The Effect Of Row Width On Data And Models Used In The .
! '': )\.United StatesDepartment ofAgricultureNationalAgricultural5 tatisticsServiceStatisticalResearchDivisionNASS Staff ReportNumber YRB·86.Q3July 1986The EffectOf Row WidthOn Data AndModels UsedIn The SoybeanObjective YieldSurveyRobert J. Battaglia
THE EFFECT OF ROW WIDTH ON DATA AND MODELS USED IN THE SOYBEANOBJECTIVE YIELD SURVEY by Robert J. Battaglia,StatisticalResearch Division, National Agricultural Statistics Service, U. S.Department of Agriculture, Washington, D. C. 20250. July 1986.Staff Report No. YRB-86-03.ABSTPACTThe effect of row-width on soybean objective yield forecast modelswas determined to be negligible. Covariance analysis techniqueswere used to determine if row-width affects forecast modelparameters.Yield components from narrow- and wide-row sampleswere examined.Narrow-row samples have more plants, smallerweight per pod, less pods per plant, and more pods per 18 sq. ft.than wide-row samples.Imputation of average values in earlyseason forecasts should reflect the domain differences.KEYWORDSCovariance analysis, row width, forecast models, soybean *********************************This paper was prepared for limited distribution to theresearch community outside the U. S. Department ofAgriculture. The views expressed herein are notnecessarily those of NASS or **************************ACKNO EDGMENTSThe author thanks Michael House, Ben Klugh, Mark Harris and RonFecso for their helpful comments and Bessie Johnson for typing thepaper.Washington, D. C.July 1986i
CONTENTSSU 1MARYiiiINTRODUCTION1ANAL YSES2Frequencies of Row Width types.Row Width effect on forecast models.Methods Results .Components of yield .Methods.Results . Count unit size in narrow row soybeans.Methods.Results . .RECOMMENDA T I ON S . . . . . . . . 2 3.3.4 5 5 6 9.9.910REFERENCESII s15APPENDIX3Model Diagnostics from full and reducedforecast modp.ls. . 17of row types by yearof yieldiiforecast14
SUr RYThe purpose of this research was to investigate the effects ofrow-width on soybean objective yield forecasts to determinewhether alternative models or procedures are necessary. Thefrequencies of wide-row, narrow-row and broadcast units wereexamined over an 8 year period. The data showed that four states- Illinois, Minnesota, Missouri and Ohio - had 10 percent or moresoybean objective yield units classified as narrow-row. The statewith the largest percentage of narrow-row units was Ohio, with 40percent.In Louisiana 26 percent of the soybean units wereclassified as broadcast.Models were constructed using data from 1977-84 with row-widthtype as a treatment. Results showed that separate forecast modelsfor wide-row, narrow-row, and broadcast units are not necessary.Yield components from the two row types were examined. Narrow-rowsamples have more plants, smaller weight per pod, fewer pods perplant and more pods per 18 sq. ft. than wide-row samples. Theseresults indicate that average number of pods per plant and theaverage weight per pod components of yield imputed in early seasonforecasts should be computed separately for narrow-row and widerow samples.Finally, count unit size in narrow-row soybeans was examined.Acomparison of 6-inch and 12-inch count units showed no differencein the number of plants per 18 sq. ft. Row-width does affect thenumber of plants in the 6-inch section. Narrow-row soybeans havefewer plants per 6-inch section because of seedingrates.Recommended changes in the imputation procedures could reduce theneed for changes in the data collection procedures.iii
THE EFFECTS OF ROWWIDTH ON DATA AND HODELS USED IN THEOBJECTIVE YIELD SURVEYSOYBEANBy Robert J. Battaglia1INTRODUCTIONThe National Agricultural Statistics Service (NASS) began researchon the soybean objective yield survey in 1956. Nine of thefifteen states currently in the program were involved in theinitial research. The survey became operational in 1967. At thetime soybean objective yield procedures were developed, soybeanswere customarily planted using a corn plan er; row-widths averagedaround 3 feet in all nine states [11, 12].The survey developersrecognized the potential for an insufficient number of plants inthe 6-inch sections of narrow-row units.Detailed plant countsobtained from the 6-inch sections are used to forecast the numberof pods with beans per plant at harvest.However, acreage ofnarrow-row soybeans was concentrated in Ohio and justification fordevelopment of an alternative procedure was not deemed