Feed Efficiency Estimation from Stochastic Feed Requirement Frontier

Feed accounts 40 to 70 percent of livestock production cost. Therefore, improving feed efficiency of production animals will promote farm profitability. To this end, precise estimation of animal level feed efficiency is important. Considering some limitations of residual feed intake (RFI) as indicator of animal level feed efficiency, an alternative approach is suggested. The approach involves estimation of a stochastic feed requirement frontier (SFRF), which explicitly allows for feed efficiency and statistical noise in the same specification. As a result, a SFRF naturally leads to feed efficiency indicator free from statistical noise. Furthermore, the feed efficiency indicator it generates is nonnegative and it can easily be expressed in terms of surplus feed intake (SFI) caused by feed inefficiency. Simulation experiment was used to illustrate the problems that arise from RFI-based feed efficiency estimation and the improvements that can be expected from the alternative approach. The experimental results showed that RFI tends to overestimate feed efficiency of animals. The overestimation gets worse as the contribution of feed inefficiency to feed intake variation increases. The results from the experiment also showed SFRF provides consistent feed efficiency estimates and associated SFI. Finally, further benefits of the alternative approach in feed efficiency estimation at animal level are presented.


INTRODUCTION
proposed residual feed intake (RFI), a widely used indicator of feed efficiency at animal level.RFI is the difference between the observed feed intake of an animal and its expected feed intake required for maintenance and production.The expected feed intake is estimated either from a feed intake regression function or from feeding standards (Connor 2015).Computed as deviation, RFI can therefore be positive, negative, or zero; and the smaller it is the higher feed efficiency it is believed to indicate.While simplicity of the approach is appealing, the association of RFI with feed efficiency at animal level has been under scrutiny lately (Connor 2015;Martin et al., 2021[AU1: The in-text citation "Martin et al., 2021" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]).
When RFI is computed as a residual of a feed intake regression function, several problems can arise.First, equating RFI with the residuals amounts to specifying the expected feed intake function as a deterministic function that makes no allowance for statistical noise.This may be difficult to justify in practice for example if there is difficulty to measure individual feed intake accurately or if approximation error arises when one makes the inevitable choice of a parametric form to estimate an unknown expected feed intake function.Furthermore, the residuals can also contain the effects of factors such as omitted variables, like what Martin et al. (2021)[AU2: The in-text citation "Martin et al. ( 2021)" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]refer to as unidentified energy sinks, and animal specific unobserved heterogeneity (e.g., feed sorting, physical activity).All these factors make RFI a less accurate estimate of feed efficiency at animal level.
Meanwhile, omitted variables contained in the residuals can create further problems if they are correlated with the regressors in the expected feed intake function.In this case, parameter estimates of the regressors from ordinary least squares (OLS) will be biased.Since RFI is a function of the estimated parameters, biased estimates compound the uncertainty regarding the relationship between feed efficiency of animals and RFI.
Second, an interest over feed efficiency presumes the presence of feed intake over a minimum amount required for maintenance and production.In other words, animals with low feed efficiency will consume more feed than justified by their production and maintenance requirements or they will exhibit surplus feed intake (SFI).If this presumption is correct, then the error term of a feed intake function is a mixture of the nonnegative SFI (given observed feed intake is compared against the minimum feed requirement) and statistical noise.As it was also noted by Martin et al. (2021)[AU3: The in-text citation "Martin et al. ( 2021)" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.],such a recognition of the error term of a feed intake function as a mixture of SFI and statistical noise implies that the error term should be specified as a composite error term.Such a composite error term can have at least 2 components: one representing SFI and another representing statistical noise.Given repeated measurements per animal, an example of a composite error specification is the mixed model in Aggrey and Rekaya (2013) and Fischer et al. (2018).With such a composite error specification, attention can be given to how feed efficiency interacts with feed intake when distributional assumptions are made for the error components during estimation as it will be discussed later.
The composite error specification has implications to the estimation of expected feed intake functions by OLS.Given that one of the components of the composite error term is nonnegative, the composite error term of a feed intake function will have a positive expected value.In this case, OLS will provide a positively biased estimate of the intercept.This positive bias in the intercept will increase the estimate of expected feed intake for each animal and accordingly reduce the RFI estimate of the animal, portraying it more feed efficient than it is.However, it must be mentioned that biased parameter estimates will not affect feed efficiency ranks based on RFI since the RFI of all animals are identically affected by the bias.Therefore, if the interest is on mere ranking of animals in terms of feed efficiency, RFI should still do the job.However, if the interest is on estimating the feed efficiency level of each animal and surplus feed intake (SFI) caused by feed inefficiency, biased parameter estimates and the resulting inaccuracy of RFI are of concern.
Third, estimating feed efficiency by comparing observed versus expected feed intake creates negative RFI values for some animals, which can be confusing and limits acceptability by producers (Connor, 2015).This problem can be solved in 2 ways.First by associating RFI with feed inefficiency than feed efficiency.Through such an adjustment, there will be a monotonic association between feed inefficiency and RFI or the former will increase when RFI increase and vice versa.In addition, the negative sign on some RFI values can be understood as absence of feed inefficiency in the sense that observed feed intake of these animals is below what was expected.Such interpretation of RFI as a measure of feed inefficiency is adopted hereafter.Second, RFI can be rescaled to nonnegative values.One approach is to shift the expected feed intake function down so that most of the observed feed intake data will be above the estimated regression line.However, the above adjustment doesn't remedy the fact that the RFI is estimated from a deterministic expected feed intake function and, in the presence of statistical noise, the adjusted RFI values are likely to be larger than they should be, leading to underestimation of feed efficiency.
This study proposes an alternative approach to animal level feed efficiency estimation based on rich literature in production economics.The approach generalizes the conventional approach based on expected feed intake function by introducing a composite error term into it.Unlike RFI, the resulting feed efficiency indicator will be nonnegative for all animals and monotonic with feed efficiency.Furthermore, the feed efficiency indicator can easily be translated into SFI.Finally, the approach makes it also possible to investigate determinants of feed efficiency variations in a statistically sound manner.

