Published December 4, 2014
An economic comparison of several models fitted
to nutritional response data
G. M. Pesti*1 and D. Vedenov†
*Department of Poultry Science, The University of Georgia, Athens 30602-2772;
and †Department of Agricultural Economics, Texas A&M University, 2124 TAMU, College Station 77843-2124
ABSTRACT: Nutritional requirements are typically
estimated based on feeding trials with animals or birds
offered several amounts of the critical nutrient(s). A
nutrient response function is then fitted to data from
the feeding trials. Modern computer techniques allow
for a variety of functional forms to be used as nutrient
response functions. However, the performance of these
models is almost undistinguishable from a purely statistical perspective. This paper approaches the issue of
determining nutrient requirements from an economic
prospective. Crude protein amounts that would maximize profits were calculated for combinations of corn,
soybean meal, and live broilers prices using several nutrient response models fitted to technical data from a
trial with several balanced CP amounts fed to broiler
chickens. Under certain combinations of input prices,
differences between the models were between 1.5 and
3.0% CP. No model consistently predicted the greatest
or least CP amounts or net profits, emphasizing that
the (tangential) slopes of the models change at different
rates over the range of nutrient (CP) amounts studied.
Models providing adequate statistical fits to research
data do not necessarily provide functions that are clearly most appropriate for maximizing producer profits.
Key words: broiler, crude protein, economics, model, profit maximization
©2011 American Society of Animal Science. All rights reserved.
J. Anim. Sci. 2011. 89:3344–3349
doi:10.2527/jas.2010-3459
INTRODUCTION
Nutritionists have estimated nutritional requirements
(or best feeding amounts) by a variety of models and
statistical methods (Vedenov and Pesti, 2008; Pesti et
al., 2009). Requirement estimates are usually been made
from feeding trials with animals offered several amounts
of the critical nutrient(s). The most common techniques
for determining requirements have been some application of a multiple-range test or the broken-line (linear
ascending) model (Pesti et al., 2009). The broken-line
(linear ascending) model assumes there is a linear ascending or descending portion of the response (depending on the nature of the variable being measured) and a
plateau where the slope is zero (the minimum or maximum response). When a requirement is determined using either of these methods, it is implicitly implied that
there is some minimum nutrient level that results in
the minimum or maximum response. Almquist (1953)
realized that the ascending or descending response to
nutrients is usually nonlinear up to the level where the
maximum response is reached. Almquist accepted that
nutritional requirements followed the law of diminish1
Corresponding author: [email protected]
Received August 26, 2010.
Accepted May 9, 2011.
ing returns, but still wanted to determine a value that
could be labeled the requirement, the level that maximizes technical animal performance.
Vedenov and Pesti (2008) compared several nonlinear
models frequently used to estimate nutritional responses from a statistical perspective and found them to fit
to response data equally well. The objective of this research is to determine if there is a difference between
the models from the economic standpoint. Results from
a recent experiment with a modern high yield broiler
strain were compared using a variety of price conditions to determine how each model affects the profitmaximizing level predictions.
MATERIALS AND METHODS
Animal Care and Use Committee approval was not
obtained for this study because the data were obtained
from an existing database (Lemme et al., 2008).
Data were taken from Lemme et al. (2008) for Ross
708 male broilers fed wheat and soybean meal-based
diets for 49 d (Figure 1). The birds were fed separate
