 
RANK ANALYSIS IN INTERCROPPING EXPERIMENT
6.1 INTRODUCTION
In Agricultural field experiments, simultaneous growing of two or more crops on
the same piece of land in separate rows is known as intercropping. The experiments
are conducted on sole crop (only one crop) and intercropping crops like combination
of Cotton with Tur or Cotton with other crops or combination of any crops. In
such a case we get data of not the sole crop but the data of intercrop from the
same plot. The data are not absolute values, it is in the proportions. Intercropping
experiment is very beneficial small scale
formers, through
maximum income. The following advantages and
experiment they get
purpose of use intercropping
experiment described by Aloke [8 ],
6.1.1
Advantages of Intercropping Experiments
The following are major advantages of intercropping experiment,
Increasing cropping intensity
Diversification of crops
Mitigating risks due whether aberrations.
Optimal use basis resources i.e. moisture , light and nutrients.
Insect, pest and weed control.
     

6.1.2 Purpose of studied Intercropping Experiment
Identify the appropriate crop combination so that the yield of the base
crops is not sacrificed.
Identify the appropriate crop combination so that the total production and
revenue is maximized.
Identify the proper geometry of planting component crops.
Evaluate the effect of singly or in combinations of several factors, such as
fertilizers, geometry, plant population, germ-plasm.
In intercropping experiment two or more crops are studied and in this study
problem arises such as appropriate statistical analysis. In the past, a good number of
attempts have been made by various research workers to identify appropriate
statistical experiment for various types of intercropping exper iment and their method
of analysis. As regards, the problem of design, it will not be out of place to
mention that they are not very different from those of sole cropping Mead &
Riley [ 64
]. The problem of designing of experiments has to be viewed more
variability in intercropping experiment than in sole cropping experiment. It is
generally accepted that more than one analysis should be applied to Intercropping
experiment Mead & Stern [65 ].
In various types of methodological problems for analysis of intercropping data are
concerned,
considerable interest
has recently been generated by advocating
different approaches. In this analysis we convert the intercropping data in Land
equivalent ratio (LER) . On the LER data, we test and compare the effect of
Parametric Two- way ANOVA and Non-Parametric Test such as Friedman [ 25 ] .
     

6.2
TWO-WAY ANOVA
INTERCROPPING
OR
RBD
PARAMETRIC
TEST
FOR
EXPERIMENT
The method of Analysis of variance (ANOVA) is usually adopted to the data
generated from the planned experiments to draw valid, reliable and unbiased
conclusions. However,
applications
of this
techniques
is
based
on the
following assumption,
Homoscadasticity of treatment variances (Comman variance ).
Independence of observations.
Additivity of the effects of components involved in the model.
Error components follows normal distribution.
In this chapter , Randomized Block Design (RBD) special case of
Two-way
ANOVA is taken as a case study.
6.2.1 Mathematical Model of Two-Way ANOVA or Randomized Block Design
Let jth individual unit in the ith treatment may be represented by the equation
yij= µi + ti + rj + eij
Where, µi = Overall mean
ti = ith treatment effect.
rj = jth replication effect
eij = error term.
     

This means that any yij is made up of overall mean + treatment effect + an error
term. Since the effects are added in this model, it is known as a linear additive model. It
is commonly known as linear additive model or an Analysis of Variance model.
6.2.2
Computational formulae for Two-way ANOVA or RBD
Correction Factor =
Total S.S. =
- CF
Block or Replication S.S. (R.S.S.) =
Treatment S.S. (T.S.S.) =
CF
- CF
Error S.S. = Total S.S - Replication S.S. - Treatment S.S.
With these results the analysis of variance table
ANOVA table for RBD with t treatments and
is completed. The form of
r replicat ions each is given in
Table 6.1
Table 6.1 General ANOVA (Two-Way Classification ) or RBD
Source of
D.F.
S.S.
M.S.S.
F ratio
Replication
r
1
RSS
RMS
RMS / EMS
Treatment
t
1
TSS
TMS
TMS / EMS
ESS
EMS
Variation
Error
Total
(t
1) ( r
rt
1
1)
Total S.S.
     

