Asymptotic nonlinear regression models for the growth of White Fulani and N'dama cattle in Nigeria

A E Salako

Animal breeding & Genetics Laboratory, Department of Animal Science, University of Ibadan, Ibadan. Nigeria.
debosalako@gmail.com   ;   debosalako@outlook.com

Abstract

Nonlinear functions of body weight at different age intervals were used to estimate the mature weight (A), shape of response (B) and maturing rate (k) parameters of asymptotic growth models for Nigeria White Fulani and N'dama cattle. Records obtained from flocks kept at the University of Ibadan teaching and research farm in Ibadan were used. Mitsherlich  Gompertz and Richard functions were fitted to the data in addition to the linear function. Age-weight records of White Fulani from birth to 4 years and from N’dama monitored to 30 months of age were used to estimate the average growth curve for each breed.

The weight difference between breeds was consistently in favor of White Fulani breed from birth to maturity during the period. Predicted A and k values for weight at fixed ages in the iterative processes indicated that Richard’s function was inadequate for both breeds. Mature weight was attained at approximately 4 years of age in N'dama from both Mitscherlich and Gompertz model but was yet to be reached at that age in the White Fulani. Mitscherlich function gave better estimates of weight at maturity, but the asymptotic residual variances were higher in N’dama because the birth weight was overestimated. Linear and nonlinear regression analyses of weight-age data and comparisons of degree of maturity at different premature ages showed that the differences in the growth patterns of the two breeds agreed with late rather than early predicted values of A and k.

Keywords: models, maturity pattern, prediction, weight-age

Introduction

The economic importance of mature size, rate of maturing and rate of gain in cattle obtainable from lifetime weight-age measurements through statistical modeling, have the potential to suggest management and genetic intervention for improved productivity. More so, model measurements by mathematical functions give the opportunity to interpolate to non-observed intervals (Tungray 1983). Two biologically relevant parameters characterize the growth models: the size parameter, most frequently evaluated as weight at maturity (A) and the growth rate relative to body size, commonly referred to as maturing rate (k). Early estimation of these parameters can be of importance for selection purposes, given their association with other traits and the economy of production (Long et al 1975; Joandet and  Cartwright 1969; Butts et al 1980a, b; Tawah and Franke 1985). Thus, strategies for altering the shape of the growth curve can be provided apart from its use for studying the pattern of growth and the relationship between growth rate and cow efficiency.

Several nonlinear models have been evaluated with regard to their goodness of fit, biological interpretation of the parameters, computational difficulty and evaluation of genetic and environmental effects on the growth curve parameters (Brown et al 1972a, b; Brown et al 1976; Fitzhugh 1976; Morrow et al 1978; DeNise and Brinks 1985; Doren et al 1989; Bullock et al 1993; Kaps et al 2000; Arango and Van Vleck 2002). Published works on the objective basis for the utilization of the White Fulani and Ndama cattle in Nigeria is scanty. In this study the fitting of four growth models to White Fulani and N’dama cattle growth data was evaluated to elicit differential growth patterns and maturity rate resulting from the discrepancies in their biology systems. Thus, different measure of size, rate of gain and rate of maturing of the genotypes furnished by the parameter estimate can be used for optimizing their utilization when kept under the same environmental condition. The economic importance of the biological interpretations of the parameters of the models remains invaluable.

The study was designed to investigate the model that can best describe the growth of White Fulani and N’dama cattle in Nigeria and compare among others the rate of maturing of the breeds

Materials and Methods

Location and animals

The farm of the University of Ibadan, Ibadan where the animals whose records were analyzed in this study were kept is situated at 200 m above sea level and lies 7⁰30’N. Ibadan, Nigeria belongs to the low land rain forest vegetation zone with two peaks of rainfall having a dry season of 3-4 months (November-March) marked by retarded vegetation growth with little variation in temperature throughout the year (Keay 1959).

The herd, which comprised White Fulani and N’dama genotypes was contributed by uncertain number of locally available extensively managed herds, institutional flocks as well as cattle markets. Animals were purchased at different times as funds were made available under the developing economy and kept for teaching and research purposes. In the course, some were culled systematically in selective breeding.