MATERIALS AND METHODS
The standard approach for estimation of efficiency in production economics starts from constructing analytical representation of a production technology.One of the methods to do so, called the primal approach, constructs a production technology as a technical relationship between inputs and outputs.A good example of this approach is estimation of a production function where output is expressed as a stochastic function of inputs used to produce it.Estimation of feed intake regression function also fall in the same category and such functions are commonly referred to as input requirement functions or inverse production functions (Diewert 1974).When these functions are estimated as a standard regression function, the error term in its entirety is interpreted either as statistical noise, implying fully efficient production units, or as an indicator of efficiency like RFI, making the functions deterministic or void of statistical noise.
However, an alternative in which both efficiency and statistical noise are entertained in the same specification is often desirable in practical applications.One such alternative is estimating a stochastic feed requirement frontier (SFRF).The SFRF represents the minimum, as opposed to the average, feed intake an animal requires for production and maintenance.Given the SFRF, therefore, an animal is feed efficient if its observed feed intake is the same as the prediction from the SFRF.On the other hand, the animal is feed inefficient if its observed feed intake exceeds the prediction, and the extent of feed inefficiency increases as the difference between observed and predicted feed intake increases.As it will be shown later, the distinguishing feature of the SFRF from the conventional expected feed intake function is the presence of a Atsbeha: Feed Efficiency from Feed Requirement Frontier nonnegative, and hence unidirectional, error term to represent feed inefficiency.The unidirectionality entertains the fact that feed inefficiency, when it exists, will only increase feed intake.Therefore, one can see the SFRF as a generalization of the expected feed intake function by decomposing its error term into 2: a part representing feed inefficiency and a part representing statistical noise.The main implication of including the unidirectional error term is to force the SFRF envelop most of the feed intake data from below (see line BB in Figure 1).On the other hand, an expected feed intake function, representing the average feed intake, will cut through the observed feed intake data with most of the data spread around it (see line AA in Figure 1).
After its estimation, the SFRF will predict the minimum feed requirement for different production and maintenance requirement levels.Given these predictions, feed efficiency of each animal can be estimated as will be shown later.
The production economics literature provides several options to construct frontier functions either parametrically (e.g., Atsbeha et al., 2012) or nonparametrically (e.g., see Atsbeha et al., 2022[AU4: The in-text citation " Atsbeha et al., 2022" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]).There are advantages and disadvantages associated with using either of the 2 approaches.For example, the parametric approach assumes production to be stochastic and therefore allows one to distinguish between inefficiency and statistical noise, which is important in the present context.However, it requires a parametric form to be assumed for the frontier.On the other hand, the traditional nonparametric approach does not require a parametric form to be assumed but offers a deterministic frontier and hence do not account for statistical noise.See Coelli et al. (2005)[AU5: The in-text citation "See Coelli et al. (2005)" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]for a more detailed comparison of the 2 approaches.
This study presents a parametric approach of constructing a SFRF.The choice is informed by the need to have both feed inefficiency and statistical noise in the same specification but also by the current practice of using a parametric approach to construct RFI.The remaining part of this section presents the theory behind construction of a SFRF, how it is used to construct a feed efficiency indicator, and empirical approaches for its estimation.

Construction of a stochastic feed requirement frontier
Inspired by Kumbhakar and Hjalmarsson (1995), the SFRF can be presented as follows: where F * is the minimum feed intake required for production and maintenance, Y 1 represents production (e.g., milk output or milk components) and Y 2 other variables that determine feed intake such as maintenance requirements, and α are parameters of the SFRF.v is a symmetrically distributed error term capturing statistical noise caused for example by measurement and approximation errors.The statistical noise term is specified with an exponential function as e v because frontier functions are usually estimated in a double-log form.That is, after a log-transformation of equation ( 1), the statistical noise term will appear in the standard additive format as it will be shown later in equation ( 6).The same reason underlies the later specification of feed inefficiency in a similar way.However, observed feed intake (F) of an animal can exceed the minimum feed intake required for its production and maintenance levels if the animal is feed inefficient.Therefore, the observed feed intake of an animal can be written as: where u ≥ 0 measures the percentage by which the observed feed intake of an animal exceeds the minimum feed intake required for production and maintenance.When u = 0 for a given animal, then e 0 1 = and F F = * , indicating that the animal is feed efficient.But if u > 0, then e u > 1 and F F > * , indicating a feed inefficient animal.

