Flexible alternatives to the Gompertz equation for describing growth with age in turkey hens

PRODUCTION, MODELING, AND EDUCATION Flexible alternatives to the Gompertz equation for describing growth with age in turkey hens T. Porter,* E. Kebreab,†1 H. Darmani Kuhi,‡ S. Lopez,§ A. B. Strathe,# and J. France* *Department of Animal and Poultry Science, University of Guelph, Guelph, Ontario, N1G 2W1, Canada; †Department of Animal Science, University of California, Davis 95616; ‡Animal Sciences Group, Faculty of Agriculture, University of Guilan, Guilan 41635-1314, Iran; §Departamento de Producción Animal, Universidad de León, E-24007 León, Spain; and #Section of Nutrition, Department of Basic Animal and Veterinary Sciences, Faculty of Life Sciences, University of Copenhagen, DK-1870, Frederiksberg, Denmark considered further. Inclusion of an autoregressive process of the first order rendered a substantially improved fit to data for the 3 growth functions. The Morgan equation provided the best fit to the data set and was used for characterizing mean growth curves for the 7 yr of production. It was estimated that the maximum growth rate occurred at 3.74, 3.65, 3.99, 4.18, 4.05, 4.01, and 3.77 kg BW for production years from 1997 to 2003, respectively. It is recommended that flexible growth functions should be considered as an alternative to the simpler functions (with a fixed point of inflection) for describing the relationship between BW and age in turkeys because they were easier to fit and very often gave a closer fit to data points because of their flexibility, and therefore a smaller residual MS value, than simpler models. It can also be recommended that studies should consider adding a first-order autoregressive process when analyzing repeated measures data with nonlinear models.

Key words: growth, modeling, turkey hen, repeated measure 2010 Poultry Science 89:371–378 doi:10.3382/ps.2009-00141

INTRODUCTION

BW and age. Although growth curves of the same or different species were not necessarily best described by 1 equation (Ricklefs, 1967), historically, the Gompertz equation has been the function of choice for describing growth in poultry (Ricklefs, 1985; Bowmaker and Gous, 1989; Hancock et al., 1995; Wiseman and Lewis, 1998). The Gompertz equation, however, has the limitation of a fixed point of inflection that occurs at 1/e (= 0.368) times the final BW (Thornley and France, 2007). This limitation has been demonstrated with both meat and egg strains of poultry, so that for most time course profiles, the relationship between BW and age, described traditionally by the Gompertz equation, was very often described better using a flexible growth function (Darmani Kuhi et al., 2002, 2003a,b). It was expected that the growth curve of birds deviates from one year to the next, and this must be

Growth functions have been shown to be valuable tools for analyzing growth responses to genetic selection and environmental change. Although there have been various definitions of growth given by different biologists, for the purposes of quantitative analysis, growth was defined as the process of an animal gaining weight with time until it reaches maturity. From the beginning of the 20th century, different growth functions were introduced by growth modelers in an attempt to describe the relationship between changes in ©2010 Poultry Science Association Inc. Received March 16, 2009. Accepted September 20, 2009. 1 Corresponding author: [email protected]

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ABSTRACT A total of 49 profiles of growing turkey hens from commercial flocks were used in this study. Three flexible growth functions (von Bertalanffy, Richards, and Morgan) were evaluated with regard to their ability to describe the relationship between BW and age and were compared with the Gompertz equation with its fixed point of inflection, which might result in its overestimation. For each function, 4 ways of analysis were implemented. A basic model was fitted first, followed by implementation of a first-order autoregressive correlation structure. A model that considered only mature BW varied with year and another that considered only the rate coefficient varied with different years were applied. The results showed that the fixed point of inflection of the Gompertz equation can be a limitation and that the relationship between BW and age in turkeys was best described using flexible growth functions. However, the Richards equation failed to converge when fitted to the turkey growth data; therefore, it was not