sufficientat the time. Since then the acreage of narrow-row plantings hasincreased.Agronomic research shows that decreasing row widthwhile increasing the space between plants in a row can increaseyield [14,15]. The purpose of this study was to (1) investigatethe effects of narrow-row soybeans on objective yield forecastsand (2) to determine whether alternative procedures are necessary.2The study had 4 objectives.First, determine which states hadsignificant numbers of narrow-row units. Second, test the effectof row-width on the slope and intercept of forecast models.Third, compare differences in number of plants per unit, number ofpods with beans per plant, and weight of beans per pod betweenwide- and narrow-row soybeans. Finally count unit size in narrowrow soybeans was examined.1The author is a mathematical statistician with the NationalAgricultural Statistics Service, U. S. Department of Agriculture, ashington, D. C . Numbers in brackets refer to literature cited in the referencesat the end of the report.1
ANALYSESFrequenciesof Row Width TypesThe first step in this analysis was to determine which stateshadsignificant numbers of narrow-row units.Narrow-row soybeans weredefined to possess row widths of less than18 inches.Soybeanobjective yield data from 1977-84 were used to construct Table1.This table shows frequenciesand percentages of soybeanunits byrow-width typefor eachOY state.Onlyfour states (Illinois,Minnesota, Missouri, Ohio) show 10 percent or more of thesampleunits as narrow-row units.The state with the largestnumber ofnarrow-row units was Ohio with 40 percent.Table 1: percentage11 of Soybean Objective Yield Unitsby row width category, October survey data, oisIndianaIowaMinnesotaMissouri ississippiMissouri (2)N. CarolinaS. 987382818899792111410705II2131891261495113Straight percentage average over the 8 years of survey data.Missouri Soybeans are divided into northern and southerndistricts.Northern soybeans are usually indeterminatevarieties while southern soybeans are usually determinatevarieties.Wide, width 18 inches, narrow, width 18 inches, broadcast,width 18 inches (by definition) .2
Appendix 1 contains bar charts for these 4 states showingfrequencies of row-width types by year. These charts reveal thatthe number of narrow-row units have increased sharply the past fewyears except in Ohio.The chart indicates that there has alwaysbeen a higher percentage of narrow-row units in Ohio than in theother states.Row Width Effect on Forecast ModelsThe next step was to compare a forecast model using row-width asan additionalindependent variable(full model)with theoperational forecast model(reduced model) where the row-widthtype was ignored. The purpose of this analysis was to determinewhether separate forecast models for narrow-row, wide-row andbroadcast soybeans are necessary.MethodsSoybean objective yield data (1977-84) from all fifteen OY stateswere used for this analysis.Outliers and leverage points wereremoved from the operational (reduced) models using Studentized Tand Cook's D statistics [8]. This procedure removed similar datawhen applied to the full models. Residual plots of the forecastmodels were examined. The residual plots were near normal,witha slight negative skew for all states.These results make thealpha levels for any hypothesis tests approximate but stilluseable.Two models were used to determine if separate forecast models areneeded for each of the row-width types. As an example the modelsused in the analysis for October are described below.Model fitwas compared using sums of squared errors (SSE) from models withseparate and combined row-width types. Relative efficiency(RE)is defined as the ratio of the sum of squared errors from acovariance model based on separate row-type treatments vs a modelwhere row-types were combined.Three row-width categories wereused; narrow-row (less than 18"), wide-row (greater than 18") andbroadcast (no rows, defined as row-width 18").An F statistic was not valid since the full and reduced modelswere built using the same data.Therefore the models were notindependent. RE values less than one indicated there was a lossin model fit associated with combining row-width types. If the REwas close to one the row-width parameters in the full model didnot account for much more of the variability of the dependentvariable (Y's) than the reduced model [10].3