Construction of a feed efficiency score
For simplicity, assume there is no statistical noise in the feed intake data and feed intake is predicted with production only.Then Figure 2[AU6: No figure matches the in-text citation "figure 2".Please supply a legend and figure or delete the citation.]illustrates the construction of a feed efficiency score from a deterministic feed requirement frontier.Observed feed intake (F) is displayed on the vertical axis and observed production (Y) on the horizontal axis.Then the deterministic feed requirement frontier is presented as an increasing function with an increasing slope to indicate that feed requirement will increase with production and at an increasing rate.However, as pointed out above, it is possible for animals to be feed inefficient and consume more than the minimum feed required for a given production level.For example, point 'a' represents a feed inefficient animal producing a production level Y a with a corresponding observed feed intake of F a .However, the feed requirement frontier predicts that the minimum feed intake required to produce the production level Y a is F a * .Given the prediction of the minimum feed intake required for the production level of the animal represented by point 'a', the feed efficiency score of the animal is obtained as: That is, the feed efficiency score is defined as a ratio of the minimum feed intake requirement to the observed feed intake of an animal.Defined as such for all animals, 0 1 < ≤ θ and it indicates increasing feed efficiency as it approaches 1 and vice versa.Compared to RFI that takes differences and confusingly (Connor 2015) equates high feed efficiency with smaller negative RFI values, θ associates increasing feed efficiency with larger values.This makes θ more intuitive for practical applications.For example, if an animal has a feed efficiency score of θ = 0.4, or 40 percent after multiplying by 100, then it implies that the minimum feed intake required to support the animal's production and maintenance requirements is just 40 percent of its observed feed intake.In other words, 60 percent of the observed feed intake is SFI.
Since all feed inefficient animals are represented in a similar manner, i.e., by points above the graph such as point 'a', the SFRF is said to envelop all feed inefficient points from below.With an estimated SFRF, the above illustration can also be generalized to all animals as  , ; .
In other words, we can get the minimum feed requirement (F * ) at any production and maintenance requirement level by setting feed inefficiency to zero (i.e., u = 0) in the estimated SFRF.Then an estimate of θ is obtained for all animals by equation ( 4).

Computation of surplus feed intake
As shown by equation ( 3) and ( 4), the feed efficiency score θ is a relative measure.However, feed efficiency can also be expressed in absolute terms or as SFI measured in DMI (or equivalent) units.This will make comparison with RFI easy after a minor adjustment to RFI values, as will be discussed later.As the name suggests, the SFI is defined as the difference between the observed feed intake F and the minimum feed intake required for production and maintenance or F * .To derive an expression for SFI, note from equation ( 4 Like θ, SFI is always nonnegative, creating an intuitive association with feed intake increasing feed inefficiency.Note however that SFI behaves in a similar manner as RFI in the sense that it represents high feed efficiency when it gets smaller and vice versa.Guan et al. (2009) used the same approach to measure excess capital use by Dutch cash crop farms.

Empirical specification of a stochastic feed requirement frontier
The abovementioned construction of feed efficiency scores and SFI requires estimation of a SFRF.To do so, we need to first assume a functional form for H.In production economics, it is customary to estimate frontier functions in double log forms such as the Cobb-Douglas pr translog functional forms.For example, the SFRF in Cobb-Douglas form is given as , where A, α 1 , and α 2 are parameters.Applied to a production function, the constant term of the Cobb-Douglas form A is interpreted as total factor productivity in economics or as output growth not explained by input growth.Theoretically, a feed intake function should not have a constant term.However, its inclusion is important from an empirical point of view (see Martin et al. 2021[AU7: The in-text citation "Martin et al. 2021" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]).
If we take the logarithm of the Cobb-Douglas functional form on both sides, the SFRF will have the usual additive form given as where 3 illustrates the SFRF.As in figure 1, assume the minimum feed requirement of each animal is predicted with production only.That is, equation ( 6) can be written as f . The graphical representation of this simplified SFRF is given in figure 3 with logarithm of feed intake on the vertical axis and logarithm of production on the horizontal axis.Let the deterministic prediction of the minimum feed requirement from the function is f With this setup, it is possible to define feed inefficiency as u f f = − * , statistical noise as and the sum of the two as u v f f + = − ˆ. Figure 3 shows these variables for two animals.For example, if we take the animal represented by point 'a', the animal is producing at production level y a and the corresponding observed feed intake is f a .When we plug the observed production level into the estimated SFRF, the deterministic prediction of the minimum feed requirement is given by fa and the stochastic prediction of the minimum feed requirement is given by f a * .Using these, it's possible to see in figure 3  When it comes to estimation of equation ( 6), it is possible to use OLS.However, as indicated above, the estimate of the intercept term will be biased upwards since E ε ( ) > 0, given the nonnegativity of one of the error terms in equation ( 6).It can be shown that and with E v ( ) = 0, the posi- tive bias in the intercept will be E(u).On the other hand, OLS will estimate the slope parameters consistently.Meanwhile, the most popular method to estimate frontier functions like the SFRF in production economics is maximum likelihood (ML).Pioneered by Aigner et al. (1977) and Meeusen and van Den Broeck (1977), a key feature of this estimator is the assumption one must make about the distribution of each of the error components and especially for the feed inefficiency component.It is common to assume v follows a symmetric normal distribution with zero expectation.For the inefficiency term u, the production economics literature provides different suggestions that can accommodate its unidirectional nature.For example, Aigner et al. (1977) assumed a halfnormal distribution truncated at zero from above and Meeusen and van Den Broeck (1977) assumed an exponential distribution.Both distributions work in the present context as well to ensure the unidirectionality of u, implying the feed increasing effect of feed inefficiency.
Other distributional assumptions proposed for the purpose are the truncated-normal (Stevenson 1980) and gamma distribution (Greene 1980).However, the halfnormal and truncated-normal distributions are popular in empirical work.Given the assumptions for the univariate distributions of the two error terms, their joint distribution is derived to form the basis for a log-likelihood function.Then, the log-likelihood function is optimized to estimate the unknown parameters of the SFRF.Further details of this estimation procedure can be found in Kumbhakar et al. (2015, p. 55).
Once parameter estimates of the SFRF are obtained, feed inefficiency is estimated as a conditional expectation of u given the joint distribution of u and v or E(u|ε).The exact formula for the conditional expectation is given in Jondrow et al. (1982).An important note here is that E(u|ε) was found to be a monotonic function of ε by Jondrow et al. (1982).This means feed inefficiency increases as total error increases and vice versa.Based on results in Bera and Sharma (1999), the implication of this finding in the present context is that the ranking of animals in terms of estimated feed efficiency from the SFRF will be largely similar to the rankings that will be obtained using classical RFI.Therefore, the primary advantage from adopting the approach proposed in this study emerges from the precision in estimating the feed efficiency of individual animals and associated SFI.

