35, 41-54
March,1979
BIOMETRICS
CompetitionExperiments
R. MEAD
Departmentof AppliedStatistics,Universityof Reading,Whiteknights,Reading,England
Summary
forms of competitionexperimentis surveyedand some of the
A wide range of diffierent
shoulddevote
biometricproblemsencounteredare discussed.It is suggestedthat biometricians
more attentionto the practicalaspects of experimentaldesign.A majorarea of interestfor
research,wherethereare majordimcultiesnot onlyin
biometriciansshouldbe in intercropping
the designof practicallyegeient experimentsbutalso in the analysisof yield datafrom two or
morecrops.
I. lntroductionand Structure
There are many diSerent kinds of competition experimentbecause the idea of competition is interpretedin many diSerentways. In attemptingto discuss the presentstate of
developmentof experimentalmethodsfor investigatingcompetitionit is necessaryto define
the scope of the paperand also to providea structurewithin whichto organisethe various
areasof investigation.This paperis verymuchmy personalview of competitionexperiments
and I shall not pretendto be objective.
In this paper I shall be concernedwith designedexperimentsto investigatecompetition
eSects between plants in field crops or in more controlledconditionsin the laboratoryor
glasshouse.I shall referonly brieflyto investigationson patternsof competitionin naturally
occurringplant communities,and I shall completelyignorecompetitionbetweenanimals.To
providea structurefor the paper I shall classify experimentsin three ways:
(a) Bythe numberof cropspeciesinvolved.
Here we have a rangeof increasinglycomplexsituations.The simplestis that wherea singlecrop
(monocrop)is grown under a numberof treatments.Differentvarietiesof the crop species may be
includedin the set of treatmentsand comparisonsmade between the varieties as between other
treatments.
The next level of complexityinvolvesthe growingtogetherof pairsof genotypesof a given crop
species to examinethe competitiveeffectsand the combinedyield of differentgenotypes.Genotype
competitionexperimentscan involve other treatmentfactors (such as nutrientsand spacing)but in
practicethey rarelyhave.
A furtherlevelof complexityis introducedby consideringcompetitionbetweena cropspeciesand
a weed species.The work in this area will not be consideredin this paper,except for some detailed
physiologicalexperimentsin Section 2.2. One importantcharacteristicof crop-weedcompetition
experimentsis that the two componentsarenot consideredsymmetrically.The primaryinterestis in the
cropyield as affectedby the competitionfrom the weed species.
The most complexlevel, which will be consideredextensivelyin this paper,since I believeit is
both importantand interesting,is the growth of two or more crop species together.This is usually
of the
referredto as intercropping(in whicheach crop is sown in rows and variousinter-arrangements
Key Words.Competition;ExperimentalDesign;Plant Interactions;Intercropping.
41
BIOMETRICS,MARCH 1979
42
rows are possible) or mixed cropping(in which the crops may be mixed within rows or broadcast
together).In intercroppingthe interestis in the yield of eachcomponentcrop, but also in the combined
crop yield. Intercroppingexperimentsalmost alwaysinvolvecomparisonof diSerentspatialarrangements of the two crops but also include other factors such as nutrientsor comparisonof diCerent
genotypesof one or both crops.
(b) By the treatmentfactorsbeing investigated.
The primarytreatmentfactor is that of spatialarrangementof plantsor of crops.This involves
the crop density,or in intercroppingthe densityof each componentcrop, and the arrangementof the
plantsor cropsrelativeto each other.Spatialarrangementfactorsmay be thoughtof as intrinsicto the
crop, whereasother factors involvingthe environmentunderwhich the crop is grown, are extrinsic.
Includedin these other factors are nutrientand shadingfactorsand also the simple comparisonof
diCerentvarietiesof genotypes.
(c) Accordingto whetherthe experimentis concernedonly with crop meanyields or with the yieldsof
individualplants.
In Table 1 the various aspects of competitionexperimentsincludedin the paper are listed
togetherwith the section numberof the paperin which each aspect is discussed,in accordance with this three-waystructure.
2. InvestigationsConcernedwithIndividualPlants
to
of competition
thephysioJDgy
Theseinvestigationsrangefromattemptsto disentangle
the use of empiricatmeasuresof competitionintendedto providethe basis for measuringthe
intensityof competitionand thus to enablediSerentcompetitivesituationsto be compared.
