TRAINING WORKING DOCUMENT
sections 2, 3 ano 4 are extracted with permission of the pubLishers from oo 319-323 ano 325-341 of "MuLtiole Cropping Systems" C.A. Francis (ed). Macmillan. New York, 19K,.
[' r 77
A REVIEW OF METHODOLOGY FOR
THE ANALYSIS OF
Training Working Document No. 6
Prepared by Roger Mead Consultant
with CIMMYT staff
CIMMYT Lisboa 27
Apdo. Postal 6-641,
06600 M6xico, D.F., Mexico
This is one of a new series of publications from CINMY entitled Training Working Documents. The purpose of these publications is to distribute, in a timely fashion, training -related materials developed by CIMMYT staff and colleagues. Some Training Working Documents will present new ideas that have not yet had the benefit of extensive testing in the field while others will present information in a form that the authors hav tested and found useful for teaching. Training Working Documents are intended for distribution to participants in courses sponsored by CIMMYT and to other interested scientists, trainers, and students. Users of these documents are encourage to provide feedback as to their usefulness and suggestions on how they might be improved. These documents may then be revised based on suggestions from readers and users and published in a more formal fashion.
CIMMYT is pleased to begin this new series of publications with, a set of six documents developed by Professor Roger Mead of the Applied Statistics Department, University of Reading, United Kingdom, in cooperation with CIMMYT staff. The first five documents address various aspects of the use of statistics for on-farm research design and analysis, and the sixth addresses statistical analysis of intercropping experiments. The documents provide on-farm research practitioners with innovative information not yet available elsewhere. Thanks goes out to the following CIMfMYT staff for providing valuable input into the development of this series: Mark Bell, Derek Byerlee, Jose Crossa, Gregory Edmeades, Carlos Gonzalez, Renee Lafitte, Robert Tripp, Jonathan Woolley.
Any comments on the content of the documents or suggestions as to how they might be improved should be sent to the following address:
CIMMYT Maize Training Coordinator
Apdo. Postal 6-641
06600 Mexico D.F., Mexico.
REVIEW OF INTERCROPPING ANALYSIS METHODOLOGY
1. Measurements and Analysis
The first point to recognise is that there is not a single form of statistical analysis which is appropriate to all forms of intercropping data. Even for a single set of experimental data it will be important to use several different forms of analysis. For the two components of an intercropping system the data may occur in different structural forms. In general, data structures from intercropping experiments will be complex with different forms of yield information available for different subsets of experimental units.
1.1 Valid Comparisons
In considering alternative possibilities for the analysis of data from intercropping experiments it is essential that the principle of comparing "like with like" is obeyed. If yields are measured in different units, or over different time periods, or for different species, then in general comparisons will not be valid and should not be attempted. To illustrate the difficulties and possibilities we consider a set of ten "treatments". Any actual experiment would be unlikely to include such a diverse set of treatments though there would typically be several representatives of some of the "treatment types" illustrated. The structure for the ten treatments is as follows.
Legume Crop Cereal Crop Monetary Relative
Species Yield Species Yield Value Performance
I) I y r2) II Z2 r
3) A a3 r3
4) - B b4 r4
5) I Y5 A a5 r5 ys/yI + a5/a3
6) I Y6 A a6 r6 y6/y I + a6/a3
7) I y7 B b7 r7 y71Y I + b7/b4
8) I y8 B b8 rg y8/y I + bs/b4
9) II z9 A a9 r9 zg/z2 + aq/a3
10) 11 zlo B blo rio zIo/z2+blo/b4
A comparison is valid only when the units of measurement are identical. Thus it is valid to investigate the effect of different cereal crops on legume yields of one species (y 1, ys, y6. y7. ygx or of the other species (z2, z9, zmo). Similarly the effect of different legume environments on crop yield (a3, a5, a6, a9) or (b4, b7, bs, bIo). The effects of different treatment systems on pairs of yields may be assessed by comparing the pair (Y5, a5) with (Y6, a6) or (y7, b7) with (y8, b8). Particular combinations of the pair of yields may also be compared so that (y5/yj + as/a3) may be compared (y6/yl + a6/a3). However it is not valid to compare
(y5/y I + a5/a3) with (y7/Y I + b7/b4) because the divisors are different. In interpretati on of these sums of ratios as "Land Equivalent Ratios" (Willey 1979, Mcad and Riley 1 981 ) the sum of ratios is thought of in terms of land areas required to produce equivalent yields through sole crops. However land areas required to grow crop A ate not comparable with land areas to grow crop B. Comparison of biological efficiency through LER's cannot be valid for different crop combinations.
The only measure by which all different component combinations can be compared must be a variable, such as money, to which all component yields can be directly converted, and which has a practical meaning.