Empirical specification of an expected feed intake function
To compare estimation of feed efficiency using SFRF with the conventional way of estimating feed efficiency using RFI, an expected feed intake function is also estimated.The expected feed intake function estimated here has the same functional form as in equation ( 6).This will ensure identical specification of variables for the two specifications during a data generating process in a simulation exercise.Given the double log specification, the conventional feed intake function is a special case of equation ( 6) when v = 0.However, the conventional specification of the feed intake function does not require u to be unidirectional and treats it like statistical noise during estimation.As explained above, the consequence is positive bias in the intercept term.
Apart from making comparison with SFRF easier, the double-log specification of the expected feed intake function has other advantages.First, log-transformation is known to alleviate the effect of nonlinear relationships among regressors (Darlington and Hayes, 2017, p 371), one of the challenges in the conventional specification (Martin et al., 2021[AU8: The in-text citation "Martin et al., 2021" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]).
The second advantage is in the computation of RFI.Note that under the double-log specification of equation ( 6), the composite error term ε = − f f ˆ can no longer be interpreted as RFI in the usual sense.This is because which implies that ε is not the difference between observed and expected feed intake like RFI but a natural log of their ratio.If we take the antilog of ε as e F F ε = * , we get a ratio indicator that can be referred to as ratio RFI (R-RFI).Just as RFI, R-RFI indicates high feed inefficiency when it gets larger and vice versa.However, one advantage of the R-RFI over RFI is that it is always nonnegative.Second, R-RFI is more discriminating of animals in terms of feed efficiency than RFI is.For example, an animal with observed and expected feed intake values of 2kg and 1 kg per day, respectively, will have RFI = 1 kg per day just like an animal that has observed and expected feed intake values of 4kg and 3kgs, respectively.By the RFI criterion, these two animals are equally feed efficient.The corresponding R-RFI values for these two animals are 2 and 1.33, correctly suggesting that the animal that consumed twice its expected feed intake is less feed efficient than the other animal that has consumed just 33 percent more.One disadvantage with a double log specification like equation ( 6) is that it does not allow variables that can have negative values as observations such as average daily gain before some data transformation is undertaken or change variables such as average daily gain are measured differently Although the expected feed intake function in this study has the same functional form as SFRF, there are 2 important differences.First, in the way the error term is treated.When used in the context of SFRF, the error term in equation ( 6) is a composite error term combining feed inefficiency and statistical noise while in the context of expected feed intake functions the composite nature of the error term is ignored.Instead, the whole error term is treated as feed inefficiency and hence the expected feed intake function is a deterministic function that does not allow for statistical noise.
Second, the expected feed intake function predicts average feed intake while the SFRF predicts minimum feed intake.This makes the comparison of the 2 difficult.However, the expected feed intake function can be easily transformed into a feed requirement frontier, though a deterministic one.This can be achieved by shifting the estimated expected feed intake function downwards until it envelops most of the feed intake data from below, as shown in Figure 1.This can be achieved by using a procedure proposed by Winsten (1957), which involves subtraction of the smallest residual from all residuals or as The above modification transforms equation ( 6) into a deterministic feed requirement frontier since it will bound most feed intake data from below.The process of estimating a deterministic frontier by equation ( 7) is Please correct the citation, add the reference to the list, or delete the citation.]).Given the above transformation of the expected feed intake function, feed efficiency score of each animal θ can be computed by equation ( 4) as well as the SFI by equation ( 5).The MOLS adjustment has several advantages.First, it allows a simple and widely used estimation technique viz OLS to be used for estimating a deterministic feed requirement frontier.Second, since min , ε ( ) < 0 the adjustment has the tendency to reduce the upward bias in the intercept when OLS is used to estimate the expected feed intake function.Third, keeping in mind one is deterministic and the other is stochastic, it makes comparison of the expected feed intake function with SFRF easier since, after transformation of the former, both approaches will measure feed efficiency against a feed requirement frontier.
However, the MOLS adjustment also has disadvantages.First, it assumes the error term in feed intake represents feed inefficiency in its entirety and makes no allowance for statistical noise.Therefore, the estimated feed efficiency scores could be lower than they should be.Second, the adjustment assumes that the feed conversion technology of the average is the same as the top producers, which is a strong assumption (Kumbhakar and Wang 2010).Third, feed efficiency estimation becomes very sensitive for outliers in the data.