1.J Y . B Z J. E7 1.JUA. (Z . ) E .whereJ1.J1.J. h b eans 1.th un1.t. 0b servat1.on. 1.n J,thFinal number of po d s W1.trow-width type.Overall meanthTreatment effect of jrowtwidth type.Regression coefficient of jrow-width type.October number of pods with beans.Error termFull modelReducedY .model1.JU A.Y . 1.JA and B are constantModel diagnosticsAppendix 3.J (B.)(U A) B (Z .) E .1.J1.Jfor all row-widthfor the Octoberwheretypes.full and reducedmodelsareinResultsThe RE's for the15 states are listedin Table 2.The RE's forall states are near one.This indicates thatseparate forecastmodelsfor narrow-row,wide-rowand broadcastunitsare notnecessary at t e state. level.4
Table2:State2/Relative efficienciesfrom combinedmodels vs. separate row-type modelsSoybean Objective Yield, 1977-84Relative Efficiency3/Aug & S tSept.Mat 3-5Mat 6-9IllinoisIndianaIowaMinnesotaMissouri amaArkansasGeorgiaLouisianaMississippiMissouri (2)N. CarolinaS. 9601/2/3/4/5/row-type1/Oct.Mat 983.989.996.997.981.997.984.956.999.987Wide, width greater than 18 inches, narrow width less than18 inches, Broadcast 18 inches (by definition) .Missouri Soybeans are divided into northern and southerndistricts.Relative efficiency is defined as the ratio of the sum ofsquared errors from a covariance model with separate row-typetreatments vs a model with a combined row-type.RE's 1represent a decrease in model fit by combining units withdifferent row types.Forecast Models are developed by maturity category.RE was calculated using less than 250 observations.Componentsof YieldMethodsThe purposeof thissection is to evaluatecomponents usedtoforecast and estimate soybean yield to see if they differbetweennarrowand wide-rowsamples.Componentsof yieldused in5
objective yieldforecasts are;(1) numberof plantsper 18 sq.ft., (2) numberof pods withbeans per plantand (3) weightofbeans per pod.Components used to estimate yield at harvest are;(1) number of pods with beans per 18 sq. ft. and (2) the weight ofbeans per pod.ResultsTable 3 contains the meanvalues of these components for narrowand wide-row samples.These means were computed using 5 years ofobjective yield data (1980-84).The states were selectedbecauseat least 10 percent of the soybean samples were defined as narrowrow insuring adequate data to compute robust means for narrow-rowunits.Formulas used to compute values in Table 3 can be found inSection 15 of the OY S&E manual [13].Table 3: Mean values of variables used to forecast andestimate yield from narrow- and wide-row samplesSoybean Objective Yield, December data, 1980-84variable2/Plants/18 sqftRow n SE66.650.9MissouriMeanSE2.50.761.043.1OhioMean SE2.10.762.546.31.70.9Pod WtGramsW.332 .007.358 .003.311 .007.334.003.323 .007.340 .004.373 .005.382.004Pods 9.91.90.641.636.11.20.8N141655Npods/18 sqftGrossYld bu.SamplesW883861/ These146520228312states have 10% narrow-row soybean yield samples.for these variables can be found in section 15 ofthe OY S&E manual [13].3/ N narrow-row, W wide-row.4/ Number of pods with beans per plant.2/ Formulas6
The first variable examined in Table 3 is the number of plants per18 sq. ft .These are the plant counts from the 42-inch rowsexpanded to 18 sq. ft. using the sample row-width. Differences inplant numbers between narrow- and wide-row samples were consistentacross all four states.Narrow-row samples averaged about 17additional plants. The second variable in the table is weight ofbeans per pod (labeled pod wt. grams). This value is determinedin the lab from pods harvested in the 3-foot section and is usedin both forecast and estimation procedures. Weight of beans perpod was lower in the narrow-row samples for all states althoughthe difference in Ohio was negligible. Currently, a state averageweight per pod is used in the operational forecast procedure.This procedure can cause an upward bias in narrow-row yieldforecasts except in Ohio where the weight per pod between narrowand wide-row samples is not different.Counts of number of pods with beans per plant are made on plantsin the 6-inch sections. The table shows wide-row soybeans averagemore pods per plant than the narrow-row beans.If there are noplants in a 6-inch section a state average number of pods withbeans per plant is substituted into the forecast equation fof thatunit.For narrow-row units this substitution also causes anupward bias in forecasted yield since the average number of podswith beans for narrow-row units will be less than the stateaverage.Appendix 2 and Table 4 show examples of how yieldforecasts are affected when state average pod weight and number ofpods per plant are used in forecast models.The