Revista de Biología Tropical, ISSN: 2215-2075, Vol. 73: e2025-171, enero-diciembre 2025 (Publicado Set. 30, 2025)

Performance of the Weibull individual

growth model for weight at age data

Enrique Villa-Diharce1; https://orcid.org/ 0000-0002-0483-9546

Evlin A. Ramírez-Félix2; https://orcid.org/0000-0002-5136-5283

Juan Antonio García-Borbón3; https://orcid.org/0000-0002-6458-0388

Miguel Á. Cisneros-Mata4*; https://orcid.org/0000-0001-5525-5498

1. Centro de Investigación en Matemáticas, 36240, Guanajuato, Guanajuato, México; villadi@cimat.mx

2. Instituto Mexicano de Investigación en Pesca y Acuacultura Sustentables, 82112, Mazatlán, Sinaloa, México; evlin.

ramirez@imipas.gob.mx

3. Instituto Mexicano de Investigación en Pesca y Acuacultura Sustentables, 23020, La Paz, Baja California Sur, México;

antonio.borbon@imipas.gob.mx

4. Instituto Mexicano de Investigación en Pesca y Acuacultura Sustentables. 85400, Guaymas, Sonora, México; macisne@

yahoo.com (*Correspondence)

Received 29-X-2024. Corrected 25-VI-2025. Accepted 12-IX-2025.

ABSTRACT

Introduction: Knowledge of different functions associated with a probability distribution, as well as their prop-

erties, can be translated into functions that provide information about different characteristics of the growth

process under study.

Objectives: To analyze the relationship between individual growth models and the cumulative distribution func-

tions of continuous random variables.

Methods: We compare the flexibility and goodness of fit of the Weibull-type model against the von Bertalanffy

weight growth model. We fit these two growth models to two very different sets of age-weight data taken from the

literature; the first comprises 22 pairs of Pacific halibut mean weight at age, and the second 900 pairs of striped

bass weight-age-data.

Results: The Weibull-type growth model had greater flexibility and neglected less information available in the

data sets than the von Bertalanffy model.

Conclusions: The Weibull model, derived from cumulative probability distribution, is a good choice to fit

weight-at-age data as it is more flexible than the commonly used von Bertalanffy model.

Keywords: cumulative distribution function; age-weight relationship; model comparison; Weibull growth model;

von Bertalanffy growth model.

RESUMEN

Eficacia del modelo de crecimiento individual de Weibull para datos de peso por edad

Introducción: El conocimiento de diferentes funciones asociadas a una distribución de probabilidad, así como

sus propiedades, se puede traducir en funciones que proporcionen información sobre diferentes características

del proceso de crecimiento en estudio.

Objetivos: Analizar la relación entre los modelos de crecimiento individuales y las funciones de distribución

acumulativa de variables aleatorias continuas.

https://doi.org/10.15517/cy079f12

AQUATIC ECOLOGY

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INTRODUCTION

Growth of fishery production or biological

production in general, implies an increase or

decrease in production in weight / biomass /

yield. However, mathematical biological mod-

els for assessing growth are commonly length

growth models (von Bertalanffy, Gompertz,

Logistic, Richards) (Katsanevakis, & Marave-

lias, 2008). Length is one of the most com-

monly used indicators for assessing growth,

since it is less expensive, faster, and simpler to

obtain from a sampling perspective. However,

it is necessary to translate this information into

weight by evaluating the relationship between

length and weight (L-W) to return to assess

biological production. In several studies growth

is expressed in weight (Botello-Ruvalcaba et al.,

2010; Luquín-Covarrubias et al., 2022).

Individual growth in fish is based in physi-

ological processes and is the net result of two

opposing processes, catabolism, and anabolism

(von Bertalanffy, 1938). Specifically (von Berta-

lanffy, 1957), the origin of the von Bertalanffy

(vB) model can be expressed as

where m represents body weight, t is time; n

and κ are constants of anabolism and catabo-

lism, respectively. Here, α and β indicate that

the rates of anabolism or catabolism are pro-

portional to body weight m to the power of α

or β, usually with α ≤ β and β = 1. The abil-

ity to model growth in fish has a wide range

of applications in population dynamics. For

example, growth models are a vital compo-

nent of many stock assessments as they reflect

potential production of the species (Juan-Jordá

et al., 2015). The most common practice in fish

growth modelling is to select a priori a single

model, generally the vB growth model (e.g.,

Haddon, 2011). Inference and estimation of

parameters and their precision are based solely

on that fitted model (Katsanevakis & Marave-

lias, 2008). An important restriction of the vB

model when age-length data are available is

that individual growth rate is monotonically

decreasing. The fit is poor in the case of hav-

ing age-length or age-weight data that have

more than one inflection point (Knight, 1968;

Knight, 1969).

