Analysis of growth performance data in sheep using fixed and random regression models

Kefelegn Kebede, Tsegay Teklebrhan, Mengistu Urge and Yoseph Mekasha

Haramaya University, School of Animal and Range Sciences, PO Box 138, Dire Dawa, Ethiopia
kebede123@yahoo.de

Abstract

In experiments of feeding-trial in animal research, the typical repeated measures design consists of animals randomly assigned to different diet-groups and the responses being taken on each animal over a sequence of time points. In such experiments, the methods generally employed to describe the data and to account for the different effects are based on the analysis of variance and standard regressions. However, most of the time, the assumptions that data are independent are violated in repeated measured data that are taken on the same subject at different time intervals. As a result, analysis of variance and regression methods may produce invalid results. In addition, the presence of intra- and inter-observers variability can potentially obscure significant differences. The linear mixed models is an extended multivariate linear regression method  that have been proposed to circumvent these limitations, by adding random effects aimed at modelling the variability due to peculiarity of the observed subjects.  The objectives of this study were twofold: (a) to compare fixed and random regression models, for evaluating body-weight gain of lambs and (b) to test and quantify the difference in body-weight gains of animals of different breed-groups exposed to different diet-groups.  In this experiment, ram lambs consisting of four breed-groups, i.e., black head ogaden, hararghe highland, ½Dorper ½black head ogaden and ½Dorper ½haraghe highland sheep were used. Up to nine ‘repeated records’ of  body-weight gains per lamb, measured between 10^th and 90^th day of age post the time of initial body weight recording, were available.

 Results showed that a linear regression on age modelled changes in variation of body weight adequately. As compared to fixed regression models, random regression models delivered better estimates of ∆-2logL, ∆AICC and ∆BIC. Furthermore, significant results were found for the comparison of the four breed-groups exposed to different diets.

Key words: body-weight gain, breed, diet, model selection

Introduction

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 resource-poor 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). Thus, to be profitable and satisfy market demands performance evaluation of growth and carcass traits improvement of sheep is required.

Quality and quantity of feeds and fodders are the major constraints in increasing ruminant's productivity under tropical conditions. Existing feedstuffs in the country often provide inadequate nutrients and vitamins to support optimum growth and development (Seare et al. 2007; Kassahun 2008).

Based upon the species of farm animals, several traits such as milk yield, body-weight 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, body-weight 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 body-weight 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 diet-groups and the responses being 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 the different effects are based on the analysis of variance (ANOVA) and standard regressions. These methods estimate the effects of the parameters for the underlying models, and 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). ANOVA and standard regression 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 inter-observers 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) has the capability to address these issues directly by using the MIXED MODEL procedure.

As mentioned by Meyer (1999 and 2004) 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. 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 (Kefelegn et al 2010).

Accordingly, the objectives of this study were twofold: (a) to compare fixed and random regression (FR and RR) models, for evaluating body-weight gain of lambs and (b) to test and quantify the difference in body-weight gains of lambs in the different breed-groups exposed to different diet-groups.

Materials and methods

Description of the study area

The experiment was conducted at Haramaya University Sheep Farm. The University is located on the eastern escarpment of the Rift valley at about 520 km East of Addis Ababa, at latitude of 9^o 26’, longitude of 42^o 03’ and altitude of 1980 m.a.s.l. It has 780 mm of rain fall during the cropping season and the mean annual maximum and minimum temperature are 23.4 and 8.25 ^oC, respectively AUA (1998). 

Experimental animals and feeds

In the present experiment a total of 48 ram lambs of age about one year old were used (see Table 1). These lambs consisted of four breed-groups based on their blood-level, i.e., B1: black head ogaden lambs, B2: hararghe highland lambs, B3: ½Dorper ½black head ogaden lambs and B4: ½Dorper ½haraghe highland lambs. B1 and B2 are pure local breeds, while B3 and B4 are crosses of the two local breeds with Dorper sheep from South Africa. For each of the four breed-groups, i.e., B1 up to B4, twelve experimental ram lambs were available for the study.

Experimental ram lambs 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 also vaccinated against common diseases (anthrax and pasteurellosis) during the quarantine period.  

In this experiment, grass hay harvested from the University campus was used as the basal diet, while Noug seed cake (NSC) and wheat bran (WB) were used as supplements. The proportion of the supplement in the rations was in the ratio of 2:1 (i.e. 2 parts of WB to 1 part of NSC). Supplements were offered to ram lambs daily in two equal meals at 08:00 and 16:00. Clean drinking water and mineralized salt-licks were made available to the individual ram lambs at all times.

The twelve ram lambs in each of the four breed-groups were divided randomly into two groups of six lambs each, where one-half was assigned to a supplement diet-group of S150 (ad libitum hay + 150 g/head/day) and the other-half to a supplement diet-group of S350 (ad libitum hay + 350 g/head/day). The table below describes these arrangements.

Body-weight measurements

Following a 14-day acclimatization period, each growing lamb was weighed at the beginning of the experiment (initial body weight, IBW) and every successive ten days thereafter. All lambs were weighed during morning hours after overnight fasting using suspended weighing scale having sensitivity of 100 g. The experiment was initiated in February and ended in April 2010.

