Accounting for individual variability in the von Bertalanffy growth model

Estimation of von Bertalanffy growth parameters has received considerable attention in fisheries research. Since Sainsbury (1980, Can. J. Fish. Aquat. Sci. 37: 241-247) much of this research effort has centered on accounting for individual variability in the growth parameters. In this paper we demonstrate that, in analysis of tagging data, Sainsbury's method and its derivatives do not, in general, satisfactorily account for individual variability in growth, leading to inconsistent parameter estimates (the bias does not tend to zero as sample size increases to infinity). The bias arises because these methods do not use appropriate conditional expectations as a basis for estimation. This bias is found to be similar to that of the Fabens method. Such methods would be appropriate only under the assumption that the individual growth parameters that generate the growth increment were independent of the growth parameters that generated the initial length. However, such an assumption would be unrealistic. The results are derived analytically, and illustrated with a simulation study. Until techniques that take full account of the appropriate conditioning have been developed, the effect of individual variability on growth has yet to be fully understood.

Maximum likelihood estimation of mortality and growth with individual variability from multiple length-frequency data

We consider estimation of mortality rates and growth parameters from length-frequency data of a fish stock and derive the underlying length distribution of the population and the catch when there is individual variability in the von Bertalanffy growth parameter L-infinity. The model is flexible enough to accommodate 1) any recruitment pattern as a function of both time and length, 2) length-specific selectivity, and 3) varying fishing effort over time. The maximum likelihood method gives consistent estimates, provided the underlying distribution for individual variation in growth is correctly specified. Simulation results indicate that our method is reasonably robust to violations in the assumptions. The method is applied to tiger prawn data (Penaeus semisulcatus) to obtain estimates of natural and fishing mortality.

Spatial variability of soil carbon in forested and cultivated sites: Implications for change detection

The potential to sequester atmospheric carbon in agricultural and forest soils to offset greenhouse gas emissions has generated interest in measuring changes in soil carbon resulting from changes in land management. However, inherent spatial variability of soil carbon limits the precision of measurement of changes in soil carbon and hence, the ability to detect changes. We analyzed variability of soil carbon by intensively sampling sites under different land management as a step toward developing efficient soil sampling designs. Sites were tilled crop-land and a mixed deciduous forest in Tennessee, and old-growth and second-growth coniferous forest in western Washington, USA. Six soil cores within each of three microplots were taken as an initial sample and an additional six cores were taken to simulate resampling. Soil C variability was greater in Washington than in Tennessee, and greater in less disturbed than in more disturbed sites. Using this protocol, our data suggest that differences on the order of 2.0 Mg C ha(-1) could be detected by collection and analysis of cores from at least five (tilled) or two (forest) microplots in Tennessee. More spatial variability in the forested sites in Washington increased the minimum detectable difference, but these systems, consisting of low C content sandy soil with irregularly distributed pockets of organic C in buried logs, are likely to rank among the most spatially heterogeneous of systems. Our results clearly indicate that consistent intramicroplot differences at all sites will enable detection of much more modest changes if the same microplots are resampled.

An overview of physical growth and maturation

An understanding of physical growth and maturation is relevant to many disciplines, including exercise and sport science, anthropology, human biology, medicine, psychology and education. Growth and maturation is governed by a complex interaction between genetic and environmental factors. There is increasing evidence that physical activity plays an important role in normal growth, development, health and well-being of children and youth, however, caution is required in the activity setting so that growth and maturation is not jeopardised. To appreciate the impact of physical activity and/or exercise on growth and maturation, a thorough understanding of the general principles of auxology is useful. Following an introduction to terminology, an overview of physical growth and development is provided in the context of morphological changes. Detailed information is provided regarding individual variability in growth and development along with sexual dimorphism. A small degree of sexual dimorphism exists at birth however striking differences develop during the pubertal years. Sexual dimorphism in body composition is largely regulated by endocrine factors with critical roles played by growth hormone and gonadal steroids.

An exploratory study of the factors enhancing and inhibiting export growth in the Chilean wine industry

The objective of this exploratory study is to investigate the main drivers that enhance and inhibit the export performance of Chilean wineries. Based on survey data collected from Chilean wineries, the findings of this study suggest that the main constraints within the Chilean wineries in developing exports is the lack of financial resources, limited quantities of stocks for market expansion, management’s lack of knowledge and experience, and the high cost of travelling and participating in trade shows. The main drivers of wine export performance according to the respondents are high quality of the wines, well established network of international distributors, and marketing skills. The major inhibitors of developing wine exports are exchange rate variability, problems in selecting a reliable international distributor, and limited government support to promote wine exports. This study also shows that export managers of Chilean wineries have high educational levels and have international experience. The findings have important implications for export development efforts of both governments and managers.

