Hanoverian, Brazilian Showjumper,
English Thoroughbred and Crossbred horses reared by the Brazilian Army
were weighed and measured from six months of age to adult. In total
4,860 measurements on 1,445 horses were available. General curves were
estimated as a function of time by the Gompertz, Brody, Logistic,
Weibull and Richards curves, using PROC NLIN procedures of SAS ®. The
Richards Curve did not converge for weight or height of any of the
genetic groups or sexes. The logistic curve did not converge for any of
the weight traits while the Gompertz also did not converge for height
in several groups. R² varied between 0.55 for weight in females of the
crossbred group to 0.92 for males of the same group. For the height
traits the highest R² (0.66) was found for female Hanovarian horses and
lowest for males of the same breed (0.12). In general the curves
estimated similar values for asymptotic height and weight, except for
Logistic curve, which also showed lowest R² and highest error. Results
for the Weibull and Brody curves were similar in all cases so where
possible the Brody curve was selected as the best curve as it had less
parameters. The Gompertz curve tended to underestimate mature weights
and height. Estimates for both weight and height were in general higher
in males than for females. In most cases the b parameter was shown to
account for < 0.0001% of the variation in the curve shape. The k
parameters, which indicate maturity, were of similar magnitude for the
Brody, Gompertz and Weibull curves, for both height and weight within
breed. This parameter indicated that there is little difference in
maturation rates between males and females.
CURVAS DE CRESCIMENTO NÃO-LINEARES PARA PESO E ALTURA EM QUATRO GRUPOS GENÉTICOS DE CAVALOS
Cavalos das raças Hanoveriana (HA),
Brasileiro de Hipismo (BH), Puro Sangue Inglês (PSI) e mestiços
(PSIxBH) criados pelo Exército brasileiro foram pesados e medidos de
seis meses de idade até adulto. Realizaram-se 4.860 medidas em 1.445
cavalos. Estimaram-se curvas de crescimento gerais usando os modelos de
Gompertz, Brody, Logistic, Weibull e Richards, segundo o procedimento
PROC NLIN do programa SAS ®. A curva de Richards não convergiu para
peso nem altura para nenhum dos grupos ou sexos avaliados. A curva
logística não convergiu para os pesos, enquanto o modelo de Gompertz
não convergiu para altura em vários grupos. R² variou entre 0,55 para
peso em fêmeas mestiças até 0,92 em machos do mesmo grupo. Para altura,
o maior R² (0,66) foi para machos Hanovarianos e o menor para fêmeas da
mesma raça (0,12). Em geral, as várias curvas estimaram a mesma altura
e peso adulto, exceto a curva logística, que teve o menor R² e mais
alto erro dentro de cada grupo. Resultados para as curvas de Weibull e
Brody foram similares em todos os casos. A curva de Brody foi
selecionada como a melhor, porque possui menos parâmetros. A curva de
Gompertz teve a tendência de subestimar pesos e alturas adultos.
Estimativas para ambos as características foram mais altas em machos
que fêmeas. No maior parte dos casos, o parâmetro b levou em
consideração menos que 0,0001% da variação em forma da curva. Os
parâmetros k, indicando maturidade, foram de magnitude similar para as
curvas de Brody, Gompertz e Weibull, para altura e peso dentro de raça,
o que indicou pouca diferença entre machos e fêmeas para taxa de
maturação entre sexos.
PALAVRAS-CHAVES: Brody, Gompertz, Logistic, Richards, Weibull.
Little is known about growth rates of
horses and tracking growth means that informed decisions can be taken
about growing horses’ nutrition program and accurate assessments growth
progress can be made. Rapid and/or irregular growth rates may be linked
to the incidence of developmental orthopedic diseases and therefore
growth tracking and accurate diet formulation are useful (KEELE et al.,
1992; WILLIAMS et al., 1992) to reduce the on-farm incidence of
developmental diseases (WEBSTER et al., 1982).