1.2 The Variety of Forms of Analysis
The only form of analysis which retains all the available information is multivariate, When the performance of each component crop may be sumnmarised in a single yield then a bivariate analysis of variance is the most powerful technique available. However only those experimental units for which both yields may be measured can be included in a bivariate analysis.
Analysis of each crop yield separately is also like y to be useful, though it is important to check that the variability for monocrop yields is the same as that for intercrop yields. Analysis of crop indices may also be useful.
2. General Principles of Statistical Analysis
2.1 Analyisi of Variance
The initial stage for most analyses of experimental data is the analysis of variance for a single variate, or measurement. The analysis of variance has two purposes. The first is to provide, from the error mean square, an estimate of the background variance between the experimental units. This variance estimate is essential for any further analysis and interpretation. It defines the precision of information about any mean yields for different experimental treatments. One major requirement often neglected is that the error mean square must be based on variation between the experimental units to which treatments are applied. If treatments are applied to plots 10 x 3 m, then the variance estimate used for comparing treatments must be that which measures the variation between whole plots. Measurements on subplots or on individual plants are of no value for making comparisons between treatments applied to whole plots.
The second purpose of the analysis of variance is to identify the patterns of variability within the set of experimental observations. The pattern is assessed through the division of the total sum of squares (SS) into component sums of squares and the interpretation of the relative sizes of the component mean squares. To illustrate the simple analysis of variance, and for illustr-ation of other techniques, later in this chapter, I shall use data from a maize/cowpea ( Vigna unguiculata) intercropping experiment conducted by Dr. Ezumah at IITA, Nigeria. The experimental treatments consisted of three maize varieties, two cowpea varieties, and four nitrogen levels (0, 40, 80, 120 kg/ha) arranged in three randomized blocks of 24 plots each. The data for cowpea and maize yields are given in Table 1. The analysis of variance and tables of mean yields for the cowpea yields are shown in Table 2. The analysis of variance shows that there is very substantial variation in cowpea yield for the different maize varieties: there is also a clearly significant (5 percent) interaction between cowpea variety and nitrogen level and a nearly significant variation between mean yields for different nitrogen levels. The tables of means for cowpea yield that should be presented are therefore for (1) maize varieties and (2) cowpea variety x nitrogen levels, with the mean yields for nitrogen levels as a margin to the table. The analysis of variance implies strongly that no other means should be presented.
The interpretation indicated by the analysis and mean yields is as follows. Yield of cowpea is substantially determined by the maize variety grown with the cowpea. Higher cowpea yields are obtained when maize variety I is grown. For cowpea variety B, cowpea yield is reduced as increasing amounts of nitrogen are applied (presumably because of correspondingly improved maize yield). Yields for cowpea variety A are not affected in this manner.
2.2 Assumptions in the Analysis of Variance
The interpretation of an analysis of variance and of the subsequent comparisons of treatment means depends critically on the correctness of three assumptions made in the course of the analysis. If the assumptions are not valid, the conclusions drawn may also be invalid and, therefore, misleading. Evidence available from the analyses of intercropping experiments suggests that failure of the assumptions is at least not less frequent than in monoculture experiments. It is therefore vital that the experimenter deliberately consider the assumptions before completing the analysis. The three assumptions are:
1 That the variability of results does not vary between treatments
2 That treatment differences are consistent over blocks
3 That observations for any particular treatment for units within a single block would be approximately
Table 1. Cowpea and Maize Yields in intercrop Trial at 11TA, Nigeria
1 2 3
Cowpea Nitrogen ________ _________ _______variety level 1 I1 I11 1 1 III I H II
A No 259 645 470 523 540 380 585 455 484
A NI 614 470 753 408 321 448 427 305 387
A N2 355 570 435 311 457 435 361 586 208
A N3 609 837 671 459 483 447 416 357 324
B No 601 707 879 403 308 715 590 490 676
B Nj 627 470 657 351 469 602 527 321 447
B Ni 608 590 765 425 262 6 12 259 263 526
B N3 369 499 506 272 4211 280 304 295 357
A No 2121 2675 3 162 22 54 3628 4069 2395 2975 4576
A Ni1 3055 3262 3749 3989 3989 4429 4429 4135 4429
A N2 3922 3955 4095 4642 4135 4642 5589 4429 5156
A N3 4129 4129 4 022 3975 4789 4282 5990 5336 5663
B No 2535 2535 2288 4209 3989 2321 2901 4429 3482
B Ni1 2675 3402 3122 4789 4936 3342 3555 4936 4135
B N2 3855 3815 3535 5083 4496 3702 60231 5296 4069
B N3 3815 4202 3749 5656 5516 5223 5516 5083 5369
aYields grouped by maize variety (1, 2, 3) and planting block (1, 11, 111). Source: Data from Dr. Ezumab, UTA, unpublished.
