Comparison of nonlinear and spline regression models for describing ...

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INRA, UR 631 Station d'Amélioration Génétique des Animaux, F-31326 Castanet-Tolosan, France; ..... tic, Gompertz, von Bertalanffy, Morgan-Mercer-Flodin,.
Comparison of nonlinear and spline regression models for describing mule duck growth curves z. G. Vitezica,*†‡1 C. Marie-Etancelin,§ M. D. Bernadet,# X. Fernandez,*†‡ and C. Robert-Granie§ *INRA, UMR 1289 Tissus Animaux Nutrition Digestion Ecosystème et Métabolisme, F-31326 Castanet-Tolosan, France; †Université de Toulouse, Institut National Polytechnique de Toulouse, UMR 1289 Tissus Animaux Nutrition Digestion Ecosystème et Métabolisme, Ecole Nationale Supérieure Agronomique de Toulouse, F-31326 Castanet-Tolosan, France; ‡Ecole Nationale Vétérinaire de Toulouse, UMR 1289 Tissus Animaux Nutrition Digestion Ecosystème et Métabolisme, F-31076 Toulouse, France; §INRA, UR 631 Station d’Amélioration Génétique des Animaux, F-31326 Castanet-Tolosan, France; and #INRA, UE 89 Unité Expérimentale des Palmipèdes à Foie Gras, F-40280 Benquet, France ABSTRACT This study compared models for growth (BW) before overfeeding period for male mule duck data from 7 families of a QTL experimental design. Four nonlinear models (Gompertz, logistic, Richards, and Weibull) and a spline linear regression model were used. This study compared fixed and mixed effects models to analyze growth. The Akaike information criterion was used to evaluate these alternative models. Among the nonlinear models, the mixed effects Weibull model had the best overall fit. Two parameters, the asymptotic weight and the inflexion point age, were considered random variables associated with individuals

in the mixed models. In our study, asymptotic weight had a greater effect in Akaike’s information criterion reduction than inflexion point age. In this data set, the between-ducks variability was mostly explained by asymptotic BW. Comparing fixed with mixed effects models, the residual SD was reduced in about 55% in the latter, pointing out the improvement in the accuracy of estimated parameters. The mixed effects spline regression model was the second best model. Given the piecewise nature of growth, this model is able to capture different growth patterns, even with data collected beyond the asymptotic BW.

Key words: mule duck growth, nonlinear growth model, spline regression model, fixed effect, random effect 2010 Poultry Science 89:1778–1784 doi:10.3382/ps.2009-00581

INTRODUCTION In France, about 95% of the fatty liver (foie gras) production comes from the mule duck, an infertile hybrid duck between a female common duck (Anas platyrhynchos) and a Muscovy drake (Cairina moschata). Magrets, the breast muscles of mule ducks, rank among the most valuable co-products. Therefore, knowledge about the growth pattern of mule duck is necessary to optimize the duck production system (e.g., by selection, feeding management, and marketing strategies). Growth models describe BW changes over time, allowing information from longitudinal measurements to be combined into a few (usually 3 or 4) parameters with biological interpretation. Growth curves are sigmoidal with an inflection point where the rate of growth is maximum and an upper asymptote (or mature) weight. These models (e.g., logistic, Gompertz, and Richards) ©2010 Poultry Science Association Inc. Received November 29, 2009. Accepted April 24, 2010. 1 Corresponding author: [email protected]

have been used extensively in different species to describe the development of BW (Aggrey, 2002, in chicken; Schinckel and Craig, 2002, in pigs; and Perotto et al., 1992, in dairy cattle). Different nonlinear models (NLM) have been used to describe growth curves in ducks (Knizetova et al., 1991; Maruyama et al., 1999). To describe duck and geese growth, the Richards model is often said to be the suitable model; however, this function can be difficult to fit (Mignon-Grasteau and Beaumont, 2000). Another function that has been found to be interesting to fit to duck growth data is the Weibull model (Maruyama et al., 1999, 2001). Recently, a new function, the generalized Michaelis-Menten, was proposed (López et al., 2000), achieving a goodness of fit similar to the Richards function. In the NLM framework, other possible models are the mixed effects nonlinear functions, which are able to deal with correlated errors and heterogeneous variances (heteroscedasticity) among the BW measures (Pinheiro and Bates, 2000). These models have recently been used in ducks (Schinckel et al., 2005) and in Japanese quail (Aggrey, 2009).