Simulation experiment
This section presents a simulation experiment that was conducted to examine the difference in estimating feed efficiency using two approaches.The first one was the conventional approach pioneered by Koch (1963)[AU10: The in-text citation " Koch (1963)" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]involving estimation of an expected feed intake function.The second one was the alternative approach suggested in this study using feed requirement frontiers: MOLS and SFRF.For the simulation experiment, the case of dairy cows was considered, and it was assumed that feed intake of dairy cows is determined by milk yield (Y 1 ) and metabolic weight (Y 2 ).To allow for the time lag between the time feed is consumed and it gets expressed in milk yield and bodyweight changes, the variables were measured on a weekly basis.
It was assumed that the feed inefficiency term in equation (6) u follows a half-normal distribution as σ with the pre-truncation mean and variance are given in the parenthesis.Also, the statistical noise term v was assumed to follow a symmetric normal distri-bution as v N v ~, .0 2 σ ( ) To evaluate the performance of the two approaches under different scenarios regarding the contribution of feed inefficiency to total feed intake variation, the share of feed inefficiency from total feed intake variance γ * ( ) was controlled in the experiment.
When γ * = 0, feed intake variation will be entirely due to statistical noise and hence equation ( 6) reduces to an expected feed intake function where the error term will represent statistical noise only.On the other hand, when γ * = 1, the feed intake variation will be entirely due to feed inefficiency and hence equation ( 6) reduces to a deterministic feed requirement frontier like the one obtained with MOLS adjustment.To facilitate such a setup of the experiment, it was assumed that σ γ u 2 = and 2 .Despite its appearance, γ does not represent the share of feed inefficiency from feed intake variance since σ u 2 is the pre-truncation variance.After the truncation of u at zero, the variance of u is given as Then, and as shown in Coelli (1995), the value γ must take so that γ * will have a certain value is obtained from . For example, if the share of feed inefficiency from total feed intake variance must be 10 percent, then γ = 0.23.The performance of the 2 approaches of estimating feed efficiency was evaluated using 2 performance metrics.The first one was bias or the difference between an estimate of a parameter and the true value of the parameter.The bias is suitable to evaluate the error direction of an estimator if there is à priori belief that it may over or underestimate a parameter.For example, it is argued in this study that OLS tends to overestimate the intercept of an expected feed intake function in the presence of feed inefficiency.In this case, it is expected that OLS will exhibit positive bias in its estimate of the intercept.However, the bias is not a good indicator of estimator accuracy or how close an estimator gets to the true parameter after it is implemented on several samples.This is because an estimator can err in either side of the true parameter value and some of these errors may cancel each other out in the averaging process of the bias.This also makes it difficult to evaluate the consistency of an estimator or its tendency to approach the true parameter as sample size increases.Therefore, mean absolute error (MAE) was used as a second performance metrics.The MAE is computed as the average of the absolute differences between estimates of a parameter and the true value of the parameter.The smaller the MAE of an estimator, Atsbeha: Feed Efficiency from Feed Requirement Frontier the closer its estimate is to the true value of the parameter on average and vice versa.An estimator is also consistent if its MAE declines as sample size increases.
Given the above setup, the simulation experiment was conducted as follows: 1.A random sample of 10,000 dairy cows was generated, assuming the distribution of milk yield (in ECM kgs) and metabolic weight (in kgs  % share of feed inefficiency from total feed intake variance.Here it must be noted that multiplying the observations by σ u changes both the E(u) and its variance σ u 2 by γ and γ, respectively.This implies that that the intercept bias of OLS will also proportionally increase with γ * .Finally, the observations for v were obtained by multiplying the second set of random numbers by σ σ γ ) ., ., . .

( )= (
) As suggested in Kumb-  hakar et al. (2015, p. 56), the stochastic feed intake data is then regressed on its covariates and estimated by OLS at first.The residuals are then checked if they are right skewed as they should be in the presence of feed inefficiency by computing the skewness coefficient and testing its statistical significance at 5%.If the residuals are not cor-rectly signed and/or do not have a statistically significant skewness coefficient, the samples are drawn again.Olson et al. (1980)[AU12: The intext citation "Olson et al. ( 1980)" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]showed that the choice of the parameter vector has no effect on bias indicators.4. Using the simulated data from step 1-3, the feed efficiency indicator given by equation ( 4) and the SFI given by equation ( 5) were computed for each cow i where i = … 1 10 000 , , , . These values represent the true values of the feed efficiency score (θ) and SFI of each cow in the data.Furthermore, the true values of R-RFI were computed for each cow a s 5.Then, equation ( 6) was first estimated as an expected feed intake function by OLS for five different sample sizes: n = 30, n = 100, n = 1,000, n = 5,000, and n = 10,000.Each sample size was established by the first n observations of the total sample of 10,000 cows.This ensures the same sample of cows was used while evaluating the effect of changing γ * .Then bias of OLS in estimating the intercept of an expected feed intake function from each sample was computed as In this step, the expected feed intake function estimated in step 5 was transformed into a deterministic feed requirement frontier by MOLS adjustment described in equation ( 7).Then the bias of MOLS in estimating the intercept term was estimated as in step 5, using ˆmin α ε 0 + ( ) as the new intercept.
Moreover, using the transformed residuals min , the feed efficiency indicator given by equation ( 4) and the SFI given by equation ( 5) were computed for each cow.Then the mean biases of MOLS in estimating θ and SFI in each sample were computed as  6) was estimated as SFRF by ML.
Then the bias of ML in estimating the intercept was computed as in step 5.Moreover, the feed efficiency indicator θ ( ) and SFI were estimated ac- cording to equation ( 4) and ( 5).Then the mean biases and MAEs of ML in the estimation of θ and SFI were computed as in step 7. 9.The simulation experiment was undertaken by repeating Step 1-8 for S = 1,700 times using the statistical software Stata (StataCorp 2023) and a seed number of 2024.
10. Finally, the average of the bias indicators from step 5-8 of the first successful 1,000 replicates are reported in tables 1 and 2 for the different combinations of γ * and n.

