries over time. Our results indicate that the total number of age and growth studies increased annually since 1988 with a slight proportional increase in the use of multi-model frameworks. Information theoretic approaches are replacing goodness-of-fit and a priori model selection in fisheries studies; however, this trend is not reflected in NOAA stock assessments, which almost exclusively rely on the von Bertalanffy growth model. Covariates such as system (e.g., marine or fresh), location of study, diet, family, maximum age, and range of age data used in model fitting did not contribute to which model was ultimately the best fitting, suggesting that there are no large-scale patterns of specific growth models being applied to species with common life histories or other attributes. Given the importance and ubiquity of growth modeling to fisheries science, a historical and contemporary understanding of the practice is critical to evaluate improvements that have been made and future challenges.
Length-weight relationships were calculated for 9 chondrichthyes fish species totalizing 284 inviduals from Edremit Bay (North Aegean Sea) caught with bottom trawls between June 2007 and June 2009. It was calculated as W=0.1306 TL2.1701 (r²=0.8645) for Dasyatis pastinaca, W=0.1139 TL2.7088 (r²=0.9768) for Mustelus mustelus, W=0.0014 TL3.1795 (r²=0.8742) for Myliobatis aquila, W=0.0322 TL2.5597 (r²=0.8836) for Raja clavata, W=0.0215 TL2.5654 (r²=0.7152) for Raja miraletus, W=0.0029 TL3.2142 (r²=0.9262) for Raja radula, W=0.000006 TL2.8817 (r²=0.8138) for Scyliorhinus canicula, W=0.0004 TL3.6397 (r²=0.6365) for Scyliorhinus stellaris, W=0.1297 TL2.4665 (r²=0.8022) for Torpedo marmorata. In addition, 157 previous studies were carried out on the characterization of L–W relationships for 12 chondricthtyes fish species in Turkey waters. The values of the b values identified by other authors varied from 2.122 to 4.15. The mean value of b was 3.165 (SE = ±0.025).
Length-weight relationships were calculated for 9 chondrichthyes fish species totalizing 284 inviduals from Edremit Bay (North Aegean Sea) caught with bottom trawls between June 2007 and June 2009. It was calculated as W=0.1306 TL 2.1701 (r 2 =0.8645) for Dasyatis pastinaca, W=0.1139 TL 2.7088 (r 2 =0.9768) for Mustelus mustelus, W=0.0014 TL 3.1795 (r 2 =0.8742) for Myliobatis aquila, W=0.0322 TL 2.5597 (r 2 =0.8836) for Raja clavata, W=0.0215 TL 2.5654 (r 2 =0.7152) for Raja miraletus, W=0.0029 TL 3.2142 (r 2 =0.9262) for Raja radula, W=0.000006 TL 2.8817 (r 2 =0.8138) for Scyliorhinus canicula, W=0.0004 TL 3.6397 (r 2 =0.6365) for Scyliorhinus stellaris, W=0.1297 TL 2.4665 (r 2 =0.8022) for Torpedo marmorata. In addition, 157 previous studies were carried out on the characterization of L-W relationships for 12 chondricthtyes fish species in Turkey waters. The values of the b values identified by other authors varied from 2.122 to 4.15. The mean value of b was 3.165 (SE = ±0.025).
Length-weight relationships in fish are important tools in fisheries management. The aim of this study was to estimate the growth curve W=aLb for European perch, Perca fluviatilis L., from the Polish coast of the southern Baltic Sea (ICES Subdivision 25). The data set comprised the total weights and total lengths of 827 specimens caught during the 2011-2013 period. The mean total length (TL) was 169.4 mm (range 100.0-310.0 mm), and the mean total weight was 83.9 g (range 11.3-553.2 g). First, a multiplicative error term and a linear regression approach to loglog-transformed data was considered. The following estimated values for the parameters were obtained: a = exp(-12.5323), b=3.25, s=0.07862. This approach was not successful in solving the common heterogeneity problem of the length-weight data. A generalized nonlinear regression approach to the original data was more suitable in our case. The estimated model was W = 3.83×10-6×L3.238 + ε, with ε~norm(0,0.0281×E[W|L]2×1.242). The estimated 95% confidence interval for b was (3.218, 3.259), and the growth was allometric. The perch from the coastal waters of the Baltic Sea (ICES Subdivision 25) was characterized by slightly better condition than that caught in inland waters and estuaries.
