SEARCHING FOR A SUITABLE FUNCTIONAL FORM IN WHEAT

I
aria Bllkilen Merkcz .·/ro,"IrIIICI Ensflliisli Dergl.'1 fJ),l.1995
SEARCHING FOR A SUITABLE FUNCTIONAL FORM IN WHEAT YIELD
RESPONSE TO FERTILIZATION
Ahmcl BAYANER 1
Vcdal UZUNLU I
AhlllCIOZCELiK:
1. Dr. Cenlwl Res. Jl1sl.fiJ,. fielJCr0l's ..·lnkol'Cl, Tllrkel"
2. Assoc. Pro( U/1/1'. o(.·!l1kw{/. 1'<ICIII'.1 of ,Ig. '( ",ke.!'.
()ZET: Dogu Anadolll 'nun dogu gc<;:il bOlgcsindc bugday \'crimi-glibrc ili~kisini dcgcrlcndinnck alllaclyla
fa rklI matcmatikscl maddclcr dcgcrlcndirilmi~tir. Sonu<;:lar hi<;:bir lllodclin digcrinc iisliinlUgli oimadlgllll.
ancak kuadratik fOrllulIl agronomik gorii~ a<;:lsll1dan \'c ckonomik dcgcrlcndirnlc bakllllludan lIygun oldugll
ortaya <;:Ikml~tlr. Bclli girdi-<;:lktI fiyat oranlannda tavsiyc cdilccck miktarlarda fa/fa bir dcgi~lllc ollllaml~tlr.
Bolgc agro-ckolojik karaktcrlcri g07-C ahndlglllda <;:if1<;:ilcrc bim/. fa/Ja giibrc kullanlllll oncrilcbilir. Cif1<;:i
vcrilcrini dcgcrlcndinncdc digcr dcgi~lllelcri dc i<;:inc alan gcnel bir model ara~ttrlcl \C polilika iircticilcr i<;:in
daha falJa bilgi \'crcbilir.
sr~IMARY: J nrious/iIl1Clio/1alfrmlls nrc uscd 10 ('mlunle lI"henl vield re,_jJOllse lo/i'rlifi:er npp/icolioll
ill1'..aslcr Margill a/Cenlral .Inalolia (EI/Cl) !?eslllis illdicn!l' Ihal {f/IV sing/c /l/odc/ is slIperior, hO\l'cl'cr,
quadralic form seems 10 }il Ihe dala beller a/1d has sO/lie agrollomic raliona/ (l/ul econo/l/ic casiness 10
inlcrprel. Given Ihe illplll - Olllplil price ralios, recolI/lI/cnded applicnllOn rales do nol \'{71Y IIl1lCh. Farmcl's
cnn be advised 10 lise s/ighl~V /l/OI'C fi'1'1i1i:cl' Ihall Iheir cUl'rell1 practices lakillg illio acco/flll Ihc agl'o­
ec%gica/ characl£'l'islics of Ihe I'egioll. . 1 morc genera/ model including, ol!Jcr mriah!es 10 el'a/I/ale
/ar//lcrs dala It'll/dill'ield //lore injiml/alionjiJr researchers and IJolin'-lI/a/':ers.
Crop rcsponsc analYsis IS an
importanl arca of rcsearch lor appropriate
fcrtilizcr recommendations \\hich ha\'e
generally been dc\dopcd using production
functions. and input-output rclationships have
been estimated by conducting fcrtilizer
experiments in thc lield (NELSON ET AL.