Modeling growth using mathematical functions summarizes growth data for
an individual or a population (BATHAEI & LEROY, 1996). Many data
points taken over time are reduced to a few parameters. The resulting
curve provides a visual description of growth, and interpretation of
the estimated parameters provides an explanation of what is occurring
biologically (KSHIRSAGAR & SMITH, 1995).
The shape of growth curves have been reported to vary according to the
species of animal, environment and trait (EFE, 1990; AKBAS et al.,
1999). An ideal equation is one which adequately predicts the overall
shape of a growth curve and, in addition, can be extended or modified
to give greater flexibility and precision (MOORE, 1985). Selection of
fast or slow growing animals can be carried out using these functions,
especially animals which are more mature at an earlier age (BROWN et
al., 1972; FITZHUGH, 1976).
Non linear functions with exponential components are most commonly used
to describe this type of growth, as these summarize a large volume of
data. These functions are easy to interpret biologically and easily
compared between different production systems (SILVA et al., 2002).
According to TEDESCHI et al. (2000), the parameters of nonlinear curves
(such as Gompertz, Brody, Logistic, Weibull and Richards) which have
biological interpretation are superior asymptotic (mature) weight and
maturing rate, which is an indication of growth rate. The other
parameters are mathematical constants which help to determine curve
shape. The objective of this study was to select a non-linear function
which best describes growth of horses reared by the Brazilian Army.
All horses in the Brazilian Army are
produced by the Coudelaria de Rincão, located in São Borja, Rio Grande
do Sul State. It is situated at latitude 55° 35’ 00” south and 28° 45’
40” West, with an altitude of 130m and the climate is Humid subtropical
Cfa according to the Köppen classification, with rainfall well
distributed throughout the year and mean 1,350 mm. The study has 206
dams and 15 stallions. Both mares and stallions were from the proper
herd or on loan from other studs. Weaning was at six months of
age. Pastures were based on temperate grasses including oats (
Avena sp), azevém (
Lolium multiflorum) and clover (
Trifolium repens). Stallions also received alfalfa hay daily (
Medicago sativa) ad libitum.
Data were also collected from Military Organizations distributed in all
of Brazilian national territory, except Amazon, where the animals are
sent after 24 months of age. In these installations they receive
concentrate with 14% crude protein (CP), divided in three meals a day
and foals received a ration with 18% CP. The horses may be stabled or
semi-stabled, depending on the station. The horses may be transferred
between stations, depending on necessity, or because they are used by a
particular officer who uses the horse in competition. Data was
collected by veterinarians in each station.
Data was available on 4,860 weight and shoulder height measurements on
1,445 Hanoverian, Brazilian Showjumper, English Thoroughbred and
Crossbred horses, measured from six months of age to adult.
Curves
Weight and height were analysed as a function of time using Gompertz,
Brody, Logistic, Weibull and Richards curves. The Richards curve
describes changes in size Y (weight or height) in relation to age
t, and is represented by the equation:
Yt = a (1±be-kt)m.
The interpretations of the parameters are as follows:
a = asymptotic value interpreted as mean weight/height at maturity;
b
= scale parameter (integration constant), indicates the proportion of
asymptotic growth to be gained after birth, and is established by
initial values of Y and
t;
k = index of maturity or precocity measure. Calculating
k-1
the time used to obtain maturity is found, which serves as a measure of
maturity (TAYLOR & CRAIG, 1965). The greater the value of
k means that animals mature earlier;
m = is inflexion parameter, which establishes the degree of maturity
u at the inflexion point
ui = [(m-1)/m]m, with ut the proportion of weight attained at age
t: ut = Yt/a = (1-be-kt)m; and
e is the Napier base for natural logarithms. The sign “±” when
m>1, is – and + when
m<0.
The parameter
m assumes the
following values for the other curves in the Richards family: Brody
(1), Gompertz (µ → ∞) and Logistic (-1). The Brody and Logistic curves
are respectively,
Yt = a-be-kt and
Yt = a(1+be-kt)-1.