We describe an information-theoretic paradigm for analysis of ecological data, based on Kullback–Leibler information, that is an extension of likelihood theory and avoids the pitfalls of null hypothesis testing. Information-theoretic approaches emphasise a deliberate focus on the a priori science in developing a set of multiple working hypotheses or models. Simple methods then allow these hypotheses (models) to be ranked from best to worst and scaled to reflect a strength of evidence using the likelihood of each model (gi), given the data and the models in the set (i.e. L(gi | data)). In addition, a variance component due to model-selection uncertainty is included in estimates of precision. There are many cases where formal inference can be based on all the models in the a priori set and this multi-model inference represents a powerful, new approach to valid inference. Finally, we strongly recommend inferences based on a priori considerations be carefully separated from those resulting from some form of data dredging. An example is given for questions related to age- and sex-dependent rates of tag loss in elephant seals (Mirounga leonina).
In this work, we propose a heteroscedastic Von Bertalany growth model considering a multiplicative heteroscedastic dispersion matrix. All estimates were obtained using a sampling based approach, which allows information to be input beforehand with lower computational eort. Simulations were carried out in order to verify some frequentist properties of the estimation procedure in the presence of small and moderate sample sizes. The methodology is illustrated on a real Kubbard female chicken corporeal weight dataset. Key word: Growth Models, Multiplicative Heteroscedasticity, Bayesian Analysis, Simulation.
The weight (W) of fishes (and other organisms) is exponentially related to their length (L) according to the equation W = aL b , where a is the intercept and b is the slope of the log-transformed relation (Le Cren 1951, Froese 2006). Based on the slope (b) of the relation between weight and length, one can check whether the growth of a fish species is isometric (b = 3, all fish dimen-sions increase at the same rate), hypoallometric (b < 3, a fish increases less in weight than predicted by its increase in length, i.e., it becomes more elongated as it grows; also termed negative allometric) or hyperallometric (b > 3, a fish increases more in weight than predicted by its increase in length, i.e., it becomes less elongated or more roundish as it grows; also termed positive allometric). Weight–length relations (WLRs) can be used for convert-ing lengths into biomass, determining fish condition, comparing fish growth among areas, and as a complement to species-specific reproduction and feeding studies (Petrakis and Stergiou 1995, Koutrakis and Tsikliras 2003, Froese 2006). Thus, they are an important compo-nent of fisheries biology and when properly calculated they can be very useful to fisheries management. Over the last decade, the number of published articles dealing with WLRs of fishes is increasing in fast rate (Fig. 1). The majority of articles have been published in special-ized fish journals, with 367 out of the 697 articles appear-ing Society: 10) and the remaining 330 articles in 144 jour-nals. With respect to environment coverage, 51% of the articles refer to marine fish, 38% to freshwater fish and 11% to brackish/lagoon fish. In this editorial note, we set some criteria and recom-mendations on important issues (i.e., number of species, sample size, length range and preservation, reporting and
ABSTRACTA growth model for the endangered cyprinid fish Tribolodon nakamurai was derived following otolith analyses of 16 wild and 53 reared specimens. The asteriscus was the most appropriate to measure size among three otolith elements, and its height OH mm was used as size index of otolith. Standard length L cm was best back-calculated using the Gompertz model, L = 70.0·exp[–exp{−0.553 (OH – 2.73)}]. Translucent zones on the lapilli, analyzed from 5-year-old-reared fish, were regarded as winter slow-growing zones. The ages of 10 wild specimens of 37.0–48.1 cm standard length were calculated as 7–10 years by counting the translucent zones on the lapilli. Age t was best back-calculated using the allometry model, t = 1.33·OH1.37. The growth trajectory of T. nakamurai followed a slender S curve, three typical growth models, von Bertalanffy, Logistic and Gompertz, and Richards’ model, which is a general formula of the above three, being fitted using the maximum likelihood method. The Gompertz model, Lt = 60.2·exp[–exp{−0.258(t − 4.68)}], was found by Akaike's information criterion (AIC) to be the statistically most acceptable growth model.