1985) "Dircctly or indirectly. dccisions
conccrning optimal ratcs of lcrtilization
il1\olvc titting somc typc 0" model to yield
data collectcd \\hcn sc\'cral ratcs of fcrtilizer
arc
applied"
(CERRATO
and
BLACKMARE. 1990), Ob\iOllsly. fcrtilizer
rccommendation should bc dcrived using thc
most appropriatc modcL Agronomist and
agricultural cconomists ha\'c spcnt more than
a ccntury in scarch of such a model (PARIS,
I(92), Data obtaincd from famlCrs havc not
bccn
usually
uscd
for
fcrtilizer
rccommcndations,
Modds
l1t
thc
expcrimental data bctter. HO\\'CVCL goodncss
of fit to lanncrs' dala is not as good, and
rcscarchcrs arc to rccommcnd also optimal
fcrtilizer usc based on lamlCrS data,
Von Licbig enunciated a conjccture
about crop rcsponsc implying a vcl)'
particular family of rcsponsc models (PARIS,
19(2). Von Licbig fomlUlated "thc low of
minimum" around 1850 (GUZEL 1(85).
Sincc thcn scvcral dillcrcnt rcsponsc models
havc bccn algcbraically fonnulatcd and uscd
INTRODUCTION
Although agriculturc has recci\'cd
less attcntion in rcccnl ycars. il still is an
important scctor in Turkish cconomy. ll1crc
is not any dcvelopcd country in the \\orld that
is not developcd in agriculture. Turkcy should
thcrcforc, put morc emphasis on agricultural
sector if it wants to bc a dc\'cloped counlry,
In reccnt ycars. agricultural support
policics havc changcd and arc less bcnefkial
to fanncrs, Transfcrs to agriculturc h3\'e
declined, and thc costs of inputs hm'c
increascd t\VO to thrcc fold. Espccially
fertilizer, \\'hich itself may increasc yield 10
% to 50 %. (ACIL 1980), became a \cry
cxpcnsivc input. 111at has had a ncgati\'c
impact on fanncr incomcs. and thc continuity
of food supply is thm:ltened. O\\ing to
increases in input priccs, f.:lnncrs hm'c to pay
morc attention to optimize input usc. ll1C aim
of this rcsearch is. thcrcfore. to rccommcnd to
fanners optimum fcrtilizer application rate to
achicv maximum cconomic yield \\ithin thc
contc:\t of currcnt practices.
Productivity 111 whcat fanning
dcpcnds on a varicty of factors such as soil.
climatc, seed quality and type, irrigation.
fcrtilizer usc. Thc objccti\'c of this stlldy is to
comparc and cvaluatc somc functional fonns
to idcntify thc economic optimum ratcs of
fertilizcr.
1
Harem",.. l'z/lnl/llmd ()z(:~llk
to idcntify cconomlc optimum ratc of
fcrtilization.
Numcrous
rcsearchcrs
hm'c
compared rcsponsc ['unctions and notcd that
thcse models ollcn disagree "hcn indentifying
thcse rates of fCltiliZc:'ltion. Among those "'ho
hm'c used diffcrcnt response models arc
Abraham and Rao, Anderson and Nelson,
Barreto and Wcstcrman, Nelson ct ai,
Blackmer and Meisinger (CERRATO and
BLACKMER. 1990, JAURAGUI and
SAIN, 1992) Some rcsponse studics arc
summarized hcrc.
ABRAHAM and RAO (1966)
comparcd se\'cral functional fonns and
concludcd, bascd on thc results as well as thc
goodness of (it. tcst ofhypothcsis about model
paramctcrs fm'or thc quadratic polinomial
function.
SANCHEZ ct al (J 981) compared
altcmati\'c cquations dcrived from field
cxpcrimcnts. and suggcsted. that thc rcsponsc
model docs not really makc much diOcrcncc.
TRONSTAD and TAYLOR
(1989)
c\'aluatcd 15 functional fonns. cxamining thc
crror stmcture and could not assert that any
singlc functional fonn was supcrior.
Most \\ idcl\" uscd functional lomls
arc quadratic squarc root. Cobb-Douglas,
transcendcntal. translog, semilog and
cxponantial functional fon1lS (HEADY.
1981. HEADY and DILLON, 19G L
HEADY et al, 196 L JAURAGUI and SAIN.
