Evaluation of Alternative Nonlinear Mixed Effects Models of Swine Growth

The Professional Animal Mixed Scientist 18 (2002):219–226 Evaluation of Nonlinear Effects Growth Models

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of Alternative Evaluation Nonlinear Mixed Effects Models of Swine Growth A. P. SCHINCKEL*,1 , PAS, and B. A. CRAIG† *Department of Animal Sciences and †Department of Statistics, Purdue University, West Lafayette, IN 47907

Abstract Alternative versions of a common three-parameter nonlinear growth function were evaluated on two groups of gilts. Gilts were randomly assigned to be reared under all-in, all-out (AIAO; n = 96) or continuous flow (CF) management (n = 96). The addition of a single random effect, in which the mature BW of each pig varies, provided a substantially better fit and smaller parameter standard errors. This model predicts a constant coefficient of variation between pigs. The addition of a second random effect further improved the likelihood statistics and reduced the residual standard deviation, although the impact was much smaller. The inclusion of the second random effect accounts for different patterns of growth between pigs. Variations in the growth patterns allow greater flexibility to describe the underlying variance/covariance structure of the serial live BW. For all mixed effects models, the mean and approximate variation in age required for pigs to reach a specific BW can be predicted. After 104 d of age, the growth of CF gilts declined more rapidly, and the standard deviation in days required to reach specific target BW (TBW) (110, 120, or

130 kg) increased more rapidly than for the AIAO gilts. These models are also easily adaptable to stochastic modeling. (Key Words: Mixed Effects Model, Nonlinear Growth Functions, Random Effects, Pig Growth.)

Introduction

Swine growth models are used to identify alternative strategies to improve the efficiency of swine production and to estimate daily nutrient requirements for pigs of various ages and genetic groups (16). For an effective application of these models, the growth potential of pigs must be accurately characterized. Several nonlinear growth equations have been used to fit BW as a function of age (3, 4, 10, 12). These equations take the form f(t:Q) + ei,t, where f is a function (with parameters Q) describing the mean BW at age t, and ei,t represents the residual deviation of the BW of pig i from this mean. These residuals are commonly assumed to be independent normal random variables with mean zero and constant variance. When fitting serial growth data, these assumptions can be troublesome because of competitive interac1To whom correspondence should be adtions and serial correlations (5). dressed: [email protected] Heavier pigs at birth and weaning

usually have a competitive advantage and remain heavier throughout their stay in the group (8). This results in increasing variance with age and correlated observations over time. The production of pigs has developed toward highly specialized coordinated production systems. Variability in the growth rates of pigs is important to the economic costs and returns of both the pork producer and processor (7, 13). The optimization of pork production systems, including the evaluation of alternative management and marketing strategies, requires knowledge of the between and within pig variation in BW (8, 16). Previously, as an alternative to relaxing the assumptions on the residuals and modeling the variance/covariance structure, we described the use of a single random effect to describe the variance/ covariance structure (5). The objectives of this study were to evaluate alternative mixed effects models of a standard growth function and to determine whether a better description of the variance/covariance structure could be found.

Material and Methods Swine Growth Model. Consider the following fixed effects growth model proposed by Bridges et al. (3):

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∆WTi,t = WTi,t – WTi,0 = C(1 – exp(–MtA)) + ei,t.

[1]