We briefly outline the information-theoretic (I-T) approaches to valid inference including a review of some simple methods
for making formal inference from all the hypotheses in the model set (multimodel inference). The I-T approaches can replace
the usual t tests and ANOVA tables that are so inferentially limited, but still commonly used. The I-T methods are easy to compute and
understand and provide formal measures of the strength of evidence for both the null and alternative hypotheses, given the
data. We give an example to highlight the importance of deriving alternative hypotheses and representing these as probability
models. Fifteen technical issues are addressed to clarify various points that have appeared incorrectly in the recent literature.
We offer several remarks regarding the future of empirical science and data analysis under an I-T framework.
KeywordsAIC–Evidence–Kullback–Leibler information–Model averaging–Model likelihoods–Model probabilities–Model selection–Multimodel inference
In situations where limited knowledge of a system exists and the ratio of data points to variables is small, variable selection
methods can often be misleading. Freedman (Am Stat 37:152–155, 1983) demonstrated how common it is to select completely unrelated
variables as highly “significant” when the number of data points is similar in magnitude to the number of variables. A new
type of model averaging estimator based on model selection with Akaike’s AIC is used with linear regression to investigate
the problems of likely inclusion of spurious effects and model selection bias, the bias introduced while using the data to
select a single seemingly “best” model from a (often large) set of models employing many predictor variables. The new model
averaging estimator helps reduce these problems and provides confidence interval coverage at the nominal level while traditional
stepwise selection has poor inferential properties.
The Akaike information criterion (AIC; Akaike, 1973) is a popular method for comparing the adequacy of multiple, possibly nonnested models. Current practice in cognitive psychology is to accept a single model on the basis of only the "raw" AIC values, making it difficult to unambiguously interpret the observed AIC differences in terms of a continuous measure such as probability. Here we demonstrate that AIC values can be easily transformed to so-called Akaike weights (e.g., Akaike, 1978, 1979; Bozdogan, 1987; Burnham & Anderson, 2002), which can be directly interpreted as conditional probabilities for each model. We show by example how these Akaike weights can greatly facilitate the interpretation of the results of AIC model comparison procedures.
In this note we explain the use of the Akiake Information Criterion and its related model comparison indices (usually derived for maximum likelihood estimator inverse problem formulations) in the context of least squares (ordinary, weighted, iterative weighted or “generalized”, etc.) based inverse problem formulations. The ideas are illustrated with several examples of interest in biology.
A method for estimating the suitable stock-recruitment equations based on a comparison of various statistical models is presented. Fourteen statistical models including the Ricker curve and Beverton-Holt curve were considered. Stock-recruitment relations were estimated by the maximum likelihood method and suitable models were selected using AIC. Thirty-five sets of stock-recruitment data derived from published papers were analyzed. The results show that the various statistical models selected as suitable models do not always coincide with the model used in the original papers. A detailed examination is needed to estimate the stock-recruitment relationship. © 1994, The Japanese Society of Fisheries Science. All rights reserved.
Model selection is difficult. Even in the apparently straightforward case of choosing between standard linear regression models, there does not yet appear to be consensus in the statistical ecology literature as to the right approach.
We review recent works on model selection in ecology and subsequently focus on one aspect in particular: the use of the Akaike Information Criterion (AIC) or its small‐sample equivalent, . We create a novel framework for simulation studies and use this to study model selection from simulated data sets with a range of properties, which differ in terms of degree of unobserved heterogeneity. We use the results of the simulation study to suggest an approach for model selection based on ideas from information criteria but requiring simulation.