1992. PARIS. 1992, CARRETO and
BLACKMER. 1990. JOHNSON. JR. ct al
1987. BEAlTIEandTAYLOR 1987).
Thc quadratic runction is prcrcrred
bccause it is casily gcncralized to models with
more thcn on nutJicnt and it allows for casy
interpration or linear. cunilineer, and
intcraction cffccts (JOHNSON. JR. ct al
1987. JAURAGUI and SAIN, 1992).
Quadratic functional fonn has been
used in numcrous studics to examine the yield
response to rertilizcr use (SEF A. 1981.
DIGDIGOGLU. 1982. BABUR., 983.
ALEMDAR. 1988. AVCl et al 1988.
CARRETO and BLACKMER. 1990,
BAnONA ct al. 1991, GRIFFIN and
HESTERMAN. 1991. MUKHOPADHYAY
et aI, 1991, OZEL and BlCER 1992,
ICARDA. 1992. DARA et al. 1992, PARIS,
1992, OZKAYA and OZDEMIR 1992,
AGARWAL et al. 1993, JERALD et ai,
1993, REEVES et al, 1993, CAMPBELL et
al. 1993, AYISI et al, 1993, EKER 1992,
BAYANER and UZUNLY, 1993, AVCI ct
al. 1993, AVCIN et al. 1993).
lnput-output relations havc also been
examined in a number or studies using Cobb­
Douglas production function (ULUG, 1973,
ZORAL. 1973, TONGISI, 1977. ACIL and
REHBER.
1978,
REHBER
1978,
SARIMESELl, 198 I, ARIKAN, 987,
OZCELIK. 1989. VURAL et aI, 1993).
MATERIALS AND METHODS
Virtually all pre\ious studies of this
type ha\'c relied on field cxpcrimcntal data.
These data fit the model well. Howcver,
fanners conditions are not the same as
cxperimcntal conditions \\·hich could be
controlled by thc rcsearchcrs and cxcept for
thc c1imatc cxcluding irrigation, rcsourccs arc
not limited. Rcsults obtaincd in such condition
arc diffcrcnt than that of farmers conditions.
Thcrcrorc it is worth to c\uluate data obtained
from thc ramlcrs. 111C data used was obtained
rrom thc famlers through a fomlal intcnic\V.
Famlcrs "'ere asked the wheat production
tcchniquc in Eastcm Margin of Central
Anatolia (EMCA). A total of 207 famcrs
wcrc intenie\\' of thcsc, 94 Ianners used
phosphorus and nitrogcn. Pre\iOllS studies
havc gcncrally used quadratic functional form
to c\aluatc )1eld rcsponse to fcrtilizer
application. This functional foml in most
cases fits thc data bcst. Howcver )1c1d data
obtaincd from the ranncrs do not exhibit a
sl1100th response to lcrtilizcr. Thcreforc, it is
il11portantto test diffcrcnt functional foms.
An ideal runctional fom is flexible
enough to capturc all thc infomation
convcyed by thc data set. However, the
quality of thc data must also be considered.
As COLWELL (1978) obscn'ed, poor data
should not deter researchers from using "the
best computing procedurcs available," but
poor data cannot support strong arguments
about whether one or another functional from
is best (JAUREGUI and SAIN, 1992).
Thcrc are altcmati\cs to the more or
less arbitra!y' choice of a pol)nomial function
to reprcscnt thc response to fcrtilization,
including thc \'ery flexible Box-Cox
transfoffi1ation. 0\\1ng to the computer
2
Searchmgfor a Sill/able Fill1c/lOnal hOIll In II hear fIeld Re.\ponse (0 rerllhz(({lOn
Transcendcntal
programming limitations, this tlexiblc Box­
Cox transfonnation was not used.