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MATERIALS AND METHODS Data Sources A total of 49 time course profiles for growing turkeys were used in this analysis. The data were collected from the years 1997 to 2003 and the profiles were composited based on year of observation. An average growth profile was calculated because information on all birds in each week of data collection was not available. Random samples of a flock of hybrid large white breeding stock turkey hens were weighed on a weekly basis. The lighting program consisted of 24 h of light from d 0 to 3, 14 h from d 4 to 16 wk, 6 h from 16 to 30 wk, and 14 h at 30 wk of age. The hens were beak-trimmed at hatch and at 21 wk of age. The flock also followed the Cold Springs Farm Limited (Ontario, Canada) vac-

cination program. Hens were provided ad libitum access to feed and water. The diets used had 2,819 kcal/ kg of ME and 27% CP fed for the first 2 wk, 2,876 kcal/ kg of ME and 27.4% CP from 2 to 6 wk, 2,787 kcal/kg of ME and 17.99% CP from 6 to 10 wk, 2,801 kcal/kg of ME and 16% CP from 10 to 12 wk, and 2,778 kcal/ kg of ME and 10.76% CP from 12 to 30 wk.

Growth Functions The functional forms, f(x), listed in Table 1 were used to investigate the relationship between BW and age. These forms are as follows: an equation describing smooth sigmoidal growth with a fixed point of inflection (Gompertz) and 3 equations (von Bertalanffy, Richards, and Morgan) describing sigmoidal growth with a variable point of inflection. A detailed description of these growth functions can be found in Thornley and France (2007).

Statistical Procedures The variable Wij denotes the BW (kg) of turkey hens in the ith year (1 ≤ i ≤ 7) and at the jth week of ageij (1 ≤ j ≤ ni), and thus the statistical model was written as follows: Wij = f(βi; ageij) + eij, where ni was number of observations within year and f was a known nonlinear function of ageij and the parameter vector βi. The effect of year was incorporated directly into the parameters of the nonlinear growth functions by considering that the asymptotic BW, Wf, and the rate coefficient b varied systematically with different years. The eij were at first instance assumed to be identical, independent, and normally distributed but that may be a simplistic assumption because the mod-

Table 1. The functional forms used to describe the relationship between BW, f(x), and age, x

Growth function1

f(x)2

Growth rate [df (x)/dx]

Time at inflection point (t*)

Gompertz

f(x) = Wf exp {[1 − exp(−bx)]ln(Wf/W0)}

W bW ln( f ) W

W 1 [ln(ln( f ))] b W0

0.368Wf

von Bertalanffy

f (x ) = [Wfv - (Wfv -W0u ) exp(-bx )]1/ u , 0 £ u £ 1/3

(bWfu / u)W (1-u) - (b / u)W

u u 1 Wf -W0 ln[ ] b uW u

Wf(1 − υ)1/υ

BW at inflection point (W*)

f

Wfn

-W

n

Richards

f (x ) = W0Wf / [W0n + (Wf n -W0n ) exp(-bx )]1/n

bW (

Morgan

f(x) = (W0Kb + Wf xb)/(Kb + xb)

æç x b -1 ö÷ ÷÷ (W -W ) b çç çè K b + x b ÷÷ø f

1

nWfn

)

Wfn

1 ln( b

-W0n

nW0n

é b - 1 ù1/b ú Kê êb + 1 ú ë û

)

Wf (n + 1)1/n 1 1 [(1 + )W0 + (1 - )Wf ] b b 2

References were Gompertz (1825), von Bertalanffy (1957), Richards (1959), and Lopez et al. (2000), respectively. Where Wf was the final BW (kg), W0 was the initial BW (kg), and K (wk), b (per wk), n, and υ (dimensionless) were constants. b and υ were positive and n ≥ −1. Age is given in weeks. 2

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taken into account. Body weight recordings taken on the same individual have been shown not to be independent. Statistical issues such as heteroscedastic and serially correlated errors in relation to modeling animal growth curves have previously been described, but solutions to overcome these problems had not been proposed (Wang and Zuidhof, 2004). Recent developments in statistical theory and improved computational power allow for specification of general nonlinear regression models (Pinheiro and Bates, 2000). These models have recently been used to describe age versus BW relations in broilers (Wang and Zuidhof, 2004). Therefore, the objectives of the present investigation were to (1) specify general nonlinear regression models that include a correlation structure for dealing with repeated measures of growth data, (2) enable comparison between different growth functions in their ability to describe turkey data, and (3) characterize patterns of turkey breeder hen growth (recorded over different years) based on the best performing function.