1
TABLE
Different Aspects of Competition Experiments
GENOTYPE
FOR
MIXTURES
CROP
SINGLE
CROP
SINGLE
SPECIES
SPATIAl
ARRANGEMENT
FACTORS
INDIVIDUAL
PLANT
OTHER
INVESTIGATIONS
FACTORS
SPATIAL
ARRANGEMENT
FACTORS
Effectsof different
numbersof neighbour
ing plants(2.2)
Effectsof local plant
density(2.3)
Patternof competingplants(2.2)
Plantinteraction
models(2.4)
Thephysiologicalbasis
of competition(2.1)
Responsemodelsfor
yield-densityrelationships(3.1)
designs
Experimental
for spatialarrangementtreatmentsin
intercropping(4.2)
Designsfor response
modelestimation(3.2)
CROP
MEAN
YlElDS
OTHER
FACTORS
TWOORMORE
CROPSPECIES
Genotypecompetition models(3.3)
Methodsfor analysing
intercroppingexperiments(4.1)
Genotypecomparison
experimentsin intercropping(4.3)
COMPETITIONEXPERIMENTS
43
All the methods of this section are directed towardsunderstandlngthe processesof
petition.
com-
2.1. Physiological Experimentation
A majorconcernamongphysiologistshas beento isolatethe competitioneSectsof above
and belowgroundenvironments.The abovegroundcompetitionis usuallyassurnedto be for
light, and the below ground competitionfor nutrients.The experimentaltechniqueswhich
have been used all derive from the method of Donald (1958) who used acrial and soil
partitionsin pots. The four basic arrangementsto investigatecompetitionbetweerltwo plant
speciesfor competitiondevisedby Donald are (a) for neitherlight nor nutrients,(b) for light
but not nutrients,(c) for nutrientsbut not light,and (d) for both light and nutrientsand these
are shown diagrammaticallyin Figure 1.
Modificationsof this basic designhave been developedby Aspinall(1960) and Schreiber
(1967) and more recentlyby Snaydon (1979). Snaydon uses alternatingrows of the two
competingspecies and aerialand soil partitionswhich may be arra2lgedperpendicularlyto
one another, or in parallel,or coincidentally.Snaydon'ssystem allows variationof overall
density,separationof root and shoot competition,variationof the relativedensitiesof the
two species, and separatevariationof root and shoot densities for either or both species.
Some of the possibilitiesof this systemaredemonstrateddiagrammatically
in Figure2 which
shows a portion of each of six differentcompetitionsituations:
(a) Equal densities, no competition between species.
(b) Equal densities, full competition between species.
(c) Equal densities, root competition only between-species.
(d) Unequal densities, shoot competition only between species.
(e) Equal densities, shoot competition only between species, greater eSective root density tha
shoot density.
(f) Equal densities, root competition only between species, effective shoot density varying between
species but eSective root density identical for the two species.
Snaydonnotes that thereare still limitationsto the technique"Firstly,all techniquesdepend
on the use of aerialand soil partitions.Secondly,these techniquesrestrictinteractionsto one
lateraldimension."Thus far the biometricianshave contributedlittle to this field of experi-
(a)
NOCOPlPETITION
(b)
COtlPETITIORJ
ABOVE
GROURJD
(c)
COhlPETITION
BELOV!
GROUND
(d)
FE'LL
GOt4PETITIOtt
Figure 1
Donald's pot design for separating above and below ground competition eSects. The two plant species
are indicated by (x) and (o); the soil partitions by solid lines and the aerial partitions by dotted lines.
_
_
_
_
_
_
_
o
_
t
j
44
BIOMETRICS,MARCH 1979
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Figure2
Examplesof Snaydon'ssystemfor investigatingroot andshoot competition.Symbolsas in Figure1.
mentationand one is left to wonderwhetherthe mathematicalingenuityof the biometrician
could be helpful.
An alternativeapproach is that of Currah (1975), who examined the eSect on size
variationwithin a crop of carrots of eliminatingthe potential competitiondue to various
possiblylimitingfactors.The crop was grownin the '6ideal"conditionsof uniformseed size,
regularplant arrangement,and amplesuppliesof light, waterand nutrients.Each factor,in
turn, was changedfrom the idealto a non-ideallevel(heterogeneousseed size, randomplant
arrangement,or limited suppliesof light, water or nutrients)and comparedwith the ideal.
2.2. Efffiects0Z1 Individual Plants of Varying the Numbers of Competis1gPlas1ts
A naturaldevelopmentof the physiologicalapproachto understandingplantcompetition
is to considerthe eSects on a single plant of the competitiondue to the presenceof diCerent
numbersof neighbours,and diSeringdistancesof the neighbours.