Routinely, the animals grazed in the early hours of the day, about 10 am depending on whether condition avoiding wet grass to avoid uptake of worms. The grazing paddock consists mainly of giant star grass, Centrosema pubescence and Cynodon spp. Cows were supplemented with concentrates made from maize, groundnut-cake, palm kernel cake and soya bean meal according to the production. One hundred and thirty-six body weight records obtained from 80 individuals comprising 57 White Fulani and 23 N'dama were analyzed. The white Fulani: 22 females and 12 males and N'dama: 13 females and 10 males cut across ten different ages: birth, 3, 6, 12, 18, 24, 30, 36, 42, 48 months. At birth, calves were weighed within 24 hours on a small platform scale and a bigger platform balance was used subsequently to take body weights at older ages. Each weight record was taken three times independently and the mean recorded. This was done for all individual animals across all ages of animals considered.

Weights at fixed ages, from birth to maturity were analyzed by least squares procedures. Complete data on 80 calves comprising male and females, born from 1982 to 1987 were available from birth, 3, 6, 12, 18, 24, 30, 36, 42 and 48 months (Data were archived until information such as investigated in this study was needed). Of these, 65 had growth data up to 36 months and 71 had up to 48 months of age.  

Experimental design and statistical model

Least square means, standard deviation (SD) and coefficient of variation (CV) of body weight of White Fulani and N'dama of different sexes and ages were computed using the descriptive statistics module of the SPSS (1989).

Analysis of variance procedure of the same package was used in a preliminary statistical treatment to detect the fixed effects of breed, sex and their interaction on body weight across the ages.

The following model was used:

Y_ijk = μ + α_i + β_j + αβ_ij + e_ijk

Where μ = general mean assume to be fixed and unknown

α_i  = fixed effect of breed ( i=1,2);

β_j  = fixed effect of sex (j=1,2);

αβ_ij  = effect of interaction between breed and sex;

e_ijk  = error term associated with the measurements and assumed to be normally and independently distributed ~(0,σ_e) where σ_e is the common variance.

Four growth models: linear, Mitsherlich, Gompertz and Richard’s models were fitted to the data using nonlinear regression program (algorithm) of the SAS (1988). The iterative procedures were selected for their efficiency in estimating the parameters compared with other algorithms (Doren et al 1989). They were compared for goodness of fit so as to choose the most appropriate model for breed comparison. The same package was used to run the linear and nonlinear regression procedure to fit modified asymptotic least square algorithm (Miller 1981). Regression equations of body weight on age and their residuals were obtained for use in a subjective test of goodness-of-fit of the models.

Model description

The model parameters of linear and non linear iterative procedures were estimated as:

Linear: Y_t  = a + bX

Model program = b1 + b2 *X

Gompert = A – Be ^–kit

Model program = b1*exp* (– b2 * exp*(– b3 * x))

Mitcherlich: Y_t = A - Be^-kt 

Model program =   b₁-b₂ *exp (- b₃ * X)

Richard’s : Y_t =A (1 – Be ^–kt)^m

Model program = b1 / ((1 + b3 * exp(-b2 * age)) ** (1 / b4))

In all the above equations, Y_ij represents the body weight of animal i at age j; a_i, b_i, and k_i, are the parameters of the model; ε_ij is the fitting error while b₁, b₂, b₃, and b₄ are equivalents of parameters A, B, k and m to be estimated. Y_t represents the body weight at time t, A is the asymptotic weight or mature weight or potential final weight which is approached as time t increases, B is a scaling parameter determining the shape of the response (relating body weight at time t = 0, i.e., at birth to mature size) and k is a function of the ratio of maximum growth rate to mature weight (maturing rate index), m being the inflection parameter in Richard’s model and e is the natural logarithm base. Age is expressed as time from birth or time from conception or time from the x-intercept (y=0). In other cases, age can be taken as t+g where g is the gestation period and then the Gompertz (1825) equation becomes Mitsherlich (1959) model where the dependent variable is not log transformed:

The convergence criterion used was as follows:

(SSE_{i -1} - SSE_i)/(SSE_i + 10⁶) < 10^-8

Where: SSE_i is the residual sum of squares for the ith iteration.  Starting values for each estimated parameter was taken arbitrarily from previous computations.

