Livestock Research for Rural Development 22 (4) 2010  Notes to Authors  LRRD Newsletter  Citation of this paper 
Most statistical approaches in experiments of feedingtrial are based on variance analysis (ANOVA). However, most of the time, the assumption that data are independent is violated since several measures are performed on the same subject (repeated measures). As a result, standard regression and analysis of variance methods may produce invalid results of repeated measures data because they require mathematical assumptions that do not hold with these data. In addition, the presence of intra and interobservers variability can potentially obscure significant differences. The linear mixed models (LMM) is an extended multivariate linear regression method of analysis that have been proposed to circumvent these limitations, by adding random effects aimed at modelling the variability due to peculiarity of the observed subjects and thus leading to more efficient parameters estimates.
In this study, repeated records of bodyweight gains on lambs were considered for analysis. Up to nine ‘repeated records’ of bodyweight gains per lamb, measured between 10^{th} and 90^{th} day of age post the time of initial body weight recording, were available. The objectives of the study were twofold: (a) to compare a twostep model, i.e. fixed and random regression (FR and RR) models, for evaluating bodyweight gain of lambs. It should be clarified, whether RR models deliver better model fitting in contrast to FR models and (b) to test and quantify the difference in bodyweight gains of animals in different feedinggroups and at different ages (daysontest).
Results showed that a linear regression on age modelled changes in variation of body weight adequately. As compared to FR models, RR models delivered better estimates of ∆2logL, ∆AICC and ∆BIC. Furthermore, significance results in LSQMeans were found for the comparison of the four feedinggroups and difference in bodyweight gains at age t=15.
Key words: bodyweight gain, feeding trial, fixed and random regressions, model selection
In Ethiopia sheep are kept for various purposes and objectives. They make a significant contribution to income generation, supply of animal source of food and serve as financial security to the resourcepoor rural households (Gryseels 1988; Zelalem and Fletcher 1991; Barrs 1998; Workneh 1998). In general, the relatively huge number of livestock resources, proximity to the export markets in neighbouring countries as well as the Middle East and the liberalization of the economy give the country comparative advantage in the trading of livestock and their products (Belachew and Jemal 2003). The rise in demand and price of sheep meat is indicated in the country (Woldu et al 2004). Thus, to be profitable and satisfy market demands performance evaluation of growth and carcass traits improvement is required.
Sheep production in Ethiopia historically has been a low labour enterprise with little emphasis on animal productivity and management practices. Maintained virtually under the traditional subsistence oriented management systems sheep constitute an important livestock component in all ecological zones and agricultural systems in the country. Quality and quantity of feeds and fodders are the major constraints in increasing ruminant's productivity under tropical conditions. Existing feedstuffs in tropical countries often provide inadequate energy, protein, minerals and vitamins to support optimum animal productivity (Reed et al 1990; Seare et al 2007; Kassahun 2008).
Based upon the species of farm animals, several traits such as milk yield, bodyweight gain, feed intake and longevity are used for selection of candidate animals as the genetic evaluation is practiced (Akbas et al 2004). In any meat producing sheep industry, bodyweight and average daily gains are considered to be important components for market lamb production (Kebede et al 1998 and 1999; Schueler et al 2001). Growth traits such as bodyweight and average daily gains are important response indicators in sheep. The typical repeated measures experiment in animal research consists of animals randomly assigned to different feedinggroups and with responses measured on each animal over a sequence of time points. In such experiments, the methods generally employed to describe the data and to account for fixed and random effects are based on the analysis of variance (ANOVA). ANOVA performs a linear regression to estimate the fixed effects parameters of the underlying models, and to reveal significant differences.
Responses measured on the same animal over a sequence of time points are correlated because they contain a common contribution from the animal. Moreover, measures on the same animal close in time tend to be highly correlated than measures far apart in time. Also, variances of repeated measures often change and increase steadily with time. Since the potential patterns of correlation and variation may combine to produce a complicated covariance structure of repeated measures, these feature of repeated measures data require special methods of statistical analysis (Littell et al 1998). Standard regression and analysis of variance methods may produce invalid results because they require mathematical assumptions that do not hold with repeated measures data. In addition, the presence of intra and interobservers variability can potentially obscure significant differences.