In this study, data were analyzed
using different functional fomls in three steps
( I), \vheat yield response to nitrogen
fertilization, (2) wheat yield response to
phosphorus fertilization, and (3) wheat yield
response to nitrogen and phosphorus
fertilization. Primarv analvsis of the data
yielded consistantlv' lower' RC. In order to
improve RC, ar~ so\\n to wheal, weed
chemical application, sowing data, so\\ing
method, and tractor 0\\ nership were all
included in the models estimated. RC was
improved only when area so\\n to wheat (A)
and weed chemical application (WCA) were
added to the models. Therefore, follo\\ing
functional fonl1S were fOffilUlated to be used
in the analysis:
y
~
Y
Linear . Y a + bN 'cl' ' d NI' ' eA ' WCA
Quadratic: Y a bN t c N' . d!" c 1" • fNP , gA +
WCA
Cubic
y. a t- bN . cN' . dN' • cl' ' ll"
gil-' t- hA
,WCA
Squarcroot: Y = a' bN tcN"~ +dl' +cl'''~ 'l\NP)')'~ + gA I
WCA
lIipcrbol: Y . a b (IN) , c (11)1 • dA ' WCA
j
t
t
Semilng (withnut quadratic tClln):
c Y
c(<i" e,\ + WC't\} N h pC NPll.
Y a' b InN· c Inl' , d InN Inl' . cA
1
In Y a ' hN
.c
cN' . dl' ' cl" . INI' ' gA . WCA
In Y , [na ' b InN' c Inl' ' dN ' cP
j
Ii' ' WCA
Transccndcntal (with interactilln):
y ~- a Nil 1)1-' C(dN ~ ell' f!\.'p ~\ ~ weAl.
In Y ~ Ina t hlnN I e1nl' • dN I cl' , IN!'
f
I
gA , WCA
Translog (without quadratic tmll):
Y ~ a N" 1" Cld •1N InP" e,'" WCAI.
In Y
cWCA
Translog : Y = aN h C'A+ eWcA.
In Y =. Ina + b InN + cA + dWCA (without quadratic tenn)
Y = aN b cl'blN idAteWCAI
c
Ina' h InN
I
c Inl' ' d InN Inl' ' cA • WCA
Translog (with quadratic tcnn):
Y a Nil IY': c(t~ln;-";I ~ cOuP, "1'L~ I"P
I
1-
~:\
--
WCA}
In Y . Ina .blnN j c1nP ~ dOnN)' c{lnl')' 'I1nNlnl' 'gA
• WCA
Cohb-Douglas : Y " a Nh 1" cdA ' wc".
In Y . Ina' b InN' c Inl' ' dA t- WCA
t
c(lnN)' ~dA -'-eWCA (with quadratic
for phosphorus usc;
Linear
Quadratic
Cubic
Squareroot :
HipcrbJI
t
Tran~cndcntal (without interaction):
Y a Nil pC ClllN 1eP ~ we.'\).
dA + cWCA (\\ith quadratic
In Y = a + bN"' cN' + dA + eWCA (with quadratic tcnn)
+
cp; dNI' ' eA . WCA
t
In Y a' bN
Nt. (N 2 t.
i
I
Exponantial (with quadratic tCl1n):
y c(r.4 hI' ~ct'\~.j dP cP~-! 11\1' I ~\-+-WCA).
Exponantial ; y .. c'" + hN + ,A t dWCAI.
In Y = a'+ bN + cAl dWCA (without quardatic tmn)
Y = e(8+bN+fN':+dA+eWCAl.