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MODELING GROWTH IN TURKEY Table 2. Fitted model comparison for data profiles (1997 to 2003) using the different growth functions and different methods of analysis1,2 Function Gompertz   −2 Log-likelihood   AIC4   BIC4 von Bertalanffy   −2 Log-likelihood   AIC   BIC Morgan   −2 Log-likelihood   AIC   BIC

P_233

P_243

Model 1

Model 2

Model 3

Model 4

−95.25 −63.25 −11.38

−167.1 −133.1 −134.0

−146.2 −124.2 −124.7

−143.6 −121.6 −122.2

0.004

0.004

−161.9 −129.9 −78.03

−194.0 −160.0 −160.9

−157.6 −135.6 −136.2

−155.7 −133.7 −134.3

<0.001

<0.001

−188.6 −154.6 −99.50

−211.6 −175.6 −176.6

−167.9 −143.9 −144.5

−165.3 −141.3 −141.9

<0.001

<0.001

eling exercise dealt with repeated measures. A basic model ignoring correlation between measurements was fitted first (model 1, Table 2). A first-order autoregressive correlation structure was then implemented and added to the basic model for all functions (model 2, Table 2, Appendix 1). A third analysis was performed by considering that only Wf varied with year of observation (model 3, Table 2). A final analysis that considered only the rate coefficient to vary with year was conducted (model 4, Table 2). The best performing model was identified based on the following goodnessof-fit indicators: Akaike information criterion, Bayesian information criterion, and residual SD. In addition, likelihood ratio tests were used to compare model 2 with either model 3 or model 4 (Table 2). All statistical computations were implemented in SAS (SAS, 2000) by means of the NLMIXED procedure and the %NLINMIX macro using method 3 estimation (Littell et al., 2006). Growth traits (i.e., points of inflection) were calculated as a function of the fixed effect estimates g(βi), and the corresponding SE was obtained by the delta method. The method approximated the SE of a transformation g(β) of a random variable β = (β1, β12, ...), given estimates of the mean and covariance matrix of β. The delta method expands the differentiable function g(β) of a random variable about its mean, with a first-order Taylor approximation, and then takes the variance (Oehlert, 1992).

RESULTS Gompertz, von Bertalanffy, and Morgan equations were fitted to the data at the basic level (model 1). However, attempts to fit the data using the Richards equation failed to converge; therefore, this function was dropped from further analysis. The results of fitting the 4 models to the data are presented in Table 2. Based on the goodness-of-fit indicators for the basic model,

the Morgan equation performed best followed by von Bertalanffy and Gompertz. The ranking of the functions in their ability to describe the relationship between BW and age was also the same order for models 2, 3, and 4 (Table 2). Likelihood ratio test showed that there was significant difference between model 2 and models 3 and 4 when using the Gompertz equation (P = 0.004). The differences between models 2 and 3 and 4 were significant at a higher level of probability when the von Bertalanffy and Morgan equations were used (P < 0.001). The best-fitting model, considering all functions and various permutations of model fitting, was the Morgan equation that incorporated Wf and K varying systematically with different years and was implemented using a first-order autoregressive correlation matrix. Therefore, full parameter estimates containing separate asymptotic BW of the birds and rate coefficients are given in Table 3. Figure 1 shows the final Morgan model fitted to turkey growth data from 1997 to 2003 simultaneously. The results showed that the birds in the year 2000 had significantly higher Wf values compared with the birds in all other years except 2001. Similarly, the rate coefficient was significantly higher in birds from 2000 than any other year. The Wf values from birds in years 1997, 1998, and 2003 were significantly different from the birds in 1999, 2000, and 2001. Similar trends were also observed for values of the rate coefficient (Table 3).