starter, grower, and finisher diets with different proportions of balanced CP. For this analysis, CP was the
weighted average of CP consumed in the starter, grower, and finisher periods. Weighting was by the amounts
expected to be consumed during the days the diets were
3344
Economics of nutritional response models
Figure 1. The performance of male Ross 708 broilers fed different
amounts of balanced CP (from Lemme et al., 2008).
offered published in the Ross 708 Management Guide [0
to 21 (18.7%), 21 to 35 (35.8%), and 35 to 49 (45.5%);
Aviagen, Huntsville, AL].
Models were fitted using the NRM 1.1 workbook
of Vedenov and Pesti (2008). Microsoft Excel 2007
(12.0.6550.5004; Microsoft Corp., Redmond, WA) does
not allow for explicit estimation of nonlinear models.
However, it includes a numerical optimization module
(Solver) that can minimize/maximize the value in a given cell by changing values in other user-identified cells.
The numerical optimization methods used by Solver
(generalized reduced gradient method) are similar to
the ones used by nonlinear regression procedures such
as NLIN in SAS (SAS Inst. Inc., Cary, NC). Therefore,
Microsoft Excel with the Solver module provides a convenient platform for development of a nonlinear regression estimation tool for the case where the same set of
known models is typically estimated. Combinations of
corn and soybean meal and BW were chosen to give
profit maximizing CP amounts of approximately 15,
17, 19, 21, and 23% using the saturation kinetics model
(Morgan et al., 1975) for comparison.
The NRM 1.1 workbook (nutritional response model;
NRM.xls) was produced with Microsoft Office Excel
2003 with Solver module (SOLVER.XLA) installed.
The NRM.xls and a tutorial on its use are available free
of charge at (http://www.caes.uga.edu/Publications/
displayHTML.cfm?pk_id = 7919, accessed March 2,
2011).
RESULTS
The technical data (Figure 1) fitted several models
(Tables 1 and 2) with goodness of fit between 96.96 and
99.56% for 49-d BW and 93.00 to 99.52% for feed consumption. The saturation kinetics model was the bestfitting model for both response variables (Table 3),
but it is practically impossible to distinguish between
the saturation kinetics, 3- or 4-parameter logistics, or
Robbins, Norton, and Baker sigmoidal curves from the
3345
statistical perspective. On the other hand, the models
result in different economic outcomes.
All fitted models were used to determine the diets
that maximize the total profit per bird under several
combinations of prices for feed components and BW
(scenarios 1 through 5 in Table 4). The broken-line (linear ascending) model maximized profits at 17.59% CP
regardless of price scenario. Although the net profits
are similar across the other models in each scenario,
the maximum profit is achieved at different amounts of
CP. Note that the broken-line (linear ascending) model
always resulted in the greatest net profit. However, this
reflects a limitation of the model rather than its superiority. Because the model forces a plateau, the predicted
BW is always the same above the CP amounts corresponding to the inflection point. Because the model
perceives the additional benefit from feeding beyond
the inflection point as zero, the optimal solution is always to feed at the inflection point. In reality, this model would most likely underestimate the actual economic
benefit of feeding greater CP amounts.
Of the models with nonlinear ascending portions, the
profit-maximizing CP amounts were within 1% CP under price conditions calling for approximately 15 to approximately 19% CP to maximize profits. However, under conditions calling for greater CP, differences were
approximately 1.5 to approximately 3.0% CP with the
extreme being the Robbins, Norton, and Baker expoTable 1. The response models fitted to the experimental data1
Saturation kinetics model2
intercept × rate constant + maximum × x kinetic order
y =
rate constant + x kinetic order
Broken-line (linear ascending) spline model (BLL)3