Statistical methods

Model selection approaches

In this study, for analyzing the data on repeated records of body-weight gains using linear mixed model a two-step modelling approach was used. The first-step models compare the expected value structure by using the ML-Method, while the second-step models compare the covariance structure by using the REML-Method (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.

Modelling the expected value structure  

Depending on the age of animals, the description of growth traits like body-weight and average daily gains can be modelled using polynomials of 1^st – 5^th degree (Albuquerque and Meyer 2000; Meyer 2000 and 2004; Nobre et al 2003).

In this first-step, 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 applied the modelling approach used by polynomials. This approach requires n covariates. Let t_max be the maximal length of the feeding-period (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 days-on-test, i.e. t = DT/t_max (with DT = days-on-test). It holds true that: x₀(t) = 1.0, x₁(t) = t; x₂(t) = t²; and x_n(t) = tⁿ. These covariates are well-suited for the description of growth curves. Furthermore, it is possible to stay within the class of linear models. 

According to the data structure and the above mentioned statements, the expected value structure for the body-weight gain of lamb m, from breeding-group i, diet-group j, on days-of-test of k, having an initial body weight of l for the standardized time t, is to be modeled for the whole observation period.

When considering all influential factors, the expected value structure of y_ijklm (t) has the following form for the model with exclusively fixed regression coefficients (fixed regression model, FRM):

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 breed and diet-groups in (1) is modelled by the respective fixed regression coefficients. Thus, differences between the breed and diet-groups are expressed by different patterns of the growth curves. The resulting flexibility can be used for the calculation of average body-weight gains for different sub-periods of the growth period. For the analysis of the data with model (1), the following SAS statement was used: 

PROC MIXED DATA=SHEEP METHOD=ML;

CLASS Breed Diet DT;

MODEL BWt = Breed Diet DT IBW X₁(Breed) X₁(Diet) X₂(Breed) X₂(Diet)...X_n(Breed) X_n(Diet) /NOINT;

In the MODEL-statement X₁, …, X_n are the covariates for the description of the different group-specific growth curves. By using the option NOINT, the general mean is estimated together with the group effects.

Modelling the covariance structure

In this second-step, 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 body-weight gains of a lamb have to be modelled with the help of covariance structures. In order to realize this, lamb-specific 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 m and let x = (x₀, x₁, …, x_n)’ be the vector of the covariates. Then, the following random regression model (RRM) for days-on-test can be written from (1):

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 body-weight gains of a lamb at time point t₁ and t₂:

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, a´_l*) = 0. The following SAS statements can be used to analyze the data with model (2):

PROC MIXED DATA=SHEEP METHOD=REML;

CLASS Breed Diet DT LAMB;

MODEL BWt = Breed Diet DT IBW X₁(Breed) X₁(Diet) X₂(Breed) X₂(Diet)...X_n(Breed) X_n(Diet) /NOINT;

RANDOM INT X₁ X₂ ... 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:

Here, ; in all other cases, . Because of the above formula, all pairs of measures on the same lamb have the same correlation. Thus, the correlation between two measures at time t₁ and t₂ is:

Up to now, the covariance structure has been modelled by using lamb-specific 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 body-weight gain on days-on-test of t. Then, let us use the following model: e(t) = e₁(t) + e₂(t). Here, e₁(t) stands for the component of the serial correlation between repeated measurements for a lamb, e₂(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 e₁(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:

Where, d = is the temporal distance between two measurement points. Frequently used functions are the Gaussian function and the exponential serial correlation function: 

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=SHEEP METHOD=REML;

CLASS Breed Diet DT LAMB;

MODEL BWt = Breed Diet DT IBW X₁(Breed) X₁(Diet) X₂(Breed) X₂(Diet)...X_n(Breed) X_n(Diet) /NOINT;

RANDOM INT X₁ X₂ ... 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 second-step, then the analysis of time trends for feeding-groups by estimating and comparing means (i.e. tests of fixed effects) can be done with the help of t- and F-tests (Tukey-Tests) (Giesbrecht and Burns 1985; Fai and Cornelius 1996; Kenward and Roger 1997; Spilke et al 2005).

Model selection methods

Likelihood-Ratio Test (LRT)

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 chi-square 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.

Information criteria

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 Maximum-Likelihood (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 Maximum-Likelihood (REML) method and the calculations of the criteria are given by:

Both criteria (i.e. AICC and BIC) use the number of variance components as penalty to balance the log-likelihood 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. 

Results and discussion

Descriptive statistics

A summary results of the data on body-weight gain over the different sub-periods, i.e. from day 10 (t₁₀) up to day 90 (t₉₀) for the different breed and diet-groups of lambs, is given in the table below.