Common variants in the region around Osterix are associated with bone mineral density and growth in childhood

Peak bone mass achieved in adolescence is a determinant of bone mass in later life. In order to identify genetic variants affecting bone mineral density (BMD), we performed a genome-wide association study of BMD and related traits in 1518 children from the Avon Longitudinal Study of Parents and Children (ALSPAC). We compared results with a scan of 134 adults with high or low hip BMD. We identified associations with BMD in an area of chromosome 12 containing the Osterix (SP7) locus, a transcription factor responsible for regulating osteoblast differentiation (ALSPAC: P = 5.8 × 10-4; Australia: P = 3.7 × 10-4). This region has previously shown evidence of association with adult hip and lumbar spine BMD in an Icelandic population, as well as nominal association in a UK population. A meta-analysis of these existing studies revealed strong association between SNPs in the Osterix region and adult lumbar spine BMD (P = 9.9 × 10-11). In light of these findings, we genotyped a further 3692 individuals from ALSPAC who had whole body BMD and confirmed the association in children as well (P = 5.4 × 10-5). Moreover, all SNPs were related to height in ALSPAC children, but not weight or body mass index, and when height was included as a covariate in the regression equation, the association with total body BMD was attenuated. We conclude that genetic variants in the region of Osterix are associated with BMD in children and adults probably through primary effects on growth.

Generalised growth models for aquatic species with an application to blacklip abalone (Haliotis rubra)

This paper presents a maximum likelihood method for estimating growth parameters for an aquatic species that incorporates growth covariates, and takes into consideration multiple tag-recapture data. Individual variability in asymptotic length, age-at-tagging, and measurement error are also considered in the model structure. Using distribution theory, the log-likelihood function is derived under a generalised framework for the von Bertalanffy and Gompertz growth models. Due to the generality of the derivation, covariate effects can be included for both models with seasonality and tagging effects investigated. Method robustness is established via comparison with the Fabens, improved Fabens, James and a non-linear mixed-effects growth models, with the maximum likelihood method performing the best. The method is illustrated further with an application to blacklip abalone (Haliotis rubra) for which a strong growth-retarding tagging effect that persisted for several months was detected

Stochastic growth models for analyzing crustacean data

The contemporary methodology for growth models of organisms is based on continuous trajectories and thus it hinders us from modelling stepwise growth in crustacean populations. Growth models for fish are normally assumed to follow a continuous function, but a different type of model is needed for crustacean growth. Crustaceans must moult in order for them to grow. The growth of crustaceans is a discontinuous process due to the periodical shedding of the exoskeleton in moulting. The stepwise growth of crustaceans through the moulting process makes the growth estimation more complex. Stochastic approaches can be used to model discontinuous growth or what are commonly known as "jumps" (Figure 1). However, in stochastic growth model we need to ensure that the stochastic growth model results in only positive jumps. In view of this, we will introduce a subordinator that is a special case of a Levy process. A subordinator is a non-decreasing Levy process, that will assist in modelling crustacean growth for better understanding of the individual variability and stochasticity in moulting periods and increments. We develop the estimation methods for parameter estimation and illustrate them with the help of a dataset from laboratory experiments. The motivational dataset is from the ornate rock lobster, Panulirus ornatus, which can be found between Australia and Papua New Guinea. Due to the presence of sex effects on the growth (Munday et al., 2004), we estimate the growth parameters separately for each sex. Since all hard parts are shed too often, the exact age determination of a lobster can be challenging. However, the growth parameters for the aforementioned moult processes from tank data being able to estimate through: (i) inter-moult periods, and (ii) moult increment. We will attempt to derive a joint density, which is made up of two functions: one for moult increments and the other for time intervals between moults. We claim these functions are conditionally independent given pre-moult length and the inter-moult periods. The variables moult increments and inter-moult periods are said to be independent because of the Markov property or conditional probability. Hence, the parameters in each function can be estimated separately. Subsequently, we integrate both of the functions through a Monte Carlo method. We can therefore obtain a population mean for crustacean growth (e. g. red curve in Figure 1). [GRAPHICS]