DISCUSSION Feed intake averages about 55 to 70% of total production cost (Leeson and Summers, 2001). Therefore, having a successful ad libitum feeding system in animal production would only be possible if the mechanisms controlling the response of the animal to feeding variables were understood. Published observations on time

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1 Model 1 was a basic model. Model 2 included a first-order autoregressive correlation structure. Model 3 assumed that only mature BW varied with year, and model 4 assumed that only rate coefficient varied with year. 2 Fitting the data using the Richards equation did not converge; therefore, details are not given. 3 P-values comparing the −2 log-likelihood of the year effect on the asymptotic BW and shape of curve parameters using the likelihood ratio test. 4 AIC = Akaike information criterion; BIC = Bayesian information criterion.

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method to recommend which model should be used for any given case (Motulsky and Ransnas, 1987). Hence, the comparison of growth functions in this study was carried out according to well-established goodness-of-fit indicators. The results showed that the different data profiles can be described by the different growth functions well (residual SD <0.04 kg). However, in spite of this similarity, the statistical significances between the calculated likelihood ratio test values indicated that there were significant differences between growth functions and various assumptions made in fitting these functions (Table 2). The Akaike information criteria and Bayesian information criteria were used to classify the models. The current analysis provided evidence that the 4-parameter functions performed better than the 3-parameter function, independent of modifications to the error structure of the models. This was because the 4-parameter functions allowed for nonsymmetric sigmoidal growth (Lopez et al., 2000). In most growth analyses, the correlation between measurements from the same animal is ignored. However, as model 2 fitting for all growth functions showed, it is essential that the variance structure is modeled properly when biological inference regarding growth profiles with different sources of variation is explored. The relevance of including a first-order autoregressive method in the analysis was further shown in Figure 2, where the fitted values were plotted against the residuals. In the absence of a first-order autoregressive structure, the residuals had a more systematic pattern that was more clearly evident for the Gompertz equation. Introducing the correct variance structure led to more uniform distribution of the residuals and improved all indicators of goodness of fit significantly. With the Morgan equation, the parameter that described the initial BW (W0) of the turkey hens in different year categories was not significantly different from

Table 3. Parameter estimates, SE, and 95% confidence limits obtained by fitting the Morgan equation (model 2 in Table 2) to the growth profiles1 Confidence limit Parameter2 W0 Wf1 Wf2 Wf3 Wf4 Wf5 Wf6 Wf7 b K1 K2 K3 K4 K5 K6 K7 1

Estimate

SE

Upper

Lower

0.12 13.84 13.50 14.76 15.51 14.99 14.84 13.96 2.12 14.17 14.03 15.31 16.93 15.55 15.26 14.48

0.011 0.143 0.137 0.172 0.206 0.178 0.172 0.150 0.037 0.152 0.148 0.188 0.234 0.193 0.186 0.164

0.10 13.55 13.22 14.41 15.10 14.64 14.49 13.65 2.04 13.87 13.73 14.93 16.46 15.16 14.89 14.15

0.15 14.11 13.76 15.09 15.91 15.34 15.17 14.25 2.19 14.47 14.31 15.68 17.39 15.93 15.63 14.80

The subscripts refer to year of observation from 1997 to 2003, respectively. Where W0 was the initial BW (kg), Wf was the final BW (kg), and b (per wk) and K (wk) were constants.