InY ~ Ina + blnN
tcnn)
WCA
Exponantial (without quadratic term):
y c(a-' \IN ' ell , d!\l' cA ~ we:\1
SClllllog
eY = elf' j 0 .... dWCAI N
Y =. a + b InN + cA i dWCA (\\'ithout quadratic tClln)
Transcendenlal; y. aNhc"N'dAteWcA/
In Y .c Ina"' b InN + cN T dA
+
Semilog (with quadratic tem! J:
eY . c'·· .lC'· WC\! Nh (N')' 1',1 (P')' (NI')f
y. a • blnN • c(lnN)' I din!' 'c( Inl')' I l1nNI' + gA I WCA
b•
+
nph c(c blP ~ tlA -+- eWCAj
For IlIlrogcn and phosphorus usc:
a ' bN I cA • dWCA
Y ~ a + blnN + c (InN)'
tCllll )
O'
In Y - Ina' binI' ' cOnl')' 'dA . cWCA (with quadratic
tClln)
Y =. a' bN + cN' + dA i cWCA
Y ~ a ~ bN + eN' ~ dN' t cA i tWCA
Y = a ~ bN + cNII~., dA. cWCA
Y =a + b(lIN) + cA + dWCA
c Y .:::. e(B+dA-...eWCAl
Y api, cl'I" .I," ,WCAI
In y. Ina' binI) 1 cl' ' dA ' cWCA
Translog
Y ,. api, c'" ',1\\'0.
In Y Ina" binI' t
cA" dWCA (without quadratic tcnn)
For nitrogen usc;
Linear
Quadratic
Cubic
Squarcroot :
Ilipcrbo[
~
Y , Whcat Yield (kgda).
a ~ Constant.
b. c. d. c. ( g. h Estimat<"d cocllicicnts.
N = Nitrogen application ratc (kg/dn).
P •. Phosphorus application rate (kg/daJ.
A ~ Area sown to wheat (da).
Y ~ a + bP T cA -+ dWCA
Y = a t bP • cP' + dA + eWCA
Y - a + bP I cP' j dP' -'- eA -'- lWCA
Y = a -+ bP + cp'J~ + dA i eWCA
Y = a + b (I IP) + cA + dWCA
c
WCA . Weed chcmical application.
WCA 1. if weed chcmical applied.
WCA = O. othcrwise.
Semilog
eY = e,a t 'A" dWCAI ph.
Y ~ a + b InP + cA " dWCA (without quadratic tcnn)
eY ~elatdAt'WCAI ph (P'l'.
Y = a + binI' + c (nl')'" dA I eWCA (wit quadratic tcnn)
0
Agricultural
economists
and
agronomists have generally judged their
empirical models onlv on the basis of
coefficient of deteffilin~tion (R\ Generallv
Exponantial ; Y ~ c,a~ hPtCAtdWCA'.
In Y = a + bP + cA + dWCA (without quadratic term)
y -= e(a;-bP+cP:!~dA-t cWCAI.
In Y= a + bP + cP' .j dA t eWCA (with quadratic tenn)
3
accepted that the higher the lit the better the
model. This criterion was cxplicitly followed
by numerous researchers. But what happens
when all the models exhibit about the same RC
(PAR[S, 1992). Consequently, the coefficient
of detcrnlination is not a relevant statistics for
selecting a model (CERRATO and
BLACKMER.. 1(90).
These discussions
were also supported by JAUREGUI and
SAIN (1992), that models used in fertilizer
response studies display positive and ncgative
features and R" commonly used docs not by
itself prO'ide su ffieient support for selecting
anyone model O\er others. and therefore
other criteria should be used for choosing the
appropriate specifkation.
11lis study im"Olvcs fitting cach
functional fonllS to data obtained from the
fanners and comparing coelllcient of
detennination. Sum of Squarcs Errors (SSE).
Log of the Likehood Functional (LLF). and
correlation coelllcients between actual and
predictcd wheat yields (rn) (KMENTA,
1986. WHITE and BUt 1988. OlCELlK.
1(94).
RESUL TS AND DISCUSSION
"nlC importance of fertilizer use is
\\e11 knO\\TI by the majority of thc fanncrs in
the rcgion, but thcy are unclcar on
appropriate ratcs and mcans of application
(SAYANER ct aL 199~) Fertilizer usc
dltTers from fanncrs to lanners. depending on
their linancial situation and fanning ability.
