Comparison of growth curves of two genotypes of dairy goats using nonlinear mixed models

SUMMARYThe objective of the current study was to assess the use of nonlinear mixed model methodology to fit the growth curves (weightv.time) of two dairy goat genotypes (Alpine, +A and Saanen, +S). The nonlinear functions evaluated included Brody, Von Bertalanffy, Richards, Logistic and Gompertz. The growth curve adjustment was performed using two steps. First, random effectsu1,u2andu3were linked to the asymptotic body weight (β1), constant of integration (β2) and rate constant of growth (β3) parameters, respectively. In addition to a traditional fixed-effects model, four combinations of models were evaluated using random variables: all parameters associated with random effects (u1,u2andu3), onlyβ1andβ2(u1andu2), onlyβ1andβ3(u1andu3) and onlyβ1(u1). Second, the fit of the best adjusted model was refined by using the power variance and modelling the error structure. Residual variance ($\sigma _e^2 $) and the Akaike information criterion were used to evaluate the models. After the best fitting model was chosen, the genotype curve parameters were compared. The residual variance was reduced in all scenarios for which random effects were considered. The Richards (u1andu3) function had the best fit to the data. This model was reparameterized using two isotropic error structures for unequally spaced data, and the structure known in the literature as SP(MATERN) proved to be a better fit. The growth curve parameters differed between the two genotypes, with the exception of the constant that determines the proportion of the final size at which the inflection point occurs (β4). The nonlinear mixed model methodology is an efficient tool for evaluating growth curve features, and it is advisable to assign biologically significant parameters with random effects. Moreover, evaluating error structure modelling is recommended to account for possible correlated errors that may be present even when using random effects. Different Richard growth curve parameters should be used for the predominantly Alpine and Saanen genotypes because there are differences in their growth patterns.

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Growth Curve of Selectively Bred Tambaqui (Colossoma macropomum) Reared in Different Environments

Journal of Agricultural Studies ◽

10.5296/jas.v8i3.16379 ◽

2020 ◽

Vol 8 (3) ◽

pp. 585

Author(s):

Rebeca Marcos ◽

Ruy Alberto Caetano Corrêa Filho ◽

Janessa Sampaio de Abreu ◽

Guilherme Do Nascimento Seraphim ◽

Ana Carla Carvalho Silva ◽

...

Keyword(s):

Growth Curve ◽

Growth Curves ◽

Growth Patterns ◽

Production Performance ◽

Environment Interaction ◽

Morphometric Traits ◽

Mato Grosso ◽

Colossoma Macropomum ◽

Genotype Environment Interaction ◽

The Family

The objective of this study was to obtain the growth curve of selectively bred tambaqui (Colossoma macropomum) reared in different environments. The experiment was carried out in the municipalities of Santo Antônio de Leverger (Mato Grosso – MT) and Campo Grande (Mato Grosso do Sul – MS), Brazil, over 431 days. Weight and morphometric traits of two families (A and B) from the second generation of selective breeding (G2) were measured every 30-45 days. The Gompertz regression model was used to obtain the growth curves. The production performance of both families and the interaction between families and locations (genotype × environment) were evaluated by analysis of variance considering the family (A and B), location (MT and MS), family × location interaction and error as variation factors. The asymptotic value (parameter A) obtained for weight and morphometric traits (except head length) was higher (P<0.05) in MT (weight of families A and B: 2279.6 g) than in MS (weight of family A: 1400.0 g; weight of family B: 1600.0 g). Family B showed better production performance in MS. There was a genotype × environment interaction effect on weight, body length and standard length. The two families have distinct growth patterns in different production environments. Family B has better growth performance in the environment with lower temperatures (MS).

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Sigmoid growth curves, a new approach to study the dynamics of the epicotyl emergence of oak

Folia Forestalia Polonica ◽

10.2478/ffp-2019-0003 ◽

2019 ◽

Vol 61 (1) ◽

pp. 30-41

Author(s):

Joanna Ukalska ◽

Szymon Jastrzębowski

Keyword(s):