number of pods with beans per 18 sq. ft. is computed using labcounts of pods harvested from the 3-foot sections at maturity andexpanded to 18 sq. ft. using the sample row-widths. The data showthat narrow-row samples average more pods per 18 sq. ft. in allstates and the difference is due to plant numbers rather than podsper plant.The last two items presented in Table 3 are theaverage gross yield per acre and the total number of samples withpositive lab data from the 5 year period.Yield is computed asthe product of; the number of pods per 18 sq. ft., weight per podand a conversion factor that converts grams per 18 sq. ft. tobushels per acre. Average yields for the narrow-row samples werehigher with differences ranging from 0.9 bushels per acre inMinnesota to 5.5 bushels in Ohio.In summary, samples located in narrow-row fields will have moreplants, smaller weight per pod, less pods per plant, and more podsper 18 sq. ft.Gross yield for narrow-row samples was higherthan average yields for wide-row samples.Differences in yieldranged from 0.9 bushels in Minnesota to 5.5 bushels in Ohio.7
Table4: The effect of using "state averages"yield forecasts for narrow-row soybeanobjective yield units, Illinois dataEstimator1/Narrow-row2 operationaMaturity O4//Yield forecast componentsPlants/pods/Bean wt.18 sqft.plantper 447.252.41/ Estimators are defined and calculationsshown inAppendix 2.2/ Pods per plant and bean weight per pod (5 yr. avg.Jwere calculated using narrow-row data only.1/ Bean weight per pod is a 5 year state averagecomputed over all units.4/ No plants were present in the 6-inch count unit sostate average pods per plant is used in addition tostate average weight per pod.Table 4 shows the resultsof how narrow-row yield forecastsareaffectedby substitutingstateaveragepod weight and stateaverage number of pods per plantinto the equations.Illinoisdata from Table 3 were usedin thisexample.Computations offinal podsper plant,state averagepods perplant, and stateaverage weight per pod areshown in Appendix 2.The narrow-rowestimator wasconsidered to be "true"since thecomponents erationalestimatorusesweightperpodaveraged over allsamples.Thisresulted in a 2.8 bushel increasein yield whencompared to the narrow-row estimate.If there were no plantsinthe6-inchcountsectiona unitis classifiedas maturitycategory zero.In this case,state average numberof pods ubstitutionresultedin a5.2 hushelincreaseovertheoperational estimate and an 8.0 bushel increase overthe "true"estimate.The state average substitutions also cause a downwardbias in wide-row forecasts.The magnitude of the biases willbeaffected by the number of wide-row and narrow-row units andthedifferencesin numberof podsper plantand weight per podbetween the two row-width types (see Table 3).8
Count Unit Size in Narrow-row SoybeansMethodsIn the final part of the analysis the effect on yield componentsof expanding the 6-inch count unit to a 12 inch count unit wasexamined.The data used were from a study conducted in Ohioduring 1983 [2].In each sample field a research soybean unitwas laid out in adnition to the two operational units. Theresearch unit was identical to the operational unit with theexception of a 12-inch count unit. In this research unit plantand plant component counts were obtained from both a 6- and 12inch section.A yield per acre is forecast using three yieldcomponents. The first two components, number of plants per 18square feet and number of pods with beans per plant, are used inmonthly regression models to forecast final yield.The thirdcomponent, weight of beans per pod, is derived using a five yearhistoric average weight [12]. If there were no plants in the 6inch section, state averages were used in the number of pods withbeans regression models to forecast yield.ResultsComparisons of yield forecast components were made to determinewhether a 12-inch count unit would be more effective in narrowrow soybeans. The first yield component tested was the number ofplants per 18 square feet. Plant numbers were calculated usingcounts from the 3-foot plus 6-inch sections and compared to thosecalculated from the 3-foot plus 12-inch section. A univariatepaired T test showed no significant differences in plant numbers(T 1.14, Pr lT: .26, n 45).This was expected since most of theplants used to estimate plants per 18 square feet are counted inthe 3-foot sections.The number of plants in the 6-inch count unit was examined next.The 1983 study conducted in Ohio reported that extension serviceseeding rate recommendations were 2.4 seeds per foot of row for7" rows and 6.1 seeds per foot in 20" rows [5]. This informationis provided to show the differences in within row plant spacingbetween the two row types.Table 5 shows the probabilities ofplants being included in one row of a 6-inch count unit in Ohio.Plants in the 6-inch count unit were numbered based on theirposition relativethe 3-foot section. For wide-row units theprobability of a 4plant being included in a 6-inch section was.62. For narrow-row units 1 plant had a .68 probability.Thisindicates that 32 percent of the narrow-row units contained noplants.tR9