It has been established that, seen as an

individual growth model, the Weibull function

is a derivation of the cumulative distribution

function of a random variable that follows a

Weibull random distribution (Swintek et al.,

2019). The Weibull function has been success-

fully applied to growth of trees (Seo et al., 2023;

Souza et al., 2021; Yang et al., 1978); as dis-

cussed below, this has not been the case for fish

growth. The nexus between individual growth

models and cumulative distribution functions

proves invaluable, enabling understanding of

distribution functions effectively. Describing

the distribution of a random variable offers dif-

ferent options, including alternative functions.

Familiarity with diverse functions –density,

cumulative distribution, survival, and more–

linked with probability distribution and their

Métodos: Comparamos la flexibilidad y bondad de ajuste del modelo tipo Weibull con el modelo de crecimiento

de peso de von Bertalanffy. Ajustamos estos dos modelos de crecimiento a dos conjuntos muy diferentes de datos

edad-peso tomados de la literatura; el primero comprende 22 pares de lenguado del Pacífico usando peso medio

por edad, y el segundo, 900 pares de corvina rayada con parejas de peso-edad.

Resultados: El modelo de crecimiento tipo Weibull tuvo mayor flexibilidad y descartó menos información dispo-

nible en los conjuntos de datos que el modelo de von Bertalanffy.

Conclusiones: El modelo de Weibull, derivado de una distribución de probabilidad acumulada, es una buena

opción para ajustar datos de peso por edad, ya que es más flexible que el comúnmente utilizado modelo de von

Bertalanffy.

Palabras clave: función de distribución acumulada; relación edad-peso; comparación de modelos; modelo de

crecimiento de Weibull; modelo de crecimiento de von Bertalanffy.

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inherent properties, facilitates to extrapolate

functions shedding light on distinct facets of

the growth process (Marshall & Olkin, 2007).

In the present work we develop a Weibull

age-weight model and compare its performance

with the vB, using two sets of age and weight

data. We compare the fit of these models

based on the Akaike information criterion

(Burnham & Anderson, 2002). The Weibull

growth model is flexible (Meeker et al., 2022)

which facilitates goodness of fit. Here, we first

analyze the origin of the vB model for age and

weight data and then develop the alternative

Weibull departing from a cumulative distribu-

tion function. Because the two data sets used

were very different in structure, it is expected

that results shown here are general and useful

in other situations.

MATERIALS AND METHODS

Theory and calculations: We caution that

data for the present work are scarce; therefore,

we used only two data sets available, as explained

below. The length growth model of vB is given

by the expression (von Bertalanffy, 1938):

where L∞ is asymptotic length, t0 is the age for

a null length and k is the intrinsic growth rate.

From equation 1 one can derive another

model for growth in weight, using the power

function relating weight (W) at length (L). The

resulting model is (Table 1):

where W∞ is the maximum possible individual

weight corresponding to asymptotic length,

and b is the same as b in the weight-length

relationship.

From model 1, one has that:

which is the proportion of the maximum theo-

retical length that an average individual has

grown at age t.

Table 1

Description of parameters used in this work for both models (vB andWeibull) and data sets.

Model Parameters Description Units

Weibull

WBody weight kg or lb

tAge Year s

to Location parameter Year s

W∞Maximum theoretical weight kg or lb

hScale parameter Year s

h - |to|Age corresponding to 63.2 % W∞Years

βShape parameter

cShape parameter

sRandom error term kg or lb

von Bertalanffy

WBody weight kg or lb

tAge Year s

W∞Maximum theoretical weight kg or lb

kBrody growth rate coefficient 1/year

to Theoretical age for t = 0 Year s

bWeight-length exponent

sRandom error term kg or lb

Because data were derived from tables, weight for Pacific halibut (Hippoglossus stenolepis) is in kg, and for striped bass

(Morone saxatilis) is in pounds. Data for halibut were obtained from Quinn II and Deriso (1999), and those of bass were

derived from Baum (2002).

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Equation 3 has the properties of a cumu-

lative distribution (Seber & Wild, 1989) and

coincides with the cumulative exponential dis-

tribution with location parameter t0 and failure

rate k (Marshall & Olkin, 2007). The vB length

growth model is widely used in fisheries, while

the exponential distribution is fundamental in

life span studies with wide application in reli-

ability and medicine (Collet, 2015; Marshall &

Olkin, 2007).

For the vB growth model in weight,

where F(t) is the generalized exponential cumu-

lative distribution (Gupta & Kundu, 1999; Mar-

shal & Orkin, 2007).

To derive a weight model which is an

extension of the vB length model one begins

with understanding the process involved in the

extension of model 1. It is easy to realize that

transforming length L(t) with a power function,

is transformed into:

A similar extension consists in transform-

ing age t with the power function t1/b, which

results in a shape function given by

This shape function coincides with one of

the formulations of the Weibull distribution

(Hallinan, 1993). It has found wide use in areas

of reliability and in the analysis of survival to

model life spans or failure times (Meeker et al.,

2022; Thach, 2022). A convenient reparameter-

ization of the Weibull distribution consists in

expressing its cumulative distribution as

where ß is the shape parameter and its value

determines the form of the distribution, while

h is the scale parameter because its location

modifies the value of t; both h and t have the

same units (time).