Two diSerentkinds of experimentsneed to be distinguished.One looks at variationof
the numberof competitors,or neighbours,in a monocropsituation.This was investigated
by Goodall (1960), who consideredindividualmangoldplants with one, two, four, or eight
neighboursarrangedround a circle centeredon the single plant to be measured.Goodall's
interpretationof his resultswas that the eSect of neighbourswas additiveup to six neigh-
COM PETITION EX PER IM ENTS
45
bours, at which stage the competingeSects were completeand the introductionof further
neighbourshad no eSect.
Goodall comparedhis resultswith those of Sakai(1955, 1956, 1957)for a ratherdiSerent
situationin which individualplants were surroundedby six neighbours,some of the neighbours being of the same species as the initial plant and the remainderof a second species.
Sakaiexaminedthe eSect of varyingthe numberof neighboursfrom the second speciesand
showedthat there was a linearrelationbetweennumberof neighboursof the second species
and the yield of the initial plant.
RecentlyMartin(1973) and Veeversand BoSey (1975) have developed"beehive"designs
in which plants of two speciesare arrangedon a hexagonalgrid such that for plantsof one
species,the numberof neighbouringplantsof the secondspeciesvariesbetweenzero and six.
These designsallow the eSects of varyingthe numberof plants of the alternativespeciesin
the set of neighbours to be investigatedin a much smaller area than is required for
experimentsin which each plant is eithera recordedplant or a competingplant. A possible
difficultywith beehivedesigns,in my view, is that it is not clearthat the eSects of varyingthe
numberof neighboursof the second species in a beehive design will be the same as the
correspondingeSects in a Sakai-typeexperiment,when each plant is eithera recordedplant
or a competitor. This difficulty of whether an ingenious and "efficient"mathematical
experimentaldesignprovidesestimatesof the eSectswhichare of relevanceto realsituations
is, of course, not new, the best known examplebeing the relevanceof resultsfrom changeoverdesignsto comparisonsof dietsfor farmanimals,and shouldnot dissuadebiometricians
from searchingfor new and more ingenious designs. However, the usefulnessof beehive
designs for answeringpracticalquestions remainsto be demonstrated.
2.S The EX7ectsof DiX7erentDistances of Neighbours
Much of the investigationof the eSects of the distanceof neighbouringplants is concernedwith crop mean yields and discussionof such experimentsis deferredto Section 3.
There are, however,some investigationsof the eSects on individualplants of the particular
arrangementof neighbouringplants.Goodall (1960) considereddiSerentdistancesof neighbours as well as diSerentnumbers.Howeverthe scope for designedexperimentsin whichthe
distancesof up to six neighboursare systematicallyvariedis limitedby the immensenumber
of possible patternsof neighbours.
Instead,the eSects of varyingthe local densityabouteach planton the yield of that plant,
has been investigatedby measuringthe position of neighboursin a designedexperimenton a
carrot crop (Mead 1966) or in naturalforest stands (Brown 1965, Jack 1971, Newnha
19669Opie 1968and Strand 1970).To examinethe eSects of varyingthe local density two
diSerentapproacheshave been tried. One is to considercirclesof influenceof each plant or
tree, the radiusof the circle possibly dependingon the size of the plant, and to define the
density at each plant in terms of the numbersof circles includingthe plant. A survey of
diSerentmeasuresof local densityof this type is givenby Nishizawa( 1968)and a comparison
of the predictingpower of diSerent measuresby Opie (1968). The other approachis to
considerthe areas of ground nearestto each plant (the plant polygon, or Dirichletcell or
Voronyipolygon) and to relateplantsize to the areaand shapecharacteristicsof the polygon
(Brown 1965,Mead 1966).A briefcomparisonof the two methodsby Mead (1971) suggests
that the simplercircleof influencemay be moreeSective.On the basis of these observational
studiesit may now be appropriateto attemptto designexperimentsto determinethe eSect of
diCerentlocal patternsof competition on the growth of individualplants in a controlled
environment.
46
BIOMETRICS,MARCH 1979
2.4. Measuresof InterplantCoznpetition
Over the last 15 years many methods of measuringor describingthe amount of competitionwithin a crop have been suggested.These are based eitheron characteristicsof the
frequencydistributionof plant sizes or on the relationshipbetween the sizes of adjacent
plants.
Ideas using the frequencydistributionare
(a) The coefficientof variation(Kira, Ogawaand Hozumi 1953,Stern 1963)
(b) Skewness,arldin particularthe log-normaldistributionto demonstratecompetition.However, the latterapproachhas beensubjectiveand no attemptshave been madeto fit the distributionor
interpretthe parametervalues. RecentlyKoch (1969) and Ford (1975) have arguedthat a log-normal
type frequencydistributiondoes not in itself provideevidenceof competition.