The Gompertz curve is a limiting case of the generalised logistic as t
becomes very small or very large, whose equation is:
Yt = a e–be ^ -kt. The Weibull curve is :
Yt = a-(be-kt^d)
Procedures described by PEREIRA & ARRUDA (1987) and SAS (1987) were
used for convergence of non-linear data. The values of sum of squares
of the residual (SQr), coefficient of determination (
r²) and divergence from regression for each function were calculated. The
r² was calculated as
r² = 1 (sqe)/(sqt), where sqe is the sum of squares of the error and sqt, is the total sum of squares (KVÅLSETH, 1985).
Individual estimates of growth parameters were obtained using the
modified Gauss-Newton method in the NLIN procedure of SAS ® (HARTLEY,
1961; LAWTON
et al., 1972). Convergence was assumed when the difference
in sum of residual sum of squares between the
ith–1 to the ith iteration was < 10
-8. The mean square of the residual was calculated for each function. The
r² and error were used to evaluate the adjust of the curve.
RESULTS
The Richards Curve did not converge for weight or height of any of the
genetic groups or sexes. The logistic curve did not converge for any of
the weight traits while the Gompertz also did not converge for height
in several groups. R² varied between 0.55 for weight in females of the
crossbred group to 0.92 for males of the same group. For the height
traits the highest R² (0.66) was found for female Hanovarian horses and
lowest for males of the same breed (0.12).
Tables 1,
2,
3 e
4 shows a summary of the analyses carried out for each breed and trait.
Figure 1 and
Figure 2
show the best fit curves for the data in this study with 95% confidence
limits. Adult height and weight is reached in all cases at about three
years of age.
In general the curves estimated similar values for asymptotic height
and weight, except for Logistic curve, which also showed lowest R² and
highest error. Results for the Weibull and Brody curves were similar in
all cases so where possible the Brody curve was selected as the best
curve as it had less parameters. The Gompertz curve tended to
underestimate mature weights and height. Estimates for both weight and
height were in general higher in males than for females.
In most cases the
b parameter was shown to account for < 0.0001% of the variation in the curve shape. The
k
parameters, which indicate maturity, were of similar magnitude for the
Brody, Gompertz and Weibull curves, for both height and weight within
breed. This parameter indicated that there is little difference in
maturation rates between males and females.
DISCUSSION
Mean adult values estimated by the adjusted curves are very close to
the mean values found by CAMPOS
et al. (2007) in the adult horses
(>48 months of age) of both sexes. The Hanoverian horse is a founder
of the Brazilian Showjumper breed (DIAS
et al., 2000), the two largest
breeds in this study.
The logistic and the Gompertz equations have three parameters, all of
which are biologically interpretable and statistically significant in
the present analysis. Parameters having a straightforward meaning are
advantageous for statistical parameterization of non-linear equations.
Parameters of such non-linear functions have to be estimated by using
an iterative regression approach, such as PROC NLIN of SAS ®, which
requires an initial estimate of the parameters. Initial values here
were taken from SANTOS
et al. (2007) for Pantaneiro horses.
Contrary to this study, experiments have shown that the Richards
function has the best for modeling this type of data (BROWN
et al.,
1976; DeNISE & BRINKS, 1985), but these studies also show that the
Brody function is better above six months of age, as with these data,
where the youngest age was 6 months. Other authors found that while the
logistic function underestimates and Brody overestimates adult weight
compared to the Richards and Von Bertalanffy functions (DUARTE, 1975;
PEROTTO
et al., 1992), although standardization has been carried out
for use of the Brody function in cattle (BROWN
et al., 1972; DUARTE,
1975; SILVEIRA JR., 1976; LUDWIG, 1977). Although it is shown to be
flexibile, the Richards equation has often been criticized as the shape
parameter has no obvious biological interpretation and is so unstable
numerically that its estimate becomes useless (ZEIDE, 1993). YIN
et al.
(2003) found that in no case did the Richards equation achieve a
statistically significant improvement over the logistic equation, and
in only two cases did it improve on the Gompertz equation. For the data
sets used in the present analysis the Richards equation did not
converge. TEDESCHI
et al. (2000) also found that the Richards curve had
the most difficulty in converging when using Brazilian beef cattle data.
