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METABOLISM AND NUTRITION |
* Institute of Animal Nutrition, University of Hohenheim, Emil-Wolff-Str. 10, 70593 Stuttgart, Germany; Bioinformatics Unit, University of Hohenheim, Fruwirthstrasse 23, 70593 Stuttgart, Germany; and Evonik Degussa GmbH, Rodenbacher Chaussee 4, 63457 Hanau, Germany
1 Corresponding author: rhmosent{at}uni-hohenheim.de
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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+ β*[1 – exp (–
*dose)] + e. Thereafter, meta-analysis was conducted by means of nonlinear mixed models, which were fitted by a full maximum likelihood method as implemented in the NLMIXED procedure. The nonlinear model used for the analysis allowed for separate plateaus or different efficiencies of the methionine sources. Mixed modeling was applied to account for heterogeneity among the studies in all regression parameters for both sources of methionine via random effects. Statistical hypotheses were tested by the asymptotic Wald test. In addition, potential co-variables were tested for inclusion as linear regressors for the nonlinear model parameters. In conclusion, the null hypothesis of equal plateaus of the 2 methionine sources was not rejected. The analyses of the response variables average daily gain (ADG) and gain to feed ratio (GF) showed a highly significant difference between the tested methionine sources. The covariate age at start of experiment significantly affected the intercept term for the response variables ADG and GF, respectively. The meta-analysis showed that biological efficiencies of DL-methionine-hydroxy-analogue-free-acid were 81 and 79% of the values for DL-methionine, on an equimolar basis, for ADG and GF, respectively.
Key Words: biological efficiency • DL-methionine • meta-analysis • methionine-hydroxy-analogue-free-acid
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Meta-analysis is an established method to combine the results of different studies in one statistical analysis to obtain broadly valid inferences (Van Houwelingen et al., 2002). Originally, meta-analysis was introduced in psychology and medicine, but today it is becoming increasingly popular in animal science as well, including different species such as poultry (Hooge, 2004), sheep (Prankel et al., 2004), cattle (Sauvant and Mertens, 2006), and pigs (Sanchez et al., 2007). For broilers, however, only few studies exist in which meta-analysis has been used to compare the biological efficiency of MHA-FA and DLM (Jansman et al., 2003; Kratzer and Littell, 2006; Vazquez-Anon et al., 2006). For example, Jansman et al. (2003) calculated, based on weight gain and feed conversion data, average biological efficiencies of 77 and 76%, for liquid MHA-FA and DLM, respectively, on equimolar basis. Vazquez-Anon et al. (2006) analyzed the predicted dose-response and relative performance of MHA-FA and DLM by linear least squares by fitting a multiple regression. They obtained no difference in growth performance between MHA-FA and DLM, supplemented to the diets on an equimolar basis. Kratzer and Littell (2006) concluded, in contrast to the study of Vazquez-Anon et al. (2006), that these 2 methionine sources have to be analyzed by means of separate plateau models. The authors analyzed individual studies using the maximum likelihood method and combined results by a sign test, based on the estimated relative efficiencies of MHA-FA and DLM in each study. Piepho (2006) pointed out in a letter to the editor the importance of including more studies in the statistical analysis and using more powerful methods of meta-analysis. In this context, the general linear mixed model with an approximate likelihood approach has proven to be a very useful and convenient framework for meta-analysis (Van Houwelingen et al., 2002). The authors concluded that linear mixed models can be used for simple structured and more complex meta-analysis, which include multivariate treatment effect measures and explanatory variables.
The aim of this study was to perform a nonlinear mixed model meta-analysis on the relative biological efficiency of MHA-FA compared with DLM in broilers, based on average daily gain (ADG) and gain to feed ratio (GF) as response variables. Another objective was to investigate if these methionine sources reach different plateaus with increasing dietary levels of methionine.