There is an element of subjectivity about the assessment of these assumptions. For a more extensive discussion the reader is referred to Chap. 7 of Mead and Cumow (1983). In brief, the experimenter should ask:
1 Does it seem reasonable, and do the data appear to confirm that the ranges of values for each
treatment are broadly similar and that there is no trend for treatments giving generally higher yields to
display a correspondingly greater range? In biological material it is more reasonable to suppose that
treatments with a high mean yield also have a rather higher variance of yield. and so an experimenter
should be prepared to recognize this occurrence and to use a transformation of yield before analysis.
2 Are treatment differences similar in the"good" blocks and in the "bad" blocks? Again if the pattern of
bigger differences in better blocks, which might reasonably be expected, is found, then a
transformation of yield is necessary.
3 Do I believe that an approximately normal distribution is a sensible assumption?
Table 2. Analysis of Variance and Tables of Means for Cowpea Data in Intercrop Trial at IITA, Nigeria
Analysis of variance
Source SS df MS F
Blocks 73,000 2 36,500 2.8
Maize varieties (M) 409,400 2 204.700 15.7a
Cowpea varieties (C) 6,000 I 6,000 0.5
Nitrogen (N) 113,10W 3 37,700 2.9
M x C 9,900 2 4,950 0.4
MxN 67,600 6 11,267 0.9
CxN 172,400 3 57,433 4.4b
MxCxN 135,400 6 22,567 1.7
Error 599,300 46 13.000
Table of means (cowpea yield (kg/ha)
Cowpea variety 0 40 80 120 Maize variety Mean
A 482 459 413 511 1 582
B 597 497 479 367 2 430
Mean 539 478 446 439 3 415
SE of difference for N means = 50 SE of difference 43
SE of difference for combinations = 71
aSignificant at 0.1% level
bSignificant at 5% level.
Source: Data from Dr. Ezumah, IITA, unpublished.
For the data in Table I a visual inspection reveals no reason to doubt the assumptions. The only peculiarity of the data is the repetition of some values in the set of maize yields, but since no obvious explanation could be found the data were used for analysis and interpretation as shown in Table 2.
2.3 Comparisons of Treatment Means
Many sets of experimental results are wasted through an inadequate analysis of the results. In many cases this results from the use of multiple comparison tests of which the most prevalent, and therefore the one that causes most damage, is Duncan's multiple range test. The reason that multiple comparison tests lead to a failure to interpret experimental data properly is that such tests ignore the structure of experimental treatments and hence fail to provide answers to the questions that prompted the choice of experimental treatments.
Two particular situations in which multiple range tests should never be used are for factorial treatment structures and if the treatments are a sequence of quantitative levels. In the former the results should be interpreted through examination of main effects and interactions. In the second the use of regression to describe the pattern of response to varying the level of the quantitative factor should be obligatory. Thus, for the cowpea yield example, the effect of nitrogen on yield for cowpea variety B can best be summarized by the regression equation
Yield = 591 - 1.77 N
where yield and N are both measured in kg/ha. The predicted yields for the four nitrogen levels (0, 40, 80, 120 kg/ha) are 591, 520, 449, and 379, which obviously agree very closely with the observed means.
Examples of the failure of experiments to interpret their data properly occur regularly in all agricultural research journals wherever multiple comparison methods are widely used. Examples of misuse and discussion of alternative forms of analysis are given by Bryan-Jones and Finney (1983), Morse and Thompson (1981), and many other authors. The only sensible rule to adopt when analyzing experimental data is never use multiple range tests or other multiple comparison methods.
2.4 Presentation of Results
The prime consideration in presenting experimental results should be to provide the reader with all necessary information for a proper interpretation of results, without unnecessary detail. This principle leads to some particular advice:
1 Tables of mean yields should always be accompanied by standard errors for differences between
mean yields and the degrees of freedom for those standard errors.
2 When multiple levels of analysis are used, as for split plot designs then all the different standard
errors must be given.
3 When the results are presented in graphic form the data should always be shown (plotting mean
yields). A graph showing only a fitted line or curve deprives the reader of the opportunity to assess
the reasonableness of the fitted model.
4 Standard errors are much more effective with tables of means than with graphs where standard errors
are represented by bars.
5 All standard errors or other measures of precision should be defined unambiguously. The statement
below a set of means "standard error = 11 -Y is ambiguous because it does not specify if it is for a
mean or a difference of means or, even, for a single value rather than a mean.