Growth Curve ◽

Goodness Of Fit ◽

Nonlinear Models ◽

Germination Rate ◽

Growth Curves ◽

Model Parameters ◽

Residual Variance ◽

Nonlinear Behaviour ◽

Pedunculate Oak ◽

Growth Phenomenon

Abstract Three of the most frequently used sigmoidal growth curves from the Richards family are the logistic model, Gompertz model and Richards model. They are used in the analysis of organismal growth over time in many disciplines/studies and were proposed in many parameterisations. Choosing the right parameterisation is not easy. The correct parameterisation of the model should take into account such parameters that are useful to describe the analysed growth phenomenon and are biologically relevant without additional calculations. In addition, each parameter of the model only affects one shape characteristic of each growth curve, which makes it possible to determine standard errors and confidence intervals using statistical software. Growth curves in germination dynamics studies should provide information on topics such as the length of the lag in onset of germination, the maximum germination rate and, when it occurs, the time at which 50% of seeds will germinate and the final germination proportion. In this article, we present three parameterisations of the logistic, Gompertz and Richards models and indicate two parameterisations for each model, corresponding to the above-mentioned issues. Our proposition is parameterisation by taking into account the maximum absolute growth rate. Parameterisations indicated as useful for germination dynamics are characterised by the fact that each parameter has the same meaning in every model, so its estimates can be compared directly amongst the models. We also discussed the goodness-of-fit measures for nonlinear models and in particular measures of nonlinear behaviour of a model’s individual parameters as well as overall measures of nonlinearity. All described models were used to study the dynamics of the epicotyl emergence of pedunculate oak. After checking the close-to-linear behaviour of the studied model parameters and by taking into account the criteria of model selection (AICc of each growth curve and the residual variance [RV]), the best model describing the dynamics of epicotyl appearance of pedunculate oak was the Richards curve.

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Mixed-effects growth curves in the valuation of Nellore sires

Scientia Agricola ◽

10.1590/s0103-90162009000100012 ◽

2009 ◽

Vol 66 (1) ◽

pp. 84-89

Author(s):

Suely Ruiz Giolo ◽

Robin Henderson ◽

Clarice Garcia Borges Demétrio

Keyword(s):

Random Effects ◽

Growth Curve ◽

Production Systems ◽

Growth Curves ◽

Logistic Growth ◽

Cattle Breeding ◽

Growth Curve Models ◽

Fixed And Random Effects ◽

Curve Model ◽

Breeding Programmes

Cattle breeding programmes need objective criteria in order to evaluate and subsequently improve production systems. This work uses a logistic growth curve model for evaluating sires based on their progeny weight measured repeatedly over time. The parameters of the curve are described as a linear function of fixed and random effects. A Bayesian approach is used for the estimation. Analysis of the weights recorded on animals of the Nellore breed shows that growth curve models with fixed and random effects can be useful to evaluate and selecting sires.

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Analysis of Synergistic and Antagonistic Effects of Herbicides Using Nonlinear Mixed-Model Methodology

Weed Technology ◽

10.1614/wt-03-047r1 ◽

2004 ◽

Vol 18 (2) ◽

pp. 464-472 ◽

Cited By ~ 17

Author(s):

David C. Blouin ◽

Eric P. Webster ◽

Wei Zhang

Keyword(s):

Repeated Measures ◽

Mixed Model ◽

Block Design ◽

Nonlinear Function ◽

Nonlinear Mixed Model ◽

Antagonistic Effects ◽

Randomized Block Design ◽

Using Data ◽

Mixed Model Methodology ◽

Proc Nlmixed

When herbicides are applied in mixture, and infestation by weeds is less than expected compared with when herbicides are applied alone, a synergistic effect is said to exist. The inverse response is described as being antagonistic. However, if the expected response is defined as a multiplicative, nonlinear function of the means for the herbicides when applied alone, then standard linear model methodology for tests of hypotheses does not apply directly. Consequently, nonlinear mixed-model methodology was explored using the nonlinear mixed-model procedure (PROC NLMIXED) of SAS System®. Generality of the methodology is illustrated using data from a randomized block design with repeated measures in time. Nonlinear mixed-model estimates and tests of synergistic and antagonistic effects were more sensitive in detecting significance, and PROC NLMIXED was a versatile tool for implementation.

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Selection of nonlinear mixed models for growth curves of dairy buffaloes (Bubalus bubalis)

The Journal of Agricultural Science ◽

10.1017/s0021859620000325 ◽

2020 ◽

Vol 158 (3) ◽

pp. 218-224

Author(s):

F. R. Araujo Neto ◽

D. P. Oliveira ◽

R. R. Aspilcueta-Borquis ◽

D. A. Vieira ◽

K. C. Guimarães ◽

...