(c) Bimodality(Ford and Newbould1970,Ford 1975).Again this has beenentirelysubjectiveso
far, the appearanceof bimodality in frequencydistributionsbeing interpretedas demonstrating
competition.
Plantinteractionmodelshave beendevelopedfor plantsin regulararrays.The earlyideas
of Kira et al. (1953) involved inter-plantcorrelationsand these were extended by Mead
(1967, 1971).A muchmorecomprehensiveattackon describinginter-plantinteractionsusing
conditionalprobabilitymodelsis providedby Besag(1974).An alternativeapproachconsidering the plants dividedinto two classes(big/small or present/absent)and usingthe probabilitydistributionsof adjacencymeasuresdevelopedby Krishna-Iyer(1949, 1950)is used by
Ford (1975).
There has been far more activity in the area of suggestingnew methods of describing
competitionthan in the practicalareaof usingthe measuresto obtain a betterunderstanding
and knowledgeof competition.An interestingexception is the paper by Cannell Njugma,
Ford and Smith, with an appendixby Ross-Parker(1977), on the analysisof an experiment
on the selectionof tea bush genotypes.
Each of the ideas basedon the frequencydistributionof plantsize suSersfromthe defect
that it is not demonstrablymeasuringcompetition.Also the frequencydistributionignores
the relativepositions of the individualsmaking up the distributions.The plant interactioll
models also have not yet been shownto be practicallyuseful.Becausethe modelscontainno
time component they do not have predictivevalue, and their use to make comparative
assessmentsof the level of competitionin diSerentsituationshas been limted (Mead 1968)
and arguablyhas added nothing to an analysisof crop mean yields.
In conclusionto this sub-section,I suggestthat thereis not yet muchevidencethatthe use
of measuresother than mean plant yield underdiSerentcompetitivepressureshas addedto
our understandngof competition.
3. CosnpetitionStudieson Mean Yieldin a Single Crop
The most substantialarea and surely the most practicallyimportantis that of yielddensity relationshipsand the models which have been found useful in describingthese
relationships.The subjectof experimentaldesignfor estimatingsuch relationshipsis not well
developed at the practical experimentallevel. The important ideas are those of optimal
designs for particularresponse models, and systematicdesigns. A separatearea of some
importanceis that of genotypecompetitionmodels. A subjectnot discussedfurtherin this
paper,which uses the relationshipbetweendensityand yield, is the adjustmentof meancrop
yields to allow for missingplants(Rayner1969,Section 18.8)or for uncontrollablevariation
in achievedplant densities(Dowker and Mead 1969).
COMPETITION EXPER1MENTS
47
3.2. ResponseModelsfor Yield-DensityRelationships
Willeyand Heath( 1969)havewrittenan importantreviewof this subjectfroma practical
viewpoint.The principalmodelswhichhave been usedto describeyield-densityrelationships
are listed below (W representsyield per unit area, w representsyield per plant and p
representsplants per area = density):
(a) The simple quadraticpolynomial
W= 0 + Fp+
ep2
used frequentlyas smoothing curves but not regardedas appropriateto describethe underlying
relationship.
(b) Mitscherlich's(1919) equation
W= oe[l - exp(-$p)]
which has been tried by various experimenters(Donald 1951, Goodall 1960, Kira, Ogawa and
Sakawaki1954)but found not to be alwaysadequate.
(c) A reciprocalrelationship
W-1 =
of +
dps
derivedby Shinozakiand Kira (1956) on the assumptionsthat the growthcurvewas logisticand yield
per area was independentof densityfor high density.
(d) Holliday's(1960) reciprocalequation,
w
=
oe +
ap
+
ep,
generalizedto the familyof inversepolynomialsby Nelder (1966).
(e) Bleasdaleand Nelder's(1960) power versionof the Shinozakiand Kira model,
w @ = a
+
dp.
Increasinglyonly (c), (d) and (e) are used, with (c) being used for asymptoticrelationships
and (d) or (e) for 'parabolic'relationships.These models are used because they usually
providean adequate (statistical)fit to the data and at least some of the parametershave
reasonablebiologicalinterpretationsand, most importantly,are often found to be invariant
over subsetsof data. Good examplesof the use of the models to summariseand interpret
substantial sets of data are given by Frappell (1973) who examined the invarianceof
parametersoveryearsfor dataon onions, and by Hearn(1972)who analyseda veryextensive
set of cotton spacing experimentsusing another modificationof (e), suggested by Berry
(1967) to allow for the separationof the effect of density into effects of the within and
betweenrow spacings(x1 and x2)
W
8
=
0e
+
dlxl-l
+
dSx2-l
+
7(XlX2)-l
Methods for fittingequationsof the form (c), (d) or (e) are describedby Nelder (1966)
and Mead (1970) based on the assumptionthat the variance of log w is homogeneous.