3. Bivariate Analysis
3.1 What is a Bivariate Analysis?
A bivariate analysis is a joint analysis of the pairs of yields for two crops intercropped on a set of experimental plots. The philosophy is that because two yields are measured for each plot. and the yields will be interrelated, they should be analyzed together. The interrelationship is important since it implies that conclusions drawn independently from two separate analyses of the two sets of yields may be misleading. There are two major causes of interdependence of yield of two crops grown on the same plot. If the competition between the two crops is intense. then it might be expected that on those plots where crop A performs unusually well, crop B will perform unusually badly and vice versa. This would lead to a negative background correlation between the two crop yields, quite apart from any pattern of joint variation caused by the applied treatments. Failure to take this negative correlation into account could lead to high standard errors of means for each crop analyzed separately, which could mask real differences between treatments.
Mternatively it may be that on apparently identical plots. the two crops respond similarly to small differences between plots producing a positive background correlation. Again looking at separate analyses for the two crops distorts the assessment of the pattern of variation.
To see how consideration of this underlying pattern of joint random variation is essential to an interpretation of differences in treatment mean yields some hypothetical data are shown in Fig. 1. Individual plot yields are shown for two intercrop, systems (X and 0), the mean crop yields for the two systems being identical for three situations. In Fig. I a the pattern of background variation corresponds to a strongly competitive situation (negative correlation), whereas for Fig. lb there is a positive correlation of yields over the replicate plots for each treatment. In Fig. Ic there is no correlation between the two crop yields. In all three cases the comparisons in terms of each crop yield separately would show no strong evidence of a difference between the two systems. However the joint consideration of the pair of yields
against the background variation shows that the difference between the systems is clearly established in Fig. I a, that Fig. l b suggests strongly that the apparent effect is attributable to random variation, and that in Fig. I c the separation of the two systems is rather more clear than could be established by an analysis for either crop considered alone.
x x 00 00
xX OX 00
x0) xx x
x 0 0 00
Figure 1. Different correlation patterns for yields with the same values of the individual crop yields:
(a) negative correlation, (b) positive correlation, (c) no correlation. The two axes are for the yields of the two crops. Two intercrop systems give yields represented by x and o.
3.2 The Form of Bivariate Analysis
The calculations for a bivariate analysis are formally identical with those required for covariance analysis. The difference is that, whereas in covariance analysis there is a major variable and a secondary variable whose purpose is to improve the precision of comparisons of mean values of the major variable, in a bivariate analysis the two variables are treated symmetrically. Bivariate analysis of variance consists of an analysis of variance for X I, analysis of variance for X2, and a third analysis (of covariance) for the products of X1 and X2. Computationally this third analysis of sums of products is most easily achieved by performing three analyses of variance for X 1, X2, and Z =X I + X2. The covariance terms are then calculated by substracting corresponding SS for X I and for X, from that for Z and dividing by 2. The bivariate analysis including the intermediate analysis of variance for Z are given in Table 3 for the maize/cowpea experiment discussed earlier.
The bivariate analysis of variance, like the analysis of variance, provides a structure for interpretation. In addition to the sums of squares and products for each component of the design. the table includes an error mean square line which provides a basis for assessing the importance of the various component sums of squares and products. The general interpretation of this analysis is quite clear and is essentially similar to the pattern of analysis of cowpea yield. There are large differences attributable to the different maize varieties and to the variation of nitrogen level; there is also a suggestion that there may be an interaction between cowpea variety and nitrogen level.
Table 3. Bivariate Analysis of Variance for Maize/Cowpea Yield Data (0.001 kg/ha) in Intercrop Trial
Maize SS Cowpea SS SS for Sum of
Source df (X1) (X2) (XI + X2) products F Correlation
Blocks 2 0.29 0.0730 0.247 -0.058 1.75 -0.40
M variety 2 17.52 0.4094 12.665 - 2. 632 11.90 -0.98
C variety 1 0.03 0.0060 0.062 0.013 0.44 1.00
Nitrogen 3 28.50 0.1131 25.081 -1.766 10.59 -0.98
MxC 2 1.11 0.0099 0.922 -0.099 0.82 -0.95
Mx N 6 1.25 0,0676 0.920 -0.199 0.64 0.93
CxN 3 0.24 0. 1724 0.152 -0.130 2.40 -0.64
MxCxN 6 1.28 0.1354 1.349 -0.033 1.40 -0.08
Error 46 15.90 0.5993 13.671 -1.414 -0.46
(MS) (0.346) (0.0130) (-0.031)
Total 71 66.13 1.5861 55.080 -6.318
Note: See Table 1
3.3 Diagrammatic Presentation
We have argued earlier that interpreting the patterns of variation in maize and cowpea yields without allowing for the background pattern of random variation can be misleading. The primary advantage of the bivariate analysis is that it leads to a simple form of graphic presentation of the mean yields for the pair of crops making an appropriate allowance for the background correlation pattern. The graphic presentation uses skew axes for the two yields instead of the usual perpendicular axes. If the yields are plotted on skew axes with the angle between the axes determined by the error correlation, and if, in addition. the scales of the two axes are appropriately chosen, then the resulting plot, such as Fig. 2. has the standard error for
comparing two mean yield pairs equal in all directions. The results in Fig. 2 are for the three maize varieties from the example, and the size of the standard error of a difference between two mean pairs is shown by the radius of the circle.