The linear mixed models (LMM) have been proposed to circumvent these limitations, by adding random effects aimed at modelling the variability due to peculiarity of the observed subjects. The general linear mixed model of the SAS System (SAS 2005) allows the capability to address these issues directly by using the MIXED MODEL procedure.
As mentioned by Meyer (1999) growth of animals is a prime example of a trait measured repeatedly per individual along a continuous scale (time) which changes gradually and continually and can be modelled using random regressions (Meyer 2004). Random regression models have been utilized to describe a linear mixed model including appropriate covariates to model the effect of time on repeated records as fixed and random terms (Schaeffer and Dekkers 1994). Using random regression models there is no need to correct towards certain landmark ages (Meyer 2004).
Accordingly, the objectives of this study were twofold: (a) to compare a twostep model, i.e. fixed and random regression (FR and RR) models, for evaluating bodyweight gain of lambs. It should be clarified, whether RR models deliver better model fitting in contrast to FR models and (b) to test and quantify the difference in bodyweight gains of animals in different feedinggroups and at different ages (daysontest).
The study was carried out at Maichew Agricultural Technical and Vocational Education and Training College which is located in southern zone of Tigray, Ethiopia. The site is situated at 12^{0} 47’N latitude and 39^{0} 32’E longitude and lies between 2396 and 2472 meters above sea level.
Twenty intact Ethiopian highland male sheep with an average body weight of 17 + 1.6 kg (mean + SD) of age about one year old were purchased from the local markets based on their dentition. The sheep were dewormed and sprayed against internal and external parasites before the beginning of the experiment and brought indoors after the pens had been properly washed and disinfected. They were vaccinated against common diseases (anthrax and pasteurelosis) during the quarantine period.
Noug seed cake (NSC) and wheat bran (WB) were used in this experiment as supplement feeds; while urea treated wheat straw (UTWS) was used as the basal diet. The proportion of the supplement feeds in the rations was at the ratio of 3:1 (i.e. 3 parts of WB to 1 part of NSC). The feed ingredients, i.e. NSC, WB and UTWS, were mixed together so that the ration formulated lied within the range recommended by the National Research Council (NRC 1985) for growth requirements of growing lambs.
The experiment was conducted using a completely randomized design fashion with five lambs randomly assigned to one of four feedinggroups: I) A basal diet of UTWS that was fed ad libitum; II) UTWS supplemented daily with 150 g/head/day of a mix (i.e. WB+NSC); III) UTWS supplemented daily with 200 g/head/day of a mix (i.e. WB+NSC); IV) UTWS supplemented daily with 250 g/head/day of a mix (i.e. WB+NSC).
Supplement feeds were offered to lambs daily in two equal meals at 08:00 and 16:00. Clean drinking water and mineralized saltlicks were made available to the individual lamb at all times. The experimental lambs were offered UTWS and the respective supplement feeds for 14 days to acclimatize them to the feeds and they were fed on the same diet for the rest of the experimental period (i.e. 90 days).
Following a 14day acclimatization period, each growing lamb was weighed at the beginning of the experiment (initial body weight, IBW) and every successive tenday thereafter. All lambs were weighed during morning hours after overnight fasting using suspended weighing scale having sensitivity of 100 grams. The experiment was initiated in April and ended in July 2008.
In this study, for analyzing the data on repeated records of bodyweight gains using linear mixed model a twostep modelling approach was used. The firststep models compares the expected value structure by using the MLMethod, while the secondstep models and compares the covariance structure by using the REMLMethod (Wolfinger 1993; Ngo and Brand 1997).
Depending on the types of problems to deal with, the number of fixed and random effects to consider and the degree of polynomial (i.e. linear, quadratic...) to fit; the number of different possible models to compare for both expected value structure and covariance structure can be very high in number.
Depending on the age of animals, the description of growth traits like bodyweight and average daily gains can be modelled using polynomials of 1^{st} – 5^{th} degree (Albuquerque and Meyer 2000; Meyer 2001; Nobre et al 2003).