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MATERIALS AND METHODS |
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A data set, containing in total 46 dose-response studies, extracted from 27 published studies, was selected and subsequently submitted to a meta-analysis. Data included in this set were selected on the basis of the following selection criteria (Table 1
):
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Statistical Analysis
For each study, the method of nonlinear least squares (PROC NLIN) was used to fit the model
 | ([1]) |
where x is the dose in percent of active supplemental methionine and e is a residual error term, assumed to be independently and normally distributed with constant variance. Studies, for which proper convergence was achieved and the parameter estimates satisfied the constraints > 0, β > 0 and > 0, were selected for meta-analysis. Finally, the meta-analysis was performed on 40 (ADG) and 38 (GF) studies (Table 1
), respectively, using the nonlinear mixed model
 | ([2]) |
where h = 1 for DLM and h = 2 for MHA-FA, k is an index for doses, x1ik is the dose of DLM, and x2ik is the dose of MHA-FA for the hkth treatment and ith study. For the control, we set h = 0. Note that for the control x1ik = x2ik = 0. The model was fitted jointly for both sources using all data of a study. Heterogeneity among studies was modeled by assuming that study-specific deviations ai, bi, c1i, and c2i follow a multivariate normal distribution with zero mean and variance-covariance matrix (Figure 1).
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). The random effects account for heterogeneity among studies, and they model the correlation of observations from the same study. The model was fitted by the maximum likelihood method as implemented in the NLMIXED procedure of the SAS Institute Inc. (1999). Gauss-Hermite quadrature (Pinheiro and Bates, 1995) was used to integrate random effects out of the joint log-likelihood. The number of quadrature points was gradually increased until the relative change in 2 successive log-likelihood evaluations was less than 
= 10–4. Between 7 and 11 quadrature points were used for all analyses. To facilitate convergence, parameter values were rescaled resulting in values of about equal magnitude. Random effects were added successively, starting with ai, then adding bi, and finally adding c1i and c2i. We chose starting values in the same range as the parameter estimates from individual studies. The parameter for the second source was re-parameterized as
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The null hypothesis H0: 1 = 2 resulted in 
= 0 so that the null hypothesis could be tested by a Wald test for H0: = 0.
To test the assumption of a common plateau, the model was extended to allow for heterogeneity among plateaus. This was achieved by replacing β with βj (j = 1, 2) and setting
 | ([4]) |
The assumption of common plateaus corresponded to H0: β = 0. No significant differences in plateaus were found. Thus, in further calculations a common plateau was assumed.
Finally, correlations of parameters with study-specific covariates were explored. Graphical inspection of scatter plots of study-specific parameter estimates as well as significance testing revealed that only the covariate age at start of experiment (ASE) was associated with . To accommodate the covariate, was replaced by + øzi, where zi was the value of the covariate for the ith study.
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RESULTS |
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and Table 3, respectively. For both, ADG and GF, differences in efficiency of MHA-FA and DLM were determined, followed by analysis for different plateaus of the 2 methionine sources and integration of the covariate ASE.
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Comparison of methionine sources revealed a significantly lower efficiency for MHA-FA compared with DLM. On an equimolar basis, the relative biological efficiency of MHA-FA to that of DLM was estimated at 81%, using ADG as response variable. The 2 methionine sources shared a common plateau; the estimate of β in [4] was not significantly different from 0 (P = 0.91). The age at the start of the experiment had a significant effect on , indicating that it should be retained in the model. Even with this inclusion of ASE in the model, the efficiency of MHA-FA was lower than that of DLM, and the relative efficiency differed very little from that obtained with the model that did not include the covariate. Finally, the following nonlinear equation was obtained:
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where D and H are the (0,1) dummy variables of each methionine source and doseDLM and doseMHAFA are the amounts of the methionine additive.
Meta-Analysis of the Response Variable GF
In general, the results for GF are in good agreement with those obtained for ADG as response variable. The biological efficiency of MHA-FA on an equimolar basis in comparison with DLM amounts to approximately 79% for the response variable GF. In addition, there exists no plateau difference (P > 0.05) between MHA-FA and DLM. The covariate ASE shows a significant effect on the intercept . For GF the following nonlinear equation was calculated:
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where D and H are the (0,1) dummy variables of each methionine source and doseDLM and doseMHAFA are the amounts of the methionine additive.