Figure 2. Bivariate plot of pairs of mean yields for three maize varieties (1,2.3). Maize and cowpea yields are in kilograms per hectare. (Data from Table 1).
Construction of the skew axes diagram is based on the original papers of Pearce and Gilliver (1978, 1979) and detailed instructions for construction are given by Dear and Mead (1983, 1984). The form of the diagram given in Fig. 2 treats the two crops symmetrically, in contrast to the original suggestion of Pearce and Gilliver, in which one yield axis is vertical and the other is diagonally above or below the horizontal axis, depending on the sign of the error correlation. A summary of the method for construction of the symmetric diagram is as follows:
If the error mean squares for the two crops are VI (= 0.346 in the example) and V2 (= 0.0130), and
the covariance is V12 (= -0.0310), then the angle between the axes 0 is defined by
Cos 0 = V2
If the range of values for the two yields XI and X2 are (Xo, Xi ) and (X2o, X2) respectively, then
we define two new variables y1 and y1,
Y1 = x = Kjx2
x2 - V12x/V1=2, ( Lx
2 - V(,/V,)"2 x2 )
y10 = klxO
yll = klxl )
Y2o = k, X20 - VL2)
Plot the four pairs of y values O(ylo. y2o), B(y1o, y21) and Cyl. y2) on standard rectangular axes,
using the same scale for yI and for Y2. The xI axis is constructed by joining the points 0 and A. the x2
axis by joining O and B. The xi scale is defined by O(xl - xlo) and A(x! = x; I); the x2 scale is
defined by (0x2 = x20) and B(x2 = x2I). Further points on both axes may be marked using a ruler and
the two defining points. The rotation of the x! and x2 axes to achieve symmetry can be performed
subjectively or by simple trigonometry. Individual points for pairs of mean yields may be plotted by
first measuring x1 along the xi axis, and x2 parallel to the x2 axis. More details of the diagram
construction are given by Dear and Mead (1983, 1984).
The interpretation of the diagrams is extremely straightforward. The results in Fig. 2 show that the differences among the three maize varieties are important for both maize and cowpea yields, with the difference between varieties 2 and 3 clearly less than between either variety and variety 1. There is a clear consistency through the sequence of varieties I to 2 to 3, with the increase in maize yield being directly reflected in a decrease in cowpea yield. The three points fall nearly on a line illustrating the strong relation between the two crop yields over the three varieties. (Note also that the correlation for maize varieties, shown in Table 3, is -0.98). Remember that random correlation between the two yields has been allowed for by the skewness of the axes and the displayed pattern is additional tot he background correlation pattern.
The results for nitrogen main effects and the interaction of cowpea variety with nitrogen are shown in Figs. 3 and 4. The four nitrogen levels produce four pairs of mean yields in an almost straight line. The dominant effect is on the yield of maize which increases consistently with increasing nitrogen. In addition there is a clear pattern of compensation between the two crop yields with cowpea yield decreasing as maize yield increases. The pattern of yields for the cowpea variety/nitrogen interaction emphasizes the two effects of yield increase for one crop and compensation between crops. For variety A the effect of increasing nitrogen is simply an increases of maize yield, the "line" of the nitrogen level means being almost exactly parallel to the maize yield axis. In contrast for variety B the dominant effect is the change in the balance of maize/cowpea yields with the maize yield increasing consistently with increasing nitrogen and the cowpea yield showing a corresponding decline.
3.4 Significance Testing
There are two forms of test that are useful in bivariate analysis, and these correspond to the t and F tests used in the analysis of a single variate. We have already mentioned in the discussion of the skew axes plot that the standard error of a difference is the same in all directions in these diagrams. because of the scaling
of axes which is part of the instruction of the diagram the standard error per observation is I (measured in the units of yl and Y2). The standard error of a mean of n observations is therefore 14n and the standard error of a difference between two points is 4I(21n.)
Figure 3. Bivariate plot of pairs of mean yields for four nitrogen levels (0, 40, 80. and 120 kg/ha). Maize and cowpea yields are in kilograms per hectare. (Data from Table 1).