6.2.3 Using Purpose of Analysis of Variance (ANOVA)
The ANOVA has two main purposes for use in analysis of experimental
design. First is to provide, from the error mean square (EMS), 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. The
second purpose the ANOVA 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 (TSS) into component sums of squares and the interpretation
of the relative sizes of the component mean squares Mead R. [ 63 ].
6.3 FRIEDMAN NON-PARAMETRIC TEST ALTERNATIVE FOR TWO-WAY
ANOVA INTERCROPPING EXPERIMENT
Analysis of experimental data is based on the assumptions like normality,
independence and homoscadicity of the observations. However, there may arise
experimental situations where these assumptions, particularly the assumption of
normality, may not be satisfied. In such situations non-parametric test procedures may
become quite useful. A lot of attention is being made to develop non-parametric tests for
analysis of experimental data. Most of these non-parametric test procedures are based on
rank statistic or rank analysis. The rank statistic has been used in development of these
tests as the statistic based on ranks is 1) Distribution free
2) Easy to simplified
     

3) Simple to explain and understand. The another reason of use of the rank statistic is
due to the well known result that the average rank approaches normality quickly a n
(number of observations) increases, under the rather general conditions, while the same
might not be true for the original data .Some non-parametric test are available for Twoway ANOVA or Randomized Block Design (RBD) also use this test in case RBD
Intercropping experiment Rajender [ 79 ].
Friedman Test
The K-W test is alternative Non-Parametric test for Parametric CRD. A CRD is used
when experimental units are homogeneous in a blocks. However , there do occur
experimental situations where one can find a factor , which, though not of interest to the
experimenter , does contribute significantly to the variability in the experimental
material. Various levels of this factor are used for blocking. For the experimental
situations where there is only one nuisance factor, the block designs are being used. The
simplest and most commonly used block design by the agricultural research or any
industrial experimental workers is Randomized Block Design (RBD). The problem of
non-normality of data may also occur in RCB design as well.
Friedman test is useful for non-normality data in case of Two-way ANOVA or
RBD situations , Friedman [ 25 ] . Let there are v treatments that are arranged in N = vb
experimental units arranged in b blocks of size v each. Each treatment appears exactly
once in each block.
     

The data generated though a RCB design can analyzed by the following linear model
Yij
i
i
ij
v; j
b
Where Yij is yield (response) of the i th experimental unit receiving the treatment in j th
block.
i
is the effect due to ith treatment.
is the effect of jth block.
ij
is random error
in response. Now we are interested to test the equality of treatment effect. In other
words, we want to test the null hypothesis Ho :
the alternative H1 : at least two of the
1
=
v=
2
(say) against
. Friedman test statistic is expressed
i
in following two different forms as follow Rangaswamy [ 78 ].
2
distribution :
This test statistic are classified in following two ways , which is described by Rajender
[ 79 ],
Without tie observations in block :
Arrange the observations in v rows (treatments) and b columns (blocks). The
observations in the different rows are independent and those in different columns are
dependent. Rank all the observations in a column (block) i.e. ranks are assigned
separately for each block. Let Rij be the
rank of the observations pertaining to i th
treatment in the jth block .
from 1 to v, therefore sum of ranks in j
has been done within blocks
th
v
block is Rj
Rij
i 1
v(v 1)
v 1
and Rj =
2
2
v2 1
and the variance is
. Sum of ranks for each treatment is Ri
12
b
Rij . If the
j 1
     

treatment effects are all the same then we expect each Ri to be equal b(v+1)/2, that is,
under Ho,
E( Ri)
1 bv(v 1)
v
2
b(v 1)
2
differences in the treatment effects.
v
Let
2
b(v 1)
Ri
2
S
i 1
The Friedman test statistic is then defined as
12S
bv (v 1)
T
12
bv(v 1)

