1204 Souza et al.

Ciência Rural, v.44, n.7, jul, 2014.

Positioning and number of nutritional levels in dose-response
trials to estimate the optimal-level and the adjustment of the models

Posição  e  número  de  níveis  nutricionais  em  experimentos 
dose-resposta  na  estimativa  do  nível-ótimo  e  ajuste  dos  modelos

Fernando  Augusto  de  SouzaI*   Euclides  Braga  MalheirosII   Paulo  Roberto  Oliveira  CarneiroI

ISSN 0103-8478
Ciência Rural, Santa Maria, v.44, n.7, p.1204-1209, jul, 2014                                                         

Received 10.15.13      Approved 12.29.13      Returned by the author 05.08.14
CR-2013-0694.R2

 http://dx.doi.org/10.1590/0103-8478cr20130694

ABSTRACT

The aim of this research was to evaluate the infl uence 
of the number and position of nutrient levels used in dose-response 
trials in the estimation of the optimal-level (OL) and the goodness 
of fi t on the models: quadratic polynomial (QP), exponential 
(EXP), linear response plateau (LRP) and quadratic response 
plateau (QRP). It was used data from dose-response trials realized 
in FCAV-Unesp Jaboticabal considering the homogeneity of 
variances and normal distribution. The fi t of the models were 
evaluated considered the following statistics: adjusted coeffi cient 
of determination (R²adj), coeffi cient of variation (CV) and the sum 
of the squares of deviations (SSD).It was verifi ed in QP and EXP 
models that small changes on the placement and distribution of 
the levels caused great changes in the estimation of the OL. The 
LRP model was deeply infl uenced by the absence or presence of 
the level between the response and stabilization phases (change 
in the straight to plateau). The QRP needed more levels on the 
response phase and the last level on stabilization phase to estimate 
correctly the plateau. It was concluded that the OL and the adjust 
of the models are dependent on the positioning and the number of 
the levels and the specifi c characteristics of each model, but levels 
defi ned near to the true requirement and not so spaced are better 
to estimate the OL.

Key words: regression models, chicken nutrition, dose-response trials, 
linear response plateau, quadratic response plateau.

RESUMO

O objetivo deste trabalho foi avaliar a infl uência 
do número e posição de níveis nutricionais utilizados em ensaios 
dose-resposta na estimativa do nível-ótimo (OL) e ajuste dos 
modelos polinomial quadrático (QP), exponencial (EXP), 
linear response plateau (LRP) e quadratic respose plateau 
(QRP). Utilizaram-se dados provenientes de ensaios dose-
resposta realizados na FCAV-Unesp Jaboticabal, atendendo as 
pressuposições de homocedasticidade e normalidade. O ajuste 

dos modelos foi avaliado considerando as seguintes estatísticas: 
coefi ciente de determinação ajustado (R²adj), coefi ciente de 
variação (CV) e soma dos quadrados dos desvios (SSD).Verifi cou-
se que, nos modelos QP e EXP, pequenas mudanças na localização 
e distribuição dos níveis ocasionam grandes alterações na 
estimativa do OL. O modelo LRP foi infl uenciado pela ausência 
ou presença do nível intermediário às fases de resposta e 
estabilização (mudança da reta crescente para platô). O modelo 
QRP precisou de um número maior de níveis na fase de resposta e 
o último nível da fase de estabilização para estimar corretamente 
o platô. Pôde-se concluir que a determinação do OL e o ajuste dos 
modelos dependem da posição e quantidade de níveis, além das 
características específi cas de cada modelo, mas níveis defi nidos 
próximos do verdadeiro requerimento e não muito espaçados são 
melhores para estimar corretamente o OL.

Palavras-chave: modelos de regressão, dose-resposta, modelo 
linear com resposta em platô, modelo quadrático 
com resposta em platô.

INTRODUCTION

Dose-response trials have been widely 
used to determine nutrient optimum-levels (OL) 
for many livestock species of interest based on 
their performance responses through application of 
regression models.

According to DRAPER &SMITH (1966) 
in regression models when there are a high number of 
levels (points) for the exploratory variable, the model 
will better represent the factor studied, but in general, 
few levels have been used in experiments, but never 
less than the number of parameter of the model, 

IPrograma de Pós-graduação da Faculdade de Ciências Agrárias e Veterinárias (FCAV), Universidade Estadual Paulista (UNESP), 14884-900, 
Jaboticabal, SP, Brasil. E-mail: fasouza84@gmail.com. *Autor para correspondência.