Confidence regions for individual treatment means can be constructed as circles with radius 'I(2FIn), where F is the appropriate percentage point of the F distribution on 2 and e degrees of freedom (e is the error degrees of freedom).
The analogue of the F test in a univariate analysis of variance is also an F test. The basic concept on which the test is based is the determinant constructed from the two sums of squares and the sum of products. Suppose that the error SSP are E,1, E2, and El 2, then the determinant is
El x E2 - E~2
and it reflects both the sizes of El and E2 and the strength of the linear relationship between x 1 and x2. To asses the treatment variation for a treatment SSP with values TI, P), and T12 we calculate a statistic, L, which compares the determinant of treatment plus error with that for error
L =(TI + E1)(T2, + ED) - (T12 + E12)' E1E2 -E~2
The test of significance then involves comparing
Pages 12 13 M 0 0 missing From
Treatment 1:1 1:2 1:3 1:4
The relative comparisons are identical for the two equivalents.
4.3 Biological Indices of Advantage or Dominance
The most important index of biological advantage is the relative yield total (RYT) introduced by de Wit and van den Bergh (1965) or land equivalent radio (LER) reviewed by Willey (1979). The index is based on relating the yield of each crop in an intercrop treatment mixture to the yield of that crop grown as a sole crop. If the two crop yields in the intercrop, mixture are MA, MB, and the yields of the crops grown as sole crops are SA, SB, then the combined index is
L = MA + MB = LA + LB SA SB
The most frequently used value index is that of financial return. Other value indices include protein and dry matter. The main criticism made specifically of financial indices is that prices fluctuate and hence the ratio of KI to K2 may vary considerably. A partial answer to this criticism is to employ several price ratios. Thus the results for the four treatments discussed earlier in this section might be presented for five price ratios as follows:
Price Ratio for Maize/Cowpea
5642 4722 1490
6599 4 7 22 5960
While some comparison patterns, such as (2 vs. 1) or (2 vs. 3), remain consistent for this range of price ratios others, such as (I vs. 3) or (2 vs. 4), do not.
One other form of single measurement comparison which is exactly equivalent to the financial value index is the crop equivalent. In calculating a crop equivalent, yield of one crop is "converted" into yield equivalent of the other crop by using the ratio of prices of the two crops. The exact equivalence of crop equivalent yield to financial index is immediately obvious algebraically but may be perceived clearly also by considering the four treatments for a I : 3 price ratio. This ratio implies that a unit yield of cowpea is worth 3 units of maize. We can therefore calculate yields as maize equivalents or cowpea equivalents as follows:
458 + 2653/3 = 1342 319 + 5323/3 = 2093
4721/3 = 1574 1490 = 1490
2653 + 3(458) = 4027 5323 + 3(319) = 6280 4722 = 4722
3(1490) = 4470
The interpretation embodied in LER is that L represents the land required for sole crops to produce the yields achieved in the intercropping mixture. A value of L greater than I indicates an overall biological advantage of intercropping. The two components of the total index, LA and LB represent the efficiency of yield production of each crop when grown in a mixture, relative to sole crop performance. For the maize/cowpea yields treatment 2 may be assessed relative to treatments 3 and 4 to give an LER
L = 533+ 39= 1. 13 + 0.21 = 1.34
Other indices have been proposed as measures of biological performance. There are two different objectives for which such indices have been proposed. The first is the assessment of the benefit, or overall advantage, of intercropping, or mixing. The second is the assessment of the relative performance of the two crops, the concept of dominance or competitiveness. It is important not to confuse these two objectives, which should be quite separate conceptually.
The RYT or LER is the main index of advantage currently used. The other index which has been used is the relative crowding coefficient (de Wit, 1960), which can be defined in terms of the LER components as
LA X LB
1- LA I -LB
The two main indices of dominance are the aggressivity coefficient. introduced by McGilchrist and Trenbath (197 1) defined essentially as
LA - L
and the competition ratio proposed by Willey and Rao (1980) and defined essentially as
The full definition of each index as originally given involves proportions of the two crops in the mixture. However, for applications in intercropping, this masks the underlying concepts involved in the ideas of advantage or dominance. Each of these four indices is based clearly on the LER components LA and LB. [Indeed since there are only four simple arithmetical operations (+, -, x, t-) it could be argued that the set of possible indices is now complete!] Crucially, however, the components LA and LB are ratios, and the value of a ratio is determined as much by the divisor as by the number divided. Hence the interpretation of LA and LB, and therefore of any index based on LA and LB, depends on the choice of divisor.