Summary of the Meta-Analyses for the Response Variables ADG and GF
The baseline model (no covariate, common plateau) showed for the response variables ADG and GF a significant difference in relative biological efficiency of about 81 and 79% for MHA-FA and DLM, respectively, based on equimolar comparisons. Furthermore, no plateau differences (heterogenous β model) could be established for these response variables (P > 0.05 for ADG and for GF, respectively). Consequently, it was assumed that there was one common plateau for each methionine source. Finally, the covariate ASE (common plateau with covariate model) affected the intercept term significantly for the response variables of ADG (Figure 2) and GF (Figure 3), respectively. The inclusion of the covariate only had a marginal effect on estimates of efficiencies. This was mainly due to the fact the regression coefficients for the methionine sources were virtually unaffected by the covariate.
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DISCUSSION |
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The standard estimation method for nonlinear models is based on least squares (Seber and Wild, 1989), as implemented in the procedure NLIN of the SAS system (SAS Institute Inc., 1999). This method was used for parameter estimation within each study taken independently. Studies not reaching proper convergence were rejected for the meta-analysis. In the analysis of individual studies, the NLIN procedure was preferred over maximum likelihood because it properly accounts for the error degrees of freedom (Piepho, 2006). As with nonlinear least squares methods for fixed effects models, successful fitting of a nonlinear mixed model by maximum likelihood requires suitable starting values [i.e., starting values close to the maximum likelihood solution (Lesaffre and Spiessens, 2001)]. With poor starting values, there is a risk of not achieving convergence to a local maximum of the likelihood (Vonesh and Chinchilli, 1997), and this problem was encountered in the present meta-analysis. Consequently, different starting values were applied to fit the model.
For meta-analysis, it is recommended to employ the most powerful procedure. According to Van Houwelingen et al. (2002), mixed modeling provides a very useful and convenient framework for both simple and complex structured meta-analyses. In the present meta-analysis, the procedure NLMIXED was used. This procedure allows for modeling the heterogeneity between studies by fitting random effects at the study level. By using the full maximum likelihood, an increased type I error rate can be expected because the loss of degrees of freedom due to fitted fixed effect is not taken into account. However, due to the relatively large number of studies being included in the present meta-analysis, the increase of the type I error rate is expected to be negligible (Piepho, 2006). Gaussian quadrature, as used here to integrate random effects out of the likelihood, is not a fool-proof method. The number of quadrature points needed to achieve sufficient accuracy of the approximation to the likelihood may easily become prohibitively large. The number of quadrature points chosen adaptively by the default settings of the NLMIXED procedure was between 7 and 11, which is in line with recommendations for generalized linear models (Lesaffre and Spiessens, 2001). One particular problem associated with successive optimizations is that starting values very close to the optimum cause the NLMIXED procedure to terminate with an error message that no valid data points exist. When setting starting values of some parameters to values more removed from the final maximum likelihood solution, this problem could be avoided. Another possibility, that was not used here, is to use options that affect the sub-options for the estimation of empirical Bayes estimates (EBSUBSTEP=, EBSTEP=, EBSFRAC=, EBSSFRAC=, EBSTOL=, EB-SSTOL=, and EBSOPT=; O. Schabenberger, SAS Institute, Cary, NC; personal communication). The empirical Bayes estimates need to be computed internally at the starting values, and convergence of the suboptimization involved may not occur for all subjects. Experimenting with the EBS options may resolve convergence problems. Generally, convergence for nonlinear models is difficult to achieve due to parameter curvature (Ratkowsky, 1983). In the case of fixed effects models, it is often possible to find an equivalent re-parameterization that reduces this problem (Ratkowsky, 1983, 1990), but performance is data-dependent (Schabenberger and Pierce, 2002). The curvature problem persists in mixed nonlinear models, but this issue is difficult to resolve because random effects are usually not invariant to re-parameterization. For example, in the case of fixed effects, a re-parameterization of the model (1), which has good properties in terms of curvature (Schabenberger and Pierce, 2002), is given by + β
x. When introducing random effects in the exponential term, there is no problem for , whereas adding a random normal effect is not possible for because this has to satisfy the constraint > 0. Another re-parameterization is + βexp[–exp()x] (Davidian and Giltinan, 1995). Here, adding a random effect at the level of is possible, but the model is no longer equivalent to (1) because the variance-covariance structure is altered.