Mitscherlich’s (1959) and Gompertz (1825) three-parameter functions, {Y_t = A(l - Be^-kt), lnY_t = A(l - Be^-kt)} respectively, and Richards’(1825) four-parameter function, Y_t = A(1 -Be-kt)^m, frequently have been used to describe weight-age relationships. The difference between the three- parameter and Richard’s (1825) models is the points of inflection (m) which in the Mitscherlich and Gompertz models is fixed. The biological interpretation of these growth curve parameters and their functions have been provided by Brown et al (1976), Fitzhugh (1976) and Doren et al (1989)

Mature weight (A), which represents the asymptote of the model, represents weight at a constant condition and not the heaviest weight attained as time ‘t’ approaches infinity. It is an averaging-out of the short term fluctuations in body condition that caused the within animal weight variation and provides the value for individual’s normal body composition under a defined production system or environment. Rate of maturing (k) stands for the constant rate of growth maintained until the animal attained the achieved mature weight and can also earliness/lateness of maturity. When ‘k’ is small, it means the animal is late maturing but when it is large, it is early maturing.  Any circumstance that influence the curve up to the asymptotic weight will also affect the maturing rate ‘k’ and hence earliness/lateness to maturity (asymptote). ‘b’ is the rate of gain. A further differentiation (from calculus) of the model enhances the biological interpretability of the parameter by further relating ‘k’ to gain and ‘b’ to early weight and maturity changes. Point of inflection ‘m’ in the Richard’s function is where the estimated growth rate changes from an increasing to a decreasing function since the rate of change is maximum at the point of inflection. The Gompertz model, as also the Von Bertalanffy has a fixed point of inflection relative to mature size but the Richards has a flexible point of inflection although more difficult to compute. Degree of maturity was estimated by the fraction of mature weight attained at various ages before maturity. The least squares means obtained from analyses of weight at fixed ages were used to calculate observed degrees of maturity, whereas the predicted weights were used to calculate estimated degrees of maturity. For Richards' model the degree of maturity at the point of inflection is a function of the inflection parameter.

Tests of goodness of fit and homogeneity of the growth curves for the two breeds could be biased due to correlated errors due to repeated measurements on the same individual, even though the residual correlations may have been reduced, because the regressions were made from several animals (Doren et al 1989). Comparison of fits was thus achieved by mere comparing the observed and predicted plots.

Growth patterns were compared by the degree of maturity attained at a given growth stage. Following the notation used by Fitzhugh and Taylor (1971), U_t = Y_t/A, where the degree of maturity, u, is expressed as the ratio of weight, Y, at a given age, t, on mature weight, A.

Results

Least square means, standard deviation and coefficient of variation of body weight of White Fulani and N'dama of different sexes and ages are presented in Table 1. It showed  a general increase in body weight from birth to 30 months in White Fulani and to 48 months in the N’dama cattle. Body weight increased in both breeds but values were generally higher for White Fulani cattle than N’dama: significant difference was observed at birth, 6 and 30 months of age (Table 2). Up to those ages, gain fell off with growth rate towards maturity. Standard deviations and coefficients of variation associated with mean values were generally large for both breeds at all ages. Ignoring gender, birth weight of White Fulani differed (P<0.05) from that of N'dama by an average of 6.40 kg. (Table 2). A superiority of 29.2 kg was recorded by the N'dama over the White Fulani at 6 months of age with higher coefficient of variation in N'dama and at 30 months. White Fulani was superior (P<0.05) at 30 months.

Sex differences were observed at 3 and 6 months of age for White Fulani and birth, 12, 24 and 30 months for N’dama.  At birth, the weight of the N’dama males did not differ from that of White Fulani. At 3 and 6 months, the male animals differed from females in White Fulani but not in N’dama. However, at 12, 24 and 30 months sex difference was observed to the advantage of the male. (Table1). The difference between the mean values of male and females at birth was 0.5kg and 4.6kg for White Fulani and N'dama respectively. From 12 months, none of the sexes was consistently superior. It was expected that the relative ratio of each sex within age could bias the significance of the difference between sex means.