IIDepartamento de Ciências Exatas, FCAV, UNESP, Jaboticabal, SP, Brasil.



Positioning and number of nutritional levels in dose-response trials to estimate the optimal-level and the adjustment of the...

Ciência Rural, v.44, n.7, jul, 2014.

1205

because more levels increases the cost of the trials 
and often makes their achievement impracticable.

The animal’s response to the increase 
of a limiting nutrient consists of four distinct 
phases: 1. Initial - the necessary level to attend the 
maintenance; 2. Response - the animal shows growth 
and production; 3.  Stabilization - no response due 
the addition of nutrients; 4. Toxic - the addition of 
the nutrient induces adverse effects (SAKOMURA & 
ROSTAGNO, 2007; REZENDE et al., 2007). Thus, it 
is suggested by several authors that nutritional levels 
used in dose-response trials should be distributed 
between the response and stabilization phases and 
in an interval that allows evaluation of the effect 
of its increase (GOUS, 1986; MORRIS, 1999; 
LAMBERSON & FIRMAN, 2002).

In dose-response trials the equations of 
the models most commonly used are: Quadratic 
Polynomial (y=ax²+bx+c); Linear Response Plateau 
(y=ax+b, for x<OL and y = plateau, for x≥OL); 
Quadratic Response Plateau (y=ax²+bx+c, for 
x<OL and y = plateau, for x≥OL) and Exponential 
(y=a+b[1-e-c(x-d)]). 

In the EXP model the parameters have 
biological meanings, where: a, represents the 
performance of animal at the basal level of the diet; b, 
the difference between the minimum and maximum 
responses to nutrient addition; c, the curve slope and 
d, the level of nutrient in the basal diet (SAKOMURA 
& ROSTAGNO, 2007).

The main diffi culty in the exponential 
model is the establishment of OL because the curve 
has an asymptotic response and never overtakes a 
maximum level. Many researches (D’MELLO & 
LEWIS, 1970; ROBBINS et al., 1979; MORRIS, 
1989; BAKER, 1986; ROSTAGNO et al., 2007) use 
99% or 95% of the asymptotic response to estimate 
the OL. It was used the equation y=a+b[1-e-c(x-d)] 
to represent the exponential model and the OL 
was obtained by deduction, considering 95% of 
the asymptotic response [0,95(a+b)]. The OL was: 

The OL in the quadratic polynomial model 
was obtained by the fi rst derivative of the model equal 
to zero with the exploratory variable x representing 
the OL, according to the formula:                  

The aim of this study was to investigate 
the infl uence of the number and the position of levels 
used to estimate the optimal-level and the adjustment 
in statistical models used in dose-response trials.

MATERIALS   AND   METHODS

The data used in this paper (Figure 1) were 
based on dose-response trials with broilers conducted 
in the poultry sector of the FCAV – Unesp Jaboticabal 
by SIQUEIRA et al. (2011). It was used fi ve 
equidistant levels of lysine to evaluate the broiler’s 
weight gain of the period from 35 to42 days old.

Three replicates with the same variance and 
standard deviation, according to the presuppositions 
of normality of errors and homocedasticity that was 
confi rmed by Cramer-Von Mises criterion and Levene 
test (P≤0.05), respectively.

The dependent variable used in the models 
was the weight gain and the exploratory variable were 
the levels of lysine. There were 17 different situations 
represented according with the acronym: 3S1-2, eg., 
the fi rst number “3” represents the number of levels 
used, the letter “S” as the abbreviation for the word 
situation and the numbers “1-2” indicate the absence 
of these levels in this situation. All the situations were 
indicated on the table 1.

The original situation 5S was used as 
a basis to compare the fi t of the models and it was 
considered the OL with 95% of confi dence interval. 
The situations with 4 and 3 levels were obtained by 
removing one or two levels of the original situation.

The models used with 5 and 4 levels 
were: QP, LRP, EXP and QRP and with 3 levels: 
QP and LRP. The fi t accuracy of the models was 
evaluated considering: the Adjusted Coeffi cient of 
Determination,                                                       in 
which n is the number of observations and p is the 
number of parameters of the model; the Coeffi cient of 
Variation,                                               , in witch MSD 
is the Mean Square of Deviations and the Sum of the 
Squares of Deviations, SSD=SST-SSM, in which 
SST is sum of the squares total and SSM  the sum of 
the squares of the model.