Here, t is the age of pig i measured in days, C is the average mature BW, M is the exponential growth decay constant, and A is the kinetic order constant (3, 15). Typically, a constant value of 1.4 kg is used for birth weight (WTi,0) such that WTi,t = ∆WTi,t + 1.4. Because the exponential decay parameter was close to zero, the model was reparameterized (M’ = log(M)) with the form ∆WTi,t = C (1 – exp(–exp (M’) tA)). In this function, any combination of C, M’, or A could be considered random. For example, the model with all three random can be expressed as ∆WTi,t = (C + ci)(1 – exp (–exp(M’ + m’i) t(A + ai) )) + ei,t, where ci, m’i, and ai would be each pig’s random effects assumed to be multivariate normal with mean 0 and variances σc2, σm’2, and σa2, respectively. Additionally, the random effects of each pig may be correlated. The ei,t are independent, normal with mean 0 and variance σ2e and assumed independent of the random effects. The simplest single random effect model has the mature weight random and form ∆WTi,t = (C + ci)(1 – exp (– exp(M’)tA) + ei,t, where ci is the random effect for each pig . This mixed effects growth function can also be written as (WT = (C + ci) f(t), where f(t) = 1 – exp (–exp(M’)tA), which increases from 0 (when t = 0) to an upper limit of 1 (5). In this form, the total variance of ∆WTi,t at time t can be separated into the between pig variance, σc2 [f(t)]2, and the within pig variance, σe2. This means the between pig coefficient of variation (σc/C) is constant, and the coefficient of variation of the total variance (σc2 [f(t)]2 + σ2e)1/2/[Cf(t)] decreases as t increases. This mixed effect growth function can be expanded to include an additional fixed effect, ∆WTi,t = C [f(t)] + ci[f(t)]D + ei,,t, in which D describes the relative change in the between pig coefficient of variation

as BW increases. With this model, the total variance at a specific age is the sum of the between pig variance, σc2 [(f(t)]2D, and the within pig variance σ2 e. If D is >1, the between pig coefficient of variation increases as t increases. If D is <1, the between pig coefficient of variation decreases as t increases. The inclusion of a second random effect (m’i or ai) accounts for different patterns of growth between pigs. The addition of m’i or ai as random effects with the ci effect allows increased flexibility in fitting of the between pig variance. However, the formulas for the approximate between pig variances are more complicated. Example Data. The data are from a trial in which gilts were randomly assigned to be reared from 49 d of age in either a grow-finish facility that had been completely emptied, cleaned, and disinfected [all-in, allout management (AIAO)] or a continuous flow (CF) grow-finish facility in which every third pen consisted of older pigs. Gilts were allotted by weight into pens with 6 gilts per pen with 3 m2 per gilt in the AIAO environment and 2.4 m2 in the CF environment. A total of 96 AIAO gilts were weighed at 49, 70, 104, 132, and 153 d of age. Also, 42 AIAO gilts had an additional BW measurement taken at 174 d of age. A total of 96 CF gilts were weighed at 49, 70, 104, 132, 153, and 174 d of age. The AIAO and CF gilts had similar mean weights up to 104 d of age. After 104 d of age, the growth rate of the CF

gilts substantially decreased, but the growth rate of AIAO gilts decreased only a small amount (Table 1; Figures 1 and 2). Estimation. The BW data from each group of pigs were fitted to the reparameterized fixed effects Bridges function using the non-linear mixed (NLMIXED) procedure of SAS (14). The SAS code for the fixed and mixed models is presented in Figure 3. The three alternative single random effects models were evaluated based on the Akaike’s Information Criteria (AIC). Then, additional random effects were added in a step-wise order based on AIC values. The R2 values were calculated as squared correlations between the predicted and actual observations. The residual standard deviation (RSD) was calculated with the equation RSD = 2

[ ∑∑ (e i, t ) / (n − p )]1 / 2 T

I

t =1 i =1

where ei,t is the residual value of pig i at age t, n is the number of observations, and p is the number of parameters in the model. The significance of the D parameter was evaluated based on its impact on the AIC statistics. Also, correlations were calculated between the actual and predicted BW at each age for each mixed model. The NLMIXED procedure provided predicted values for the random effect of each pig, variance estimates