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course data for growing turkeys are few and largely confined to short-term studies. Many equations have been proposed to describe the relationships between changes in BW and time under ad libitum feeding conditions. Age (time) per se does not cause BW to increase and then plateau but provides opportunity for the inherent potential of an individual for growth and maturation to be examined, as was demonstrated by Lister et al. (1966). Although growth curves have been shown to be useful tools for analyzing the growth response to genetic selection and environmental change, they have the disadvantage of looking at the animal as a system with output only. Because animal production systems have been demonstrated to be input-output systems, such data give no information on the mechanisms involved in animal growth. Therefore, in the study presented here, an evaluation of different growth functions as candidates to describe the relationship between change in BW and age in growing turkey hens was used. Traditionally, the Gompertz equation has been the function of choice for describing growth in poultry. The Gompertz equation, however, has the possible limitation of a fixed point of inflection that occurs at 1/e ( = 0.368) times the final BW (Thornley and France, 2007). Therefore, 3 flexible growth functions, von Bertalanffy, Richards, and Morgan, were evaluated with regard to their ability to describe the relationship between BW and age in growing turkey hens and were compared with the Gompertz equation with its fixed point of inflection. The 3 functions remove the limitation of the Gompertz equation and allow for a possible better fit of the growth function to observed data. The purpose of fitting a growth function has always been to describe the course of mass increase with age by a simple equation with few parameters that in the most useful models may be biologically interpretable (Ricklefs, 1985). However, there was no single, simple

MODELING GROWTH IN TURKEY

zero (P = 0.97). Therefore, a common W0 was fitted to all of the data. The exponent (b) that determined the sigmoidal shape in the Morgan equation was also unaffected by the year-to-year variation (P = 0.97). The year-to-year variation in the parameter value that described the mature BW (Wf) was highly significant (P < 0.0001) and was quantified by the variance com-

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ponent. Similarly, age at approximately half maximum BW (K) varied from year to year (P < 0.0001). Therefore, the model that best fitted the data considered separate Wf and K estimates (Tables 2 and 3). The analysis confirmed that growth profiles from different years were significantly different from each other in mature size and the rate at which mature BW was achieved.

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Figure 1. Data profiles of BW (kg) against age (wk). The lines indicate fitted values based on the Morgan equation.

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Therefore, it can be recommended that studies that are conducted in different years with the same types of bird consider the fixed effects of year and flock. In the current analysis, the birds in year 2000 grew faster and reached significantly higher BW at 30 wk of age compared with birds in other years, although the same strains of turkey hens were used in all years. There was no evidence that mature BW and rate to maturity improved with year of observation. In fact, birds in year

2003 did not achieve a higher mature BW or rate coefficient compared with birds in years 1999 to 2002. This must have been due to various factors specific to the year the turkey hens were reared. Based on the best-fitting model for the 3 functions that converged, growth traits were calculated for the data profiles (Table 4). Methods for calculating the inflection points (times where maximum growth occurs and corresponding BW) are given in Table 1. It was in-

Table 4. Calculated growth traits for data profiles (1997 to 2003) using the Gompertz, von Bertalanffy, and Morgan equations1,2 Function Gompertz   t*   W* von Bertalanffy   t*   W* Morgan   t*   W* 1

1997

1998

1999

2000

2001

2002

2003

9.94 (0.003) 4.44 (0.016)

9.90 (0.003) 4.36 (0.015)

10.71 (0.014) 4.70 (0.020)

11.67 (0.025) 4.87 (0.025)

10.93 (0.016) 4.80 (0.021)

10.67 (0.014) 4.73 (0.020)

10.25 (0.009) 4.50 (0.018)

8.71 (0.002) 3.72 (0.014)

8.65 (0.002) 3.64 (0.014)

9.39 (0.011) 3.95 (0.019)

10.37 (0.020) 4.15 (0.023)

9.58 (0.010) 4.03 (0.019)

9.35 (0.011) 3.97 (0.019)

8.91 (0.005) 3.76 (0.016)

8.73 (0.070) 3.74 (0.027)

8.64 (0.071) 3.65 (0.027)

9.43 (0.061) 3.99 (0.023)

10.43 (0.052) 4.18 (0.018)

9.58 (0.060) 4.05 (0.023)

9.40 (0.062) 4.01 (0.024)

8.92 (0.067) 3.77 (0.025)

t* and W* are inflection points [i.e., the time (wk) and BW (kg) at maximum growth rate calculated from the fixed effect parameter estimates]. Standard errors are given in parentheses.