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An alternative to complicated NLM is spline curves. Splines are usually defined as piecewise polynomials of different degrees that join at selected points, called knots (Ruppert et al., 2003). Recently, spline linear regression models have received major attention for fitting poultry growth data (Aggrey, 2002, 2004) as well as in animal breeding (Sánchez et al., 2008). An extension of these regression models is the use of splines to model individual curves, by including individual splines as random effects in mixed model analysis. These more flexible models allow describing individual curves modeled as splines (Ruppert et al., 2003; Sánchez et al., 2008). In ducks, there is little recent information in the literature on the modeling of growth curve variables. The best growth models for mule ducks are yet to be found. The aims of this study were to use different nonlinear and linear spline models (fixed and mixed effects) to describe the growth of mule ducks and to compare them using criteria related to model selection.

MATERIALS AND METHODS The present study was carried out in agreement with the French National Guidelines for the Care and Use of Animals for Research Purposes (certificate no. 7740, Ministry of Agriculture and Fish Products, Paris, France).

Birds From 2005 to 2006, eight hundred male mule ducks were hatched each year in 2 pedigree batches of 400 ducklings. The analyzed hybrid mule duck data were originated in an experimental design to detect and localize genetic regions (QTL) responsible for variation in the phenotypes. These 1,387 male mule ducks belong to 7 maternal grandfather families (F1, ..., F7; Figure 1). The founder males of each family, with origins as dissimilar as possible, were chosen to maximize phenotype

Figure 1. Quantitative trait loci experimental design.

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diversity. Birds were bred under natural conditions of light and temperature at the Experimental Farm for Waterfowl Breeding (INRA-Unité Expérimentale des Palmipèdes à Foie Gras, Benquet, France). From hatching to 6 wk of age, mule ducks were fed ad libitum with a starter diet containing 173 g of protein and 12 MJ/kg. From 6 to 10 wk of age, ducks were fed with a diet restricted in the protein content (155 g of protein and 11.9 MJ/kg) to avoid excessive fatness. During this period, the daily intake of the diet was not reduced. In preoverfeeding (from 10 to 12 wk), the daily food intake reached 320 g per day. At 12 wk, the overfeeding period began for 2 wk (80 g of protein and 13.10 MJ/kg). The food intake weight during this period was equal to 16,640 g in the 2 wk (daily intake of 1,200 g/d). At the end of the overfeeding period (104 d of age), the weight of the birds was recorded. After slaughtering, the carcass, the abdominal fat, and the pectoralis major muscle (breast meat) were weighed for each bird. Thus, birds were weighed at 12, 28, 42, 70, 91, and 104 d of age. In growth analysis, only the preoverfeeding period (from 12 to 91 d) was considered.

Fixed Effects Growth Models To estimate the BW at a certain age, two 3-parameter and two 4-parameter NLM as well as a spline linear regression model were fitted to the mule duck BW data. Each family was fitted separately. Gompertz-Laird Model. The following equation describes the Gompertz-Laird growth curve (Laird et al., 1965):

üï ïìæ L ö Wt = W0 exp ïíççç ÷÷÷ éê1 - exp (-Kt )ùú ïý , ûï ïïçè K ÷ø ë ïþ î

where Wt is the BW of duck at time t, W0 is the initial (hatch) BW, L is the instantaneous growth rate (per d), and K is the rate of exponential decay of the initial

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specific growth rate. The age at the inflexion point where the maximum rate of growth is achieved, ti, and æL ö 1 the asymptotic BW, WA, are equal to ti = log ççç ÷÷÷ çè K ÷ø K æ L ö÷ and WA = W0 exp ççç ÷÷, respectively. çè K ÷ø Logistic Model. The well-known logistic growth model (Robertson, 1923) was fitted to the data. The growth curve was:

Wt =

WA , 1 + exp éê-K (t - ti )ùú ë û

where Wt is the BW at time t, WA is the asymptotic BW, K is the exponential growth rate, and ti is the age at the inflexion point. The logistic and the Gompertz functions have no flexible point of inflection. Richards Model. The BW of each duck after hatching was described by the growth curve modified from Richards (1959) function (Sugden et al., 1981): 1



ìï m ùü é ïï1-m ïï ê æ ö 1-m ú ï t t ï úï i÷ ÷÷ Wt = WA í1 - (1 - m ) exp êê-K ççç , ú ýï ïï çè m ÷÷ø ê úï ïï êë úû ïï ïî ïþ

where Wt is the BW of duck at time t, WA is the mature BW, and K is the maximum relative growth (per d), ti is the age at the inflexion point, and m is a shape pa1

rameter. The expression m 1-m determines the proportion of the final size at which the inflection point occurs (Thornley and France, 2007). The BW at the inflection point is calculated from the parameters of the curve as Wt = WA/m1/(1 − m). Weibull Model. The growth curve parameters were estimated using the Weibull function, according to Maruyama et al. (1999, 2001):