TABLE 1. Means and standard deviations of the live weight data (kg). All-in all-out Age

n

Mean

49 70 104 132 153 174

96 96 96 96 96 42

20.88 36.85 67.28 93.00 115.14 120.29

Continuous flow

SD

CV

n

Mean

2.34 3.76 5.34 7.99 8.52 9.03

11.2 10.2 7.9 8.6 7.4 7.5

96 96 96 96 96 96

20.74 36.72 67.88 90.05 106.08 113.65

SD

CV

2.26 3.18 5.32 8.10 9.05 9.96

10.9 8.7 7.8 9.0 8.5 8.8

Evaluation of Nonlinear Mixed Effects Growth Models

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Figure 1. Relationship of average daily gain ADG to age.

for each random effect, residual variance, and the estimated covariance between the random effects. Approximate standard errors of the function parameters, variance estimates, and covariance estimates are based on the second derivative matrix of the likelihood function. These approximate standard errors are based on large sample inferences (9, 11). Bootstrap estimates of precision and confidence intervals are better in these cases but may require extensive computer calculations (6, 11).

Results and Discussion The parameter estimates and nonlinear regression statistics are presented in Table 2 for the AIAO gilts and in Table 3 for the CF gilts. The fixed effects model had a greater VAR(e) for the AIAO pigs (51.4) than for the CF pigs (43.3). The R2 and RSD values were similar for the fixed effects functions for both groups of pigs. In each environment, the addition of a random effect ci for each pig provided the best single random parameter addition. For the AIAO pigs, the value of C increased

Figure 2. Relationship of BW (kg) to age.

Figure 3. SAS code for PROC NLMIXED assumes data set variables are ID (pig ID number), age (days), and MWTG (weight gain from birth, BW – 1.4, in kg). Correlation option requests the parameter estimate correlation matrix. Parms statement provides initial model estimates (strongly encouraged); u2, u3, and cu23 are used for the random variables.

from 148.7 to 225.2 kg. Of the 96 pigs, 54 were removed at 153 d of age based on the fact they had achieved 112 kg as an end BW; 42 remained to 174 d of age. This likely caused a downward bias in the magnitude of C in the fixed effect model (5, 15). The mixed effect model has been shown to decrease the magnitude of

such bias (5) produced by the early removal of pigs. The addition of the random effect ci increased the magnitude of C from 132.28 to 149.85 kg for the CF pigs. The VAR(e) values were reduced (93% for the CF and 84% for the AIAO gilts), likelihood statistics improved, and RSD values were reduced (70% for the CF and

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TABLE 2. Parameter estimates and approximate standard errors for the Bridges Function (all-in, all-out management)a.

Parameter Fixed effects model C M’ A VAR (e) Mixed effects model with ci random C M’ A VAR (e) VAR (c) Mixed effects model with ci random and D coefficient as a fixed parameter C M’ A D VAR (e) VAR (c) Mixed effects model with c and m’ random C M’ A VAR (e) VAR (c) VAR (m’) COV (c, m’) Mixed effects model with c and m’ random and D coefficient as a fixed parameter C M’ A D VAR E VAR (c) VAR (m’) COV (c, m’)

Estimate

Approximately SE

148.68 –9.930 2.039 51.36

5.68 0.272 0.069 4.06

2,168.6

2,176.0 7.20 0.9641

225.15 –9.066 1.728 7.418 301.39

15.08 0.082 0.033 0.767 61.8

1,903.8

1,913.8 2.17 0.9967

226.75 –9.050 1.725 1.024 7.404 308.62

15.74 0.083 0.034 0.090 0.765 80.4

1,904.2

1,917.1 2.17 0.9968

216.27 11.47 –9.12 0.077 1.755 0.028 5.395 0.622 831.23 223.8 0.02725 0.0051 –3.880 0.967

1,883.0

1,897.0 1.70 0.9980

217.14 12.01 –9.11 0.081 1.761 0.031 1.024 0.093 5.329 0.621 833.21 296.2 0.02267 0.0112 –2.991 1.41