maximum,
if x > requirement
y = 
maximum + rate constant ×(requirem
ment − x ), if x ≤ requirement
Broken-line (quadratic ascending) spline model (BLQ)4
if x > requirement
maximum,

y = 
2
maximum + rate constant×(requirem
m
ent
)
,
if
−
x
x ≤ requirement

Three-parameter logistic model4
maximum
y =
1 + (maximum / intercept − 1)× e −scale×x
Four-parameter logistic model5
maximum + [intercept ×(1 + shape) − maximum]× e −scale×x
y =
1 + shape × e−scale×x
4
Compartmental model
−b×x
y = a × e
(1 - e−c×(x −d) )
Sigmoidal model 16
range
y = lower asymptote +
1 + er+s×x
Exponential model 26
c×x
y = intercept + range ×(1 − e
)
1
In all models, y = response variable, x = nutrient level, e = base of
natural log; other variables are parameters.
2
Morgan et al. (1975).
3
Pesti et al. (2009).
4
SAS Inst. Inc., Cary, NC.
5
Gahl et al. (1991).
6
Robbins et al. (1979).
3346
Pesti and Vedenov
Table 2. Regression coefficients for models used to estimate the response to balanced CP at 49 d of age using the
data of Lemme et al. (2008)
Item
Maximum
Rate constant
Intercept
Kinetic order
1
Saturation kinetics model
BW
Feed intake
Broken-line (linear ascending) spline model2
BW
Feed intake
Broken-line (quadratic ascending) spline model3
BW
Feed intake
Three-parameter logistic model3
BW
Feed intake
Four-parameter logistic model4
BW
Feed intake
Compartmental model3
4,080.7675
6,752.8355
Maximum
44,823
39,332,633,835
Rate constant
3,982.6668
6,728.5353
Maximum
Rate constant
4,019.4626
6,728.5281
Maximum
Maximum
0.5957
0.9558
–32,452.2198
–43,817.0051
b
Intercept
Scale
48.3917
10,541.7382
0.4993
0.8829
c
d
0.0283
0.0384
Range
r
19,145.56
21,480.51
–0.3974
–0.3973
1
Morgan et al. (1975).
Pesti et al. (2009).
3
SAS Inst. Inc., Cary, NC.
4
Gahl et al. (1991).
5
Robbins et al. (1979).
2
Table 3. Comparisons of model fits using the response to balanced CP data at 49 d of
age from Lemme et al. (2008)1
Model
Saturation kinetics2
Broken-line (linear ascending) spline3
Broken-line (quadratic ascending) spline4
Three-parameter logistic4
Four-parameter logistic5
Compartmental4
Sigmoidal model 16
Exponential model 26
1
R2 = goodness of fit.
Morgan et al. (1975).
3
Pesti et al. (2009).
4
SAS Inst. Inc., Cary, NC.
5
Gahl et al. (1991).
6
Robbins et al. (1979).
2
–0.5090
–0.8968
c
393,871.2540
396,254.4537
BW
9.5714
7.9343
s
4.7158
10.3551
Range
 
 
Shape
0.0651
0.0572
–15,090.5108
–14,735.1178
 
 
Scale
1.6768
0.0249
a
–389,767.8515
–389,384.9659
BW
Feed intake
20.1549
17.9790
Intercept
4,056.4421
6,745.7937
 
 
Requirement
Intercept
Lower asymptote
BW
Feed intake
Exponential model 25
17.5850
17.0683
–35.6936
–106.7950
4,044.3841
6,743.7141
7.5274
13.3755
Requirement
–362.5119
–473.1280
57,742.0739
57,761.9223
BW
Feed intake
Sigmoidal model 15
–15,918,714
–126,957,810
Feed consumption
Sum of square
residuals
R2, %
Sum of square
residuals
R2, %
3,439
23,573
17,777
6,664
4,952
26,358
5,113
8,654
99.56
96.96
97.71
99.14
99.36
96.60
99.34
98.88
4,119
8,314
8,314
5,306
4,920
72,748
4,998
59,909
99.52
99.03
99.03
99.38
99.43
91.50
99.42
93.00
 
 
3347
Economics of nutritional response models
Table 4. Economic implications of common models used in animal nutrition studies using the response to balanced
CP data at 49 d of age from Lemme et al. (2008)
Scenario
Item
Item
Price of corn, $/t
Price of soybean meal, $/t
Value of BW, $/kg
Cost of 14.8% CP feed, $/t
Cost of 24.4% CP feed, $/t
Model (profit maximizing conditions)
Saturation kinetics1
 
 
 
 
 