2

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Figure 2. Residuals and first-order autoregressive [AR(1)] innovations when data were fitted using Gompertz, von Bertalanffy (Von B), and Lopez equations.

MODELING GROWTH IN TURKEY

ACKNOWLEDGMENTS This research was undertaken, in part, thanks to funding from the Canada Research Chairs Program (Ottawa, Ontario, Canada) and Manitoba Rural Adaptation Council (Winnipeg, Manitoba, Canada).

REFERENCES Bowmaker, J. E., and R. M. Gous. 1989. Quantification of reproductive changes and nutrient requirements of broiler pullet at sexual maturity. Br. Poult. Sci. 30:663–675. Darmani Kuhi, H., E. Kebreab, S. Lopez, and J. France. 2002. A derivation and evaluation of the von Bertalanffy equation for

describing growth in broilers over time. J. Anim. Feed Sci. 11:109–125. Darmani Kuhi, H., E. Kebreab, S. Lopez, and J. France. 2003a. An evaluation of different growth functions for describing the profile of live weight with time (age) in meat and egg strains of chicken. Poult. Sci. 82:1536–1543. Darmani Kuhi, H., E. Kebreab, S. Lopez, and J. France. 2003b. A comparative evaluation of functions for the analysis of growth in male broilers. J. Agric. Sci. Camb. 140:451–459. Gompertz, B. 1825. On the nature of the function expressive of the law of human mortality, and on a new method of determining the value of life contingencies. Phil. Trans. R. Soc. 36:513–585. Hancock, C. E., G. D. Bradford, G. C. Emmans, and R. M. Gous. 1995. The evaluation of the growth parameters of six strains of commercial broiler chickens. Br. Poult. Sci. 36:247–264. Leeson, S., and J. D. Summers. 2001. Nutrition of the Chicken. University Books, Guelph, Ontario, Canada. Lister, D., T. Cowen, and R. A. Mccance. 1966. Severe under-nutrition in growing and adult animals. Br. J. Nutr. 20:633–639. Littell, R. C., G. A. Milliken, W. W. Stroup, R. D. Wolfinger, and O. Schabenberger. 2006. SAS System for Mixed Models. 2nd ed. SAS Inst. Inc., Cary, NC. Lopez, S., J. France, M. S. Dhanoa, F. Mould, and J. Dijkstra. 2000. A generalized Michaelis-Menten equation for the analysis of growth. J. Anim. Sci. 78:1816–1828. Motulsky, H. J., and L. A. Ransnas. 1987. Fitting curves to data using nonlinear regression: A practical and nonmathematical review. FASEB J. 1:365–374. Oehlert, G. W. 1992. A note on the delta method. Am. Stat. 46:27– 29. Pinheiro, J., and D. M. Bates. 2000. Mixed Effects Models in S and S-PLUS. Statistics and Computing. Springer, New York, NY. Richards, F. J. 1959. A flexible growth function for empirical use. J. Exp. Bot. 10:290–300. Ricklefs, R. E. 1967. A graphical method of fitting equations to growth curves. J. Ecol. 48:978–983. Ricklefs, R. E. 1985. Modification of growth and development of muscles in poultry. Poult. Sci. 64:1563–1576. SAS. 2000. SAS/STAT User’s Guide. Version 8 ed. SAS Inst. Inc., Cary, NC. Thornley, J. H. M., and J. France. 2007. Pages 136–169 in Mathematical Models in Agriculture. 2nd ed. CABI Publ., Wallingford, UK. von Bertalanffy, L. 1957. Quantitative laws for metabolism and growth. Q. Rev. Biol. 32:217–231. Wang, Z., and M. J. Zuidhof. 2004. Estimation of growth parameters using a nonlinear mixed Gompertz model. Poult. Sci. 83:847– 852. Wiseman, J., and C. E. Lewis. 1998. Influence of dietary energy and nutrient concentration on the growth of body weight and of carcass components of broiler chickens. J. Agric. Sci. Camb. 131:361–371.