In this firststep, the selection of an optimum model for the expected value structure is found by incorporating fixed effects to be tested in the different models through the use of fixed regression model where selection of an optimum model is carried out by considering uncorrelated residual effects and homogenous variance assumptions.
In order to realize this, we make use of the modelling approach used by polynomials. This approach requires n covariates. Let t_{max} be the maximal length of the feedingperiod (in this study, t_{max }= 90 days). In an attempt to achieve better convergence properties, the covariates were expressed as a function of the standardized daysontest, i.e. t = DT/t_{max} (with DT = daysontest). It holds true that: x_{0}(t) = 1.0, x_{1}(t) = t; x_{2}(t) = t^{2}; and x_{n}(t) = t^{n}. These covariates are wellsuited for the description of growth curves. Furthermore, it is possible to stay within the class of linear models.
Accordingly, the bodyweight gain of lamb l, from feedinggroup i, on daysontest of j, having an initial body weight of k for the standardized daysontest t = t_{ijkl}/t_{max} is given by:
(1)
In this working model:
b_{i0}
to b_{in} stand for fixed regression coefficients of
feedinggroup i,
is
the vector of covariates formed according to polynomials, and
represents
random residual effects.
By the comparison and finally selection of an optimum model using the above polynomial models, the aim will be to find the best order of fit for the fixed regression on age to model population trajectory and to determine the degree of the polynomial that leads to optimum results.
In contrast to the usual modelling of experimental effects, the effect of feedinggroup in (1) is modelled by the respective fixed regression coefficients. Thus, differences between the feedinggroups are expressed by different patterns of the growth curves. The resulting flexibility can be used for the calculation of average bodyweight gains for different subperiods of the growth period. For the analysis of the data with model (1), the following SAS statement was used:
PROC MIXED DATA=LAMB2 METHOD=ML;
CLASS FG DT;
MODEL BWG = FG DT IBW X_{1}(FG) X_{2}(FG)...X_{n}(FG)/NOINT;
In the MODELstatement X_{1}, …, X_{n} are the covariates for the description of the different groupspecific growth curves. By using the option NOINT, the general mean is estimated together with the group effects.
In this secondstep, the selection of an optimum model for the covariance structure is done by using the optimum model chosen for the expected value structure (section “Modelling the Expected Value Structure”) and then by incorporating the random effect part (i.e. animal) in the different models to be tested. This is realized by using random regression model (RRM) where selection of an optimum model for the covariance structure is accomplished by considering correlated residual effects and heterogeneous variance assumptions.
The classical random regression model involves a random intercept and slope for each subject. Random regression models provide a valuable tool for modelling repeated records in animal experiments, especially if traits measured change gradually over time like in this study.
Dependencies between repeated bodyweight gains of a lamb have to be modelled with the help of covariance structures. In order to realize this, lambspecific random regression coefficients a_{l0} – a_{ln} are introduced as deviations from the fixed regression coefficients, where the value of n should not be greater than that found for the optimum model of the expected value structure.
Let a_{l} = (a_{l0}, …, a_{ln})’ be the vector of the random regression coefficients of lamb l and let x = (x_{0}, x_{1}, …, x_{n})’ be the vector of the covariates. Then, the following random regression model for daysontest can be written from (1):
(2)
In model (2), all random effects are assumed to be normally distributed with a mean value of 0. Additionally, we assume that all random effects associated with different lambs are generally independent of each other. By using the covariate matrix A_{l} and the residual variance between bodyweight gains of a lamb at time point t_{1} and t_{2}:
(3)
Furthermore, all lambs in model (2) were treated as unrelated. For the vectors of the random regression coefficients of two lambs l and l^{*}, it follows: cov (a_{l}, aŽ_{l*}) = 0. The following SAS statements can be used to analyze the data with model (2):
PROC MIXED DATA=LAMB2 METHOD=REML;
CLASS FG DT LAMB;
MODEL BWG = FG DT IBW X_{1}(FG) X_{2}(FG) ... X_{n}(FG)/NOINT;
RANDOM INT X_{1} X_{2} ... X_{n}/SUBJECT=LAMB TYPE=UN;
If n=0 in model (2), this means that one random effect per lamb is included and that the covariance function in (3) is of the form:
Up to now, the covariance structure has been modelled by using lambspecific random effects only. In the following, the residual covariance structure of model (2) will be extended further (Verbeke et al 1998; Lesaffre et al 2000).