This question of interpretation is extremely important. and becomes even more important when comparison of LERs is considered in the next section. For the LER to be interpreted as the efficiency of land use the sole crop yields, SA and SB must represent some well-defined, achievable, optimal yields. It is therefore necessary that the choice of sole crop yield used in the calculation of the LER be clearly defined and justified as appropriate to the objective that the LER is intended to achieve. To illustrate this argument consider the yields for several intercrop and sole crop treatments in the maize/cowpea experiment. The mean yields for two maize varieties, two cowpea varieties and two nitrogen levels are shown in Table 4.
If we consider a particular intercrop combination, for example M, C I No, we could assess the biological advantage of intercropping as
L = + --= 1.03 + 0.44 = 1.47
This is simply interpretable as the benefit in the situation where the only varieties available are M, and C1 and no nitrogen is available. It also implies that the sole crop yields of 2568 and 1036 could not be improved by modifying the spatial arrangement or the management of the sole crop since we are assessing the intercrop performance in relation to the land required to produce the same yields by sole cropping. No one would deliberately use an inefficient method of sole cropping to try to match the intercropping performance. Suppose the combination MICINo is now considered. Since the sole crop yield for C1 is
Table 4. A Subset of Yields from the Maize/Cowpea Experiment
Intercrop yields Sole-crop yields
Treatment Maize Cowpea M1 M1 C1 C2
M1CINo 2653 458
M3CiNo 3315 508
MIC2No 2453 731 2568 3555 1036 787
M3C2No 3604 585
MICIN3 4093 706
M3C1N2 5663 366 3651 4722 1795 1490
MIC2N3 3922 458
M3C2N3 5323 320
Note: Data from intercrop trial (Table 1).
better than that for C-2, the advantage of intercropping might be argued to be overestimated if we compare MIC2No with MI and C2 for which the LER would be
L = 2 + 73 = 0.96 + 0.93 = 1.89
If we measure MI C2 No against MI and C1 we obtain and LER value
L = - + -- = 0.96 + 0.71 1.67
We could go further and argue that if M3 is available as an alternative to MI then we should compare MI C2 No with the best available varieties, M3 and C1, which could be used as a sole cropping alternative. We would then have
L = + - = 0.69 + 0.71 = 1.40
This last L value represents the most stringent assessment of advantage of the intercropping combination MIC2No and alternative forms of L could all be criticized as presenting an illusory benefit of intercropping as compared with sole cropping.
What about using sole crop yields for N3 rather than for No? Here the argument becomes more complicated. It may well be that in the farming situation for which the conclusions drawn are to be relevant, there is no real possibility of using extra nitrogen as required in N3. The advantage of 1.40 would then be assessed in the most stringent manner possible for the practical situation considered.
The purpose of this example is not to define rules for calculating LER measures of advantage but to demonstrate that the choice of divisors for the LER is a matter requiring carefulI thought. The divisor in LER calculations cannot be assumed to be obvious, and discussions about LER values when the choice of divisor is not clearly defined should be treated with suspicion.
One distinction that might usefully be made is between the LER or RYT as a measure of biological sufficiency of a particular combination without any implications of agronomic benefit and the use of the LER to assess the greater efficiency of the use of land resources. The former concept developed naturally from competition studies and is a strictly nonagronomic idea. The latter is an inherently more complex measure. Perhaps we should use RYT for the non-practical biological concept and LER for the agronomic concept!
4.4 Comparison and Analysis of LER Values
The assessment of advantage of a single intercrop combination requires careful thought. When it is desired to compare different intercrop treatments using LER values, the need to calculate the LER to produce meaningful comparisons is accentuated. There are now two problems.
The first is the choice of divisor, and I believe that comparisons of LER values are valid in their practical interpretation only if the divisors are constant for all the values to be compared. If different divisors are used for different intercrop treatments then the quantities being compared may be considered as
SA I SBI
S A2 SB2
The interpretation of any difference between L I and L2 cannot be assumed to be the advantage of intercropping treatment I compared with intercropping treatment 2, since the difference could equally well be caused by differences between sole cropping treatments SB I and SB2 Or between SA I and SA2.
Although LER values using different divisors are often compared, the concept 'that is being used as the basis for comparison is the vague one of efficiency which is not interpretable in any practically measurable form of yield difference between different intercropping treatments. We should recognize that such comparisons are of a theoretical nature only and are not practically useful.
The form of the LER which is the sum of two ratios of yield measurements has prompted concern about the possibility of using analysis of variance methods for LER values. More generally the question of the precision and predictability of LER values has been felt by some to be a problem.