12
bv (v 1)
v
i 1
b (v 1)
Ri
1
2
v
Ri 2 3b(v 1)
2
( v 1)
i 1
The method of determining the probability of occurrence when Ho is true of an observed
values of T depends upon the sizes of v and b . For large b and v, the associated
2
distribution with v-1 d.f.
With tie observations in block :
When there are ties among the ranks for any given block, the statistics T must be
corrected to account for changes in the sampling distribution. So if ties occur then we use
following statistic
v
Ri 2 3b2 v(v 1)
12

2
i 1
b
bv
bv(v 1)
( v 1)
gj
t
j
3
js
s
(v 1)
     

Where gj is the number of sets of tied ranks in the jth block and tjs is the size of the jth set
of tied ranks in the ith block.
Pair-wise Comparisons
When the Friedman test rejects the null hypothesis that the all treatment effects are not
the same, it is of interest to identify significant difference between the paired treatments.
therefore , a test procedure for making pair wise comparisons is needed. The null
hypothesis Ho :
i
=
i

against H1 :
Ri
Ri '
i
Zp
i
for all
bv (v 1)
i, i1 1, 2,........, v, i
6
i'
-1) and Zp is the quantile of order 1-p under the standard normal
distribution. From the above, we can say that the least significant difference between the
c
Zp
bv(v 1)
6
the help of
following example.
6.4
LAND EQUIVALENT RATIO (LER)
In case of intercropping experiment, the form of underlying distribution of
the data is not known, arises while handling the data from intercropping
experiments. In such experiments the physical yields of both the crops are
     

converted into univariate variable and this variable known as Land Equivalent
Ratio (LER) Gandhi Prasad [ 71 ].
6.4.1 The computational Procedure of LER
Where,
YA = The yield of base crop grown singly
YB = The yield of companion crop grown singly
YAI = The yield of base crop gown in intercropping
YBI = The yield of companion crop gown in intercropping
According to Wiley [ 96 ] , the most generally useful single index for
expressing the yield advantage is probable the Land Equivalent ratio (LER),
defined as the relative land area required as sole crops to produce the same yields
as intercropping. The advantages of LER ( Mead & Wiley [ 66 ] ) are that it
provides standardized basis so that crops can be added to form combine yield.
6.4.2 Illustration makes the LER clear
Table 6.2 : Yield of Sole and intercropping crops with Partial LER
Treatment (Crop )
Cotton (Sole)
Tur (Sole)
Intercrop Cotton + Tur (yield of single
plot)
Cotton Partial LER
Tur Partial LER
Yield (Kg/Ha)
34
18
24 + 10
24/34 = 0.70
10/18 = 0.55
     

Where,
YA = The yield of base crop grown singly ( i.e. Cotton ) = 34
YB = The yield of companion crop grown singly (i.e. Tur ) = 18
YAI = The yield of base crop gown in intercropping (i.e. Cotton) = 24
YBI = The yield of companion crop gown in intercropping (i.e. Tur)
= 10
PLER(Cotton) =
and PLER (Tur) =
LER = PLER(Cotton) + PLER (Tur)
LER = 0.70 + 0.55 = 1.25
The LER of the system is 1.25. This means that 25 % more land would be
required as sole crops to produce the same yields as intercropping.
6.5 INCOME EQUIVALENT RATIO (IER)
Income Equivalent Ratio (IER) is similar in concept of LER , except that yield
is measured in terms of net income, rather than plant product productivity.
Because income is a function of both yield and crop price, even if the agronomic
response is consistent, IER for intercrops may vary in different years as crop
prices fluctuate.
IER ( or LER) can be determined for systems involving more than two crops by
summing the intercrop to sole crop yield ( or net income) ratios of each crop
included in the intercropping system.
     