We find that the relative predictive performance of model selection by different information criteria is heavily dependent on the degree of unobserved heterogeneity between data sets. When heterogeneity is small, AIC or are likely to perform well, but if heterogeneity is large, the Bayesian Information Criterion (BIC) will often perform better, due to the stronger penalty afforded.
Our conclusion is that the choice of information criterion (or more broadly, the strength of likelihood penalty) should ideally be based upon hypothesized (or estimated from previous data) properties of the population of data sets from which a given data set could have arisen. Relying on a single form of information criterion is unlikely to be universally successful.
Using Akaike's information criterion, three examples of statistical data are reanalyzed and show reasonably definite conclusions. One is concerned with the multiple comparison problem for the means in normal populations. The second is concerned with the grouping of the categories in a contingency table. The third is concerned with the multiple comparison problem for the analysis of variance by the iogit model in contingency tables, Finite correction of Akaike's information criterionis also proposed.
This study presents a historical review, a meta-analysis, and recommendations for users about weight–length relationships, condition factors and relative weight equations. The historical review traces the developments of the respective concepts. The meta-analysis explores 3929 weight–length relationships of the type W = aLb for 1773 species of fishes. It shows that 82% of the variance in a plot of log a over b can be explained by allometric versus isometric growth patterns and by different body shapes of the respective species. Across species median b = 3.03 is significantly larger than 3.0, thus indicating a tendency towards slightly positive-allometric growth (increase in relative body thickness or plumpness) in most fishes. The expected range of 2.5 < b < 3.5 is confirmed. Mean estimates of b outside this range are often based on only one or two weight–length relationships per species. However, true cases of strong allometric growth do exist and three examples are given. Within species, a plot of log a vs b can be used to detect outliers in weight–length relationships. An equation to calculate mean condition factors from weight–length relationships is given as Kmean = 100aLb−3. Relative weight Wrm = 100W/(amLbm) can be used for comparing the condition of individuals across populations, where am is the geometric mean of a and bm is the mean of b across all available weight–length relationships for a given species. Twelve recommendations for proper use and presentation of weight–length relationships, condition factors and relative weight are given.
Akaike’s information criterion (AIC) is increasingly being used in analyses in the field of ecology. This measure allows one
to compare and rank multiple competing models and to estimate which of them best approximates the “true” process underlying
the biological phenomenon under study. Behavioural ecologists have been slow to adopt this statistical tool, perhaps because
of unfounded fears regarding the complexity of the technique. Here, we provide, using recent examples from the behavioural
ecology literature, a simple introductory guide to AIC: what it is, how and when to apply it and what it achieves. We discuss
multimodel inference using AIC—a procedure which should be used where no one model is strongly supported. Finally, we highlight
a few of the pitfalls and problems that can be encountered by novice practitioners.
KeywordsAkaike’s information criterion–Information theory–Model averaging–Model selection–Multiple regression–Statistical methods
The performance of six statistical approaches, which can be used for selection of the best model to describe the growth of individual fish, was analyzed using simulated and real length-at-age data. The six approaches include coefficient of determination (R
2), adjusted coefficient of determination (adj.-R
2), root mean squared error (RMSE), Akaike’s information criterion (AIC), bias correction of AIC (AICc
) and Bayesian information criterion (BIC). The simulation data were generated by five growth models with different numbers of parameters. Four sets of real data were taken from the literature. The parameters in each of the five growth models were estimated using the maximum likelihood method under the assumption of the additive error structure for the data. The best supported model by the data was identified using each of the six approaches. The results show that R
2 and RMSE have the same properties and perform worst. The sample size has an effect on the performance of adj.-R
2, AIC, AICc
and BIC. Adj.-R
2 does better in small samples than in large samples. AIC is not suitable to use in small samples and tends to select more complex model when the sample size becomes large. AICc
and BIC have best performance in small and large sample cases, respectively. Use of AICc
or BIC is recommended for selection of fish growth model according to the size of the length-at-age data.