Thc most commomly used fcrtilizer typcs
wcre Diamllloniulll Phosphatc (DAP) and
Amonium Nitratc (AN) TI1C a\'crage nitrogcn
usc \vas 6.~9 kg/da (== AN * 0.26 + DAP *
O. [8), and phosphonls use 7. D kg/da (==
DAP * 0.46) in EMCA
The estimates of the models gi\'en in
mcthodolo,b'Y for \\hcat yield rcsponse to
nitrogcn (N). phosphorus (P) and nitrogen +
phosphorus (N-P) \\cre obtained by Ordinary
Least Squarc (OLS) technique. The selection
criteria for thcse models are gi\"CI1 in Table l.
2 and 3 for N, P and N-P models respcctively.
Criteria taken into acccount arc RC• F-\alue.
Log of the Likelyhood Function (LLF) and
the correlation coellicient between obser,ed
and predicted yield va[ues (rn).
\
The coefficients of detennination for
models including nitrogen varied bet\\een
O. [98 and 0.230 including phosphorus 0.189
and 0.209 and for the full models 0.119 and
2
0.246 All the R valucs for the models are
statistically significant at the level of 0.01.
The data consistanly exhibited a lower R2
because the data were obtained from the
fanners with a low variation. Most of the
fanners use about the same amount of
fcrtilizer. This typc of data generally exibite
the second stage of the classical production
function.
SSE \alues are higher for
transfonned data for the models including
nitrogen and phosphorus, indicating that
models are not better. and LLF values are
some\\hat highcr howcvcr. the ditTerences are
not important. rH . on the other hand, does not
\'aI} in a grcat detail.
TIle models including nitrogen
phosphonls are statisticallly proven to be
somcwhat bcttcr or superior to those only
including nitrogcn or phosphorus in ternlS of
SSE, LLF. and rYY criteria. Users are
suggested to include both fertilizer types in
their analysis for fanner recommendations.
Results rcported here indicate that no
singlc model can be rccommcnded over others
for all situations, rcgardless of the selection
criteria used, "TIle researchcrs can only hope
that thc bcst model has some agronomic
rational, and produces estimates of economics
optimum that are rcasonably frcc of bias
(PARIS, 19(2)
For the purpose of calculating
economcis application rates of fertilizer for
recommondation in Eastern Margin of
Central Anatolia. among the models
estimated, quadratic model was selccted. The
results of thc estimatcs are gi\'cn in Tablcs 4,
). and 6.
TIlere are three major reasons for
using the quadratic. The first. and perhaps
most prominent reason is to capture turning
points Turning points occur when the elTcct
of an additional unit of X causes a change in
thc direction of the etTcct of X on Y, i.e.,
Marginal Product (MP) of X is zero. The
qiadratic fonn can easily be generalized to
models with more than one nutrients, and it
allO\\'s for casy interpretation of linear,
cunilinear. and interaction effects.
Optimum fertilizer rates for the
models were calculated. given the input­
output pricc ratios (rip) (Table 7) Maxinlllm
Searching/or a Sllita"'e Fimcl/onal hOI/lin 1I71C0I fld" ResliOnse ro FerrilIzaTion
Models of this sort should include
not only fertilizer but also other variables such
as. pre\ious crops. soi I nutrient content,
available moisture. rainfall received and
temperaturc during the gro\\ing period,
irrigation if applied. so\\ing equipment, weed
and pest chemicals etc.
Sush a general model would capture
the changcs in whcat yield that is of intercst to
both rcscarchers and policy makers.
Results obtained arc in famr of
quadratic rOml. becausc it has some
agronomic rational and economic easiness to
intcrpret and secms to l1t the data better.