RESULTS AND DISCUSSION
In this section the results from the simulation experiment are presented.All the discussion hereafter is based on the mean values of the different indicators from step 5-8 over the 1,000 replicates.

Performance of OLS
Figure 4 summarizes the results from the OLS estimator of the expected feed intake function and the R-RFI computed from it.As it can be seen in the chart on the top-left of figure, the mean bias of OLS in estimating the intercept of the expected feed intake function was always positive.This implies overestimation of the intercept in the presence of feed inefficiency and that remained to be the case for all sample sizes as well as values of γ * .As expected, the mean positive bias increased with γ * and it approached the E(u) at each value of γ * , which were 0.38, 0.68, and 0.78 for γ * ., = 0 1 γ * ., = 0 5 and γ * ., = 0 9 respectively, when sample size increased.The top right chart of figure 4 shows the MAE of OLS with respect to the intercept.It appears from this chart that the estimation accuracy of OLS was positively related to γ * (i.e., OLS had the lowest estimation accuracy when γ * was small and vice versa).This relationship is more pronounced for the two smaller sample sizes.However, this result had more to do with high share of statistical noise in feed intake variance than low γ * since the MAE declined quickly as sample size increased.Eventually, the above result was reversed for the largest sample size where OLS exhibited the highest accuracy when γ * is small.
Using the expected feed intake function estimated by OLS, R-RFI was computed and compared to its true value as well.As can be seen in the bottom-left chart of figure 4, the overestimation of the intercept of the expected feed intake function by OLS had the anticipated effect on R-RFI.In particular, the mean bias of OLS in estimating R-RFI was negative for all values of γ * .This implies the R-RFI was biased downwards and hence underestimated feed inefficiency.Reflecting the higher intercept bias as γ * increased, the bias from OLS in estimating R-RFI also increased with γ * .On the other hand, increasing sample size had an insignificant effect on R-RFI, indicating OLS estimates of the parameters of the expected feed intake function largely remained the same across sample sizes on average.The MAE of OLS in estimating R-RFI is shown in the bottom-right chart of figure 4 and it is almost a mirror reflection of the chart for the mean bias of OLS in estimating R-RFI over the x-axis, indicating that the mean bias was negative for all animals as it should be.Therefore, just as the mean bias, the MAE of OLS largely remained unaffected by sample size changes.
The above results have implications for feed efficiency estimation using R-RFI (or RFI) as obtained from a deterministic expected feed intake function.First, the re-  sults show that RFI will be approach its true value when γ * gets smaller.However, as γ * gets smaller, feed inefficiency becomes less important source of feed intake variations.On the other hand, as γ * increases, R-RFI (RFI) will increasingly underestimate each animal's feed inefficiency.Therefore, R-RFI (RFI) will fail to reflect the true individual feed inefficiency when feed inefficiency matters the most for feed intake variations.Second, the results showed a weak ameliorating effect of increasing sample size on the bias magnitude of OLS in estimating R-RFI irrespective of the size of γ * .Therefore, it can be concluded that estimating feed efficiency based on a different modelling approach has a better chance of improving precision in feed efficiency estimation than a costly increase in sample size.