The comparison of LER values within an analysis of variance is. I believe, usually valid provided that a single set of divisors is used over the entire set of intercropping plot values. Some statistical investigations of the distributional properties of LERs were made by Oyejola and Mead (198 1) and Oyejola (1983). They considered various methods of choice of divisors including the use of different divisors for observations in different blocks. Allowing divisors to vary between blocks provided no advantage in precision or in the normal distributional assumptions: variation of divisors between treatments was clearly disadvantageous. The recommendation arising from these studies is therefore that analysis of LER is generally appropriate, provided that constant divisors are used, and with the usual caveat that the assumptions for the analysis of variance for any data should always be checked by examination of the data before, during and after the analysis.
The question of precision of LERs and, by implication, their predictability, is an unnecessarily confusing one. If LERs are being compared within experiments that standard errors of comparison of mean LERs are appropriate for comparing the effects of different treatments. Experiments are inherently about comparisons of the treatments included rather than about predictions of performance of a single treatment. The precision of a single LER value must take into account the variability of the divisors used in calculating the LER value. However a more appropriate question concerns the variation to be expected over changing environments and this must be assessed by observation over changing environments. No single experiment can provide direct information about the variability of results over conditions outside the scope of the experiment. This, of course, does not imply that single experiments have no value since we may reasonably expect that the precision of estimation of treatment differences will be informative for the prediction of the differential effects of treatments.
4.5 Extensions of LER
In the last section it was mentioned that there were two problems in making comparisons of LER values for different intercropping treatments. The second problem is that the concept of the LER as a measure of advantage of intercropping assumes that the relative yields of the two crops are those that are required. The calculation of the land required to achieve, with sole crops. the crop yields obtained from intercropping makes this assumed ideal of the actual intercropping yields clear. However with two (or more) intercropping treatments the relative yield performance LA :LB will inevitably vary and hence the comparison of LER values for two different treatments can be argued to require that two different assumptions about the ideal proportion LA :LB shall be simultaneously true.
This difficulty led to the proposed "effective LER" of Mead and Willey (1980) which allows modification of the LER to provide the assessment of advantage of each intercropping treatment at any required ratio X LA(LA + LB). The principle is that to modify the achieved proportions of yield from the two crops we consider a "dilution" of intercropping by sole cropping. The achieved proportion of crop A could be increased by using the intercropping treatment on part of the land and sole crop A on the remainder, the land proportions being chosen so as to achieve the required yield proportions. Details of the calculations are given in Mead and Willey (1980). It is important if the use of a modification of the LER is proposed that the reason for using the effective LER is clearly understood. It is not primarily a form of practical adjustment but arises from the philosophical basis of the LER.
It may be that in using the LER as a basis for comparison of different treatments the emphasis is not on the biological advantage of intercropping but on the combination of yields onto a single scale, in terms of yield potential. In this view the LER becomes another form of value index, the two values being the reciprocals of the sole crop yields. When a range of price ratio indices is used, it is almost invariably found that the ratio of the LER values is well in the center of the price ratio range. The principle of the argument for using an effective LER is no longer essential but there may still be advantages, in making practical comparisons or treatments in terms of performance at a particular value of k. There are, however, other possible ways of modifying the LER, and the most important of these is the calculation of combined yield performance to achieve a required level of crop yield A. Arguments for, and details of, this alternative modified LER are given by Reddy and Chetty (1984) and Oyejola (1983).
4.6 Implications for Design
The particular implications to be considered here concern the use of sole Crop Plots. If the arguments about the choice of divisors are followed then it will not be necessary to include many sole crop treatments within the designed experiment. The investigation of the agronomy of monocropping has been extensive and in most intercropping experiments there should rarely be any need for an experimental investigation of the optimal form of monocropping. Therefore, there should often be no need for more than a single, sole cropping treatment for each crop.
The reduction in the number of sole crop plots in intercropping experiments would be of great benefit because it would enable a greater part of the resources for an experiment on intercropping to be used for investigating intercropping. Many intercropping experiments which I have seen have used between onethird and one-half of the plots for sole crops. To some extent this reflects a propensity for continuing to ask whether intercropping has an advantage, when this is widely established, instead of asking the practically more important question of how to grow a crop mixture.
It is possible to take the reduction of sole crop treatments further. The analysis in this chapter and the previous chapter do not require sole crop treatments within the experiment to be treated Like other treatments. For the bivariate analysis no sole crop information is essential though sole crop information does provide a standard against which to compare the pairs of yields. For the analysis and interpretation of LERs, estimates of mean yields fo the two sole crops are needed as divisors. However there is no need for the sole crops to be randomized and grown on plots with the main experiment. Sufficient information for the calculation and interpretation of LERs can be obtained from sole crop areas alongside the experimental area. This will tend to improve the precision of the experiment by reducing block sizes and also simplifies the pattern of plot size.
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