Recommcndcd application rates did not
change much. gi\Cll the price ratios. As input
prices increase. or output prices decline. input
usc reduces or \icc ,'crsa. Dcpcnding on a
numbcr of factors. ho\\'c\er. operating arca is
bounded \\ith thc sccond stagc of the
production function. Fanncrs in EMeA can
be recommcnded to use more fertilizer than
thcir currcnt application rates \\ith cautious.
becausc the region is dry recci\ing around
300 mm rainfall a year.
yield is optained by only applying 7. TTl kg/d
nitrogcn or 10.041 kg/da phosphorus. Whcn
both fertilizers arc applied. maximum yield is
reached at 7.830 kg/da nitrogen and 9.190
kg/da phoshorus.
Optimum ratcs of nitrogen \aried
from 7.200 to 7.625 kg/da, and phosphorus
from 8. IO 1 to 9.653 kg/da when only using
nitrogcn or phosphorus. For the model
including nitrogcn + phosphorus, optimum
nitrogcn rates wcre bct\V'ccn 7.72 and 7.8 I
kg/da, and phosphorus rates between 9. I I and
9.18 kg/da.
SUMMARY AND CONCLUSIONS
The objectives of this study \\ere to
investigate a suitable functional fonn in '\heat
yield response to fertilizer application and to
rccommend to fanners optimum fertilizer
application rate \\ithin the conte:-.1 of currcnt
practices. Because of the nature of the data
used, strong econometric e\idence is unlikely
that any of the functional forms il)Vestigated
can be rccommended O\'cr others. In addition,
this type of data docs not fit the models well.
comparing
\\ith
experimelltal
data.
Nevcrthcless, fanners' data should also be
used to recommend fertilizer application rate.
5
BayuniJl'. UZlln!1i and (jz<;xhk
Table -t; Wheat yield response to Nitrogen
fertili/.ation
Variables
Intercept
N
NC
A
WCA
Statistics
R:
F
N
Parameters
9.ol567
36.320
- 2.3ol87
0.2 J 573
IS.SolS
Standard Error
89.378
28.lIol
2.1536
O.06ol212··
13.S57
1>.221
6.32
90l
** Significilnt allhe 0.01 Ic\d
Table 5. Wheal yield response to phosphonJs fertili7;ltion
Variables
Intercept
P
pc
A
WCA
Parameters
83.69ol
12.9ol1
- 0.6olHO
0.2 JOO2
17.7H
Stilndard Error
68.olS7
18.997
1.2737
0.06ol973
lol.3ol3
..
Statistics
RC
F
N
0.205
5.7ol
90l
** Significant at the 0.0 I Ic\d
Table 6. Wheal yield response to nilrogen and phosphonJs
fertiliz;llion
Variables
Intercepl
N
N:
P
pc
NP
A
WCA
Paramelers
29.6S3
28.706
ol.1716
0.26332
ol.369ol
-10.229
0.20885
16.553
Slanrulrd Error
93.5ol3
57A18
8.3olol7
38. 196
3.ol1n
9.09H
IHl67olol
93.5ol3
..
Statistics
RC
F
N
0.237
3.818
30l
6
SC(lrchlng/or (I SIII/"nl" !';lnclJon(l!l;;-olllln II 71e(l/ rl,,[L! Re.,vms" to F"I'1I[IZ(l11011
Table 7. Optimum fertilizer application rales (kg/da)
rip
N usc
Technical opt.
7.732
7.625
7.520
0.5
1.0
1.5
2.0
2.5
?A 13
7.306
7.200
N + P use
P usc
N
1O.0-l1
9.653
9.265
8.877
8A89
8.101
9.190
9.180
9.160
9.1-l0
9.120
9110
P
7.830
7.810
7.790
7.760
7.7-l0
7.720
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Seoreilmg/or a SlIIr"hle fimetlOnol J"i'olll In IIh"ot }',eld R"sponsc to rCrfJlizollOn
Table 1. Selection criteria for the functional relationships betwecn
whcat yield and nitrogcn usc
Models
RC
F
LLF
SSE
ryy_ _
Lincar
0.211
8.02
305
:no
-513.422
0.459
Quadratic
0.221
6.32
301 340
-5/1.798
0.471
Cubic
0.230
5.25
298070
-5/1.285
0.479
Squarcroot
0.117
6.18
302 900
-513.040
0.466
Hiperbol
0.114
8.16
304170
-513.151
0.461
Transcendental
0.198
5.51
I. 8 l.\: 10()
-504.358
0.458
Translog!\\;thout
quadratic tcnn
0.198
7.41
1.83:-.:10/'
-504.381
0.457
Translog/with
quadratic tcnn
0.198
5.50
182" Hi'
-504.379
0.457
Semilog!\\'ithoul
quadratic tcnl1
0244
8.17
304 150
-5Il233
0.463
Scmilog/\\ith
quadratic tenll
0.215
6.09
303840
-51l185
0.463
Exponantial/\\il.hOlil.