Cù é ê æçC - 1ö÷ æç t ö÷ ú ÷ ç ÷ ú , Wt = WA - (WA - B ) exp ê- çç ê çè C ÷÷ø ççèti ÷÷÷ø ú êë úû

where WA is the asymptotic weight and ti is the age at the inflexion point. The parameters B and C have no biological interpretation. The Richards and the Weibull functions are flexible in their point of inflexion and are mainly used on birth growth data. Spline Linear Regression Model. The spline model fitted was of the general form:

m

Wt = W0 + b1 (t - t1 ) + å bk (t - Tk ) + e, k =1

where Wt is the BW at time t, W0 is the initial weight, t1 is the age at which the first growth period starts (12 d), Tk are the so-called knots (age at which the growth rate changes), (x)+ = max(0, x) equal to x if x is positive and equal to 0 otherwise, and e is the residual error. This gives a slope of the first segment, that is, for t ≤ Tk = 1; a slope of β1 + βk = 1 for the second segment with Tk = 1 ≤ t ≤ Tk = 2; and a slope of b1 +

m +1

å bk

for

k =m

the segment between bordered by Tm and Tm+1 (Ruppert et al., 2003; Meyer, 2005). This model is, after appropriate linear transformation, equivalent to the piecewise form, often presented in the literature (Aggrey, 2002): Wt = W0 + b1 (t - T1 ) + b2 (t - T2 ) +  + b j (t - Tj ), where T1…Tj are the knots and the slope of the first segment is β1, the slope of the second segment is β2, and so on. In this model, the β coefficients are as the within-segment slopes. The data were plotted to select the number of segments. The segments were joined at 28, 42, and 70 d of age.

Mixed Effects Growth Models Weibull Model. The parameters WA (the asymptotic weight) and ti (inflexion point age) were considered random variables associated with an individual in Weibull mixed effects model. The model can be written as: Cù é ê æC - 1ö÷ æç t ö÷ ú ÷÷ ú , ÷÷ çç W t = (WA + u1 ) - éê(WA + u1 ) - B ùú exp ê- ççç ë û ê çè C ÷ø çèti + u2 ÷÷ø ú êë ûú

where u1 and u2 are the random parameters with mean equal to zero and an unstructured covariance matrix. Spline Linear Regression Model. Random regression models using linear splines have been extensively applied in genetic analysis of longitudinal data (Misztal, 2006). Because parameter estimation is time-consuming and to compare this model with the Weibull mixed effects model, 2 parameters (W0 and β1) were considered random. The intercept for each duck (W0) corresponds to the weight at 12 d. The slope of the first segment (β1) is the growth rate at an age lower than 28 d.

Statistics and Model Comparison Fitted models using the different growth functions and different methods of analysis (basic and mixed models) were compared by using Akaike’s information criterion (AIC; Akaike, 1974). The AIC was estimated as:

AIC = -2L - 2p,

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37.38 104.25 162.36 225.05 246.34 ± ± ± ± ± 274.21 1,261.14 2,165.48 3,281.17 3,784.32 36.96 119.96 189.00 279.99 325.84 ± ± ± ± ± 284.06 1,306.19 2,279.03 3,340.34 3,879.31 37.02 129.75 198.43 271.88 299.79 ± ± ± ± ± 282.06 1,285.83 2,241.52 3,340.44 3,874.39 38.40 105.93 169.57 282.68 303.19 ± ± ± ± ± 414.99 1,483.81 2,420.02 3,534.55 3,863.20 40.94 108.35 166.92 274.29 283.94 ± ± ± ± ± 385.11 1,429.99 2,281.48 3,413.18 3,758.56 34.52 112.23 162.97 265.24 295.72 ± ± ± ± ± 383.96 1,478.78 2,451.49 3,547.27 3,900.40 36.52 106.32 171.86 271.63 307.65 *P < 0.05.