With repeated measures data, an obvious first statistics to consider is descriptive statistics results of body-weight gain of animals against time. Table 2 shows among others the means and standard deviations for individual ages at recording in 10-days intervals. Generally, the mean and standard deviations of the uncorrected data in the above table increased slowly and steadily with age. This pattern of increase over time could be explained in terms of variation in the growth rates of the individual animals. In addition, the results shown in the table reveal differences in body-weight gain among the different breed and diet-groups. These results were expected for two reasons. First, as the proportions of feed ingredients in the two diet-groups (S150 vs. S350) were variable, this has resulted to different nutritional compositions of the supplied diets. Second, as the breed-groups studied differed, this has led to different responses in body-weight gain among the different breed-groups. The results and patterns of increase of body-weight gain found in table 2 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.

Modeling 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 will be possible only after optimization of the covariance structure has been done.

As can be seen from the above table, seven models, i.e. M1, …, M7, 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 time-dependent estimation of the model effects.

According to the ∆-2logL, ∆AICC and ∆BIC values given for the seven 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, …, M7), where the decrease in M7 is the highest so that this model is chosen to be the optimum model.

The models M1, …, M6 can be seen as special cases of M7 that can be found through substituting one or more fixed effects in M7 by zero.

The following SAS statement can be used to analyze the data with fixed regression model (7):

PROC MIXED DATA=SHEEP METHOD=ML;

CLASS Breed Diet DT;

MODEL BWt = Breed Diet Breed*Diet Time IBW X1*Breed X1*Diet / NOINT;

RUN;

Testing of additional models beyond M7 with a quadratic and above polynomials for the effect of “BREED” and/or “DIET” has led to no improvement in decreasing the values of AICC and BIC as was the case in M7. As a consequence considering quadratic and above polynomials are left out.

Modelling the covariance structure

The table below gives the results found for the different models tested to find an optimum model for the covariance structure. The summarized models (M8, …, M11) given in the table below are analysis results from the random regression model with correlated residual effects and heterogeneous variance assumptions.

As can be seen from the above table, four models, i.e. M8, …, M11, are given in increasing order of complexity. According to the ∆-2logL, ∆AICC and ∆BIC values given for the four models, M8 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 (M9, M10 and M11), where the decrease in M11 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 M8 to M11 even though the number of model parameters increased from 1 in M8 to 5 in M11.

The models M8, …, M10 can be seen as special cases of M11 that can be found through substituting one or more effects in M11 by zero. The following SAS statement can be used to analyze the data with the random regression model (11):

PROC MIXED DATA=LAMB2 METHOD=REML;

CLASS FG DT LAMB;

MODEL BWt = Breed Diet Breed*Diet Time IBW X1*Breed X1*Diet / 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.

Comparison of breed and diet-groups

The model comparison of the different breed and diet-groups on body weight gain is presented in table 5. The corresponding SAS statement for the implementation of this comparison is given by: 

LSMEANS BREED*DIET/PDIFF ADJUST=TUKEY;

A significant effect was obtained for the Breed*Diet interaction suggesting that the two effects cannot be interpreted separately as one factor depends on what the other factor is doing. The table below shows least square means of the breed by diet-groups interaction and their significance tests.

Comparison of breed and diet-groups on body weight gains

Figure 1a (left). LS Means of body weight gain (y abscise) for the different breed-groups (x abscise) of the two diet-groups	Figure 1b (right). LS Means of body weight gain (y abscise) for the two diets (x abscise) of the four breed-groups

As can be seen from the table 5 and figure 1a, diet-group S350 has resulted for all the breed-groups (i.e. B1 – B4) studied, a significantly higher body weight gain than diet-group S150. This increased body weight gain is justified by the higher crude protein content of the feeds in S350 than S150. In this regard, even though the difference in body weight gain across the breed-groups is clear to see, the rate and magnitude of change in body weight gain is different for the two diet-groups.

Figure 2b shows that there is a trend indicating breed-group differences in body weight gain along the two diet-groups. When comparing the pure local breeds (i.e., B1 and B2), B1 has a significantly higher body weight gain than B2 at S150 but this difference is not significant at S350; similarly when comparing the crosses (i.e. B3 and B4), B4 has a significantly higher body weight gain than B3 at S150 but this difference is not significant at S350.

While the comparison of B1 and B3 did not show a significant difference at S150, this difference between the two breed-groups was significant at S350, indicating the better response of the crosses (i.e., B3) to this diet-group. On the other hand the comparison of B2 and B4 was significantly different for both diet-groups applied.

Besides, as it was reflected in the results of the descriptive statistics table (see table 2), the crosses (i.e., B3 and B4) have a significantly higher body weight gain than the pure breeds (i.e., B1 and B2). This suggests the impact of cross breeding on increased body weight gain and thereby increasing meat production of growing lambs for market supply.

Conclusions

• The provision of S350 has resulted in higher body weight gain for the breed-groups studied. This higher body weight gain can be explained by the higher content of crude protein in S350 than S150, as a result of which a breed supplemented with S350 is heavier.

  
  

• From an implementation point of view, it can be seen that a significantly higher body weight gain has been achieved for the crosses than the local breeds. In this regard as B3 and B4 have no significant difference at S350 they can be equally distributed for farmers able to provide lambs with S350 diet.

  
  

• The increased body weight gain of the crosses has an impact on the profitability of sheep production on the local market or for export. 

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Received 16 April 2011; Accepted 30 July 2011; Published 10 October 2011

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