Let e(t) be the residual effect of a lamb for the bodyweight gain on daysontest of t. Then, let us use the following model: e(t) = e_{1}(t) + e_{2}(t). Here, e_{1}(t) stands for the component of the serial correlation between repeated measurements for a lamb, e_{2}(t) denotes the component for the residual error with an equal variance for all measurements. The residual effects for the latter are assumed to be independent and identically distributed. The model for the serial covariance structure is completed by adding a distance correlation function g. This function is selected in such a way that all residual effects e_{1}(t) of a lamb have the same variance and that the correlation between two such effects is always positive but decreases monotonically with increasing temporal distance between two measurements for the same lamb. Then, the variance and covariance function of the residual effects of a lamb are given by:
(4)
Where, d = is the temporal distance between two measurement points. Frequently used functions are the Gaussian function and the exponential serial correlation function:
(5)
The two functions are always positive and decrease monotonically with increasing temporal difference d. They are continuous at d = 0 and meet the requirement that g(0) = 1. In (5), r is an unknown parameter greater than 0. The smaller the value of r, the stronger the function g decreases with increasing value of d. Model (2), extended with the exponential correlation function, can be fitted with the following SAS statements:
PROC MIXED DATA=LAMB2 METHOD=REML;
CLASS FG DT LAMB;
MODEL BWG = FG DT IBW X_{1}(FG) X_{2}(FG) ... X_{n} (FG)/NOINT;
RANDOM INT X_{1 }X_{2} ... X_{n}/SUBJECT=LAMB TYPE=UN;
REPEATED/SUBJECT=LAMB TYPE=SP (EXP) (DT);
Once the optimum model for the covariance structure is found in the secondstep, then the analysis of time trends for feedinggroups by estimating and comparing means (i.e. tests of fixed effects) can be done with the help of t and Ftests (TukeyTests) (Giesbrecht and Burns 1985; Fai and Cornelius 1996; Kenward and Roger 1997; Spilke et al 2005).
The LRT is a statistical test of the quality of fit of two hierarchically nested models. A model is hierarchically subordinated to another model if the former can be reduced to a special case of the latter by setting one or more of its parameters to zero or to fixed values. Therefore, the subordinated model is denoted as restricted and the hierarchically higher model is denoted as unrestricted. The null hypothesis is that both models are the same (extra parameters do not improve the fit). If we fail to reject the null hypothesis, the restricted model, because of its simpler form (and yet, its comparable explanatory power), is to be preferred. The likelihood ratio test is used to make sure that the unrestricted model indeed returns a (significantly) better result than the restricted model.
The likelihood ratio test statistic is given by:
.
The LRT statistic approximately follows a chisquare distribution. The degrees of freedom are equal to the number of restrictions in the extended model, which means that model(s) can be obtained as a special case of (g). Thus, the degrees of freedom are given by the number of variance components that need to be set to zero in the general model (g) to obtain the restricted model (s).
There has been concern that the use of LRT to determine the “best” model to fit the data might favour over parameterised models, thus this test does not favour parsimonious models. This lead to the use of information criteria – sometimes also referred to as penalised likelihoods  which adjust for number of parameters estimated and sample size.
For the comparison of models without hierarchical structures, the information criteria of Akaike (1969, 1973 and 1974) and its modification by Hurvich and Tsai (1989) and also the criterion by Schwarz (1978) can be used.
When using the MaximumLikelihood (ML) method, the calculations of these criteria for comparing the expected value structure are given by:
and .
Where, p_{X} is the rank of the design matrix X for the fixed effects, q is the number of variance components to be estimated and n is the number of records per animal.
The comparison of the covariance structure by identical expected value structure will be done using the Restricted MaximumLikelihood (REML) method and the calculations of the criteria are given by:
and .
Both criteria (i.e. AICC and BIC) use the number of variance components as penalty to balance the loglikelihood value. The penalty imposed by BIC is more severe than that imposed by AICC. The best model (of all competing models) is the one with the lowest criterion value. The ranking of models by their AICC or BIC values assumes the existence of identical fixed parameter structures in all competing models.
A comparison of the model selection criteria shows, AICC tends to prefer complex models while that of BIC tends to prefer simpler models for selection.
A summary results of the data on bodyweight gain over the different subperiods, i.e. from day 10 (t_{10}) up to day 90 (t_{90}), is given in the table below.
Table 1. Descriptive statistic results of bodyweight gain (kg) of lambs across the four feedinggroups measured on nine consecutive times (t_{10} to t_{90}) 