6.5.1 The computational Procedure of IER
Where,
IA = The income of base crop grown singly
IB = The income of companion crop grown singly
IAI = The income of base crop gown in intercropping
IBI = The income of companion crop gown in intercropping
6.5.2 Illustration makes the IER clear
Assume price of Cotton per Kg = 45 Rs.
Assume price of Tur per Kg = 30 Rs.
Yield (Kg) of Cotton (Sole) = 18 Kg
and
Yield (Kg) of Tur (Sole ) = 10
Kg
Yield (Kg) of Cotton (Intercrop) = 15 Kg & Yield (Kg) of Tur (Intercrop) =
08 Kg
Income of Cotton (Sole )= Yield * Price = 18 * 45 = 810 Rs.
Income of Tur (sole ) = Yield * Price = 10 * 30 = 300 Rs.
Income of Cotton (Intercrop )= Yield * Price = 15 * 45 = 675 Rs.
Income of Tur (Intercrop) = Yield * Price = 08 * 30 = 240 Rs.
     

Table 6.3 : Income of Sole and intercropping crops with Partial IER
Treatment (Crop )
Income (Kg/Ha)
Cotton (Sole)
810
Tur (Sole)
300
Intercrop Cotton + Tur (yield of single
675 + 240
plot)
Cotton Partial IER
675 / 810 = 0.83
Tur Partial IER
240 / 300 = 0.80
Where,
IA = The income of base crop grown singly (Cotton) = 810
IB = The income of companion crop grown singly (Tur ) = 300
IAI = The income of base crop gown in intercropping (Cotton)
= 675
IBI =
The income of companion crop gown in intercropping
(Tur) = 240
PIER(Cotton) =
and PIER (Tur) =
IER = PIER(Cotton) + PIER (Tur)
Where, PIER = Partial Income Equivalent Ratio
IER = 0.83 + 0.80 = 1.63
     

The IER of the system is 1.63. This means that 63 % more income would be
required as sole crops to produce the same income as intercropping.
6.5.3 Interpretation of IER or LER
Table 6.4 : IER or LER Interpretation
IER or LER Value
Equal to 1 ( = 1)
Interpretation
Sole & intercrop pattern are equivalent in their
yields or income.
Greater than 1 ( > 1) The intercropping pattern provides a positive yield
or income benefit.
Less than 1 ( < 1)
The sole cropping pattern provides a greater yield
or income than does the intercropping patter.
.
6.6 EXISTING
ANALYTICAL PROCEDURE
FOR
INTERCROPPING
EXPERIMENT
On the basis of using existing Two-way ANOVA or Randomized Block Design
i.e. Parametric Test and alternative Non-parametric Friedman [ ] Test for Twoway ANOVA . Using existing analytical algorithm for solve Two-way ANOVA or
RBD Intercropping Experiment.
Step 1 : Statistical Analysis of Land Equivalent Ratio (LER)
Where,
YA = The yield of base crop grown singly
YB = The yield of companion crop grown singly
YAI = The yield of base crop gown in intercropping
     

YBI = The yield of companion crop gown in intercropping
PLER(Crop 1) =
and PLER (Crop 2) =
LER = PLER(Crop 1) + PLER (Crop 2)
Where, PLER = Partial Land Equivalent Ratio
Step 2 : Statistical Analysis of Income Equivalent Ratio (IER)
Where,
YA = The Income of base crop grown singly
YB = The Income of companion crop grown singly
YAI = The Income of base crop gown in intercropping
YBI = The Income of companion crop gown in intercropping
PIER(Crop 1) =
and PIER (Crop 2) =
IER = PIER(Crop 1) + PIER (Crop 2)
Where, PIER = Partial Income Equivalent Ratio
Step 3 : Parametric Test Two-Way ANOVA or Randomized Block Design
(RBD)
Applying Two-way ANOVA Parametric Test or RBD for univariate variable i.e.
LER & IER. Procedure of this described in 6.2.
Step 4 : Friedman Non-Parametric Test alternative of Parametric Test Two-way
ANOVA
Applyoing Friedman Non- Parametric Test for univariate variable i.e. LER & IER. .
Procedure of this test described in 6.3.
     