RecentlyGillis and Ratkowsky( 1978)have examinedthe samplingdistributionsof parameters, principallyoeand W.in models (d) and (e), when the true model is (c). The methodsof
fittingused were iterativemaximumlikelihoodusing the methodof Nelder.Gillis and Ratkowsky show that when using model (e) when model (c) is true, substantialbiases can be
found in the estimationof oeand d. This appearsto be the resultof the strong correlation
between0 and(oe,d) and showsthe desirabilityof eitherfixing0 = 1, whichhas beenfoundto
be acceptablefor many crops, or seekingan invariantvalue for 0 over a numberof sets of
data whenthe estimationof oland d will be as preciseas for the case when0 is not estimated.
I believe that Gillis and Ratkowsky'sconclusion, that the Bleasdale-Neldermodel (e) is
48
BIOMETRICS,MARCH 1979
unsuitablefor yield-densityrelationships,is not justifiedwhen the biologicaladvantagesof
the modeland the normalprocedurefor searchingfor an invariantvalueof 0 areconsidered.
I believe that the models (d) and (e) provide a good frameworkwithin which to
investigatepracticalyield-densityrelationshipsand this is one aspectof competitionexperiments which is in a satisfactorystate of development.
3.2. Designs for Response Model Estimation
Thereis little evidencethat the densitiesin experimentson yield-densityrelationshipsare
chosen on the basis of statisticaladvice. This is not surprisinggiven the concentrationof
optimal design researchon generalresults,and the resultinglack of knowledgeof statisticians about the choice of treatmentlevels for particularsituations.Mead and Pike(1975), in
a generalreviewof responsesurfacemethodology,suggestthat 'the detailedinvestigationof
experimentaldesigns for particularresponsemodels would be of more practicalvalue than
the continuedpursuitof generalresults.'
My experiencewith experimentersis that they want their design to satisfy a range of
objectives estimationof optimumdensityand the economicoptimumdensity,checkingthe
validityof theirmodel,and obtaininginformationoverthe whole rangeof possibletreatment
levels.They are not happywith the fundamentalprinciplesof designsfor parameterestimation of response functions (see Box 1968) that the number of levels should equal the
numberof parametersin the model,and that the rangeof levelsshouldbe as wide as possible.
Whatseemsto be neededis informationaboutthe behaviourof variouscriteriaIvariancesof
estimatesof parametersand of importantfunctionsof parameters,sums of squaresfor lack
of fit) for a rangeof diSerentdesigns so that experimenterscan see to what extenta design
which is optimal for criterionA is sub-optimalfor criterionB. EssentiallyI think we should
stop pretendingthat there can be only a single optimal design and we should provide
experimenterswith informationabout the meritsand defects of a range of designs so that
they can make a subjectivebut informedchoice of designs.
The systematicdesignsoriginallysuggestedby Nelder(1962) and modifiedby Bleasdale
(1967) are definitelynot optimalin any senseof formaloptimaldesigntheory.Howeverthese
designsare popularwith some agronomistsand do have potentialstatisticaladvantagesin
the sense of the efficientuse of availablematerial,throughthe reductionof non harvested
areas.They also have the advantageof not beingtied to a singleparticularobjectivewhichis
importantwhen priorinformationabout parametersof a responsemodelis slight,a situation
likely to be common in intercroppingexperimentationto be discussedin Section 4.
The disadvantageof systematicdesignsis that the lack of randomisationmeansthat any
simple analysisof variancemust relyon the homogeneityof the systematicplot togetherwith
the assumptionsthat the plants are randomlyallocated and that the errors of yields are
normallydistributedand independent.Whereindependenceappearsto be markedlyunrealistic a more complex analysis using Papadakismethod or one of the spatial interaction
alternativesto it (Bartlett 1978) may be appropriate.Howeverusuallya simple fittingand
comparisonof responsecurvesis a quite adequateform of analysis.