No mathematical model can accurately describe every biological
phenomenon (TABATABAI
et al., 2005). Many models have been developed to
deal with sigmoid growth (ZEIDE, 1993) and new ones are continuously
being proposed. While the logistic function is symmetric around the
point of inflection, the Richards function is more flexible and can fit
asymmetric growth patterns (TABATABAI
et al., 2005; ZHU
et al., 1998),
however, it has more parameters than the logistic function. The
Gompertz function has the same number of parameters as the logistic
function and the Weibull function has the same number of parameters as
the Richards function and both can fit asymmetric growth, but they are
not very flexible (YIN
et al., 2003). In the logistic model, the growth
curve is symmetric around the point of maximum growth rate and has
equal periods of slow and fast growth (TABATABAI
et al., 2005). In
contrast, the Gompertz model does not incorporate the symmetry
restriction and has a shorter period of fast growth. Both the logistic
and Gompertz have points of inflection that are always at a fixed
proportion of their asymptotic population values. The same author noted
that the description of growth by fixed-shape sigmoid models such as
logistic, Gompertz or Von Bertalanffy curves may not be adequate
because of the failure of the assumption that a constant curve shape
holds across treatment groups. While the above equations predict a
positive non-zero value for Y at time t = 0, the Weibull
function predicts
Yt = 0 when
t = 0 (YIN
et al., 2003), which is not true with growing animals.
The results found here are different from those found by SANTOS
et al.
(2007) working with Pantaneiro horses and who chose Richards and
Weibull curves for shoulder height and weight, respectively. SANTOS
et
al. (1998) with different data, also for the Pantaneiro horse, chose
the Weibull curve for shoulder height. It also differs from the results
found by FREITAS (2005), who studied growth curves in eight species of
animal, and concluded that the logistic model was the most adequate.
Figures 1 and
2 show the resultant curves and 95% confidence limits for male and female horses respectively.
The choice of the best curve using the analysis of variance residuals
is not necessarily the best option as longitudinal data (such as weight
and height on the same animal) show correlated errors between ages.
These errors are caused by fluctuations over time, due o various
factors which are not necessarily reflected in the curve. Recommended
methods for curve selection include the evaluation of the difference
between observed and predicted values at specific ages (BROWN
et al.,
1976), but these ages are not always available. Other methods include:
residual sum of squares (PEROTTO
et al., 1992), regression deviations,
determination coefficient (R²), percentage and difficulty of
convergence (BROWN
et al., 1976; OLIVEIRA
et al., 2000), curve
behaviour and evaluation of parameters by comparison and graphic
evaluation of the curves (FITZHUGH, 1976). Here R² and divergence were
used as criteria for selection of the model.
Other authors (JONES, 1987; HEUSNER, 1992; YAMAMOTO
et al., 1993;
THOMPSON
et al., 1994 and PAGAN
et al., 1996), found that males
generally presented higher growth rates than females. The males in this
study were castrated at 24 months of age, but some authors (HEUSNER,
1997) state that castration does not affect development at this age.
Growth data in the literature is limited for adult horses of the breeds
studied here. Adult shoulder height was found to be 137.65 cm for the
Pantaneiro horse in Brazil (MISERANI
et al., 2002) and 144cm for
Campeiro (McMANUS
et al., 2005). The same trait in Mangalarga Machador
horses was found to be 151.5cm and 151.6cm for males and females
respectively (CABRAL
et al., 2004) and 149 and 150cm for Arabian horses
(SADEK
et al., 2006). Using regression equations, STAINER
et al. (2004)
showed that Thoroughbred horses reached 542 ± 6.2 kg reached at 7 yrs,
somewhat heavier than found here.
CONCLUSION
The Weibull and Brody curves best fitted the increase of height and
weight for horses reared in the Brazilian Army. Other curves returned
results not consistent with the data or, in the case of the Richards
curve, failed to converge. There was little difference between sexes
for maturing rate.
ACKNOWLEDGEMENTS
The authors wish to thank the Brazilian Army for access to their data
as well as CNPq INCT-IGSPB and Finatec for financing the study.
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Protocolado em: 14 jan. 2009. Aceito em: 1o fev. 2010.