DT^{***} 
n 
Feedinggroup 

I 
II 
III 
IV 

t_{10} 
5 
17.3(1.73)* [15.2;19.5]** 
17.8(1.61)* [15.8;19.9]** 
17.8(1.55)* [15.9;19.8]** 
18.2(1.72)* [16.2;20.6]** 
t_{20} 
5 
17.2(1.68)* [15.2;19.3]** 
17.8(1.71)* [15.6;20.0]** 
17.9(1.55)* [16.1;20.0]** 
18.7(1.73)* [16.8;21.2]** 
t_{30} 
5 
17.4(1.66)* [15.3;19.4]** 
18.2(1.69)* [16.0;20.4]** 
18.5(1.61)* [16.4;20.5]** 
19.1(1.92)* [17.0;22.0]** 
t_{40} 
5 
17.6(1.65)* [15.4;19.5]** 
18.4(1.62)* [16.2;20.4]** 
18.7(1.62)* [16.6;20.8]** 
19.6(2.08)* [17.4;22.9]** 
t_{50} 
5 
17.4(1.61)* [15.4;19.4]** 
18.5(1.68)* [16.4;20.6]** 
19.0(1.72)* [16.8;21.3]** 
20.3(2.22)* [17.7;23.7]** 
t_{60} 
5 
17.5(1.52)* [15.6;19.4]** 
18.4(1.54)* [16.4;20.4]** 
19.2(1.86)* [16.8;21.8]** 
21.0(2.37)* [18.0;24.5]** 
t_{70} 
5 
17.7(1.55)* [15.6;19.5]** 
18.6(1.57)* [16.6;20.8]** 
19.7(2.05)* [17.0;22.6]** 
21.7(2.55)* [18.2;25.3]** 
t_{80} 
5 
17.6(1.62)* [15.8;19.7]** 
18.9(1.93)* [16.6;21.6]** 
19.9(2.25)* [17.0;23.3]** 
22.5(2.83)* [18.3;26.2]** 
t_{90} 
5 
17.7(1.57)* [16.0;19.8]** 
19.0(2.05)* [16.8;22.0]** 
20.3(2.49)* [17.0;24.0]** 
23.2(3.05)* [18.5;27.0]** 
*** DT = daysontest, implies the number of days elapsed post IBW recording at the start of the test 
With repeated measures data, an obvious first graph to consider is the scatter plot of the weight of animals against time. Figure 1 displays the means for individual ages at recording in 10days intervals. This simple graph reveals several important patterns. All lambs gain weights. The spread among all animals was substantially smaller at the beginning of the study than at the end.


The results shown in table 1 as well as figure 1 reveal differences in bodyweight gain among the four different feedinggroups. This was expected as the proportions of feed ingredients in the different feedinggroups were variable resulting in different nutritional compositions.
The profile for feedinggroup I (the basal diet) shows bodyweight gains less than for other feedinggroups on all days. Profiles for feedinggroups II, III, and IV show increases in bodyweight gain corresponding to increases in dietary supplement feeding. The profile for feedinggroup III shows bodyweight gains greater than that of feedinggroup II on all days after day 20. Profile for feedinggroup IV shows increase in bodyweight gains compared to all other feedinggroups in response to increased amount of supplement feeding.
Figure 2 shows the standard deviations for individual ages at recording in 10days intervals post IBW recording at the start of the feedingtrial.


Phenotypic standard deviations of the uncorrected data in the above figure show a similar picture as has been explained for that of the means (figure 1). They increased slowly and steadily with age. One can see a clear difference of the bodyweight gain records at different measurement times.This pattern of increasing variance over time could be explained in terms of variation in the growth rates of the individual animals.
The above results found for the phenotypic standard deviations and means are as expected for repeated records data measured on the same individuals over a time period. Thus, these facts should be considered when modelling the expected value structure.
The table below gives the results found for the different models tested to find an optimum model for the expected value structure. The summarized models given in the table are analysis results from the fixed regression model with uncorrelated residual effects and homogenous variance assumptions.
It is to emphasize that decisions made here only apply in relation to the expected value structure but not on the confidence interval and significance tests of the fixed effects used in the model. That is possible only after optimization of the covariance structure has been done.
Table 2. Estimated error variance (_{}), restricted log likelihood (logL) multiplied by 2 and information criteria from AICC and BIC 