To get a better grasp of the meaning of ß,

cumulative relative weight was calculated using

equation 8 with t0 = 0 and t = h = 2 for five

values of ß. As observed (Fig. 1), at age t = h

= 2, the growth is equal to 0.632 W∞, regard-

less of the value of ß. The five curves, seen as

Fig. 1. Shape functions (generalized exponential cumulative distribution) of the Weibull growth model, for different values

of the parameter (= 1, 2, 3, 4, 5).

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cumulative distribution functions, coincide at

t = h = 2; that is, in reliability research, when

initial time = zero, for this model, the 0.632

quantile is equal t = h = 2.

As shown, parameter h is interpreted as

the 0.632 quantile of asymptotic weight. A

possible biological interpretation is discussed

below. This behavior is easy to see expressed

as follows:

In contrast, when one repeats the exercise

using the vB growth model, unlike the previous

model, the curves remain in lower locations

as the value of b is greater. This is because in

the vB model, the shape function F(t) takes

values in the interval (0, 1) and consequently,

when raised to an increasing exponent b it

takes decreasing values (Fig. 2). The plots are

ordered in reverse order to the values of b, and

consequently, the graphs of the shape function

do not intersect.

The density and survival functions of the

Weibull distribution are given, respectively, by

And

From the perspective of growth models,

the Weibull type model is given in terms of the

cumulative distribution function:

Interestingly, in this distribution, the shape

parameter ß relates to the form of the growth

curve; when the average individual reaches age

h - |t0| (Table 2), it will have grown 63.2 % of

its asymptotic size plus an error term e (e.g.,

Swintek et al., 2019).

Extensions of the vB length model: Thus

far we have discussed the vB length growth

model (1) and two extensions, growth in weight

(2) and the Weibull model (12). As seen, expres-

sions for the two extensions are, respectively

(eq. 7) F(t) = [1-exp(-k(t-t0))]b and (eq. 8) (t)=

[1-exp{-((t-t0) ⁄ h)ß}].

In addition to being shape functions

of individual growth models, they are also

cumulative distribution functions. There is an

additional distribution function which is a

generalization of both; this is the generalized or

exponential Weibull distribution.

The cumulative distribution of the latter is

given by (Mudholkar & Hutson, 1996; Mudhol-

kar & Srivastava, 1993; Mudholkar et al., 1995):

F(t) =

Fig. 2. Shape functions (generalized exponential cumulative distribution) of the von Bertalanffy growth model for different

values of the parameter ß (= 1, 2, 3, 4, 5).

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Table 2

Features of the von Bertalanffy, Weibull and Generalized Weibull weigth – age growth models applied to two data sets:

Pacific halibut, Hippoglossus stenolepis and striped bass, Morone saxatilis.

Model Equation Parameters Likelihood Error structure and

restrictions

Pacific halibut (Hippoglossus stenolepis)

Von Bertalanffy W(t) is weight at time t; W∞ is asymptotic

weight; t0 is the age for a null weight and k is

the intrinsic growth rate; b is a parameter of

the weight-length relationship (W(t) = aL(t)b)

Additive (Normal)

No restrictions

Weibull W(t) is the weight at time t; ß is the shape

parameter; h is the scale parameter because its

location modifies the value of t.

Parameterization of the

0.632 growth in time

quantile

t0 = 0

Generalized

Weibull

ß and c are both shape parameters; when c

= 1 the function corresponds to the Weibull

model, else is a generalized Weibull function

Parameterization of the

0.632 growth in time

quantile

t0 = 0

Striped bass (Morone saxatilis)

von Bertalanffy W(t) is weight at time t; W∞ is asymptotic

weight; t0 is the age for null weight and k is the

intrinsic growth rate; b is a parameter of the

weight-length relationship (W(t) = aL(t)b)

Multiplicative (Lognormal)

No restrictions

Weibull W(t) is the weight at time t; ß is the shape

parameter; h is the scale parameter because

its location modifies the value of t.

Multiplicative (Lognormal)

Error structure

No restrictions

Generalized

Weibull

ß and c are both shape parameters; when c

= 1 the function corresponds to the Weibull

model, else is a generalized Weibull function

Multiplicative (Lognormal)

Error structure

No restrictions

t0 = 0

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where ß and c are both shape parameters; when

c = 1 the function corresponds to the Weibull

model, else is a generalized Weibull function.