1,881.9

1,902.1 1.70 0.9980

–2 log likelihood AICb

RSDc R2

aThe fixed effects model for BW gain (kg) from birth is C(1 – exp(– exp(M’)tA)) + e, where e is normal with variance σ2e. The mixed effects model is (C + ci)(1 – exp (– exp(M’)tA)) + e, where e is normal with variance σ2e, and ci, pig i’s random effect, is normal with variance σ2c. The mixed effects model with D fixed and ci random is C(1 – exp(– exp (M’)tA)) + ci(1 – exp(– exp (M’)D + e, where e is normal with variance σ2e, and ci, pig i’s random effect, is normal with variance σ2c. The mixed effects model with c and m’ as random effects is (C + ci) [1 – exp (– exp(M’ + m’i)tA)] + e, where e is normal with variance σ2e and ci and m’i are pig i’s random effects with mean 0 and variances σc2 and σm2 and with covariance of c with m σc,m. The general model has the form C (1 – exp(– exp(M’ + mi)tA) + ci(1 – exp(– exp(M’ + m’i)tA)D + e. bAIC = Akaike’s Information Criteria. cRSD = Residual standard deviation.

60% for the AIAO gilts) with the ci random effect. The approximate standard errors of M’ and A were reduced 50 to 70% for the single random effect vs fixed effects equations of the CF and AIAO gilts. In the single random effect model, the random effect ci implies that each pig is a constant percentage (ci/C) heavier or lighter in BW than the mean BW at each age and that the between pig coefficient of variation is constant (5). The addition of the fixed D parameter to single random effects mixed model was calculated for both the CF and AIAO environments. For the AIAO pigs, D had an estimated value of 1.02 with an approximate standard error of 0.09. For the CF gilts, D had a value of 1.136 with an approximate standard error of 0.087. The AIC statistic was reduced from 3643.2 to 3643.0 for the CF gilts. The value of 1.136 suggests that the between pig coefficient of variation increases slightly as t increased, although it was not found statistically different from one with these data. The addition of a second random effect for m’i had a relatively lesser impact to reduce the residual variance and improve the likelihood statistics. The significance of the random effect for M’ indicates pig-to-pig variation in the shape of the BW growth curves. In this model, the predicted between pig variance is a function of the VAR(c), VAR(m’), and the covariance of c and m’. The random effect m’i was negatively correlated with the random effect ci (r = –0.82 and –0.76 for the AIAO and CF pigs, respectively). This correlation indicates that pigs with greater values of ci have lower values of m’i. The addition of the D parameter to the two-variable random effects model was evaluated in both datasets. For the AIAO pigs, D had an estimated value of 1.024 with an approximate standard error of 0.093. For the CF pigs, D had a value of 1.125 with an approximate standard error of 0.23. The AIC value increased from 3580.7 to 3581.5 with the addition of D. Thus, D is not

Evaluation of Nonlinear Mixed Effects Growth Models

TABLE 3. Parameter estimates and approximate standard errors for the Bridges Function (continuous flow management)a.

Parameter Fixed effects model C M’ A VAR (e) Mixed effects model with ci random C M’ A VAR (e) VAR (c) Mixed effects model with ci random and D coefficient as a fixed parameter C M’ A VAR (e) VAR (c) D Mixed effects model with c and m’ random C M’ A VAR (e) VAR (c) VAR (m’) COV (c, m’) Mixed effects model with c and m’ random and D coefficient as a fixed parameter C M’ A D VAR (e) VAR (c) VAR (m’) COV (c, m’)