6.7 NUMERICAL EXAMPLE
We taken as hypothetical data relevant to Two-way ANOVA or Randomized
Block Design Intercropping experiment. In this experiment we taken generalize
two crop
( Crop 1 & Crop 2) yield and income in case of sole and
intercropping. Separately analyze this data in Yield and Income approach.
( Crop 1 = Base Crop, Crop 2 = Companion crop or Intercrop )
Table 6.5 : Yield (Kg/ha) of Intercropping and Sole crop
Treatment
T1
T2
T3
T4
Crop
Crop
Crop
Crop
Crop
Crop
Crop
Crop
Crop
1
2
1
2
1
2
1
2
Intercropping Yield
(Kg/ha)
Replications (Block)
I
II
III
2125
2679
3166
263
649
474
3059
3266
3753
618
474
757
3926
3959
4099
359
574
439
4133
4189
4022
613
858
703
Sole Crop Yield (Kg/ha)
Replications (Block)
I
II
III
2389
2789
3100
324
743
287
3478
3607
3768
689
634
867
4056
3865
5321
432
687
532
4478
5876
5023
648
838
698
Step 1 : Statistical Analysis of Land Equivalent Ratio (LER)
Using observed yield data (Table 6.5 ), calculate LER for sole and Intercrop yield.
Table 6.6 : Rank analysis of Land Equivalent Ratio
Treatment
T1
T2
T3
T4
Land Equivalent Ratio
(LER)
Rep. - I Rep.Rep. II
III
1.70
1.83
2.67
1.78
1.65
1.87
1.8
1.86
1.6
1.87
1.74
1.81
Ranks for LER
Rank Total
Rep. - I
Rep.- II
1
2
3
4
3
1
4
2
Rep.III
4
3
1
2
08
06
08
08
     

Step 2 : Statistical Analysis of Income Equivalent Ratio (IER)
Using Analytical procedure step 4 and Table 6.5
Intercrop income.
,calculate IER for sole and
( Crop 1 Cost / Kg = 70 Rs.
and Crop 2 Cost / Kg = 48
Rs. )
Table 6.7
: Income ( Rs./ha) of Intercropping and Sole crop
Treatment
T1
T2
T3
T4
Intercropping Income (Rs.)
Replications (Block)
I
II
III
148750 187530 221620
12624
31152
22752
214130 228620 262710
29664
22752
36336
274820 277130 286930
17232
27552
21072
289310 293230 281540
29424
41184
33744
Crop
Crop
Crop
Crop
Crop
Crop
Crop
Crop
Crop
1
2
1
2
1
2
1
2
Sole Crop Income (Rs.)
Replications (Block)
I
II
III
167230 195230 217000
1552
35664
13776
243460 252490 263760
33072
30432
41616
283920 270550 372470
20736
32976
25536
313460 411320 351610
31104
40224
33504
Using income data (Table 6.7 ), calculate IER for sole and Intercrop Income.
Table 6.8 : Rank analysis of Income Equivalent Ratio
Income Equivalent
Ranks for IER
Treatment Ratio(IER)
Rep. - I
Rank Total
Rep.-
Rep. -
II
III
Rep. - I
Rep.- II
Rep.III
T1
1.70
1.83
2.67
1
3
4
08
T2
1.78
1.65
1.87
2
1
3
06
T3
1.8
1.86
1.6
3
4
1
08
T4
1.87
1.74
1.81
4
2
2
08
     