Predicted body weight from linear, Mitscherlich and Gompertz function are presented in Table 3. Accuracy of the different models was judged by comparing the observed with estimated values. It showed that the linear function overestimated all the body weights of White Fulani except at 3 month whereas it underestimated the weights of N’dama except at 3, 42 and 48 months. However, estimated body weight increased until 48 months in both breeds. Mitscherlich model on the other hand overestimated the body weight of White Fulani cattle at birth, 6, 18, and 24 months and that of N’dama cattle at birth, 12, 18, 24 and 42 months. The model fit of the Gompertz function indicated that body weight was overestimated at birth, 3 and 24 months in White Fulani and at birth, 24 and 42 months in N’dama. Results further showed that all the three functions overestimated the birth weights and weights at 24 months and underestimated body weights at 30 months for the two breeds. Weights at 18 months for the two breeds were overestimated by the Mitscherlich but underestimated by the Gompertz model. Both three-parameter functions underestimated N’dama weights at 36 months but overestimated them at 42 months. Table 3 shows the predicted values obtained from observed values after the iterative procedure was applied to both breeds. The range of the residuals were -30.65-18.44 and -13.08-13.59 in White Fulani and N'dama respectively. Correlations between the residuals of both breeds was not significant (0.344)

In Table 4, the parameter estimates A, B, k, their associated standard errors and coefficient of determinations were tabulated for the two breeds. The linear function showed the intercept on Y-axis and regression coefficient respectively for N’dama (63.6 versus 3.79) and White Fulani (28.7 and 7.27) regression lines. The regression coefficient was higher in White Fulani while the intercept was higher in N’dama. The parameter estimates and standard errors for the White Fulani from the Mitscherlich model were high: 21208, 21180 and 1.00 respectively. However, the associated coefficients of determination were comparable to the others: 0.89 – 0.98. A B and k parameters were higher for the White Fulani from Gompertz model.

Degrees of maturity of the two breeds estimated from the three-parameter functions are presented in table 5. Degrees of maturity shown by Mitscherlich and Gompertz models were 0.0011- 0.017 and 0.050-0.748 respectively for the White Fulani cattle. Corresponding range for the N’dama cattle was 0.077-0.955 and 0.081-1.00. In both cases, the rate of growth fell off with the increase.

Discussion

Rate of maturing, rate of gain and mature size are directly related to economics of production and as such are important traits which have attracted attention of breeders, livestock scientists and private beef enterprise. The exploitation of these parameters in growth models through curve fitting using growth-age data could alter economic returns in a positive direction. Asymptotic models have been useful for generating information that are used for improving the efficiency of production in different cattle breeds across different production environments. In Nigeria, White Fulani (WF) and N’dama cattle are traditionally used as beef animals but their relative efficiencies and potentials for beef production has been based upon some traditional considerations rather than empirical evidences emanating from objective experimentation. 

Result of this study has shown that the White Fulani cattle are generally bigger than the N’dama cattle and this difference is reflected at most ages (Tables 1 and 2).  Body weight ranges reported in literature for these cattle at different ages are characterized with substantial variation. However, the figures are comparable to those obtained in this study. For example, at birth, Rege et al (1993) and Robert and Gray (1973) reported 18.2±9.2 at Madauchi and 24.2kg for the White Fulani at Vom in Nigeria respectively. Wheat and Broadhurst (1968) and Tawah and Mbah (1989) reported 23.0±4.6 and 23.1±1.8 at Kabomo and Cameroon respectively while up to 27.0kg has been reported for the same breed at Shika compared with 24.30kg reported in this study. For the N’dama cattle, Olutogun, (1976) reported 17.7-17.8kg at upper Ogun ranch with no sexual dimorphism for birth weight while Ngere (1975) reported as low as15.0kg in Ghana. Theoretically, the superiority of the males has a physiological base. At 12 and 30 months of age, 82±3.4-180.52±3.2kg and 267±2.2kg - 300±9.4kg respectively for White Fulani at Birnin Kudu and Shika in Nigeria were reported against 135kg and 265kg respectively obtained in this study. On the other hand, 164 and 250.0kg have been reported for the respective ages (Olutogun 1976). Feedlot studies have shown that the White Fulani cattle are able to achieve growth performance of up to 1kg per day (Olayiwola et al 1975 1981, 1986; Ngere 1985) Data were not available on the White Fulani cattle for further growth comparison of the two breeds after 36 months in this study. However, marked variation observed from birth has been seen to characterize the breed at older ages. Apart from variation within localities where the cattle are kept, genotype-environment interaction might be responsible for the high variation reported in literatures among others. Comparison based on size alone can be very misleading since efficiency of production depends on many other factors. Comparison on the basis of cow productivity index that takes into consideration reproductive rate, cow and calf viability, milk yield, calf growth and calf weight have therefore been proposed. It was concluded that relatively smaller breeds such as N’dama could be equally efficient for beef production as the larger breeds such as the White Fulani, but optimum weight at slaughter for maximum efficiency will be lower for smaller breeds.  More so, smaller breeds could even have advantage under conditions of extreme fluctuations in feed supply because of their ability to lay down fat on a lower plane of nutrition which probably explains their ability to remain in a better condition than larger breeds during the dry season when breed is scarce.