Performance of MOLS
As explained above, the MOLS adjusts the expected feed intake function, and the adjustment process will reduce the positive bias of OLS in estimating the intercept of the expected feed intake function.However, given the adjustment is not necessarily meant to correct the bias, there is no guarantee it will eliminate it.In fact, it can create a downward bias if the absolute value of the minimum residual is larger than the magnitude of the positive bias.The results of the simulation experiment in table 1 showed that the mean bias of MOLS in estimating the intercept was mostly negative.This implies that the MOLS generally overcompensated for the positive bias of OLS in estimating the same parameter.The negative bias of MOLS also declined with γ * .This is as expected since the MOLS attribute to ascribe all deviations from the deterministic feed requirement frontier to feed inefficiency becomes more fitting as γ * increases.
The MAE of MOLS in estimating the intercept are reported in table 2 and the bias magnitude of MOLS is larger than that of OLS except for the largest γ * in the experiment.Furthermore, the consistency of MOLS improved as γ * increased.For example, for the smallest γ * , increasing sample size decreased the MAE initially although MAE started to increase as sample size increased further.On the other hand, the MAE declined continuously as sample size increased for the largest γ * in the experiment.This result supports the abovementioned conclusion that the MOLS will perform best when γ * is large.
With the MOLS estimated deterministic feed requirement frontier, feed efficiency scores (θ) and SFI were also computed and compared to their true values from step 4. Reflecting the effect of its underestimated intercept, the results in table 1 show that MOLS underestimated the true feed efficiency scores as expected.However, unlike what was found for R-RFI, underestimation of feed efficiency scores decreased as γ * increased.That is, MOLS performed better in estimating the true feed efficiency scores as the value of γ * increased.The MAE of MOLS in estimating θ reported in table 2 also reflect the same result regarding how the estimation accuracy of MOLS behaves with respect to the size of γ * .However, table 2 shows that the MAE of MOLS in estimating θ increased with sample size.This can be explained by looking at how the mean bias of MOLS in estimating the intercept was behaving as sample size increased.This mean bias of MOLS was increasing in absolute terms, which implies the residuals of MOLS were increasing.Then, the feed efficiency scores computed according to equation ( 4) would be decreasing with sample size.Therefore, the difference between the feed efficiency scores estimated by MOLS and their true values will increase with sample size.This implies that MOLS is an inconsistent estimator of feed efficiency even when feed inefficiency is the main contributor of feed intake variation.
The results from computation of SFI based on MOLS are reported in table 1 and 2 as well.Given its underestimation of feed efficiency, the MOLS overestimated the SFI caused by feed inefficiency.The overestimation magnitude was severe for the lowest value of γ * when MOLS reported SFI values that were three to four times larger than when γ * was at its largest value in the experiment.The inconsistency of MOLS is also apparent in its estimation of SFI since the MAE of MOLS in estimating SFI increased with sample size.
Overall, the results for MOLS showed that estimation of a deterministic feed requirement frontier is a sensible choice only when feed inefficiency is believed to be the main source of feed intake variation and the contribution of statistical noise to feed intake variation is very low.In such cases, MOLS provides a better estimate of the SFI caused by feed inefficiency than R-RFI.However, it should be noted that MOLS is an inconsistent estimator of feed efficiency scores, and its biases are unlikely to disappear by increasing sample size.

Performance of ML
Finally, the performance of the ML estimator of SFRF is assessed.Regarding the estimation of the intercept, table 1 shows that mean bias of ML in estimating the intercept is mostly negative, although there is no clear pattern with respect to γ * .A pattern emerges when these biases are seen in absolute terms in table 2. The MAE of ML in estimating the intercept declined as γ * increased.Furthermore, the MAE of ML in estimating the intercept were mostly smaller than the comparable MAE of the other two estimators, clearly suggesting a better performance in terms of estimating the intercept.
When it comes to estimation of feed efficiency scores, it can be seen in table 1 that the mean bias of ML in estimating θ was mostly positive for the two smallest sample sizes, indicating overestimation of feed efficiency, and entirely negative for the largest three sample sizes in the experiment or underestimation of feed efficiency.However, it is also noticeable that the mean biases were close to zero.For the first three sample sizes, the mean bias of ML in estimating θ increased with γ * initially though it declined as γ * increased further.For the large two samples, the mean bias declined consistently with γ * .
The bias magnitude and its relationship with γ * as well as sample size becomes clearer in table 2. First, table 2 shows that the accuracy of ML in estimating θ is generally better in comparison to MOLS.Second, the MAE of ML in estimating θ declines consistently with γ * as well as with sample size.It is also interesting to notice that MAE of ML in estimating θ reached its bottom more quickly for the sample sizes considered in this study.The exception is when γ * was the smallest, in which case increasing the sample reduced the MAE further.This result indicates the ML is not only a consistent estimator of θ but also it doesn't need very large sample sizes to provide reliable estimates of θ unless γ * is small.
Finally, table 1 and 2 also show the performance of ML in estimating SFI.The patterns of the mean bias of ML in estimating SFI reflect the behavior of ML mean bias in estimating θ.That is wherever ML overestimated θ, ML underestimated SFI and vice versa.The behavior of mean bias of ML in estimating SFI with respect to γ * is also similar to what was observed for θ.