Number 
Models for the EVS 
P 

2logL 
2logL

DF 
AICC 
BIC 
M1 
MODEL BWG = FG /NOINT; 
4 (3) 
3.80 
751 
488 (<0.001) 
14 
465 
436 
M2 
MODEL BWG = FG DT /NOINT; 
13 (12) 
3.23 
722 
459 (<0.001) 
5 
452 
443 
M3 
MODEL BWG = FG DT IBW/NOINT; 
14 (13) 
0.57 
411 
147 (<0.001) 
4 
143 
137 
M4 
MODEL BWG = FG DT IBW X1(FG)/NOINT; 
18 (17) 
0.25 
263 
0 
0 
0 
0 
M1, …, M4 = Model 1, …, Model 4; EVS = Expected Value Structure; FG = FeedingGroup; DT = DaysonTest; IBW = Initial Body Weight; p = Number of fixed effects; p_{X }= Rank of the design matrix for the fixed effects; DF = degrees of freedom for the likelihoodratio test (LRT); ∆2logL, ∆AICC and ∆BIC = Differences of the respective 2logL, AICC and BIC to Model M4. 
As can be seen from the above table, four models, i.e. M1, …, M4, are given in increasing order of complexity. In this regards M1 is only given as a demonstration purpose to show the development of the different models tested. This model does not allow a timedependent estimation of the model effects.
According to the ∆2logL, AICC and BIC values given for the four models, M1 is found to be the least chosen, as it has the largest values for ∆2logL, AICC and BIC. A substantial decrease in the values of ∆2logL, AICC and BIC can be seen for the other models (M2, M3 and M4), where the decrease in M4 is the highest so that this model is chosen to be the optimum model.
The models M1, …, M3 can be seen as special cases of M4 that are found through substituting one or more fixed effects in M4 by zero.
The following SAS statement can be used to analyze the data with model (4):
Fixed Regression Model, FRM 
PROC MIXED DATA=LAMB2 METHOD=ML; CLASS FG DT; MODEL BWG = FG DT IBW X1(FG)/NOINT; RUN; 
Testing of additional models beyond M4 with a quadratic and above polynomials for the effect of feedinggroup has led to no improvement in decreasing the values of AICC and BIC as was the case in M4. As a consequence considering quadratic and above polynomials are left out.
The table below gives the results found for the different models tested to find an optimum model for the covariance structure. The summarized models (M5, …, M8) given in the table below are analysis results from the random regression model with correlated residual effects and heterogeneous variance assumptions.
Table 3. Restricted log likelihood (logL) multiplied by 2 and information criteria from AICC and BIC 