This is the shape function of the generalized

Weibull model, which is expressed as follows

when errors are additive:

Derivation of confidence intervals for

parameter values: We obtained confidence

intervals for parameters using a method based

on the likelihood function L(θ) or loglikelihood

function l(θ) = ln[L(θ)]. We first obtained an

information matrix I of parameters given by the

negative values of the second derivatives of the

loglikelihood l(θ) evaluated at the maximum

likelihood estimates of the model parameters,

= ( 1, 2, … , k). For example, for the Weibull

model we have the following parameter esti-

mates 1 = ∞, 2 = , 3 = 0, 4 = , 5 = . The

information matrix is given by:

The variance-covariance matrix Σ = l-1 for

parameters is obtained by inverting the infor-

mation matrix, or:

Estimates of the model parameter varianc-

es are the terms of the diagonal of the variance-

covariance matrix. So, the variance of the i-th

term of the parameter vector is:

Similarly, the covariance between terms i

and j of the parameter vector is term (i, j) of the

variance-covariance matrix:

This is a squared, symmetric matrix, i.e.,

In the case of the Weibull model, the

variances of the model parameter estimates

are the terms of the diagonal of the variance-

covariance matrix:

Once we have the variances of the param-

eter estimates, the process of calculating the

confidence intervals (CI) has two stages. First,

the parameters are estimated by maximum

likelihood, and then the limits of the CI are

obtained by the Normal approximation (Hoel et

al., 1971; Meeker et al., 2022) as follows:

All calculations were made using the sta-

tistical computing language R (R Core Team,

2023). Table 2 shows the features of the models

contrasted and applied to the next two different

data series, as well as the rationale of the analy-

sis involved (parameters, likelihood function

and type of error used).

First data set-small sample case: Firstly,

using Pacific halibut (Hippoglossus stenolepis)

22-pair data set in Quinn II and Deriso (1999),

a comparison was done of the vB and Weibull

growth models for age-weight data (Table 2).

Pacific halibut (H. stenolepis) are demersal spe-

cies of flatfish that range throughout the North

Pacific Ocean. They are among the largest of all

flatfish: females may reach up to approximately

227 kg in weight and nearly three meters in

length (International Pacific Halibut Commis-

sion [IPHC], 1998); they are relatively long-

lived with the oldest individuals estimated to be

55 years of age (Forsberg, 2001). Maturity varies

with sex, age, and size of the fish. Females grow

faster but mature slower than males (IPHC,

1987). The average age at 50 percent maturity

is eight years for males and 12 years for females

(St-Pierre, 1984). The vB growth model param-

eter estimates for female ranged from: L∞, 1.32

m to 2.39 m; k, 0.109 to 0.039 and t0 0.418 to

-1.538 (Perkins, 2015). According to the IPHC

(1987), Pacific halibut can reach a maximum

length of nine feet (~2.7 m) and maximum

weight of 500 pounds (~227 kg).

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It is important to note that parameter b was

originally known (b = 3.24) from previously

reported work (Quinn II et al., 1983). While

adjusting the Weibull model, all parameters are

estimated using equation 12, where, in terms

of probability, h - |t0| corresponds to quan-

tile 0.632 or 63.2 % of the asymptotic weight

(Table 2).

To select the “best” of both models (i.e., vB

and Weibull), we used the Akaike (1973) infor-

mation criterion:

where l() is the maximized log-likelihood and

p is the number of model parameters. The value

of AIC is interpreted as the loss of information

available in the data set resulting in the model

fit; lower values indicate better fits (Burnham &

Anderson, 2002).

In the case of additive errors, we have the

following expression for the log-likelihood:

where weight µi at time ti for the vB is given by

equation (2), whereas for the Weibull model µi

is given by equation (12). Log-likelihood values

were obtained numerically using statistical soft-

ware in R language (Ogle, 2016).

Parameters of vector θ estimated are vec-

tors which maximize the likelihood or log

likelihood, that is:

where Q is the parametric space where search

is done to find the maximum (log) likeli-

hood. Although in some problems not too

complicated one can obtain closed forms for

the maximum likelihood estimator , common-

ly numerical procedures are used to maximize

likelihood or log likelihood functions.

Second set-extensive data: For the sec-

ond example, we used 900 pairs (age-weight)

of striped bass Morone saxatilis from Arkan-

sas, USA derived from an age (years)-weight

(pounds) key (U.S. Fish & Wildlife Service

[USFWS], 2010). Briefly, 900 individuals were

distributed in a matrix, 100 per each of nine

weight categories (4.5 to 39.5 kg) and from

three to 15 years of age. The striped bass is a

temperate anadromous fish species that is the

basis of an important recreational and com-

mercial fishery in the Eastern United States

(Gervasi, 2015). Female striped bass mature

between 5-6 years, and fecundity increases

asymptotically with age (Brown et al., 2024).