DISCUSSION
In animal science, feed efficiency has been commonly estimated using RFI.When RFI is computed from an expected feed intake regression function, it implies the function is deterministic.In other words, no allowance is made for statistical noise.There are 2 major problems arising out of this.First, it is difficult to justify the exclu-Atsbeha: Feed Efficiency from Feed Requirement Frontier sion of statistical noise in practice.For example, one of the difficulties of feed efficiency estimation is measuring individual feed intake.Therefore, it is reasonable to expect measurement error in feed intake data.Furthermore, the mathematical relationship between feed intake and energy sinks is usually unknown, which means one is very likely to commit approximation error while estimating an expected feed intake function.
Second, even if on is willing to accept the exclusion of statistical noise, estimation of the expected feed intake function by OLS disregards the fact that the effect of feed inefficiency is to increase feed intake.Therefore, the error term of the deterministic expected feed intake function is unidirectional rather than symmetric as estimation by OLS implies.The consequence of estimating the expected feed intake function by OLS is then overestimation of the intercept, which will translate into underestimation of RFI.These effects were confirmed in the simulation results reported in this study.
The abovementioned problems were addressed in 2 ways.First, by acknowledging the unidirectionality of the error term in a deterministic expected feed intake function through the MOLS specification.However, the MOLS specification remained deterministic, making no allowance for statistical noise.As the simulation results showed, this deterministic nature of MOLS compromised its estimation accuracy when the share of statistical noise in the feed intake data increased.A natural second step toward improvement of feed efficiency estimation was therefore to employ an approach that makes allowance both for feed inefficiency and statistical noise in the same specification.This was offered by the SFRF, which was estimated by ML.The simulation results for ML showed SFRF offered better performance in estimating feed efficiency as well as the SFI caused by feed inefficiency.According to the results from the simulation, the bias magnitude from SFRF in estimating SFI were on average 85 percent smaller than what was observed under MOLS.
The literature in production economics provides further extensions of the SFRF approach that could prove useful in feed efficiency estimation at animal level.First, to the extent there is a goal to improve feed efficiency, the interest in the topic rarely ends at its estimation.Rather, researchers seek to understand what determines feed efficiency variations (e.g., Xi et al., 2016).The classical methodology of computing RFI is not up to such a task (Martin et al., 2021[AU13: The in-text citation "Martin et al., 2021" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.])and existing attempts on this line rely on a 2-stage kind of analysis.That is, feed efficiency is initially estimated through RFI and then a second stage ensues using estimated RFI values, or categorization of animals based on estimated feed efficiency, to understand the sources of its variation (e.g., Xi et al., 2016).This 2-stage approach is however heavily criticized in the production economics literature (see Battese and Coelli 1995;Wang and Schmidt 2002).The main problem is the exclusion of the determinants of feed efficiency in the first stage can create an omitted variables problem (Kumbhakar and Lovell 2000, p 264).This is especially problematic if the determinants of feed efficiency and the covariates of feed intake regression are correlated, which leads to biased parameter estimates of the expected feed intake regression (Kumbhakar and Lovell, 2000, p 264).Even when no such correlation exists, the feed efficiency estimates from the first stage will have low variance, which will have consequences for the second stage (see Kumbhakar et al., 2015, p 72).The framework for estimating SFRF can be extended to integrate determinants of feed efficiency by allowing the researcher to specify conditional distributions of feed inefficiency.Depending on the distributional assumption selected for feed inefficiency, the determinants of feed inefficiency can be specified either through their effects on mean feed inefficiency or on the variance of feed inefficiency or on both.More details can be found in Kumbhakar et al. (2015).
Second, since feeding experiments are usually conducted with repeated measurements per animal, there are possibilities to isolate the contribution of feed inefficiency not only from statistical noise but also from unobserved individual heterogeneity such as feeding behavior.This will provide an extension of the mixed model framework used in animal science (e.g., Fischer et al., 2018) to isolate animal specific feed intake variations, which includes those from feed efficiency variations.Furthermore, given large data sets that span over a whole production cycle (e.g., lactations), the introduced approach can be extended to distinguish between short-term and long-term feed efficiency as well as the determinants of their trajectories (e.g., see Kumbhakar et al., 2014[AU14: The in-text citation " Kumbhakar et al., 2014" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.]).This can be crucial for considering sustainable efficiency as a breeding goal (Martin et al., 2021[AU15: The in-text citation "Martin et al., 2021" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.])

CONCLUSION
With increasing interest to integrate feed efficiency in breeding goals, improving the precision of feed efficiency estimation will not only assist breeding efforts but also facilitate a better perspective to the expected gains from feed efficiency improvements.Adopting approaches from production economics, this paper suggested alternatives Atsbeha: Feed Efficiency from Feed Requirement Frontier to estimation of expected feed intake functions in the form of deterministic and stochastic feed requirement frontiers (SFRF).Both types of feed requirement frontiers offer a feed efficiency indicator that is always nonnegative and moves monotonically with feed efficiency, as opposed to the RFI that increases as feed efficiency decreases.Moreover, both frontiers make it possible to estimate the SFI caused by feed inefficiency.The SFI makes objective assessment of the cost implications of feed inefficiency and its improvement possible.Meanwhile, the SFRF fine tunes the frontier approach better by allowing of feed inefficiency and statistical noise in the same specification.Therefore, by distilling statistical noise, it creates more confidence in what is being estimated as feed efficiency.Simulation experiment was used to investigate the performance of the different approaches.The results show that RFI tends to overestimate feed efficiency of animals.The SFRF on the other hand provided the least biased and more consistent estimation of feed efficiency, especially when feed inefficiency contributes a larger share of the total feed intake variation.

Figure 1 .
Figure 1.Expected feed intake function versus feed requirement frontier.

Atsbeha:
Figure 2. [AU34: No figure matches the in-text citation "Figure 2".Please supply a legend and figure or delete the citation.]Estimatingfeed efficiency (θ) for animal "a." how the total error term is decomposed into feed inefficiency ( The same analysis can be done at point 'b' except at this point v

Figure 3 .
Figure 3. Stochastic feed requirement frontier, adapted from Coelli et al. (2005)[AU35: The in-text citation "Coelli et al. (2005)" is not in the reference list.Please correct the citation, add the reference to the list, or delete the citation.].

Atsbeha:
Feed Efficiency from Feed Requirement Frontier sometimes referred to as modified OLS or MOLS in production economics (Parameter, 2023[AU9: The in-text citation "Parameter, 2023" is not in the reference list.
estimate of R-RFI was obtained from the expected feed intake function estimated in step 5

Atsbeha:
Feed Efficiency from Feed Requirement Frontier

Figure 4 .
Figure 4. Mean bias and mean absolute error (MAE) of ordinary least squares (OLS) for estimating the intercept of an expected feed intake function and associated ratio-residual feed intake (R-RFI).

Table 1 : Mean bias of feed efficiency score and surplus feed intake (SFI) estimated from feed requirement frontier
Atsbeha: Feed Efficiency from Feed Requirement Frontier