Weight-length regressions are presented for 33 fish species caught during 1992–1993 in the South Euboikos Gulf, Aegean Sea. Samples were collected with beach seine and gill and trammel nets. The values of the exponent b in the weight-length relationship W = aLb ranged from 2.320 to 3.521 and the median value was 2.987, whereas 50% of the values ranged between 2.840 and 3.140. The application of these regressions should be limited to the observed length ranges.
Variations in environmental variables and measurement errors often result in large and heterogeneous variations in fitting fish stock–recruitment (SR) data to a SR statistical model. In this paper, the maximum likelihood method was used to fit the six statistical SR models on six sets of simulated SR data. The best relationships were selected using the Akaike information criterion (AIC) and Bayesian information criterion (BIC) methods, respectively. Which have the advantage of testing the significance of the difference between the functions of different model specifications. The exercises were also conducted on eight sets of real fisheries SR data. The results showed that both AIC and BIC are valid in selecting the most suitable SR relationship. As far as the nested models are concerned, BIC is better than AIC.
Although it is often assumed that birds strongly prefer tailwinds for their migratory flights, we predict that a strategy of no wind selectivity (traveling independently of winds) may be more favorable than wind selectivity (traveling on tailwind occasions but stopping to rest under headwind occasions) for birds with low energy costs of travel relative to rest and for birds that cannot use stopover time for efficient fuel deposition. We test this prediction by analyzing the daily traveling or stopping as recorded by satellite tracking of five ospreys Pandion haliaetus, a species often using energy-saving thermal soaring, during their migration between northern Europe and Africa. Besides wind, precipitation is another weather factor included in the analyses because thermal soaring migrants are expected to stop and rest in rainy weather. In logistic regression analyses, taking into account the effects of latitude, behavior on previous day, season, date, and individual for discriminating between traveling and stopping days, we found a lack of influence of winds, suggesting that the ospreys travel or stop without regard to wind. This lack of wind selectivity under light and moderate winds is in agreement with our prediction. We expect a low degree of wind selectivity and thus regular flights under headwinds also among other types of birds that cannot use stopping time for efficient foraging and fuel deposition. We also found an unexpected lack of influence of precipitation, possibly because of relatively few instances with rainfall in combination with poor geographic precision for estimates of this weather variable. Copyright 2006.
Ecologists are increasingly applying model selection to their data analyses, primarily to compare regression models. Model selection can also be used to compare mechanistic models derived from ecological theory, thereby providing a formal framework for testing the theory. The Akaike Information Criterion (AIC) is the most commonly adopted criterion used to compare models; however, its performance in general is not very well known. The best model according to AIC has the smallest expected Kullback-Leibler (K-L) distance, which is an information-theoretic measure of the difference between a model and the truth. I review the theory behind AIC and demonstrate how it can be used to test ecological theory by considering two example studies of foraging, motivated by simple foraging theory. I present plausible truths for the two studies, and models that can be fit to the foraging data. K-L distances are calculated for simulated studies, which provide an appropriate test of AIC. Results support the use of a commonly adopted rule of thumb for selecting models based on AIC differences. However, AICc, a corrected version of AIC commonly used to reduce model selection bias, showed no clear improvement, and model averaging, a technique to reduce model prediction bias, gave mixed results.
We develop a small sample criterion (AICc) for the selection of extended quasi-likelihood models. In contrast to the Akaike information criterion (AIC). AICc provides a more nearly unbiased estimator for the expected Kullback-Leibler information. Consequently, it often selects better models than AIC in small samples. For the logistic regression model, Monte Carlo results show that AICc outperforms AIC, Pregibon's (1979, Data Analytic Methods for Generalized Linear Models. Ph.D. thesis. University of Toronto) Cp*, and the Cp selection criteria of Hosmer et al. (1989, Biometrics 45, 1265-1270). Two examples are presented.
Introductory fisheries analysis with R-Chapman and Hall/CRC
- D H Ogle
Model selection in ecology and evolution
- Johnson
Selecting the best growth model for fish using Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)
- Panhwar
On the use of log-transformation vs. nonlinear regression for analyzing biological power laws
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