0.196
quadratic tenn
7.31
l. 94" I0
-504.500
0.453
Exponantia[/\\;th
quadratic tern1
5.62
1.64" lOb
-504.161
0.464
0.202
6
11
BilJ'(mar. [Jzllnl" Clnd ()n;:~1ik
Table 2. Selection criteria for the functional relationships betwccn
wheat yield and phosphorus use
Mood
R2
Linear
ryy_ _
F
SSE
LLF
0.203
7.64
308490
-513.899
0.450
Quadratic
0.205
5.74
307610
-513.764
0.453
Cubic
0.209
4.66
306060
-513.527
0.457
Squareroot
0.204
5.72
301510
-51:".801
0.452
Hiperbol
0.204
7.70
307980
-513.821
0.452
Transcendental
0.192
5.29
2.18xI0/;
-504.735
0.445
Translog/\\ithout
quadratic tenn
0.191
7.09
2.23xI0
6
-504.776
0.445
Translog/\vith
quadratic tenn
O.J92
5.28
2.20x10
6
-504.751
0.445
Semilog/\vithoul
quadratic tenn
0.204
7.69
308010
-513.826
0.452
SemilogMith
quadratic ternl
0.204
5.72
307920
-513.812
0.452
Exponantial/\\ithout
0.189
quadratic tcnn
7.00
2.8h106
-504.897
0.444
Exponantial/\\ith
quadratic tern1
)
-...,.,'.'
2.IOxI0()
-504.662
0.446
0.193
J2
Searehmgfor a SI/i/anle 1'llne/lOnal h'alllln II heo/ Yield JI,'sl'0nse to Fer/J117011OrJ
Table :i. Selection criteria for the functional relationships bet\\een
\vheat yield and nitrogen + phosphonts usc
Model
R~
F
SSE
LLF
fu_
Lincar
0.222
5.mS
~00910
-512.nO
OA72
Quadratic
0.237
3.818
29S240
-51 uns
OAX7
Cubic
0.246
3.040
291920
-311.304
0,4%
Squareroot
0.2:i5
3.767
2961XO
-511.9X6
0,4X4
Hipcrbol
0.21()
6.I~G
~0:n40
-SI~.108
OA(»)
Transcendental!\\ithout
Interaction ten11 0.199
3.594
1.79,,10(>
-504.345
0.457
Transccndcntal!\\ith
Interaction ten11 0.212
:U04
1.2lx 10"
-)O~.557
0,477
4.3()~
1.29" 10"
-50-U44
0,459
:i.2X7
1.24,,10"
-50~.G
II
0.469
0.217
4.S82
~02960
-SI3.048
0.4()6
0.233
3.729
2%8XO
-512.0%
0.483
E"ponantiall\\;thout
quadratic tern)
0.204
4.512
1.52,,10';
-504.025
O.4G~
Exponantial/with
0.209
quadratic tern)
3.254
Uhl({'
-503.70S
0.47()
Translogl\\ithout
quadratic ten11
0.199
Translog/\\ith
0.211
quadratic tern)
Scmilogl\\ithout
quadratic tern)
Semilogl\vith
quadratic ten11
13