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 BAYANER1 Vcdal UZUNLUI
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: Jnrious/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.
INTRODUCTION Crop rcsponsc analYsis IS an
importanl arca of rcsearch lor appropriate Although agriculturc has recci\'cd fcrtilizcr recommendations \\hich ha\'e less attcntion in rcccnl ycars. il still is an generally been dc\dopcd using production important scctor in Turkish cconomy. ll1crc functions. and input-output rclationships have is not any dcvelopcd country in the \\orld that been estimated by conducting fcrtilizer is not developcd in agriculture. Turkcy should experiments in thc lield (NELSON ET AL. thcrcforc, put morc emphasis on agricultural 1985) "Dircctly or indirectly. dccisions sector if it wants to bc a dc\'cloped counlry, conccrning optimal ratcs of lcrtilization
In reccnt ycars. agricultural support il1\olvc titting somc typc
0"
model to yield policics havc changcd and arc less bcnefkial data collectcd \\hcn sc\'cral ratcs of fcrtilizer to fanncrs, Transfcrs to agriculturc h3\'e arc applied" (CERRATO and declined, and thc costs of inputs hm'c BLACKMARE. 1990), Ob\ iOllsly. fcrtilizer increascd t\VO to thrcc fold. Espccially rccommendation should bc dcrived using thc fertilizer, \\'hich itself may increasc yield 10 most appropriatc modcL Agronomist and % to 50 %. (ACIL 1980), became a \cry agricultural cconomists ha\'c spcnt more than cxpcnsivc input. 111at has had a ncgati\'c a ccntury in scarch of such a model (PARIS, impact on fanncr incomcs. and thc continuity I(92), Data obtaincd from famlCrs havc not of food supply is thm:ltened. O\\ing to bccn usually uscd for fcrtilizer increases in input priccs, f.:lnncrs hm'c to pay rccommcndations, Modds l1t thc morc attention to optimize input usc. ll1C aim expcrimental data bctter. HO\\'CVCL goodncss of this rcsearch is. thcrcfore. to rccommcnd to of fit to lanncrs' dala is not as good, and fanners optimum fcrtilizer application rate to rcscarchcrs arc to rccommcnd also optimal achicv maximum cconomic yield \\ithin thc fcrtilizer usc based on lamlCrS data,contc:\t of currcnt practices. Von Licbig enunciated a conjccture Productivity 111 whcat fanning about crop rcsponsc implying a vcl)' dcpcnds on a varicty of factors such as soil. particular family of rcsponsc models (PARIS, climatc, seed quality and type, irrigation. 19(2). Von Licbig fomlUlated "thc low of fcrtilizer usc. Thc objccti\'c of this stlldy is to minimum" around 1850 (GUZEL 1(85). comparc and cvaluatc somc functional fonns Sincc thcn scvcral dillcrcnt rcsponsc models to idcntify thc economic optimum ratcs of havc bccn algcbraically fonnulatcd and uscd
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 (SEFA. 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,
2
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
Searchmgfor a Sill/able Fill1c/lOnal hOIll In II hear fIeld Re.\ponse (0 rerllhz(({lOn
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
RC
improve , ar~ so\\n to wheal, weed
chemical application, sowing data, so\\ing method, and tractor 0\\nership were all
RC included in the models estimated. 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:
For nitrogen usc;
Linear y ~ a ' bN I cA • dWCA Quadratic Y =. a' bN + cN' +dA i cWCA Cubic Y ~ a ~ bN + eN' ~ dN' t cA i tWCA Squarcroot : Y = a ~ bN + cNII~., dA. cWCA Ilipcrbo[ Y=a + b(lIN) + cA + dWCA
SClllllog eY = elf' j 0 .... dWCAI Nb •
Y =. a + b InN + cA dWCA (\\'ithout quadratic tClln)i
c Y .:::. e(B+dA-...eWCAl Nt. (N2t.
Y ~ a + blnN + c (InN)' + dA+ cWCA (\\ith quadratic tCllll )
Exponantial ; y .. c'" + hN + ,A t dWCAI.
In Y = a'+ bN +cAl dWCA (without quardatic tmn) Y = e(8+bN+fN':+dA+eWCAl.
In Y = a + bN"' cN' + dA + eWCA (with quadratic tcnn) Transcendenlal; y. aNhc"N'dAteWcA/
In Y .cIna"' b InN + cN TdA cWCAi
Translog : Y = aNhC'A+ eWcA.
In Y =. Ina + b InN + cA + dWCA (without quadratic tenn)
Y = aNb cl'blN idAteWCAI
InY ~ Ina + blnN + c(lnN)' ~dA -'-eWCA (with quadratic tcnn)
for phosphorus usc;
Linear Y ~ a + bP T cA -+ dWCA Quadratic Y = a t bP • cP' + dA + eWCA Cubic Y - a + bP I cP' j dP' -'- eA -'- lWCA Squareroot : Y = a -+ bP + cp'J~ + dA eWCAi HipcrbJI Y = a + b (I IP) + cA + dWCA 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) 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)
Transcendcntal ~ Y O' api, cl'I" .I," ,WCAI
In y. Ina' binI) cl' ' dA ' cWCA1
Translog Y ,. api, c'" ',1\\'0.