Land Equivalent Ratio (LER) and Income Equivalent Ratio have the same results ,
so use Parametric test and Non-Parametric test on any one of LER or IER data.
Here we used LER data for further analysis.
Step 3 : Parametric Test Two-Way ANOVA or Randomized Block Design
(RBD)
Compute this Two-way ANOVA or RBD using MINITAB software.
Table 6.9 : Two-way ANOVA or RBD for
LER data
Source of
Variation
D.F.
S.S.
M.S.S.
F ratio
P value
Replication
or Block
Treatment
2
0.1154
0.0577
0.68
0.542
3
0.2077
0.0669
0.79
0.544
Error
6
0.5101
0.0850
Total
11
0.102
p value of treatment greater than 0.05 ( p>0.05), so all the treatment effects are
not statistically significant ( Non-Significant ). The effect of all treatment have the
same.
Step 4 : Friedman Non-Parametric Test alternative of Parametric Test Two-way
ANOVA
Friedman Non- Parametric
Test
for
univariate variable i.e. LER or
IER ranks.
Procedure of this test described in 6.3.
     

Using MINITAB Software , calculate T value of Friedman Test
12S
bv (v 1)
T
12
bv(v 1)
12
bv (v 1)
v
i 1
b (v 1)
Ri
1
2
v
Ri 2 3b(v 1)
2
( v 1)

i 1
T = 0.60
p = 0.896
p value this test greater than 0.05 ( p>0.05), so all the treatment effects are not
statistically significant ( Non-Significant ). The effect of all treatment have the
same.
6.8 PROPOSED ANALYTICAL PROCEDURE
By using existing analytical procedure most of the times some
problems arises in
analysis of
by researchers
two-way ANOVA intercropping experimental data
agricultural. Therefore, we proposed some important analytical procedure for
in
non-
normal data in Two-Way ANOVA or RBD Intercropping Experiment. Generally
researcher use parametric ANOVA for intercropping by taking total of monetary
returns of both crops. Some times they also use rank analysis of the data of monitory
returns. This is not an appropriate procedure and results of experiment may vary. In
many cases. Therefore,
we
propose
following procedure for the analysis of
intercropping experiments using rank analysis and also Friedman NP test using LER
and IER .
     

Generally in intercropping experiment the data are
not analysed by usual
procedure as in the same plot we take two crops. So the analysis
become
complicated. The researcher most of the time do not know the proper procedure.
Most of the times the data received from these experiments are heterogeneous.
Therefore the rank analysis is appropriate in this case.
Two-way
ANOVA or
RBD
Intercropping experiment Friedman
Non-
Parametric Test is appropriate and this test used only for Land Equivalent
Ratio (LER) and Income Equivalent Ratio (IER) which is in the form of
ranks. So convert the two or more crops data with considering sole and
intercrop case into LER and IER. This propose procedure help for researcher
in case of using non-parametric with the special reference to LER and IER.
Here Non-Parametric test appropriate in such cases.
In some situation may be LER and IER rank have same. In this chapter such type
of case is arise, rank analysis of Land Equivalent Ratio (LER) and Income
Equivalent Ratio (IER) data have the same ranks , so use Friedman NonParametric test on any one of LER or IER rank.
Commonly researchers are going for LER, but they may also go for IER
as it is also easy and convenient. Because income is a function of
both
yield and crop price.
When researcher find which treatment is recommended
for getting more
yield or cash benefit so they should give ranks LER and IER. So for such
rank analysis Friedman Non-Parametric test is appropriate.
     

Generally in
Intercropping
agricultural
experiment,
most
of the
time
observed data of sole crop and intercrop does not follows non-normal
distribution. Proposed analytical procedure is
helpful for non-statistician
agricultural researcher to use proper method of analysis such as applicat ion of
LER and IER and when to use Non-Parametric Friedman test.
Using Rank based
analysis and Non-Parametric test on Land Equivalent
Ratio (LER) and Income Equivalent Ratio (IER) , when data shows all values
of
LER and IER greater than one ( > 1) , so the intercropping pattern
provides a positive yield or income benefit.
There may be adverse consequences in the recommendations of the research
experiments if researchers do not follow the proposed procedures.
     