± ± ± ± ± 390.95 1,465.39 2,409.01 3,469.10 3,813.26

F5 (n = 231) BW, g

F4 (n = 175) F3 (n = 169) F2 (n = 185) F1 (n = 166) Age, d

Table 1. Means and SD for BW at different ages in each family of mule ducks

For each family of mule ducks, overall means and SD for BW at different ages are presented in Table 1. Among families, differences for BW were found at each age from 12 to 70 d of age (P < 0.05). The withinfamily SD for BW increased with age. A duck that is heavier at one age will be heavier at the next measure. These observations point out the problem of heteroscedasticity and correlated errors, which are ignored in fixed effects models. The NLM have been used extensively to model animal growth (Thornley and France, 2007). Assuming an appropriate growth function, the accuracy of model parameters depends on the accuracy of the data. The data set of the current study is unique in mule ducks, based on a very large number of birds with BW recording over different phases of the growth curve. However, the BW data were collected every 2 wk. In a recent study, Aggrey (2008) found a reduction in growth rate and in the rate of decay with this frequency of data collection using a Gompertz model. It is, however, possible that our results are affected by the frequency of data collection. To deal with the limitation of our data set and avoid an inaccurate prediction of BW at different ages, in this study we also used the nonlinear mixed effects models, which accounted for the variance-covariance structure, thereby improving the accuracy of prediction (Aggrey, 2009) over using traditional NLM, which are fixed effects models. Growth curves are often nonlinear sigmoidal functions parameterized to include an asymptote and an inflection point. The AIC values are used as criteria for the goodness of fit (Table 2). Parameter estimates for the different nonlinear and linear models with fixed and mixed effects are presented in Tables 3 and 4. Among the fixed effects NLM, the best models based on AIC values for ducks of different families were the Richards and the Weibull models. This is in agreement with Maruyama et al. (2001), who applied the logistic, Gompertz, von Bertalanffy, Morgan-Mercer-Flodin, and Weibull functions to selected duck lines. Maruyama et al. (2001) also noted that the Weibull function is best for fitting duck BW data. In the Weibull model, the average values of the age at the inflexion point (ti) confirm the earliness of growth in duck in respect to

F6 (n = 229)

RESULTS AND DISCUSSION

12 28 42 70 91

F7 (n = 232)

P-value

where L is the maximized log likelihood and p is the number of parameters in the model. The AIC takes into account both the statistical goodness of fit and the number of parameters that need to be estimated in the model. The model with the smallest AIC value is the best for fitting BW data. Body composition data of the families of birds (F1 to F7) with different values of the shape parameter m from the Richards model were compared using the Tukey test. All statistical analyses were performed using packages nlme and lme in R (http:// www.r-project.org/) by maximum likelihood.

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MODELING GROWTH CHARACTERISTICS OF MULE DUCK

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Table 2. Values of Akaike’s information criterion (AIC) for each family of mule ducks AIC Model Gompertz Logistic Richards Weibull Weibull_MM1 Spline Spline_MM3

F1 (n = 166)

F2 (n = 185)

F3 (n = 169)

F4 (n = 175)

F5 (n = 231)

F6 (n = 229)

F7 (n = 232)

−262.01 −131.88 −268.72 −264.94 −1,267.242 −268.37 −1,081.852

−348.79 −185.87 −357.57 −352.51 −1,477.612 −358.15 −1,292.342

−316.81 −177.09 −328.20 −330.85 −1,035.182 −330.84 −913.512

−275.30 −152.90 −278.01 −276.45 −1,096.062 −279.29 −959.862

−225.70 48.02 −296.01 −286.37 −1,181.582 −321.34 −1,161.322

−135.98 125.59 −208.51 −198.46 −1,263.552 −244.87 −1,397.022

−668.82 −311.65 −762.54 −753.52 −1,758.542 −774.58 −1,638.282

1Weibull_MM

= Weibull mixed effects model. smallest values among the nonlinear and linear models. 3Spline_MM = spline mixed effects model. 2The

chicken and turkeys. In all families, the average values of ti (23.1 d) show the fast-growing duck against slower growing in chicken and turkeys (47.7 and 74.0 d of age, respectively; Knizetova et al., 1995). The proportion of the total growth at the inflexion point was equal to 0.26 on average. This value is considerably different from 0.50 and 0.37 fixed in logistic and Gompertz models. This result explains that models with fixed shape are not flexible enough to correctly reflect the growth pattern of mule ducks. Using the Richards model, estimates of the effects of m and the associated carcass components of families are shown in Table 5. Comparing 2 families (F4 and F7) with different values of m (0.82 and 0.47, respec-

tively), BW at 104 d, carcass weight, and carcass yield were similar between these families. The proportion of the major breast muscle and the abdominal fat differed according to the growth trajectory of the birds. These results were consistent with the study of Aggrey (2002). The family F7 reduced abdominal fat, in both amount and percentage of carcass weight, when it was compared with the family F4 (P < 0.05). This pattern was not observed in F6 (m = 0.43). From Table 5, it is likely that the shape of the growth curve, through the parameter m, could be involved in the determination of the proportion of abdominal fat. However, further studies will be needed to confirm the effect of m on carcass components.