Estimate

Approximate –2 log SE likelihood AICb

RSDc R2

132.28 –9.538 1.984 43.30

2.81 0.178 0.045 2.45

4,115.6

4,123.6 6.60

0.9648

149.85 –9.096 1.846 9.228 120.25

2.94 0.082 0.023 0.628 15.06

3,633.2

3,643.2 2.60

0.9946

151.02 –9.072 1.836 9.140 140.67 1.136

3.12 0.083 0.024 0.622 21.76 0.087

3,631.0

3,643.0 2.58

0.9946

151.15 –9.120 1.851 6.219 280.0 0.0267 –2.0914

2.91 3,567.7 0.071 0.019 0.476 52.7 0.00565 0.494

3,581.7 1.97

0.9969

151.86 –9.103 1.845 1.125 6.153 279.9 0.02260 –1.811

3.14 3,565.5 0.0732 0.0200 0.232 0.468 96.7 0.0123 1.18

3,581.5 1.95

0.9969

aThe fixed effects model for BW gain (kg) from birth is C(1 – exp(– exp(M’)tA)) + e, where e is normal with variance σ2e. The mixed effects model is (C + ci)(1 – exp (– exp(M’)tA)) + e, where e is normal with variance σ2e, and ci, pig i’s random effect, is normal with variance σ2c. The mixed effects model with D fixed and ci random is C(1 – exp(– exp (M’)tA)) + ci(1 – exp(– exp (M’)D + e, where e is normal with variance σ2e, and ci, pig i’s random effect, is normal with variance σ2c. The mixed effects model with c and m’ as random effects is (C + ci) [1 – exp (– exp(M’ + m’i)tA)] + e, where e is normal with variance σ2e and ci and m’i are pig i’s random effects with mean 0 and variances σc2 and σm2 and with covariance of c with m σc,m. The general model has the form C (1 – exp(– exp(M’ + mi)tA) + ci(1 – exp(– exp(M’ + m’i)tA)D + e. bAIC = Akaike’s Information Criteria. cRSD = Residual standard deviation.

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helpful in fitting these data. The addition of the D parameter slightly reduced the estimate of the VAR(e) from 6.219 to 6.153 for the CF pigs. The VAR(c) was not affected (280.0 vs 279.9). However, the VAR(m’) was reduced from 0.0267 to 0.0226. In some cases, where the initial variation in BW has been artificially reduced (sorting), such uniformity in BW may result in increased aggressive behavior and a rapid substantial increase in the between pig variance during the post-weaning period (7, 8). The significance of D to be >1 or <1 indicates that the random effects terms included in the nonlinear regression equation are not sufficient by themselves to account for the changes in the between pig coefficient variation as BW or age increases. The SAS algorithms were not able to solve all three parameters as random effects. For the CF pigs, the equation with ci and ai as random terms resulted in predicted correlation of –1.00 between the ci and ai random effects. It appears that a form of multicolinearity existed between these two random effects. The multicolinearity may be caused by the relationship of ai and ci with the BW at which maximum growth rate is achieved (BWMAX). The relationship is expressed through the equations A = 1/[log (1 – F) + 1] and F = BWMAX/C, the fraction of mature weight (C) in which maximum growth rate is achieved. At constant C, F and, thus, BWMAX, increase as A increases. At constant A, BWMAX increases with C; thus, pigs with a higher BWMAX can have either a higher A or C value. In the CF environment, the gilts had a more distinct inflection point, the age and BW at which maximum growth rate occurred, than did AIAO gilts. The ADG of the CF gilts increased and reached a plateau of 0.89 kg from 87 to 92 d of age and then declined. The ADG of the AIAO gilts increased from 49 to 104 d. The predicted ADG was almost constant (0.93 kg/d) from 104 to 127 d of age and then declined slowly after 128 d of age.

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Comparison of Rearing Programs. The parameters of the Bridges function differed between the two environments. The predicted mature BW parameter, C, was smaller for the CF pigs (151.86 kg) than for the AIAO pigs (216.27 kg). The maximum growth was predicted to occur at a BW of 55.7 kg at 89.5 d for the CF pigs vs 75.6 kg and 111.7 d for the AIAO pigs. This indicates that the growth parameters are substantially affected by the environmental conditions. It is important to recognize that the growth rates and growth parameters are a combined result of genetic and environmental conditions including health status, social interactions, stocking density, and air quality (15, 16). Growth parameters of different genetic populations cannot be compared unless the pigs are evaluated under the same environmental conditions. The correlations between the predicted and actual values at each age evaluate the precision with which the pigs’ BW were predicted in each production environment (Table 4). The single and two random variable equations (r = 0.953 to 0.991) accurately predicted BW from 132 to 174 d of age. The two random variable equations had 0.067 to 0.197 greater correlations between the predicted and actual BW from 49 to 104 d of age. Inclusion of the additional m’i random effect appears to allow each