3.3. Genotype Competition Models
The main formalapproachto analysingexperimentsin whicha numberof genotypesare
grown in all possiblepairsand in purestandshas been a seriesof papersstartingwith Sakai
(1961) and Williams(1962) and continuingthroughMcGilchrist(1955) to McGilchristand
COMPETITIONEXPERIMENTS
49
Trenbath(1971). In this last paper the analysis is based on De Wit and Van den Bergh's
(1965) conceptof Relative Yield Total, which is obtained by consideringthe yield of each
componentof the mixturerelativeto the purestand yield of that componentand averaging
thesestandardisedyields over the components.A second importantconceptin their work is
the aggressivityof one species with respectto the other which is definedas the diSerence
betweenthe two standardisedyields. Both these basic concepts are expressedas a combination of main eSects and interactionsand a substantialand complex analysisis built up.
McGilchristand Trenbathlist a numberof papersin which other forms of analysisare
proposedand they argue that these other analysesare all empiricalwhereastheiranalysisis
not. I find this argumentdifficultto accept.I think that what McGilchristand Trenbathare
actuallydoing is to take a very particularview of what a genotypecompetitionexperimentis
designedto achieve, and then building a model to embody that aim. Clearly when it is
assumedthat the experimentis designedto investigateaggressivityand depressioneSects of
individualgenotypes, then an analysis couched in terms of aggressivityand depression
parametersis less empiricalthan other analyses.However,for other objectives,such as the
agronomicone of selectingthe best combination,this model may not be appropriateand
may actually obscurepatterns.SometimesI wonderif the terms 'more empirical'and 'less
empirical'shouldnot be bannedfroma comparativediscussionof modelsbecausethey imply
an objectivitywhich is spurious!
4. IntercroppingExperiments
Intercroppingcan be definedas the growingof two or more crops simultaneouslyon the
samepieceof land. It has beenrecognisedfor a long time that intercroppingwas importantin
the developingtropicsbut in the past it seemsto havebeenassumedthat it wouldgive way to
monocroppingas a consequence of agriculturaldevelopment.It is now clear that intercroppingcan give substantialyield advantagecomparedwith monocroppingin the sense of
requiringless land to producethe sameyields of the componentcrops and it is also clearthat
intercroppingwill continue as a common practiceand that there is a need for a substantial
experimentalprogrammeto investigateagronomicpracticein intercropping.I believe this
developmentin experimentationoSers the most interestingresearcharea in experimental
designand analysis for statisticiansfor many years, and I am fortunateto be involvedin a
veryextensiveintercroppingresearchprogrammeat the InternationalCrop ResearchInstitute in the Semi-AridTropics(ICRISAT)whichwill formthe basis of muchof the comment
in this section. A general view of the presentdevelopmentof intercroppingis providedby
Willey(1978).
In consideringthe agronomy of intercroppingit is necessaryto considerthe intercrop
mixtureas 'a crop',just like a monocrop,and to discussexperimentsin termsof the levels of
the varioustreatmentfactors.The reasonswhy intercroppingexperimentationis so statistically interestingare:
(1) The largenumberof faetorsof interest,whichis morethan for a sole erop becauseof all the
differentquestionsabout spatialarrangementof the two separatecomponentcrops.
(2) The degreeof ignoraneeabout optimal levels of spatialarrangementand other factors.For
example,very mueh higherdensitiesmay be appropriatefor some eomponenterops in intereropping
than would be optimal for the same erops grownas sole erops.
(3) The very substantialproblemsof analysis, both what to analyze and how to interpretthe
results.
A separatearea of considerableimportanceis the screeningof genotypesfor use in intercropping(Section4.3).
50
BIOMETRICS,MARCH 1979
4.1. Methods of Analysis of IntercroppingExperiments
Therehas been much argumentabout how to combinethe yields of the componentcrops.
Probablythe single most popular measureis the Land EquivalentRatio (LER) which is
definedas the relativeland area requiredfor sole crops to produce the yields achievedin
intercropping(Willey and Osiru 1972). This is essentiallythe sarlleas the Relative Yield
Total of De Wit and Van den Bergh(1965), thoughthe standardisingsole cropyieldsused as
denominatorsin calculatingthe LER's arenot regardedas necessarilybeingthe yieldsfor the
sole crop underthe same conditionsas the intercropbut ratheras a measureof the maximum
achievablesole crop yield (Huxley and Maingu 1978).With this approachthe values of the
standardisingmonocrop yields need not be based solely on within experimentinformation
and consequentlythe variabilityof this compositeyield is likelyto be less than that used by
McGilchristand Trenbath.
The usual reason advanced for preferringthe LER to a composite yield based on a
comparisonwith the yield, based on equal areasof the two crops, is that at harvestthereis a
higherproportionof the more competitivecrop than is indicatedby the sown proportions
and that the LER representsa measureof biologicalefficiencyin achievingthe sameyieldsby
intercroppingrather than monocroppingeHowever the LER does have a corresponding
disadvantagein that it representsthe biologicalefficiencyof intercroppingconditionalon the
harvestproportionsof yield being those which are required.