The statistical analyses were performed on 
SAS software (SAS System, version 9.1) using Proc 
GLM and ANOVA to test the presuppositions. In order 
to adjust the models it was used Proc REG to QP and 
Proc NLIN (Gauss-Newton) to LRP, QRP and EXP.

RESULTS   AND   DISCUSSION

The data presented in tables 2 and 3 report 
the OL variation and the statistics used for measuring 

{((-1)⁄c)[ln(0.05(a+b)⁄b) ] }+d.

OL= (-b)⁄2a.

R²adj=1-[((n-1)×(1-R²))⁄((n-p))], 

CV= (√MSD⁄mean)×100



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Ciência Rural, v.44, n.7, jul, 2014.

the adjustment in each case. QP, LRP, EXP and QRP 
model adjusted with 4 levels are presented in table 2 
and the QP and LRP models adjusted with 3 levels are 
presented in table 3.Both situations were compared 
to 5S and the models showed signifi cant F value in 
all cases.

For the QP and EXP models, the 
situations with none or few initial levels in the 
response phase (4S1, 3S1-2, 3S1- 3 and 3S2-3) 
showed underestimated values for OL and the worst 
adjustment. Situations without the fi nal levels in the 
stabilization phase, especially without the last level 
(4S5, 3S1-5, 3S2-5, 3S3-5, 3S4-5), showed opposite 

results, i.e. overestimated values of OL and better 
adjustments than in other situations.

Without the 3rd and 4th levels (4S3 and 
4S4) they observed the best results for these models. 
The OL considered appropriated to compare all 
situations was the level established in the 95% 
range of confi dence interval in the 5S situation. The 
adjustment of EXP was better without the 5th level, 
but this model (y=a+b[1-e-c(x-d)]) does not predict a 
decreasing response (MORRIS, 1999). These results 
corroborate with RODEHUTSCORD and PACK 
(1996) and PACK et al. (2003) regarding to the 
sensitivity of the curvature o f the EXP model for the 
distribution of levels and its direct infl uence in the 
estimation of OL.

The only satisfactory situation to QP with 3 
levels was 3S2-4 because the OL was in the estimated 
range and the fi t was appropriated. In situations 
with only three levels they should be established in 
accordance with the observations of LAMBERSON 
& FIRMAN (2002), i.e. equally above and below of 
the OL with particular attention to the extreme values.

The OL in QP and EXP had a direct 
relation with the curvature defi ned by the quadratic 
coeffi cient “a” in the QP and by the parameter “c” 
in the EXP. The higher values for the parameter 
a=-1375.85 and parameter c=-4.17were observed 
when it was removed the last level in the stabilization 
phase and it caused a super estimation in OL values. 
Lower values were observed without fi rst levels and 
provided models with narrow curvature (a=-2067.67 
and c=-11.24), underestimating the OL.

It was possible to verify in QP and EXP 
models that small changes on the placement and 
distribution of the levels caused great changes in the 
estimation of the OL, even though the adjustment 

Figure 1 - Experimental levels average with standard deviations and repetitions.

Table 1 - All the situations studied and the levels (represented by
L1, L2, L3, L4 e L5).

L1 L2 L3 L4 L5
Situation

0.786 0.872 0.958 1.044 1.130

5S x x x x x
4S1 x x x x
4S2 x x x x
4S3 x x x x
4S4 x x x x
4S5 x x x x
3S1-2 x x x
3S1-3 x x x
3S1-4 x x x
3S1-5 x x x
3S2-3 x x x
3S2-4 x x x
3S2-5 x x x
3S3-4 x x x
3S3-5 x x x
3S4-5 x x x



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Ciência Rural, v.44, n.7, jul, 2014.

1207

seemed to be appropriate with high value for R²adj 
and low for the CV and the SSD, agreeing with 
MORRIS (1989) that described the sensibility of 
the curvature about the variations of the treatments 
intervals, which can lead to an estimated of optimum 
values outside the studied interval.

The LRP model had a direct infl uence of 
the straight line slope that is defi ned by the coeffi cient 
“a” of the linear component. It was observed that the 
highest value of parameter “a” occurred without 3rd 
level and the lowest without 1st level. The 4S3, 3S3-4 
and 3S3-5 presented overestimated OL and highest 
value of a. The opposite situation, 3S2-4 (with the 
1st, 3rd and 5th level) presented the best values of 
statistics R²adj, CV and SSD with an adequate OL. 