 
CP2
Net profit3
CP2
Net profit3
CP2
Net profit3
CP2
Net profit3
CP2
Net profit3
CP2
Net profit3
CP2
Net profit3
CP2
Net profit3
Broken-line (linear ascending) spline4
Broken-line (quadratic ascending) spline5
Three-parameter logistic5
Four-parameter logistic6
Compartmental5
Sigmoidal model 17
Exponential model 27
1
2
3
4
5
197
707
0.75
325
498
197
529
0.75
293
418
197
707
1.98
325
498
197
357
1.98
265
340
338
357
1.98
372
408
14.83
0.40
17.59
0.47
14.83
0.41
14.83
0.40
14.83
0.40
15.35
0.41
14.83
0.40
15.94
0.43
17.00
0.65
17.59
0.77
18.52
0.65
17.38
0.65
17.17
0.65
16.82
0.65
17.19
0.65
16.85
0.68
19.00
5.06
17.59
5.36
19.29
5.18
19.02
5.09
19.05
5.07
19.83
5.08
19.05
5.07
19.17
5.05
21.00
5.82
17.59
5.97
19.77
5.91
20.50
5.84
20.76
5.83
21.29
5.90
20.73
5.83
21.31
5.83
23.00
5.28
17.59
5.31
19.98
5.32
21.81
5.28
22.32
5.28
22.02
5.35
22.26
5.28
23.08
5.29
1
Morgan et al. (1975).
CP amount that maximizes returns over feed costs.
3
Net profit = total revenue less feed cost.
4
Pesti et al. (2009).
5
SAS Inst. Inc., Cary, NC.
6
Gahl et al. (1991).
7
Robbins et al. (1979).
2
nential model (23.08%) vs. the broken line (quadratic
ascending; 19.98%) with low soybean meal and greater
corn and BW prices. No model consistently predicted
the greatest or least CP amounts or net profits, emphasizing that the (tangential) slopes of the models change
at different rates over the range of nutrient (CP) studied here.
DISCUSSION
With the advent of modern computers, a variety of
methods have been developed to fit the nonlinear ascending portion of the nutrient response function to
data from feeding trials. These include the saturation
kinetics model of Morgan et al. (1975), the broken-line
spline with ascending quadratic segment model, 3-parameter logistic model (SAS Inst. Inc.), the 4-parameter logistic model of Gahl et al. (1991), compartmental
model (SAS Inst. Inc.), sigmoidal model 1 of Robbins et
al. (1979), the exponential (model 2) of Robbins et al.
(1979), and the sigmoidal model of Pilbrow and Morris
(1974). A common approach to determining an optimal
feeding level for these models is to set the requirement
at the nutrient level corresponding to an arbitrary fraction of the maximum response (usually 0.95 or 0.99 of
the maximum).
Although the requirement determined in this way
may result in the best physical characteristics of the
bird, it largely ignores the economic aspect of the problem. Depending on the relative price of feed and finished product, feeding up to the maximum response
level may not necessarily result in the maximum economic benefit. Granted, feeding levels that maximize
profits will be unique for each combination of feed and
product prices and thus cannot be expressed as absolutes, but they do have an advantage of being based
on a solid economic foundation rather than arbitrarily
selected.
When methods of determining requirements other
than multiple-range tests or the broken-line linear model are used, some method of choosing the requirement
or most appropriate amount to feed must be selected.
Quite different approaches were taken by Pilbrow and
Morris (1974) and Robbins et al. (1979). Robbins et al.
(1979) arbitrarily chose the nutrient level resulting in
95% of the maximum response to be the requirement.
In a classic study that demonstrated the appropriate
use of nonlinear regression in poultry nutrition, Pilbrow
and Morris (1974) fitted a nonlinear, sigmoidal, model
(Curnow, 1973) to data on the responses of laying hens
to dietary Lys concentration. In an economic evaluation of their technical response model, Pilbrow and
3348
Pesti and Vedenov
Morris (1974) estimated the optimum dose of lysine
based on the costs of Lys and value of eggs. However,
their model assumed that the feed consumption of hens
laying different amounts of eggs was constant, with the
Lys intake being the only independent variable in their
model and its cost being the only input cost considered against the value of egg output. Despite noting
several differences in feed intake, they apparently did
not attempt to model feed intake as a function of either
dietary Lys or egg production. Clearly, ME and feed
intakes do vary with egg output, and therefore, the
Pilbrow and Morris (1974) economic interpretation was
greatly oversimplified.
Another oversimplification of nonlinear modeling and
econometrics methods in AA nutrition modeling can be