I believethat no singlemeasureof overallyield will be completelysatisfactoryand that in
some form it will be necessaryto use a two-dimensionalrepresentationof yield advantage.A
method of displayingthe biological advantageobtained by intercropping,for a range of
proportionsof the two harvestedcrop yields, is proposedby Mead and Willey(in preparation) and it is hoped that this will providea usefulmethodof data presentation.For formal
analysisI suspectthat it will usuallybe usefulto analyseyields for each crop separatelyand
also to analyse one or more forms of compositeyield.
4.2. Experimental Design for Spatial A rrangement Treatments
The problems of designing experimentsto investigatespatial arrangementeffects in
intercroppingare particularlycomplex becauseit is necessaryto considerfive factors the
plant density for each crop; the spatial arrangementin terms of within and between row
distancesfor each crop; and the inter-relationshipof the two spatial arrangements,sometimes referredto as the degreeof intimacyof the two crops.
Most of the experimentsused previouslyhave held constant or confounded severalof
these factors.Thus most of the experimentsof the De Wit school (De Wit 1960,1961,De Wit
and Van den Bergh 1965)have workedon the replacementprincipleusing varyingproportions of the two cropsbut keepingthe overallcrop densityconstant.More recentexperiments
(Huxley and Maingu 1978, Wahua and Miller 1978)have used a modificationof Nelder's
( 1962)systematicfan designin whichplantsarespacedalong the radiiof a circle.Wahuaand
Millerhave some quadrantsof the circleplantedwith mixedcrops and otherquadrantswith
sole crops. Huxleyand Mainguvary the ratio of the two componentcropsbetweendifferent
segments of the circle as well as varying the density along each spoke. At ICRISAT a
systematic row design of the type advocated by Bleasdale (1967) was used to vary the
population density of safflowerin steps of 10%within each of four constant chickpea
densities,and the resultsshowed a most strikingresponseto safflowerdensity.
I believethat systematicdesignswill be extremelyusefulin intercroppingexperimentsand
anticipate that many more variationson systematicdesigns will be devised. Two possible
designs,the first of whichis currentlybeingtriedat ICRISAT,are shown in Figures3 and 4.
COM PETITION EXPERIM ENTS
(9)
(c)
(c)
51
(b)
(b)
(a)
(a)
(a)
(e)
(e)
(d)
(d)
(f)
G G G G G G G C G G F1C G G C t1 G G G G ;1 G G G M G G t1 G G M C G M G G (W1
t1 M G G G G t1 t' C G G M M G G G M M G G G M ; " M MM
G G C G G G G G G G M G G G G M G G G G t1 G G G M G G t. G G t1 G C1h1G G G t1 F1G G G C. F1M C G G tl M G G G M t1 G G G t1 M M tt
M
G C C G G G G G G G F1G G G c t! G G G G t' G G G M G G t: G G ' C G Fl G G G F11 G G.G G1M F1G1C G H M G G G li M G G G M F: M M lT M
G G G G G G G G G G M G G G C t1 G G G G lI G G G M G G tt G G t; G G F1G G G M F: G G C C M Fs G G G t: $' G G G M O;G G G M M M M M M
G G G G G G G G G G tl G G G G t: G G C G M G G G M G G 1 G G tx'G G tl C. G1G M M G G G G t! M G G C .: ' G G G M M G G G M M M M M M
r.
Figure3
Systematicrow design to investigatevariationof proportionsand intimacyin a groundnut(G)-Millet
(M) intercrop.[See text for definitionsof (a) to (g)].
In Figure 3 a systematicdesign to compare various density ratios and intimaciesof two
crops, PearlMillet and Groundnut,is shown. Rows run verticallyand five plants are shown
in each row. The row arrangementtreatmentsrequiredwere (a) 1 row Millet, 2 rows
Groundnut,(b) 1 row Millet, 3 rows Groundnut,(C) 1 row Millet, 4 rows Groundnut,(d) 2
rows Millet, 3 rows Groundnutand (e) 2 rows Millet, 4 rows Groundnut,plus sole crops of
(f) Millet and (g) Groundnut.By arrangingthe rows systematicallyas shown only two rows
need be discardedand the designis 20%or 30%more efficientin usingplant materialthan a
randomiseddesign would be. This is one systematicdesign for which a standardanalysis
ignoringthe non-randomnesswill probably be appropriate(the experimenthad not been
harvestedat the date of writing).Figure4 shows,in abbreviateddiagrammaticform,a simple
modificationof Bleasdale's(1967) systematicrow design which allows the densitiesof the
two crop componentsto be variedindependently,the density for one componentcrop (x)
changingsystematicallyalong each row, with the same patternfor each row, and the density
for the other componentcrop (o) changingsystematicallyfrom row to row with a uniform
densityin eachrow. The analysisof this designwoulddefinitelybe by fittingresponsecurves.