Table 3. Parameter values of the fixed effects Gompertz, logistic, Richards, Weibull, and spline regression model growth in families of mule ducks Model

F1 (n = 166)

F2 (n = 185)

F3 (n = 169)

F4 (n = 175)

F5 (n = 231)

F6 (n = 229)

F7 (n = 232)

Gompertz   Age of maximum growth (ti)   Asymptotic weight (WA)   Hatching weight (W0)   Initial growth rate (L)   Rate of decay (K) RSD1 Logistic   Age of maximum growth (ti)   Asymptotic weight (WA)   Exponential growth rate (K) RSD Richards   Age of maximum growth (ti)   Asymptotic weight (WA)   Relative growth (K)   Shape parameter (m) RSD Weibull   Age of maximum growth (ti)   Asymptotic weight (WA) RSD Spline   Initial weight – 12 d (W0)   Regression coefficients    β1    β2    β3    β4 RSD

  28.08 3,945 62.29 0.210 0.050 0.206   34.73 3,763 0.082 0.222   25.39 4,038 0.018 0.754 0.205   23.94 3,946 0.205   390.95   67.15 0.25 −29.55 −20.26 0.205

  28.34 4,036 58.86 0.215 0.051 0.199   34.96 3,848 0.083 0.218   25.73 4,130 0.018 0.759 0.198   24.38 4,034 0.199   383.96   68.42 1.06 −30.36 −21.02 0.198

  28.93 3,942 73.97 0.189 0.048 0.200   35.82 3,739 0.077 0.217   25.37 4,075 0.017 0.695 0.198   24.33 3,956 0.198   385.11   65.30 −4.48 −20.43 −22.68 0.198

  28.23 4,026 70.87 0.199 0.049 0.206   34.99 3,833 0.080 0.221   26.30 4,094 0.017 0.820 0.206   25.21 3,997 0.206   414.99   66.80 0.07 −27.17 −22.69 0.205

  31.33 4,047 51.72 0.205 0.047 0.219   37.79 3,794 0.078 0.246   24.05 4,435 0.015 0.456 0.212   21.62 4,303 0.213   282.06   62.73 5.53 −29.03 −12.61 0.209

  30.85 4,025 50.26 0.209 0.048 0.227   37.25 3,782 0.080 0.255   23.06 4,426 0.016 0.426 0.220   19.85 4,317 0.221   284.06   63.88 5.60 −31.06 −10.95 0.217

  31.54 3,973 52.05 0.201 0.046 0.181   38.09 3,724 0.077 0.211   24.47 4,344 0.015 0.469 0.174   22.60 4,193 0.174   274.21   61.68 2.91 −24.75 −14.76 0.173

1RSD

= residual SD.

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MODELING GROWTH CHARACTERISTICS OF MULE DUCK Table 4. Parameter values of the mixed effects Weibull and spline regression model growth in families of mule ducks Model Weibull   Age of maximum growth (ti)   Asymptotic weight (WA)   Variance (WA)   Variance (ti) RSD1 Spline   Initial weight – 12 d (W0)   Regression coefficients    β1    β2    β3    β4   Variance (W0)   Variance (β1) RSD 1RSD

F1 (n = 166)

F2 (n = 185)

F3 (n = 169)

F4 (n = 175)

F5 (n = 231)

F6 (n = 229)

F7 (n = 232)

  24.63 3,928 0.125 2.90 0.065

  24.74 4,026 0.106 2.23 0.063

  25.10 3,925 0.111 3.17 0.084

  25.75 3,979 0.109 2.72 0.082

  22.42 4,244 0.148 3.63 0.095

  20.39 4,268 0.179 2.51 0.091

  23.33 4,151 0.110 3.18 0.071

   

390.95 67.15 0.25 −29.50 −21.98 1.98 0.014 0.088

   

383.96 68.42 1.05 −30.32 −22.76 3.09 0.012 0.082

   

385.11 65.30 −4.48 −20.37 −24.44 1.83 0.011 0.105

   

414.99 66.80 0.07 −27.09 −24.46 2.15 0.013 0.102

   

282.06 62.73 5.53 −29.01 −13.86 3.23 0.012 0.107

   

284.06 63.88 5.60 −31.57 −12.26 3.53 0.015 0.089

   

274.21 61.68 2.91 −24.73 −15.95 3.34 0.009 0.084

= residual SD.

Mixed effects NLM are a more flexible framework to model growth because they take into account the within-individual correlated nature of repeated measures (Pinheiro and Bates, 2000). However, fitting data using the Richards mixed model did not converge, most likely due to overparameterization and sample size. For NLM, the Weibull mixed effects model provided the best goodness of fit based on AIC values. The residual SD (