For several fish species such as red snap-

per, Lutjanus campechanus (Lowerre-Barbieri

& Friess, 2022) and sablefish, Anoplopoma

fimbria (Rodgveller et al., 2018) it has been

determined that fecundity increases with fish

age. Female striped bass mature between 5-6

years, and fecundity increases asymptotically

with age (Brown et al., 2024). To add random

variability in weight-at-age from the age-weight

key, raw “observed” weights were multiplied by

Normal random shocks (1, 1). How there are

variability in weight grows as age increases;

this pattern of increasing variation led us to

consider growth models with an error term that

acts multiplicatively.

For models with multiplicative error structure, the log likelihood function is given (Table 2) by

(Quinn II & Deriso, 1999):

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weight µi at time ti for the vB and Weibull

models are given, respectively, by equations (2)

and (12).

RESULTS

First data set-small sample case

vB growth model: For the vB model fit,

Table 3 shows estimated parameter values and

their uncertainties. Except for t0, all estimates

were significant (p-values < 0.001).

Weibull growth model: Table 4 shows

that, for the Weibull model fit, parameter val-

ues for t0 and β are non-significant judging

by 1) the relatively large p values, and 2) the

confidence intervals include zero. This pro-

vides an indication that the model might be

over parameterized.

The parameterization was done in terms of

the 0.632 quantile of growth in time.

To reduce over parameterization, we con-

sidered a restriction t0 = 0, which yielded results

shown in Table 5, where the rest of parameter

values were significant. This is because by elim-

inating the need to obtain a fit of t0, the relative

standard error of parameter β decreased (from

55.6 to 7.2 %). The other estimated parameters

(W∞ and σ) both have small relative standard

errors values; hence their confidence intervals

are unchanged.

Table 6 shows results for the fit of the

generalized Weibull growth model 16 to data

shown in Table 2.

It should be noted that the lowest AIC

value corresponds to the third model adjust-

ment. In general, performance was better using

the Weibull than the vB growth model.

Second data set-large sample size

vB growth model: Table 7 shows the esti-

mates that result in the fit of the vB model to

Table 3

Estimated parameter values for the von Bertalanffy growth model using Pacific halibut, Hippoglossus stenolepis, data shown

in Fig. 3A assuming additive errors.

Parameter Estimate Std error Confidence interval t-value p-value

W∞68.60 4.92 (58.96, 78.25) 13.94 < 10-20

k 0.122 0.02 (0.09, 0.15) 8.16 < 10-8

t03.037 0.91 (1.25, 4.83) 3.33 0.002

b3.24

σ3.06 0.52 (2.06, 4.08) 5.96 < 10-6

logLikMx -54.80

AIC 119.60

Table 4

Maximum likelihood of parameter values for the Weibull growth model using 22 pairs of age-weight data for Pacific halibut,

assuming additive errors.

Parameter Estimate Std error Confidence interval t-value p-value

W∞56.79 1.80 (53.26, 60.33) 31.50 < 10-50

h24.22 10.19 (4.25, 44.19) 2.38 0.017

t0-7.32 10.22 (-27.32, 12.67) -0.72 0.473

β4.50 2.19 (0.21, 8.79) 2.05 0.040

σ2.05 0.24 (1.60, 2.52) 8.59 < 10-8

logLikMx -53.55

AIC 117.10

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the sample using 900 pairs of age-weight data

for striped bass (M. saxatilis) shown in Fig. 3B,

we observe in the age-weight data of this figure

a reduced variation in the last two age classes

(14 and 15 years); and a possible structure

related to selectivity, i.e., larger weights seem to

be capped between 40 and 50 pounds.

Estimates of all the parameters of this

model are statistically different from zero,

although the estimate of the parameter t0 is

very close to being non-significant because the

standard error of this estimate has a high rela-

tive standard error or a small t-value (t-value =

-6.66). This is because, can be seen in the scat-

terplot of the data (Fig. 3B), there is a reduced

amount of information around zero, which

is where we have the information about t0. It

is interesting to compare the estimate of this

Table 5

Maximum loglikelihood estimates of the Weibull parameter values using 22 pairs of age-weight data for Pacific halibut

assuming additive errors.

Parameter Estimate Std error Confidence interval t-value p-value

W∞59.78 2.84 (54.22, 65.35) 21.06 < 10-50

h17.62 0.66 (16.34, 18.90) 26.89 < 10-50

t00

β2.80 0.20 (2.42, 3.19) 14.16 < 10-20

σ2.54 0.38 (1.80, 3.28) 6.72 < 10-8

logLikMx -51.89

AIC 113.77

Two restrictions were used: 1) parameterization of the 0.632 growth in time quantile, and 2) t0 = 0.

Table 6

Maximum loglikelihood parameter estimates for the generalized Weibull model using 22 pairs of age-weight data for Pacific

halibut assuming additive errors.