4.3. Genotype Comparison Experiments
Among intercroppingexperimentsother than those primarilyconcerned with spatial
arrangementfactors, the most important are probably those concerned with screening
x
x
x
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Figure4
Two-waysystematicspacing design for two crops (x and o) with a constantbetween-rowdistance.
52
BIOMETRICS,MARCH 1979
genotypesbecauseagain the experimentsare particularlycomplicated.This is due simplyto
havingtwo componentcropsso that not only are theretwo genotypefactors,but also it may
be necessaryto include all the sole crops for each genotypeof each crop. Some interesting
problemsare:
(i) How many intercroppingsituations should one impose on a comparisonof a numberof
genotypesof one crop intendedfor use in intercroppingand how should an experimentto compare
genotypesof a single crop, undera rangeof diSerentintercroppingsituationsbe designed?
(ii) Is it necessaryto includesole crop plots of all genotypesincludedin a genotypetrial?
(iii) How manygenotypesshouldbe includedwhenvariouscombinationsof genotypesof eachof
two crops are being compared?Should all possiblecombinationsbe used?And again should all sole
cropsbe included?What kind of experimentaldesignsare useful?
In most situationsthe numberof experimentaltreatmentswill be large,thoughplots neednot
be, and variabilityin intercroppingexperiments(in randomisedexperiments)is frequently
very high, often with coefficientsof 25%to 30%.Thus the need to deviseefficientdesignsis
extremelyimportantand again systematicdesignsmay have much to offer.
5. Epilogue
My prejudiceswill havebecomeincreasinglyclearto the readerof this paperbut I believe
that it is usefulto drawmy conclusionsfrom the variousaspectsof competitionexperiments
together. In generalbiometricianshave, I believe, failed to concernthemselvessufficiently
with the practicalaspectsof experimentaldesign.The objectiveof obtainingthe maximum
amount of informationfrom a given amountof experimentalmaterialdoes not end with the
formulafor the efficiencyof an incompleteblockdesign.Systematicdesignsofferone tool for
the biometricianto use in improvingexperimentaldesign. Good designs for the detailed
physiologicalexperimentsare needed.ThereareprobablyotherareasI havenot identified.
I am sure that intercroppingis the most importantarea of agriculturalresearchfor the
next decade. Will biometriciansbe involvedin the intellectuallyfascinatingand practically
importantproblems of experimentaldesign and analysis?Since most of the researchon
intercroppingwill be done at researchinstitutesin developingcountries,wherethe biggest
benefitsof intercroppingseem to lie, and wheretherearegenerallyveryfew biometricians,it
will be necessaryfor biometriciansto make a deliberateeffortto be involved. If we do not
makethat effortwe will be clearlyshownto be interestedonly in our mathematicsand not in
the real world.
The only area where I have suggestedthat mathematicalwork is necessary,is that of
optimaldesignfor variousestimationcriteriawith yield densityresponsemodels.Two areas
urgentlyneedingjustificationby practicalapplicationare those of plant interactionmodels
(Section 2.4) and the varietalcompetition'bee-hive'designs(Section 2.2).
Acknowledgments
I have enjoyedstimulatingdiscussionsand advicefrommanypeople and am particularly
gratefulto Dr. R. W. Willeyat ICRISAT,ProfessorR. N. Curnow,Dr. R. W. Snaydonand
Dr. R. D. Sternat Reading and ProfessorA. A. Raynerat the Universityof Natal.
Resume
Apresune revuedes difJerentstypesd'experiencesde competition,quelquesproblemesbiometriquesengendrespar ces expe'riencessont discutes. On suggere que les biometriciens
devraientaccorderplus d'attentionaux aspectspratiquesduplan d'experience.Les experiences
COMPETITIONEXPERIMENTS
53
mettanten jeu plusieursculturesdoEventetre le champd'actionprioritairedes bioss1etriciens.
Ce sont celles,en effet,quicreentles plusgrandesdifficultesnonseulementdansl'etablissement
d'unplan d'experiencepratiqueet efficacemais aussi dans l'analysedes donnees.
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