Keyword(s):

Growth Curve ◽

Mixed Models ◽

Logistic Models ◽

Growth Curves ◽

Information Criterion ◽

Growth Patterns ◽

Instantaneous Growth Rate ◽

Growth Curve Analysis ◽

Nonlinear Mixed Effects Models ◽

Nonlinear Mixed Models

AbstractThe determination of livestock growth patterns is important for meat or milk production systems, and nonlinear models are used to summarize and interpret the information. The aim of this study was to more accurately estimate growth curve parameters in buffalo cows by evaluating and selecting nonlinear mixed models that employ different types of residuals and include or not contemporary groups (CG) as a covariate. Weight records from 720 animals obtained over a period of 60 months were used. The growth curves were fit using nonlinear mixed-effects models. The Bertalanffy, Gompertz and Logistic models were evaluated. Modelling residuals using four structures (constant, combined, exponential and proportional) and the inclusion or not of CG in the models were also evaluated. The Akaike information criterion (AIC) and Bayesian information criterion (BIC) were used to select the model. In addition to estimating the parameters of the nonlinear growth models and their correlations, the instantaneous growth rate and inflection point were obtained. The Bertalanffy model with a combined residual structure and CG exhibited the lowest AIC and BIC values. Asymptotic weight (A) estimates ranged from 621.8 to 742.1 kg, and the maturity rate (k) ranged from 0.068 to 0.115 kg/month. The correlation between A and k ranged from −0.32 to −0.82 among the models evaluated. The selection criteria indicated that the Bertalanffy model was the most suitable for growth curve analysis in buffaloes.

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Genome-Wide Association Studies for Growth Curves in Meat Rabbits Through the Single-Step Nonlinear Mixed Model

Frontiers in Genetics ◽

10.3389/fgene.2021.750939 ◽

2021 ◽

Vol 12 ◽

Author(s):

Yonglan Liao ◽

Zhicheng Wang ◽

Leonardo S. Glória ◽

Kai Zhang ◽

Cuixia Zhang ◽

...

Keyword(s):

Mixed Model ◽

Growth Traits ◽

Association Studies ◽

Growth Curves ◽

Single Step ◽

The Body ◽

Genome Wide Association ◽

Genome Wide Association Studies ◽

Nonlinear Mixed Model ◽

Genome Wide

Growth is a complex trait with moderate to high heritability in livestock and must be described by the longitudinal data measured over multiple time points. Therefore, the used phenotype in genome-wide association studies (GWAS) of growth traits could be either the measures at the preselected time point or the fitted parameters of whole growth trajectory. A promising alternative approach was recently proposed that combined the fitting of growth curves and estimation of single-nucleotide polymorphism (SNP) effects into single-step nonlinear mixed model (NMM). In this study, we collected the body weights at 35, 42, 49, 56, 63, 70, and 84 days of age for 401 animals in a crossbred population of meat rabbits and compared five fitting models of growth curves (Logistic, Gompertz, Brody, Von Bertalanffy, and Richards). The logistic model was preferably selected and subjected to GWAS using the approach of single-step NMM, which was based on 87,704 genome-wide SNPs. A total of 45 significant SNPs distributed on five chromosomes were found to simultaneously affect the two growth parameters of mature weight (A) and maturity rate (K). However, no SNP was found to be independently associated with either A or K. Seven positional genes, including KCNIP4, GBA3, PPARGC1A, LDB2, SHISA3, GNA13, and FGF10, were suggested to be candidates affecting growth performances in meat rabbits. To the best of our knowledge, this is the first report of GWAS based on single-step NMM for longitudinal traits in rabbits, which also revealed the genetic architecture of growth traits that are helpful in implementing genome selection.

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The generalizability crisis

Behavioral and Brain Sciences ◽

10.1017/s0140525x20001685 ◽

2020 ◽

pp. 1-37

Author(s):

Tal Yarkoni

Keyword(s):

Quantitative Analysis ◽

Statistical Inference ◽

Random Effects ◽

False Positive ◽

Mixed Model ◽

Linear Mixed Model ◽

Random Effect ◽

Statistical Procedures ◽

Subject Variability ◽

Replication Crisis

Abstract Most theories and hypotheses in psychology are verbal in nature, yet their evaluation overwhelmingly relies on inferential statistical procedures. The validity of the move from qualitative to quantitative analysis depends on the verbal and statistical expressions of a hypothesis being closely aligned—that is, that the two must refer to roughly the same set of hypothetical observations. Here I argue that many applications of statistical inference in psychology fail to meet this basic condition. Focusing on the most widely used class of model in psychology—the linear mixed model—I explore the consequences of failing to statistically operationalize verbal hypotheses in a way that respects researchers' actual generalization intentions. I demonstrate that whereas the "random effect" formalism is used pervasively in psychology to model inter-subject variability, few researchers accord the same treatment to other variables they clearly intend to generalize over (e.g., stimuli, tasks, or research sites). The under-specification of random effects imposes far stronger constraints on the generalizability of results than most researchers appreciate. Ignoring these constraints can dramatically inflate false positive rates, and often leads researchers to draw sweeping verbal generalizations that lack a meaningful connection to the statistical quantities they are putatively based on. I argue that failure to take the alignment between verbal and statistical expressions seriously lies at the heart of many of psychology's ongoing problems (e.g., the replication crisis), and conclude with a discussion of several potential avenues for improvement.