Parameter Estimate Std error Confidence interval t-value p-value

W∞57.37 1.98 (53.49, 61.25) 28.97 < 10-50

h21.04 0.86 (19.35, 22.73) 24.43 < 10-50

t00

β6.42 1.59 (3.28, 9.52) 4.02 < 10-4

c0.33 0.10 (0.14, 0.53) 3.39 < 10-4

σ2.53 0.42 (1.70, 3.36) 5.97 < 10-6

logLikMx -50.40

AIC 112.80

Two restrictions were used: 1) parameterization of the 0.632 growth in time quantile, and 2) t0 = 0.

Table 7

Results of fitting the von Bertalanffy parameter to 900 pairs of age-weight data for striped bass.

Parameter Estimate Std error Confidence interval t-value p-value

W∞51.56 2.26 (47.13, 55.99) 22.80 < 10-50

k0.17 0.02 (0.13, 0.21) 8.16 < 10-8

t0-4.12 0.62 (-5.33, -2.91) -6.66 < 10-8

b7.43 0.85 (5.77, 9.09) 8.79 < 10-8

σ0.28 0.01 (0.27, 0.30) 42.88 < 10-50

l( ) -168.27

AIC 346.55

Revista de Biología Tropical, ISSN: 2215-2075, Vol. 73: e2025-171, enero-diciembre 2025 (Publicado Set. 30, 2025)

parameter in the case of a large data set and in

the case of the small sample (Quinn II & Deri-

so, 1999). The t-value is of different orders (3.33

and -6.66). This is further discussed below.

Weibull growth model: Table 8 shows the

estimates that result in the adjustment of the

Weibull-type model to the sample of 900 age-

weight pairs of data. In this fit, the estimates

of all the parameters are significant and the

estimate with the greatest uncertainty is t0, in a

similar way adjustment of the previous model,

in both cases, the systematic part of the model

is different, but the stochastic part (error term)

is similar. Parameters W∞ and ß have the lowest

estimation uncertainties.

Generalized Weibull model: Table 9

shows the estimates that result in the adjust-

ment of the generalized Weibull type model to

the sample of 900 age and weight data pairs.

The generalized Weibull type has one more

parameter than the Weibull type model. This

additional parameter causes this last model to

be over parameterized, which is expressed in an

estimate with a high uncertainty for parameter

Fig. 3. Graphical comparison of fits of the von Bertalanffy and Weibull models to two data sets. A. Pacific halibut 22 pairs of

mean weight-at-age. B. striped bass 900 data pairs of weight-at-age.

12 Revista de Biología Tropical, ISSN: 2215-2075 Vol. 73: e2025-171, enero-diciembre 2025 (Publicado Set. 30, 2025)

t0. The latter results in a confidence interval

for t0, that contains zero (-6.1676, 1.1276),

which indicates that the estimate of t0 is not

significant. Another parameter that has a low

significance, although it is still significant, is

parameter c, with a confidence interval (0.0731,

0.8465) very close to zero.

When comparing the values of the AIC, we

observed a ranking of the goodness of fit of the

three models, vB, Weibull type and generalized

Weibull. Table 10 shows the values of the AIC

index for the two example data sets considered

herein. In the first example, it is assumed in the

Weibull type model that parameter t0 is zero,

while in the second example this assumption

is assumed for the generalized Weibull model.

Comparison of fits: Relative performance

of both models (vB and Weibull) fit to the two

data sets can be graphically appreciated (Fig. 3).

It can be observed how the “winner” Weibull

model fits better the two data sets as compared

to the “winner” vB model. Although we observe

a reduced variation in the last two age classes

(14 and 15 years); the figure also suggests a

possible structure related to selectivity, i.e.,

larger weights seem to be capped between 40

and 50 pounds.

DISCUSSION

In the present work the case is made

of using the Weibull growth function as an

alternative to the vB model when pairs of age-

weight data are available. It is striking to realize

that often growth is analyzed in terms of length

and as mentioned, this is the result of ease to

obtaining length data. In fisheries studies it

is less common to find weight at-age data to

Table 10

AIC values for three model fits to the data sets.

Model Pacific halibut

(22 obs.)

Striped bass

(900 obs.)

von Bertalanffy 119.60 346.55

Weibull 117.10 256.46

Generalized Weibull 112.80 260.22

Table 9

Results of fitting the generalized Weibull model to the data set of 900 pairs of age-weight data for striped bass.

Parameter Estimate Std error Conf interval t-value p-value

W∞44.33 2.90 (38.64, 50.01) 15.29 < 10-50

h17.73 2.21 (13.41, 22.05) 8.04 < 10-8

t0-6.99 2.28 (-11.45, -2.53) -3.07 0.0027

β4.51 0.49 (3.54, 5.47) 9.18 < 10-8

c0.96 0.11 (0.74, 1.18) 8.49 < 10-8

σ0.27 0.01 (0.26, 0.29) 42.76 < 10-50

l( ) -124.11

AIC 260.22

Table 8

Results of fitting the Weibull model to 900 pairs of age-weight data for striped bass.