Appendix 1 The SAS code for fitting a nonlinear mixed model with a first-order autoregressive process using the Morgan function as an example (model 2 in Table 2) is given below. Comments are provided that describe the specific details of the procedures used. The SAS code explanations are given between /* and */. /* SAS code for fitting the full model */ /*nlinmix is a SAS macro for fitting nonlinear mixed models using PROC NLIN and PROC MIXED. It requires SAS/STAT Version 8 or higher*/ /* include the macro from file */ %nlinmix(data=A, /*The model = argument defines the mean growth function. Due to multiple statements ending with semicolons, the entire argument is enclosed with the %str() macro. */

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teresting to note that the inflection points were higher in the Gompertz equation compared to the other functions. The 3-parameter model predicted that turkey hens took longer time to reach maximum growth. This was mainly due to the inflexible nature of the Gompertz equation. In contrast, the two 4-parameter functions predicted inflection points that were not significantly different from each other. Both the von Bertalanffy and Morgan equations predicted lower time points for the turkeys to reach their maximum growth rate and at a significantly lower BW. This kind of information is important from a nutritional management standpoint because it allows the producer to match the requirement by adjusting what is fed to the animal when the growth rate is at its maximum. In conclusion, 4 different growth functions (Gompertz, von Bertalanffy, Richards, and Morgan) have been fitted to 49 turkey hen growth profiles. The growth analysis revealed that inclusion of an autoregressive process of the first-order considerably improved the fit to data for all models that converged. This can be attributed to removal of the serially correlated errors, and thus inclusion of a first-order autoregressive process can be recommended when modeling frequently sampled growth data. The Morgan equation provided the best fit to the data and was used for estimating population growth curves related to the different years of observation.

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  model=%str(Wf = Wf1*(year=1) + Wf2*(year=2) + Wf3*(year=3) + Wf4*(year=4) + Wf5*(year=5) + Wf6*(year=6) + Wf7*(year=7);     K = K1*(year=1) + K2*(year=2) + K3*(year=3) + K4*(year=4) + K5*(year=5) + K6*(year=6) + K7*(year=7);    predv=(W0*K**b+Wf*Age**b)/(K**b + Age**b);   ), /*Starting values for fixed effects are listed in the parms argument. The model and parms arguments in %nlinmix are similar to PROC NLIN */

/*The stmts argument specifies the PROC MIXED statements to be executed for each iteration. The response variable must be declared as pseudo_y where y is the response variable in the input dataset i.e. weight. In the model statement, the options noint, notest, solution, and cl must be specified */   stmts=%str(    class year;    model pseudo_BW = d_Wf1 d_Wf2 d_Wf3 d_Wf4 d_Wf5 d_Wf6 d_Wf7 d_W0 d_c       d_K1 d_K2 d_K3 d_K4 d_K5 d_K6 d_K7/ noint notest solution cl; /* In the repeated statement, the options subject and type are used to specify an autoregressive process of first order for modeling the covariance structure of within-year residuals */      Repeated / subject=year Type=ar(1);      ods output covparms = cov2;   ), /*The expand argument is used to employ a first-order Taylor expansion around the current estimates of fixed effects and the conditional modes of the random effects. The procopt are used for numerical specifications to the PROC MIXED call */    expand=zero,   procopt=%str(maxiter=500 method=ml empirical) ) run;

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  parms=%str(Wf1=13 Wf2=13 Wf3=13 Wf4=13 Wf5=13 Wf6=13 Wf7=13      W0 =0.12 b=2.15      K1=15 K2=15 K3=15 K4=15 K5=15 K6=15 K7=15      ),