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A Hybrid Modeling Technique of Epidemic Outbreaks with Application to COVID-19 Dynamics in West Africa

Biology ◽

10.3390/biology10050365 ◽

2021 ◽

Vol 10 (5) ◽

pp. 365

Author(s):

Chénangnon Frédéric Tovissodé ◽

Jonas Têlé Doumatè ◽

Romain Glèlè Kakaï

Keyword(s):

West Africa ◽

Growth Curve ◽

Exponential Growth ◽

Latency Period ◽

Growth Curves ◽

Exponential Growth Phase ◽

Model Framework ◽

Logistic Regression Models ◽

Containment Measures ◽

Reporting Data

The widely used logistic model for epidemic case reporting data may be either restrictive or unrealistic in presence of containment measures when implemented after an epidemic outbreak. For flexibility in epidemic case reporting data modeling, we combined an exponential growth curve for the early epidemic phase with a flexible growth curve to account for the potential change in growth pattern after implementation of containment measures. We also fitted logistic regression models to recoveries and deaths from the confirmed positive cases. In addition, the growth curves were integrated into a SIQR (Susceptible, Infective, Quarantined, Recovered) model framework to provide an overview on the modeled epidemic wave. We focused on the estimation of: (1) the delay between the appearance of the first infectious case in the population and the outbreak (“epidemic latency period”); (2) the duration of the exponential growth phase; (3) the basic and the time-varying reproduction numbers; and (4) the peaks (time and size) in confirmed positive cases, active cases and new infections. The application of this approach to COVID-19 data from West Africa allowed discussion on the effectiveness of some containment measures implemented across the region.

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Growth in achondroplasia, from birth to adulthood, analysed by the JPA-2 model

Journal of Pediatric Endocrinology and Metabolism ◽

10.1515/jpem-2020-0298 ◽

2020 ◽

Vol 33 (12) ◽

pp. 1589-1595

Author(s):

Mariana del Pino ◽

Virginia Fano ◽

Paula Adamo

Keyword(s):

General Population ◽

Growth Curve ◽

Growth Velocity ◽

Adult Height ◽

Growth Curves ◽

Peak Height ◽

Height Growth ◽

Individual Growth ◽

Height Velocity ◽

Peak Height Velocity

AbstractObjectivesIn general population, there are three phases in the human growth curve: infancy, childhood and puberty, with different main factors involved in their regulation and mathematical models to fit them. Achondroplasia children experience a fast decreasing growth during infancy and an “adolescent growth spurt”; however, there are no longitudinal studies that cover the analysis of the whole post-natal growth. Here we analyse the whole growth curve from infancy to adulthood applying the JPA-2 mathematical model.MethodsTwenty-seven patients, 17 girls and 10 boys with achondroplasia, who reached adult size, were included. Height growth data was collected from birth until adulthood. Individual growth curves were estimated by fitting the JPA-2 model to each individual’s height for age data.ResultsHeight growth velocity curves show that after a period of fast decreasing growth velocity since birth, with a mean of 9.7 cm/year at 1 year old, the growth velocity is stable in late preschool years, with a mean of 4.2 cm/year. In boys, age and peak height velocity in puberty were 13.75 years and 5.08 cm/year and reach a mean adult height of 130.52 cm. In girls, the age and peak height velocity in puberty were 11.1 years and 4.32 cm/year and reach a mean adult height of 119.2 cm.ConclusionsThe study of individual growth curves in achondroplasia children by the JPA-2 model shows the three periods, infancy, childhood and puberty, with a similar shape but lesser in magnitude than general population.

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Response transformations for random effect and variance component models

Statistical Modelling ◽

10.1177/1471082x20966919 ◽

2020 ◽

pp. 1471082X2096691

Author(s):

Amani Almohaimeed ◽

Jochen Einbeck

Keyword(s):

Maximum Likelihood ◽

Random Effects ◽

Mixed Model ◽

Linear Mixed Model ◽

Random Effect ◽

Statistical Technique ◽

Response Distribution ◽

Level Data ◽

Variance Component Models ◽

Response Transformation

Random effect models have been popularly used as a mainstream statistical technique over several decades; and the same can be said for response transformation models such as the Box–Cox transformation. The latter aims at ensuring that the assumptions of normality and of homoscedasticity of the response distribution are fulfilled, which are essential conditions for inference based on a linear model or a linear mixed model. However, methodology for response transformation and simultaneous inclusion of random effects has been developed and implemented only scarcely, and is so far restricted to Gaussian random effects. We develop such methodology, thereby not requiring parametric assumptions on the distribution of the random effects. This is achieved by extending the ‘Nonparametric Maximum Likelihood’ towards a ‘Nonparametric profile maximum likelihood’ technique, allowing to deal with overdispersion as well as two-level data scenarios.

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