pig’s growth function to better fit the data from 49 to 104 d of age. The correlations were not affected by the addition of the D parameter to the single random variable equation for the CF pigs. The correlations only increased 0.015 and 0.012 at 49 and 70 d of age, respectively, with the addition of the D parameter to the two random variable equations for the CF pigs. Researchers have used alternative approaches to evaluate the between pig variation in BW growth or pigs assigned to different experimental treatments. One alternative is to compare the standard deviation or coefficient of variation in BW at a specific age. The CF pigs and AIAO pigs have similar amounts of variation based on these statistics from 49 to 132 d of age. At 153 and 174 d of age, the standard deviation and coefficient of variation were only slightly greater for the CF pigs than for the AIAO pigs (Table 1). The nonlinear models can be used to predict the age required for each pig to reach a specific target BW (TBW). Using the two parameter mixed model and 1.4 kg as the initial BW, the age at a specific TBW for pig i is ti = [(log (1 – (TBW – 1.4))/(C + ci))/(–exp (M’ + m’i))]1/a. The mean and variance of ti can then be investigated under the different rearing programs. Note that our distribution of ti is based on the set of estimated

parameters (and effects) and does not incorporate estimation uncertainty. The difference in the predicted days required to reach TBW increased between the AIAO and CF pigs as TBW increased from 100 to 130 kg (Table 5). The variability in the predicted age to reach a TBW increased as the TBW increased and increased more rapidly for the CF gilts than for the AIAO gilts. Days of age to reach specific TBW were predicted for the five fastest gaining and five slowest gaining gilts to each environment based on the predicted age required to reach 120 kg (Table 5). The difference in age required to achieve 100 kg between the five fastest and five slowest growing gilts was similar for the two environments (38.8 and 36.6 for the AIAO and CF pigs, respectively). The difference in age predicted to achieve 120 kg between the fastest and slowest growing gilts increased to 61 d in the CF environment (146.4 vs 207.4 d) and 53 d in the AIAO environment. The five fastest and five slowest growing AIAO gilts required 18.4 and 32.6 d, respectively, to grow from 100 to 120 kg. The fastest and slowest growing CF gilts required 23.2 and 47.6 additional d, respectively, to grow from 100 to 120 kg live weight. The increased standard deviation and greater age difference between the AIAO and CF gilts are partially caused by the

TABLE 4. Correlations of predicted and actual BW at each agea. All-in, all-out

Age

n

ci random

49 70 104 132 153 174

96 96 96 96 96 42

0.574 0.825 0.844 0.955 0.978 0.975

aP

<0.001 for all correlations.