A maximum likelihood approach for estimating growth from tag–recapture data

The Fabens method is commonly used to estimate growth parameters k and l infinity in the von Bertalanffy model from tag-recapture data. However, the Fabens method of estimation has an inherent bias when individual growth is variable. This paper presents an asymptotically unbiassed method using a maximum likelihood approach that takes account of individual variability in both maximum length and age-at-tagging. It is assumed that each individual's growth follows a von Bertalanffy curve with its own maximum length and age-at-tagging. The parameter k is assumed to be a constant to ensure that the mean growth follows a von Bertalanffy curve and to avoid overparameterization. Our method also makes more efficient use nf thp measurements at tno and recapture and includes diagnostic techniques for checking distributional assumptions. The method is reasonably robust and performs better than the Fabens method when individual growth differs from the von Bertalanffy relationship. When measurement error is negligible, the estimation involves maximizing the profile likelihood of one parameter only. The method is applied to tag-recapture data for the grooved tiger prawn (Penaeus semisulcatus) from the Gulf of Carpentaria, Australia.

Estimation of growth parameters from multiple-recapture data

This article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown. The existing approaches either impute the initial ages or model them as random effects. Assumptions about the initial age are not verifiable because all the initial ages are unknown. We present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual. Optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions. Explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity. Simulation studies indicate that the proposed method works well. Two real data sets are analyzed for illustration, one from whelks (Dicathais aegaota) and the other from southern rock lobster (Jasus edwardsii) in South Australia.

Estimating equations for parameters in stochastic growth models from tag-recapture data

James (1991, Biometrics 47, 1519-1530) constructed unbiased estimating functions for estimating the two parameters in the von Bertalanffy growth curve from tag-recapture data. This paper provides unbiased estimating functions for a class of growth models that incorporate stochastic components and explanatory variables. a simulation study using seasonal growth models indicates that the proposed method works well while the least-squares methods that are commonly used in the literature may produce substantially biased estimates. The proposed model and method are also applied to real data from tagged rack lobsters to assess the possible seasonal effect on growth.

Effect of individual variability on estimation of population parameters from length-frequency data

We consider estimation of mortality rates and growth parameters from length-frequency data of a fish stock when there is individual variability in the von Bertalanffy growth parameter L-infinity and investigate the possible bias in the estimates when the individual variability is ignored. Three methods are examined: (i) the regression method based on the Beverton and Holt's (1956, Rapp. P.V. Reun. Cons. Int. Explor. Mer, 140: 67-83) equation; (ii) the moment method of Powell (1979, Rapp. PV. Reun. Int. Explor. Mer, 175: 167-169); and (iii) a generalization of Powell's method that estimates the individual variability to be incorporated into the estimation. It is found that the biases in the estimates from the existing methods are, in general, substantial, even when individual variability in growth is small and recruitment is uniform, and the generalized method performs better in terms of bias but is subject to a larger variation. There is a need to develop robust and flexible methods to deal with individual variability in the analysis of length-frequency data.

An improved Fabens method for estimation of growth parameters in the vonBertalanffy model with individual asymptotes

In the analysis of tagging data, it has been found that the least-squares method, based on the increment function known as the Fabens method, produces biased estimates because individual variability in growth is not allowed for. This paper modifies the Fabens method to account for individual variability in the length asymptote. Significance tests using t-statistics or log-likelihood ratio statistics may be applied to show the level of individual variability. Simulation results indicate that the modified method reduces the biases in the estimates to negligible proportions. Tagging data from tiger prawns (Penaeus esculentus and Penaeus semisulcatus) and rock lobster (Panulirus ornatus) are analysed as an illustration.

Growth curves with explanatory variables and estimation of the effect of tagging

The von Bertalanffy growth model is extended to incorporate explanatory variables. The generalized model includes the switched growth model and the seasonal growth model as special cases, and can also be used to assess the tagging effect on growth. Distribution-free and consistent estimating functions are constructed for estimation of growth parameters from tag-recapture data in which age at release is unknown. This generalizes the work of James (1991, Biometrics 47 1519-1530) who considered the classical model and allowed for individual variability in growth. A real dataset from barramundi (Lates calcarifer) is analysed to estimate the growth parameters and possible effect of tagging on growth.