Despite the simplicity and objectivity, the LRP model 
demonstrated less variation to estimate the OL and in 
the statistics to measure the fi t, except in situations 
that the 3rd level was not well defi ned. 

It was demonstrated the necessity of at 
least one well defi ned level during the stabilization 
phase associated with intermediate phase (change the 
line to plateau) and the deeply association of LRP 
with the distribution and positioning around optimal-
level and corroborate with ANDERSON & NELSON 
(1975) who described the importance of establishing 
the levels studied near to the optimal-dose in models 
with response plateau.

The QRP presented adequate values of OL 
for situations 4S1, 4S2 and 4S4, whereas in 4S4 the 

y=-1764.88x2+3876.38x-952.14
y=-2067.67x2+4492.97x-1263.61
y= -1812.09x2+3961.15x-988.35
y= -1822.38x2+3986.54x-1003.46
y= -1686.31x2+3716.37x-873.79
y= -1375.85x2+3177.84x-641.70

y=800.2+400[1-e-6.68(x-0.679) ]
y=782.5+400[1-e-11.24(x-0.740) ]
y=788.0+400[1-e-8.33(x-0.691) ]
y=801.7+400[1-e-6.46(x-0.676) ]
y=791.9+400[1-e-6.84(x-0.675) ]
y=865.2+400[1-e-4.17(x-0.682) ]

y=779.1x+397.07
y=664.6x+503.45
y=779.1x+393.79
y=893.6x+303.75
y=779.1x+397.09
y=779.1x+397.20

y=-1331.8x2+3101.8x-609.14
y= -381.3x2+1362.3x+185.03
y= -885.8x2+2324.0x-273.29
y= -1822.4x2+3986.6x-1003.56
y= -1331.8x2+3101.6x-608.91
y= -1375.9x2+3177.9x-641.84

Table 2 - Optimal-level (OL), adjusted coefficient of determination (R²adj), coefficient of variation (CV) and Sum of square deviations
(SSD) for situations with four and five experimental levels.

Models¹ Situations² Equations³ OL4 R²adj CV(%) SSD

5S 1.098±0.007 0.96 1.09 1793.25
4S1 1.086 0.88 1.19 1668.60
4S2 1.093 0.96 1.22 1690.90
4S3 1.094 0.98 0.93 965.30
4S4 1.102 0.96 1.11 1338.30

QP

4S5 1.155 0.97 1.07 1242.10

5S 0.963±0.024 0.95 1.26 2390.10
4S1 0.910 0.85 1.29 1941.70
4S2 0.921 0.95 1.33 2012.90
4S3 0.970 0.96 1.19 1555.70
4S4 0.953 0.95 1.18 1520.30

EXP

4S5 1.124 0.96 1.07 1252.80

5S 0.997±0.020 0.95 1.11 1872.2
4S1 1.009 0.89 1.18 1637.70
4S2 1.002 0.96 1.19 1620.90
4S3 0.974 0.98 0.88 863.90
4S4 0.995 0.96 1.18 1517.90

LRP

4S5 1.000 0.96 1.15 1435.90

5S 1.034±0.026 0.96 1.10 1678.10
4S1 1.014 0.87 1.18 1637.70
4S2 1.018 0.96 1.19 1620.90
4S3 1.163 0.97 0.93 965.40
4S4 1.028 0.96 1.17 1323.90

QRP

4S5 1.077 0.96 1.07 1242.20

(1) QP – Quadratic Polynomial; EXP – Exponential; LRP - Linear Response Plateau and QRP - Quadratic Response Plateau.
(2) 5S – situation with all levels; 4S1 – without 1st level; 4S2 – without 2nd level; 4S3 – without 3rd level; 4S4 – without 4th level; 4S5 –
without 5th level;
(3) For LRP and QRP these equations were valid if only x < OL, otherwise if x = OL the equation was: y = plateau.
(4) The OL for 5S is the optimal level ± confidence interval (95%).



1208 Souza et al.

Ciência Rural, v.44, n.7, jul, 2014.

adjust statistics were better than 4S1 and 4S2. The 
absence of 3rd and 5th levels caused super estimation 
and overestimation in OL, respectively, but these 
situations presented the best adjust.

In 4S3 the curve slope was so pronounced 
that there was no formation of the plateau within the 
range studied and, graphically, the adjustment of the 
model was similar to a parable. The equation of this 
situation showed the highest value of the quadratic 
coeffi cient and the best adjustment.