found in Pack et al. (2003). They fitted an exponential
model to AA response data and then determined the
AA amounts that minimized feed cost per unit of BW
gain or minimized feed cost per unit of breast meat
yield. However, the minimum cost approach is acceptable from the economic standpoint only when the revenue is held constant. Because different feeding diets
result in different BW and thus revenue from the live
bird, the minimum feed cost per unit of BW gain or
feed cost per unit of breast meat yield is not necessarily
the economically best feeding amount to choose. This is
illustrated in Figure 2, which shows cost, revenue, and
profit per bird for different amounts of CP corresponding to scenario 3 in Table 3. Note that the cost curve
is monotonically increasing (i.e., the minimum cost diet
should theoretically feed 0% of CP unless an arbitrary
minimum requirement is set: 15% in this example). On
the other hand, the net profit (revenue less cost) has a
clear maximum at 19.0% CP, which is what should be
fed to maximize profits. An alternative interpretation
of this phenomenon is that the producers are happy to
continue feeding, whereas their extra (marginal) revenue from adding more CP is greater than the extra
(marginal) expenses of feeding that CP (Figure 3).
In the same volume as Pack et al. (2003), Baker
(2003) proposed choosing AA requirements based on
fitted responses to 1) the broken-line (linear ascending)
model, and 2) a simple second order polynomial model.
He calculated the requirement by averaging amounts
where the 2 models intersect above break point and
below the maximum of the second order polynomial.
Because the polynomial coefficients will be especially
dependent on how many high amounts of the nutrient
are fed, this procedure seems to be particularly arbitrary and inappropriate. Although it is based on the realization that practical feeding levels should be greater
than those determined by the broken-line models, but
less than the maximum of second order polynomials, it
lacks any economic justification. Such models may be
useful for choosing appropriate ratios between AA and
then finding the amount of balanced AA that maximizes profits, but not for finding profit-maximizing
amounts of AA or CP per se.
If one of these response models clearly fitted the experimental data better than the others, then it could be
comfortably concluded that that model gave the best
prediction of economic realities. However, none of the
models is clearly superior from a statistical perspective,
yet result in estimates of profit maximizing CP amount
as different as 3%. It appears that greater care needs to
be taken when choosing a nutrient response model to
Figure 2. Total costs and revenues for producing broilers with a range of dietary CP. In this example, 18.2% CP maximizes net revenues per
bird.
Economics of nutritional response models
3349
Figure 3. Marginal costs and revenues for producing broilers with a range of dietary CP amounts. In this example, 19.0% CP maximizes net
revenues per bird.
be sure that it adequately represents economic realities.
Models providing adequate statistical fits to research
data do not necessarily provide functions that are clearly appropriate for maximizing producer profits.
With data from a very good experiment, with what
is generally recognized as very good replication, there
is no clear mathematical model that is superior to another from a statistical perspective. No model can be
declared the best to use. But it can be concluded from
the data in Table 4 that inefficiency of the magnitude
of $2.24 to $5.59/t of feed could result from using the
wrong model under the different price scenarios chosen. Considering that the broken-line (linear ascending) model is usually the one chosen, we think this
range of values is justified. The broiler industry in the
United States alone fed approximately 41,700,000 t of
broiler feed in 2009 (USDA-National Agricultural Statistics Service, 2010). We can positively conclude that
some large fraction of at least $93,408,000 ($2.24/t ×
41,700,000 t) could, or perhaps should, be spent each
year on research to determine the best model and ensure that profit maximizing amounts of CP and AA are
being fed to broilers. Similar calculations should result
for growing swine and other animals.
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