Continuous flow

ci random and D

ci and m’ random

ci and m’ random and D

0.575 0.825 0.843 0.955 0.978 0.976

0.748 0.931 0.911 0.962 0.989 0.991

0.750 0.933 0.911 0.962 0.989 0.991

n

ci random

ci random ci and m’ and D random

ci and m’ random and D

96 96 96 96 96 96

0.600 0.662 0.834 0.953 0.969 0.930

0.598 0.658 0.830 0.952 0.970 0.934

0.812 0.894 0.939 0.960 0.980 0.970

0.797 0.882 0.939 0.960 0.980 0.970

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specific TBW increased more rapidly in the CF gilts than in the AIAO gilts, which indicates that, as the environmental conditions worsened, the CF gilts had more variable All-in, all-out Continuous flow growth rates. This increased variation in days required to achieve Age Age market TBW has an economic cost, as Target most pork processors discount pigs BW Upper Lower Upper Lower below their specified market TBW (2, (kg) Mean 5 5 SD Mean 5 5 SD 13). The economic evaluation of rearing pigs under AIAO or CF 100 138.1 121.0 159.8 8.4 144.4 123.2 159.8 10.9 management must include both the 110 149.5 130.4 175.2 9.7 160.9 134.4 180.0 16.5 impact of the differences in the mean 120 161.5 139.4 192.4 11.3 177.4 146.4 207.4 19.3 performance and differences in the 130 174.2 149.2 211.8 13.4 199.3 154.8 258.9 22.9 variation in growth (7, 13). The benefits of implementing aThe five gilts with the least and greatest predicted days to achieve 120-kg BW in mixed effects swine growth models each environment. and the ease of their implementation bUsing Bridges mixed model equation WT = (C + c )[– exp (– exp (M’ + m’ ) tA)] + i,t i i using SAS have been discussed. 1/A 1.4; ti,TBW = [(log(1 – (TBW – 1.4))/(C + ci))/(– exp (M’ + m’i))] , where C, M’, and A Additional concerns such as estimates are mean parameter values, and ci and m’i are parameters for pig i. of standard errors of parameters (i.e., bootstrapping vs large sample apdecreasing ADG of CF gilts after 104 d increasing serial correlations among proximations) and non-normal (i.e., of age. The reduced growth of the the serial BW as age increases (5, 7). skewed) random effects are currently slower gaining CF pigs after 104 d of Simulation of these random effects being studied. age compared with the AIAO gilts nonlinear functions (i.e., stochastic makes it more difficult to achieve a modeling) is also straightforward. specific TBW of 120 kg or greater in With the single random effect mixed the CF environment than in the model, only one additional random Animal growth models have been AIAO environment. effect [ci with mean = 0 and variance developed with the goal of optimizDiscussion. The growth potential = VAR(c)] needs to be simulated to ing production systems. These model the growth of each pig. For of a population of pigs must be models require a parameterization of the two random effect models, two accurately characterized to refine animal growth and the between and random effects (i.e., ci and m’i) need alternative management, nutrition, within pig sources of variation. to be simulated with the varianceand marketing strategies. Recently, Nonlinear mixed effects models allow covariance structure predicted by the a more accurate and precise estimapork producers have become conmixed model nonlinear regression cerned with quantifying and reduction of animal growth functions analyses. Simulation can be used to ing variation in the growth of pigs than the traditional fixed effects (7, 8, 16). The AIAO management of predict the age required for each pig models. Nonlinear mixed effects to reach a specific TBW. The variaswine facilities results in improved models provide parameters needed for tion in age required to achieve a TBW stochastic modeling, which is needed health status and increased growth is affected by growth rates achieved rates. However, the need to comto evaluate the economic impact of close to the TBW and the variation pletely empty the facility is in conmanagement changes that reduce the in weight at the age at which the flict with the pork processor’s desire amount of variation. mean pig achieves the TBW. The for uniformity (13). To optimize variation in age required to reach pork production systems, swine specific TBW is more closely related to management and marketing stratemarketing decisions than the variagies must be refined (1, 2). This tion in BW at a specific age (1, 2). requires accurate estimates of both The AIAO and CF gilts had similar the between pig and within pig growth rates until 104 d of age, and, sources of variation. Mixed model nonlinear regression analyses produce subsequently, the ADG of CF gilts declined much more rapidly than the the parameters and variance-covariADG of AIAO gilts. After 104 d of ance estimates needed to reproduce 1. Boland, M. A., K. A. Foster, P. V. Preckel, and A. P. Schinckel. 1996. Analyzing pork age, the standard deviation of the the increasing variance changes in carcass evaluation technologies in a swine predicted age required to achieve the coefficient of variation and bioeconomic model. J. Prod. Agric. 9(1):45.

TABLE 5. Overall means and means for the fastest and slowest growing pigsa with standard deviations for predicted age to achieve specific BW (kg)b.

Implications

Literature Cited

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