Model Selection
For the evaluation of the biological efficiency of the 2 methionine sources in this study, a nonlinear regression model with a common plateau was assumed. This is in agreement with other studies in which this model has also been used (Lemme et al., 2002; Jansman et al., 2003; Hoehler et al., 2005). Furthermore, the nonlinear regression method has been established for determining nutrient requirements based on dose-response experiments [e.g., phosphorus (Potter et al., 1995; Dhandu and Angel, 2003; Schulin-Zeuthen et al., 2007) and vitamins (Kasim and Edwards, 2000)]. Moreover, Littell et al. (1997) used a nonlinear approach for analyzing the relative bioavailability of DLM and MHA-FA by means of the SAS procedure NLIN. Rodehutscord and Pack (1999) concluded from the results of a meta-analysis that for the calculation of estimates of amino acid requirements from dose-response studies, a nonlinear model has to be applied. Furthermore, the nonlinear model was used in the meta-analysis performed by Jansman et al. (2003) to calculate the biological efficiency of MHA-FA in comparison with DLM in broilers, pigs, laying hens, and turkeys. In summary, most of the authors use nonlinear asymptotic models to describe response curves for meta-analysis, whereas according to Vazquez-Anon et al. (2006) and Rosen (2007), the response curve for gain and feed conversion could be described by a quadratic function as well.
However, by feeding physiological dose rates of methionine sources to a healthy organism, a reduction in gain and feed conversion is not likely to occur. More recently, Kratzer and Littell (2006) concluded, in contrast to the analysis used in related studies in the literature (Lemme et al., 2002; Jansman et al., 2003; Hoehler et al., 2005) and the present analysis, that a separate plateau model should be used when comparing the 2 methionine sources. Piepho (2006) emphasized in a letter to the editor that the evidence of a difference in plateaus based on the results of the meta-analysis by Kratzer and Littell (2006) is weaker than claimed by the authors. In addition, Piepho (2006) annotated that, to more thoroughly investigate the important question of whether plateaus differ, it would have been advantageous if the authors had included data from more experiments in their model. Another approach to determine the biological efficiency of MHA-FA in comparison with DLM was proposed by Lemme et al. (2002) and Hoehler et al. (2005). These authors used multi-exponential regression to determine if the analysis with a common plateau model can be applied for the determination of efficiencies in methionine sources. They compared a diluted source of DLM with a known purity of 65% with DLM and MHA-FA. They could verify the results of the diluted DLM, and on this account they concluded that the mathematical model that was used was valid for such comparative nutritional purposes.
Comparison of Efficiency Between MHA-FA and DLM
On an equimolar basis, the biological efficiency of MHA-FA in comparison with DLM was 81 and 79% in broilers for the response variables ADG and GF, respectively. Previously, Potter et al. (1984) calculated on an equimolar basis, by means of multi-regression analysis of individual experiments with broilers, a biological efficiency of 78% for MHA-FA compared with DLM for the response variable ADG. Moreover, Jansman et al. (2003) obtained in broilers only a marginal difference of 77 and 76% in biological efficiency for the response variables ADG and feed conversion, respectively, based on exponential regression analysis of individual experiments and by computing arithmetic means of parameter estimates. However, in contrast to the results of this study and related reports in the literature (Lemme et al., 2002; Jansman et al., 2003; Hoehler et al., 2005), Vazquez-Anon et al. (2006) could not find, by using a multiple linear regression model, significant differences in performance between MHA-FA and DLM treatments, on an equimolar basis. However, because the results of this study are in good agreement with those obtained in other studies (Potter et al., 1984; Jansman et al., 2003; Hoehler et al., 2005), it is suitable to use a nonlinear common plateau model for analyzing the biological efficiency of different methionine sources.
Finally, it can be concluded that the dose-response of both methionine sources on ADG and GF could be described with a nonlinear mixed model. Moreover, it could be shown that for both methionine sources common plateaus were reached; therefore, a common plateau of both methionine sources could be assumed for the meta-analysis. In conclusion, on an equimolar basis, a relative biological efficiency of MHA-FA in comparison with DLM of 81 and 79% for the response variables ADG and GF, respectively, could be confirmed.
Received for publication December 19, 2007. Accepted for publication June 3, 2008.
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