Number 
Models for the CS 
q 
2RlogL 
2RlogL (pvalue) 
DF 
AICC  BIC 
M5 
MODEL BW = FG DT IBW X1(FG) /NOINT; 
1 
300.6 
295 (<0.001) 
5 
288 
288 
M6 
MODEL BW = FG DT IBW X1(FG) /NOINT; RANDOM INT / SUBJECT=Lamb TYPE=UN; 
2 
248.9 
243 (<0.001) 
4 
234 
237 
M7 
MODEL BW = FG DT IBW X1(FG) /NOINT; RANDOM INT X1 /SUBJECT=Lamb TYPE=UN; 
4 
97.5 
91.5 (<0.001) 
2 
91.5 
91.5 
M8 
MODEL BW = FG DT IBW X1(FG) /NOINT; RANDOM INT X1 / SUBJECT=Lamb TYPE=UN; REPEATED / SUBJECT=Lamb TYPE=SP(EXP) (DT) 
6 
6 
0 
0 
0 
0 
CS = Covariance structure; q= Number of covariance parameters; DF=degrees of freedom for the restricted likelihoodratio test (RLRT); ∆2logL, ∆AICC and ∆BIC = Differences of the respective 2logL, AICC and BIC to Model M8. 
As can be seen from the above table, four models, i.e. M5, …, M8, are given in increasing order of complexity. According to the ∆2logL, AICC and BIC values given for the four models, M5 is found to be the least chosen, as it has the largest values for ∆2logL, AICC and BIC. A substantial decrease in the values of ∆2logL, AICC and BIC can be seen for the other models (M6, M7 and M8), where the decrease in M8 is the highest so that this model is chosen to be the optimum model.
Considering tables 2 and 3 results found for ∆2logL, ∆ AICC and ∆ BIC indicate that random regression models fit the data better than fixed regression models. Random regression models gave better estimates and took into account that measurements are not all done at the same age.
Once the covariance structure has been chosen the results for the tests of fixed effects can be interpreted. As can be seen from the above table, the analysis results of the restricted likelihood ratio test shows a significant improvement as one goes from M5 to M8 even though the number of model parameters increased from 1 in M5 to 6 in M8.
The models M5, …, M7 can be seen as special cases of M8 that are found through substituting one or more random effects in M8 by zero.
The following SAS statement can be used to analyze the data with model (8):
Random Regression Model, RRM 
PROC MIXED DATA=LAMB2 METHOD=REML; CLASS FG DT LAMB; MODEL BWG = FG DT IBW x1(FG)/NOINT; RANDOM INT X1/SUBJECT=LAMB TYPE=UN; REPEATED/SUBJECT=LAMB TYPE=SP(EXP)(DT); RUN; 
Usually, when using PROC MIXED, the variation between animals is specified by the RANDOM statement, and covariation within animals is specified by the REPEATED statement.
The model (2) allows the graphical presentation of growth curves for each of the feeding groups. The expected bodyweight gain of a lamb from feeding group i on daysontest of t, averaged over all control days (n_{T}), has the form:
The calculation of mean values over all control days is necessary in order to estimate environmental effects on the individual control days. The calculation of (6) in SAS can be performed with the help of least square means (LSM). In order to do so, the covariate has to be calculated at t/t_{max}. For example, let t = 15; then, it hold true that:
X_{1}(15) = 15/90 = 0.167
The corresponding SAS statement for the calculation of (6) for t = 15 is given by:
LSMEANS FG/PDIFF AT
(X_{1}) = (0.167);
ESTIMATE ‘DIFFERENCE AT T=15’ FG 11
X_{1}(FG) 0.167 0.167;
Using the PDIFFoption results in testing the differences among the feedinggroups against 0 on daysontest of t = 15. The same test can be performed using the ESTIMATEstatement.
By using the fixed regression coefficients of model (2), any fitted bodyweight gains of different growth subperiods and the standard errors of the difference between such bodyweight gains can be calculated without difficulty. Thus, model (2) allows the comparison of four feedinggroups for different subperiods of the growth period as long as estimates of the fixed regression coefficients and their standard errors are known.
Table 4. Results of testing the differences among the feedinggroups against 0 on daysontest of t = 15 plus a comparison of the four feedinggroups 

Estimates 

Label 
Estimate 
Standard Error 
DF 
tValue 
pValue 

Difference at t=15 
0.26 
0.10 
133 
2.53 
<0.05 

Differences of Least Square Means 

FG 
FG 
IBW 
X1 
Estimate 
Standard Error 
DF 
tValue 
pValue 

1 
2 
17.28 
0.17 
0.26 
0.10 
133 
2.53 
<0.05 

1 
3 
17.28 
0.17 
0.19 
0.10 
133 
1.83 
0.07 

1 
4 
17.28 
0.17 
0.70 
0.10 
133 
6.80 
<0.01 

2 
3 
17.28 
0.17 
0.07 
0.10 
133 
0.71 
0.48 

2 
4 
17.28 
0.17 
0.44 
0.10 
133 
4.28 
<0.01 

3 
4 
17.28 
0.17 
0.51 
0.10 
133 
4.98 
<0.01 
Many statistical methods of analysis have been previously developed under balanced design assumption for the analysis of repeatedmeasures studies. At present, it’s clear that in order to develop more effective and more powerful observational studies, mixed models methods should be used more systematically.
The proposed analysis can be used to include the kind of data structures and error sources (such as different starting points of growths and changing environmental conditions on test days) that inevitably occur in similar experiments. By direct consideration of test data available on each daysontest, the model eliminates noisy variables in an improved way, and thus allows a better utilization of the inherent information of experimental data compared to cumulative bodyweight gains analysis.
From an implementation point of view, it is important to note that efficient software like SAS is available. This software does not only provide estimates for the parameters in linear mixed models, but it also supplies the user with fit statistics and tests for significances to the analysis of experimental data.
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Received 7 October 2009; Accepted 20 March 2010; Published 1 April 2010