Parameter Estimate Std error Conf interval t-value p-value

W∞44.23 2.58 (39.17, 49.29) 17.14 < 10-50

h17.89 1.63 (14.70, 21.08) 10.10 < 10-30

t0-7.21 1.77 (-10.68, -3.75) -4.08 < 10-4

β4.43 0.60 (3.26, 5.61) 7.37 < 10-8

σ0.28 0.01 (0.26, 0.29) 42.29 < 10-50

l( ) -123.23

AIC 256.46

Revista de Biología Tropical, ISSN: 2215-2075, Vol. 73: e2025-171, enero-diciembre 2025 (Publicado Set. 30, 2025)

study growth, although there are good reasons

to use weight. Allen and Hightower (2010)

point out that weight can be used as a surrogate

for fecundity or the contribution of females to

the spawning population. Harvest regulations

in some fisheries are set to allow the average

weight of fish to increase, with the expectation

that protecting large, highly fecund females will

improve recruitment; or simply as targets of

biological production expressed in biomass or

number of individuals and its relative weight

(catch quota). Information about growth also

indicates the “health” of a population relative to

its food resources and the quality of the aquatic

environment. There are many fisheries that

regulation is based on the weight of the indi-

vidual such as abalone and some clams (Botel-

lo-Ruvalcaba et al., 2010; Luquín-Covarrubias

et al., 2022). However, in aquaculture studies

age-weight data are often obtained under the

assumption of controlled conditions.

Since the Weibull distribution was con-

ceived with the probability of failure in mind,

it inherently tied to time. This intrinsic con-

nection poses a significant challenge for its

application in marine organisms. In fisheries

management, age structure is a key predic-

tor of population dynamics and is therefore

crucial for sustainable management. Age infor-

mation forms the basis for calculations of

growth rate, mortality rate, and productivity

(Campana, 2001).

Campana and Thorrold (2001) estimated

that well over 1 million fish were aged world-

wide in 1999, primarily using scales and oto-

liths. These efforts far exceed those routinely

applied to non-fish species, underscoring the

importance of age-structured information in

fisheries science. Techniques for determining

the age of organisms have not only improved in

precision but have also streamlined the research

and monitoring processes of marine popula-

tions. Among the most notable techniques are

reading growth rings in otoliths (Fairfield et al.,

2021), scales (Campana & Thorrold, 2001), and

digital imaging, the latter offering greater preci-

sion and efficiency in reading (Villamor et al.,

2016). Additionally, genetic DNA techniques

are particularly valuable when other methods

are not applicable, such as in the case of lobsters

(Fairfield et al., 2021). The Weibull model could

see more frequent use.

Comparison of t0 values for both cases.

vB model: we have two sets of data where the

ages and weights are in similar ranges, and in

both cases the data begins in age class 4. The

level of information about t0 is not similar, in

the first model the error is additive and average

age-weight data are used; while in the second

model, the error is multiplicative, and 900 pairs

of age-weight data are used. What is noticeable

in the second example is the reduced uncertain-

ty in the estimation of the error term, compared

to the fit of the model to the data set of the first

example (22 pairs of data).

We can see this in the magnitudes of the

t-values of the fits to both models, t-value =

5.96 and t-value = 42.88, respectively. Weibull

model: With both data sets, the systematic part

of the model is different, but the stochastic

part (error term) is similar. When fitting the

vB model, the parameters W∞ and k have the

lowest estimation uncertainties. When compar-

ing the quality of fit of the vB and Weibull type

models, using the AIC we found that the second

model has a better fit, since it has a lower AIC

value (256.46) than the first model (346.55).

In the Weibull type model, we note that the

age t = h corresponds to the inflection point

of the growth model, regardless of the value

of the parameter β; whereas in the vB model

the inflection point is not a parameter but is

a function of the different model parameters.

Accordingly, parameter β of the two models

acts differently. In terms of metabolic pro-

cesses, it can be thought that, for the Weibull

model, catabolic processes dominate up to the

value h = t - |t0|; thereafter, anabolic processes

dominate until asymptotic weight is reached. A

similar interpretation is not straight forward in

the vB growth model (Shi et al., 2014). In the

vB model, catabolism is assumed proportional

to body weight, and anabolism to surface area

or body weight (Gamito, 1998). For individual

growth, the Weibull model is a generalized ver-

sion of the vB model and has also been related

14 Revista de Biología Tropical, ISSN: 2215-2075 Vol. 73: e2025-171, enero-diciembre 2025 (Publicado Set. 30, 2025)

to the Schnute-Richards general growth model

(Swintek et al., 2019). The Weibull function

has also been proposed to model population

growth because it has similar properties to the

logistic, with an asymptotic size of K; in this

case t0 is not present; in other words, the model

does not depend on t0. In this case, parameter h

expresses the value of time when the population

is 0.63 K (Prager et al., 1989).