Differences in growth pattern were expressed in terms of differences in mature weight and maturing rate (Tables 4 and 5). In general, weights at early ages are influenced more by environmental effects than were weights at later ages. The White Fulani weights were predicted well by all the three models although the linear and the Mitscherlich models were better than Gompertz as indicated by the respective residuals (Table 4). But for N’dama, Mitscherlich model predicted weights from birth to 24 months better than Gompertz which also predicted weights at older ages (30-48months) better than other functions. Comparisons of asymptotic body weight obtained with different functions using the same fitting error model showed that the Mitscherlich model provided better estimates for both breeds. These results are similar to those reported by Oliveira et al (2000) and Garnero et al (2005) in studies on growth of Zebu females. Considering the same growth function however, greater adult body weights were estimated when the fitting error variance was held constant along the trajectory. Varona et al (1997) reported that different approaches to model the growth curve fitting error variance can result in differences in parameter estimates. Here, highly similar estimates of adult body weight were obtained using either linear or exponential models to describe the variance of fitting errors. From The model with smallest standard error of prediction is regarded to have provided the best fit to the data, in order that ‘A’ parameter (asymptotic weight) values offered the best opportunity to make direct comparisons among all models (Brown et al 1976). All results in nonlinear regression are asymptotic. That means the standard error, for example, is only correct if the sample size is infinitely large. For any finite sample size as it was in this study, standard errors may be biased due to correlated repeated measure made on same individual animal.

Forni et al (2009) reported that the Gompertz, Von Bertalanffy, and Brody functions adequate for establishing mean growth patterns and to predict the adult BW of Nellore females. The Brody model is more accurate in predicting the birth weight of these animals and has better overall fit. The prediction of adult BW using nonlinear functions can be accurate when growth curve parameters and their covariance components are estimated jointly (Forni et al 2009).

Differences in growth were expressed in terms of differences in mature weight and maturing rate. The regression coefficients for weight change shown in Table 4 indicate that both breeds had a different growth trend between birth and 48 months of age. After 48months, the N'dama showed full maturity, indicating that mature weight was reached at approximately at this age while the White Fulani was only 74.8% mature as estimated from predicted value at 48 months from Gompertz equation (Table 5). Differences in the growth pattern of the two breeds were expressed by their degree of maturity as calculated from the different functions (Table 4).

The estimation of potential final weight in different species is a function of an algorithm fitting and the accuracy of judging is possible only when precise final weight is available (Brown et al 1976; Lopez et al 2000). Gompertz model estimated the degrees of maturity in both breeds but the Mitscherlich model was not efficient for calculation of these values for White Fulani. It is surprising that the Mitscherlich model which fairly well predicted the growth of White Fulani cattle produced outrageous asymptote (A) and errors estimates (Table 3). We can conclude that the Gompertz model provided a better range of parameter estimate. The more accurate Gompetz model showed that the rate at which mature weight was reached was faster in N'dama than White Fulani (Table 4).

The quantitative verification of the model fits were made using error measurement indices accuracy of the model. Traditionally, selection in beef cattle has emphasized heavy weights at market ages. For the cattle feeder, this criterion is associated with faster growth rates and increased economic returns over short periods of time. However, high growth rates to market age also may result in larger mature weights of brood cows and as a consequence, increased maintenance costs (Klosteman 1972).

Conclusion

• The Gompertz model was more adequate for describing the growth patterns of White Fulani and N'dama cattle.  The White Fulani cattle were heavier but less mature than N'dama at any age before maturity.  In comparison, the time taken to reach maturity was longer for White Fulani cattle. Genetic variation in the parameter estimates was observed when model parameters were compared. N’dama cattle could be more efficient in the beef enterprise though smaller than White Fulani in size.

Acknowledgement

The effort of the farm assistants of the Teaching and Research farm, University of Ibadan and the advice of L.O. Ngere are deeply appreciated.

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Received 3 April 2014; Accepted 24 April 2014; Published 1 May 2014

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