Table 5:sectionProbabilityof plantsincluded in the first6-inchOhio data, September 1983, soybean objective yieldNo. plants in 1/6-inch 8.90.77.62.68.39.21.132341/PlantPlant1 is the plant closest to the 3-foot section with2 being the second plant from the 3-foot section.The component of yield most affected by narrow-row widths was thenumber of pods with beans per plant.This component is presentlyforecast usingplant componentcounts fromthe 6-inch section.In narrow-rowsoybeans, plantsare spacedfarther apart withinthe rowresulting in many unitswith no plants in the 6-inchsections.Number of plants per 18 square feet component was lessaffected by the plant spacing in narrow-row units since the plantcounts are estimated by adding plants from the 3-foot sectiontothe plants in the 6-inch count unit.A consequence of increasing count unit size would be a potentialincrease in counting errorsand other nonsampling errorsdue toenumerator fatigue, difficult working conditions etc.RECOMMENDATIONSAn analysisof thefrequenciesof row-width types, by state,showed that four States (Ohio, Illinois, Missouri, and Minnesota)had 10 percent or greater narrow-rowsoybean units.Of thosefourstatesOhiohadthelargestpercentage, by far, at 40percent.Louisianahad the mostbroadcast units (26 percent).Analysis on number of pods with beans forecast models usingrowtype as a treatment showed that separate models for wide,narrowand broadcastunits werenot necessary.Ananalysis of yieldcomponentsshowedthatnarrow-rowsampleshavemore plants,smaller bean weight per pod, fewer pods per plant, more podsper18 sq.ft. and higher yields.Theoverall analysis indicatesthat narrow-row soybeans causean estimation problem becauseof10
few or no plants in the 6-inch section.This problemsummarizationproceduresresultingin imputationsyield components.are basedon theseaffects thefor missingThe followingrecommendations1.Separatesoybeansforecast models for wide,are not necessary.2.State average weight per pod imputed for the forecastequations should be computed separately for narrow-rowand wide-row units.3.State average number of pods per plant used in forecastequations when no plants are in the 6-inch section should becomputed separately for narrow-row and wide-row units.narrowfindings:and broadcastMethods Staff should run parallel forecasts using theor ginal operational procedure to measure the effect ofrecommendations2 & 3 on the forecast procedures.4.If states do not have enough narrow-row units to provideadequate averages it may be necessary to increase the lengthof the count unit for narrow-row units.It is preferablehowever, not to alter the current data collection proceduressince previous research indicates counting errors areassociated with larger plant numbers in the count units.11
REFERENCES1.Battaglia, Robert J. Covariance Analysis of SoybeanObjective Yield Maturity Categories 7,8 & 9. StatisticalReporting Service, u.S.Department of Agriculture,1985.2.Battaglia, Robert J. 1983 Soybean Objective YieldNonsampling Error Research Study. Statistical ReportingService, U. S. Department of Agriculture1985.3.Battaglia, Robert J., and Benjamin F. Klugh. Assessment ofFixedVariable vs. Stepwise Forecast Models to Predict theNumber of Soybean Pods with Beans per Plant. StatisticalReporting Service, U. S. Department of Agriculture,1985.4.Battaglia, Robert J., and Benjamin F. Klugh. Fixed vs.Stepwise Forecast Models to Predict Number of Pods withBeans per Soybean Plant in Southern States. StatisticalReporting Service, U. S. Department of Agriculture.5.Beuerlein and Walker. Ohio Soybean Performance TrialsCooperative Extension Service, Ohio StatetUniversity,Agronomy Department Series 212, AGDEX 141,1983.6.Freund, Rudolf and Ramon Littell. SAS for LinearInstitute, Inc., North Carolina 1985.7.Huitema, Bradley. The Analysis of Covariance andAlternatives.J. Wiley & Sons, New York 1980.8.Klugh, Benjamin F. Regression Analysis Documentation.unpublished program documentation,SRS internal memo,9.10.1982.Models.SAS1982.Kramer, Clyde Y. A First Course in Methods of MultivariateAnalysis.Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia1972.Neter, John and William Wasserman. Applied LinearStatistical Models.Richard D. Irwin, Inc., Illinois1974.11.U. S. Department of Agriculture, Statistical ReportingService, Report on 1961 Objective Surveys for ForecastingSoybean Yield. April 1962.12.U. S. Department of Agriculture, Statistical ReportingService, Preliminary Report on 1962 Objective Surveys forForecasting Soybean Yield. May 1963.12