In growth studies, generally speaking, the

Weibull model has been found to outperform

other models including the Richards and Gom-

pertz (Dagogo et al., 2023). A simplified, one-

parameter version of the Weibull model was

found to fit better pine tree growth as com-

pared to other models, including that of Rich-

ards (Souza et al., 2021). The Weibull model

best described live weight data of partridges

compared to other eight other models com-

monly used, including logistic and Richards

(Wen et al., 2019).

To our knowledge, for striped bass there

had been no previous publications on indi-

vidual growth using the Weibull model. In

an earlier study, the vB model was used to fit

individual growth data. The oldest fish were 12

years old and differences were found in param-

eters of males and females (Collins, 1982). For

reared striped bass, the vB model was also

fitted to growth data using tagging informa-

tion. Maximum age estimated was 9 years and

analyses were not conducted for separate sexes

(Callihan et al., 2014).

As expected, the result obtained using the

vB model with the Pacific halibut was similar

to that of Quinn II and Deriso (1999). No stud-

ies using the Weibull model for this organism

were found. For the Pacific halibut, in all cases,

the Weibull model estimated lower asymp-

totic weights compared to the vB model. This

is explained by the fact that the latter model

assumes a rapid initial growth that decreases

as it approaches the asymptote, whereas the

Weibull model has a lower initial parameter

(-7.32 and zero for two scenarios) compared to

vB (3.037).

In all three cases of the Weibull model,

using the AIC criterion, the best fit was

achieved with growth parameterization and

setting t0 = 0.

The relationship between individual

growth models and cumulative distribution

functions is very useful because it allows us to

use the knowledge we have about distribution

functions (Marshall & Olkin, 2007), to obtain

new growth models or extend existing mod-

els. Knowledge of different functions (such as

density, cumulative distribution, and survival)

associated with a probability distribution, as

well as their properties, can be translated into

functions that inform about characteristics of

the growth process.

The recognition of the equivalence between

individual growth models and the cumulative

distribution functions F(t) allows the use of

knowledge about the distribution functions of

continuous random variables. Various func-

tions related to the cumulative distribution

function can be used to express some type of

information of the individual growth process.

For example, the density function f(t), which

is the derivative of the cumulative distribu-

tion, can be associated with the growth rate

W∞f(t). The survival function S(t), which is

the complement of the cumulative distribution

S(t) = 1-F(t), gives us information about the

proportion of an organism that remains to grow

at time t, W∞S(t).

In the study of lifetimes, the quantile func-

tion tp is defined as the time at which a pro-

portion p of failures will have occurred, it

can be used to know the time at which an

organism will have grown a proportion p of its

maximum asymptotic size (Meeker et al., 2022).

Knowledge of the density, survival, and quan-

tile functions, among others, of a distribution

associated with a given growth model allows

us to have a better vision of the characteristics

and properties of the individual growth model

under study.

The vB length and weight growth mod-

els are two clear examples of the relationship

between growth models and distributions. The

length growth model is associated with the

exponential distribution, and the vB weight

growth model is associated with the generalized

Revista de Biología Tropical, ISSN: 2215-2075, Vol. 73: e2025-171, enero-diciembre 2025 (Publicado Set. 30, 2025)

exponential distribution. That is, the vB weight

growth model and the generalized exponential

distribution are two simple extensions of the

vB weight growth model and the exponential

distribution, respectively.

It seems that the wide application of the vB

weight growth model is not due to its quality

of fit, since the fit of this model to growth data

does not always outperform other models with

which it competes, rather very often its use

is chosen a priori. For this reason, it is worth

asking if, in addition to the vB weight growth

model, there is another model, which is an

extension of the basic vB length growth model,

and with a similar form, but with adequate flex-

ibility in order to have a better quality of fit to

age-weight growth data.

An extension of the exponential distribu-

tion that is well known and widely applied in

the analysis of Survival and Reliability data is

the Weibull distribution. The above suggests

that a direct extension of the vB length growth

model, additional to the vB weight growth

model, is the Weibull growth model. An impor-

tant advantage of the Weibull-type growth

model is its flexibility, which yields better fits

than other models, as shown in the comparative

study shown herein.

Ethical statement: The authors declare

that they all agree with this publication and

made significant contributions; that there is no

conflict of interest of any kind; and that we fol-

lowed all pertinent ethical and legal procedures

and requirements. All financial sources are fully

and clearly stated in the acknowledgments sec-

tion. A signed document has been filed in the

journal archives.

ACKNOWLEDGMENTS

This research did not receive any specific

grant from funding agencies in the public, com-

mercial, or non-profit sectors. MÁCM, JAGB,

EVD and EARF received partial financial sup-

port from CONAHCYT.

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