In Y Ina" binI' cA" dWCA (without quadratic tcnn) t
Y nph c(c blP ~ tlA -+-eWCAj
In Y - Ina' binI' ' cOnl')' 'dA . cWCA (with quadratic tClln)
For IlIlrogcn and phosphorus usc:
Linear . Y a + bN 'cl' ' d NI' ' eA ' WCA Quadratic: Y a j bN t c N' . d!" c 1" • fNP , gA + WCA Cubic y. a t-bN . cN' . dN' • cl' ' ll" t gil-' t-hA ,WCA Squarcroot: Y = a' bN tcN"~ +dl' +cl'''~ 'l\NP)')'~ + gA I WCA
lIipcrbol: Y . a t b (IN) , c (11)1 • dA ' WCA
Semilng (withnut quadratic tClln): c Y c(<i" e,\ + WC't\} Nh pC NPll.
Y a' b InN· c Inl' , d InN Inl' . cA + WCA 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
Exponantial (without quadratic term): y c(a-' \IN ' ell , d!\l' 1 cA ~ we:\1 In Y .ca ' hN I cp; dNI' ' eA . WCA Exponantial (with quadratic tCl1n):
y c(r.4 hI' ~ct'\~.j dP t cP~-! 11\1' I ~\-+-WCA).
In Y a' bN t cN' . dl' ' cl" . INI' ' gA . WCA
Tran~cndcntal (without interaction):
Y a Nil pC ClllN 1eP ~ we.'\).
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 f ~\ ~ weAl.
In Y ~ Ina t hlnN I e1nl' • dN I cl' , IN!' I gA , WCA
Translog (without quadratic tmll): Y ~ a N" 1" Cld •1N InP" e,'" WCAI.
In Y 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, I "1'L~ I"P 1-~:\ --WCA}
In Y . Ina .blnN j c1nP ~ dOnN)' c{lnl')' 'I1nNlnl' t 'gA
• WCA
Cohb-Douglas : Y " a Nh 1" cdA
' wc". In Y . Ina' b InN' c Inl' ' dA t-WCA
Y , Whcat Yield (kgda). a ~ Constant.
b. c. d. c. ( g. h c Estimat<"d cocllicicnts.
N = Nitrogen application ratc (kg/dn). P •. Phosphorus application rate (kg/daJ. A ~ Area sown to wheat (da).
WCA . Weed chcmical application.
WCA 0 1. if weed chcmical applied. WCA = O. othcrwise.
Agricultural economists and agronomists have generally judged their empirical models onlv on the basis of coefficient of deteffilin~tion (R\ Generallv
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).
RESULTS 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 EMCAThe 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
0.246 All the R2 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
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.
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.
BayuniJl'. UZlln!1i and (jz<;xhk
Table -t; Wheat yield response to Nitrogen
fertili/.ation
Variables Parameters Standard Error
Intercept 9.ol567 N 36.320 NC - 2.3ol87 A 0.2 J 573 WCA IS.SolS Statistics R: 1>.221 F 6.32 N 90l 89.378 28.lIol 2.1536 O.06ol212·· 13.S57
**
Significilnt allhe 0.01 Ic\dTable 5. Wheal yield response to phosphonJs fertili7;ltion
Variables Parameters Stilndard Error
Intercept P pc A WCA 83.69ol 12.9ol1 - 0.6olHO 0.2 JOO2 17.7H 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\dTable 6. Wheal yield response to nilrogen and phosphonJs fertiliz;llion
Variables Paramelers Slanrulrd Error
Intercepl 29.6S3 93.5ol3 N 28.706 57A18 N: ol.1716 8.3olol7 P 0.26332 38. 196 pc ol.369ol 3.ol1n NP A -10.229 0.20885 9.09H IHl67olol
..
WCA 16.553 93.5ol3 Statistics RC 0.237 F 3.818 N 30l 6SC(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)
N + P use
rip N usc P usc N P
Technical opt. 7.732 1O.0-l1 9.190 7.830
0.5 7.625 9.653 9.180 7.810 1.0 7.520 9.265 9.160 7.790 1.5 ?A 13 8.877 9.1-l0 7.760 2.0 7.306 8A89 9.120 7.7-l0 2.5 7.200 8.101 9110 7.720 REFERENCES
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Table 1. Selection criteria for the functional relationships betwecn whcat yield and nitrogcn usc
Models
R
C F SSE LLF 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. quadratic tenn 0.196 7.31 l.94" I06 -504.500 0.453 Exponantia[/\\;thBilJ'(mar. [Jzllnl" Clnd ()n;:~1ik
Table 2. Selection criteria for the functional relationships betwccn
wheat yield and phosphorus use
R
2Mood F SSE LLF ryy_ _
Linear 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 2.23xI06 -504.776 0.445
Translog/\vith
quadratic tenn O.J92 5.28 2.20x106 -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 quadratic tcnn 0.189 7.00 2.8h106 -504.897 0.444 Exponantial/\\ith
-
.,.,quadratic tern1 0.193 ) ..
'.'
2.IOxI0() -504.662 0.446Searehmgfor a SI/i/anle 1'llne/lOnal h'alllln IIheo/ 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
SSELLF
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
Translogl\\ithout
quadratic ten11 0.199 4.3()~ 1.29" 10" -50-U44 0,459
Translog/\\ith
quadratic tern) 0.211 :i.2X7 1.24,,10" -50~.G II 0.469
Scmilogl\\ithout
quadratic tern) 0.217 4.S82 ~02960 -SI3.048 0.4()6
Semilogl\vith
quadratic ten11 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