13.U. S. Department of Agriculture, Statistical ReportingService Objective Yield Supervising and Editing Manual.1984.14.Weber, Shibles, and Byth. Effect of Plant Population and RowSpacing on Soybean Development and Production. AgronomyJournal, 58:99-102, 1966.15.Wiggins, R.G. The Influence of Spacing and Arrangement onthe Production of Soybean Plants. Agronomy Journal, 31:314321, 1939.13
APPENDIXPERCENTAGE1OF ROW TYPESBY YEARPERCENToIILLINOIS,DII1.D;cIII IIIII 10I)'"IIIIIII 197919801981198219831984 BroadcastIIWide-row0Io14II I I I I I I I I I I I I/0zj) 30J.IO ()be10II IIIIg :1- - ') D
Appendix2: Forecastsusing yield componentsunitsfrom narrow-rowINTRODUCTIONThis appendix shows howsoybean yield forecasts canchange s.Informationon the proceduresandformulascan be found insection 15 of the S&E[13]. These substitutions arethe resultofoperationalprocedures.Examplesof yieldforecastcalculations are shown using Illinoisdata from Table 3.Stateaverage weight per pod is currentlyused in the yield forecastformula for all samples.It is a 5-year average over allunits.State average podsper plant is computed over all units.Thisvalue is used as a substitute to forecast final podswhen thereare no plants in the 6-inchsection.Again, Table 3 showsthatthe valuesfor wideand narrow-rowunits are different.Theexamples belowshow how yield forecastsare affectedby theseoperational substitutions.Yield (#PLANTS/18sqft)assume fixedExampleusing IllinoisState Avg. /POD(#PODS w BEANS/PLT)forecastdatafrom Table(WT/POD) (.088918)5 yr avg.3: «141*.332) (655*.358))/796State Avg. PODS/PLANT «141*24.9) (655*28.1))/796Illinois Oct. maturity 8 forecastfor an average narrow-row unit:model .353 27.5for PODS w BEANS/PLANTFinal pods -.653 .972(24.9) 23.5 PODS with BEANS/PLANTsubstitute state avg. pods (27.5) 26.1""""Example 1. Use NARROW-ROW *PODS and NARROW-ROW WT/PODYLD (64)*(23.5)*(.332)*(.088918) 44.4 Bu/AExample2. Use NARROW-ROW #PODS and STATE AVG. WT/POD(OPERATIONAL)YLD (64)*(23.5)*(.353)*(.088918) 47.2 Bu/AExample3. Maturity O(no plants in 6" section) use STATE AVG.#POOS and STATE AVG. /POD (OPERATIONAL)15
YLD (64)*(26.1)*(.353)*(.088918) 52.4 Bu/AExample 4. Maturity O use STATE AVG. #PODS and NARROW-ROWYLD (64)*(26.1)*(.332)*(.088918) 49.3 Bu/AWT/PODConclusionThe use of state average weight per pod and state averageof pods per plant in the operational procedures causes anbias in narrow-row soybean forecasts.16numberupward
Appendix3Full Model DiagnosticsStateOverall ModelStatistics 2OFMSEPr FSignificanceROct·3PodsPr FProbability ofParametersRow lype :InteracsionPr F:Pr n square error is an estimate of the variance of the true errors.Significance probability of MSR/MSE F, indicates significance of somelinear function of parameters different than O.Significance probability that slope parameter is different than O.Significance probability that intercepts of row-width treatmentsare equal.Significance probability that slopes of row-width treatments areequal.17
ReducedOverallStateDFModelHSEIModel DiagnosticsStatisticsPr F2R2Significance Probability ofParametersIOct. odsInterc ptIIPr FPr 01.0004.27.29.43.27.29.251/2/3/4/Mean square error is an estimate of the variance of the true error.Significance probability of MSR/MSE F, indicates if some linearfunction of parameters is significantly different than o.Significance probability that the slope parameter is different thanSignificance probability that the intercept parameter is differentthan o .18o.
an additional independent variable (full model) with the operational forecast model (reduced model) where the row-width type was ignored. The purpose of this analysis was to determine whether separate forecast models for narrow-row, wide-row and broadcast soybeans are necessary. Methods Soybean objective yiel
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