This model, like a LRP, had an infl uence of 
the level between the response and the stabilization 
phases to determine the OL point where the parable 
joins the plateau. However, like an EXP, it was 
necessary the 5th level, that indicated a decrease 
on the response, to demonstrate a reduction on the 
weight gain and stabilization to form the plateau. 
The situation 4S4 indicated the necessity of a larger 

number of levels to set the curvature on the response 
phase and a well-defi ned level to form the plateau.

CONCLUSION

The number and the position of levels 
infl uence the estimation of optimal-level and the 
goodness of fi t in polynomial quadratic, linear response 
plateau, exponential and quadratic response plateau 
models. The goodness of fi t is directly infl uenced by 
the position of levels and the intrinsic characteristics 
of the models’ curvature. The OL is infl uenced by the 
number of the levels, if the distribution of the levels are 
near from the true requirement and used a small range 
is possible reduce the number of the levels according 
with the parameters of the models used and it is also 
important to estimate the appropriate optimal-level and 
to avoid overestimated or underestimated values.

y=-1764.88x2+3876.3x-952.1
y= -2715.13x2+5856.6x-1978.3
y= -2283.59x2+4917.8x-1468.9
y= -1851.75x2+4053.2x-1043.4
y= -2052.98x2+4416.4x-1196.8
y= -1721.76x2+3781.8x-902.7
y= -1591.77x2+3532.8x-787.2
y= -1419.91x2+3263.1x-682.6
y= -1390.54x2+3204.2x-653.2
y= -1361.17x2+3150.4x-629.1
y= -1331.80x2+3101.7x-609.6

y=779.1x+397.07
y=812.90x+361.40
y=713.20x+461.10
y=664.6x+503.45
y=785.70x+388.60
y=779.10x+393.82
y=893.60x+303.81
y=664.60x+503.45
y=779.10x+393.77
y=893.60x+303.79
y=779.10x+397.10

Table 3 - Optimal-level (OL), adjusted coefficient of determination (R²adj), coefficient of variation (CV) and Sum of square deviations
(SSD) for situations with three and five experimental levels.

Models¹ Situations² Equations³ OL4 R²adj CV(%) SSD

5S 1.098±0.007 0.96 1.09 1793.2
3S1-2 1.078 0.46 1.38 1556.2
3S1-3 1.077 0.93 1.01 799.3
3S1-4 1.094 0.87 1.29 1283.4
3S2-3 1.077 0.98 1.02 782.4
3S2-4 1.096 0.98 0.93 647.6
3S3-4 1.110 0.98 0.84 509.6
3S1-5 1.149 0.88 1.24 1201.5
3S2-5 1.152 0.96 1.26 1184.6
3S3-5 1.157 0.98 0.78 427.6

QP

3S4-5 1.164 0.95 1.14 911.8

5S 0.997±0.020 0.95 1.11 1872.2
3S1-2 1.000 0.53 1.29 1580.5
3S1-3 1.000 0.93 0.95 823.5
3S1-4 1.006 0.87 1.29 1283.4
3S2-3 1.000 0.98 0.96 806.7
3S2-4 0.998 0.98 0.93 647.6
3S3-4 0.972 0.98 0.85 509.7
3S1-5 1.012 0.88 1.25 1201.6
3S2-5 1.004 0.96 1.27 1184.6
3S3-5 0.976 0.98 0.78 427.7

LRP

3S4-5 1.000 0.94 1.17 1105.9

(1) QP – Quadratic Polynomial and LRP - Linear Response Plateau.
(2) 5S – situation with 5 levels; 3S1-2 - without 1st and 2nd levels; 3S1-3 – without 1st and 3rd levels; 3S1-4 – without 1st and 4th levels; 3S1-5 –
without 1st and 5th levels; 3S2-3 – without 2nd and 3rd levels; 3S2-4 – without 2nd and 4th levels; 3S2-5 – without 2nd and 5th levels; 3S3-4 –
without 3rd and 4th levels; 3S3-5 – without 3rd and 5th levels; 3S4-5 – without 4th and 5th levels;
(3) For LRP these equations were valid if only x < OL, otherwise if x = OL the equation was: y = plateau.
(4) The OL for 5S is the optimal level ± confidence interval (95%).



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Ciência Rural, v.44, n.7, jul, 2014.

1209

ACKNOWLEDGEMENTS

We thank to Coordenação de Aperfeiçoamento de 
Nível Superior (CAPES) of the Ministry of Education in Brazil by 
fi nancial aid and the Professor Nilva Kazue Sakomura who kindly 
provide de data for this research.

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