Research Journal of Mathematics and Statistics 4(1): 10-13, 2012
ISSN: 2040-7505
© Maxwell Scientific Organization, 2012
Submitted: May 27, 2011
Accepted: November 26, 2011
Published: February 25, 2012
The Effect of Blocking on the Experimental Precision of Tomato Yield
Adam Taiye Mudi and Aliyu Usman
Department of Mathematics, Statistics and Computer Science, Kaduna Polytechnic, Kaduna
Abstract: Past experiments on the effect of fertilizer levels on tomato yield have frequently used Randomized
Complete Blocked Designs (RCBD), even though the experimental areas seem relatively uniform with regard
to external variables. Consequently, Completely Randomized Designs (CRD) may have had better precision
with which to estimate treatment effects compared with RCBD. The objective of this study was to Estimate the
Relative Efficiency (ERE) of a RCBD compared with a CRD from the data sets obtained from Kwara State
Agricultural Development Project, Nigeria for three farm trials with the aim of estimating the effect with less
effort. The result of the analysis showed that ERE (RCBD to CRD) are (1.064, 0.9164 and 0.9068) respectively
for the three trials. Hence blocking is inefficient for future similar experiment under the same conditions.
Keywords: CRD: completely randomized design, ERE: estimated relative efficiency, MSB,: mean square for
blocking effect, MSE : error mean square, RCBD: randomized complete block design, SSB,: sum
of squares for blocking effect, SSE,: error sum of squares
occasional failure to adequately minimize the effect of
soil heterogeneity. Generally, the greater the
heterogeneity within blocks, the poorer the precision of
treatment effect estimates (Bisgaard and Steinberg, 2007).
But in particular, it is of interest to investigate this
precision with respect to some specific crops - tomatoes
in this study.
INTRODUCTION
When designing field experiments, blocking is often
used to increase precision on treatment effects by
reducing experimental error variance (Montgomery,
2009). Blocking stratifies experimental units into
homogenous subgroups and in field trials, plots are
normally blocked according to their proximity to each
other. Blocking will increase treatment precision only if
plots are blocked according to one or more varying
external factors (irrigation gradient, soil nutrient status,
etc.). If however, an experimental area is homogenous,
blocking may actually decrease the precision of
estimating treatment effects. This results from a Larger
Means Square error (MSE) term in the ANOVA since
error degrees of freedom are reduced without a
comparable reduction in error Sum of Squares (SSE). In
this situation, a Completely Randomized Design (CRD)
would estimate treatment effects more precisely than a
Randomized Complete Block Design (RCBD) (Anbari
and Lucas, 1994). Block designs arise in comparative
experimentation as fundamental devices for improving
precision when working with heterogeneous experimental
units. The blocks are simply a partition of the units into
(say) b sets exhibiting homogeneity within sets. Given the
blocks, of sizes k1, k2, k, ..., kb, a block design is an
assignment of v treatments to the ∑ bj =1 k j experimental
units. Optimality theory for block designs attempts to
determine which of the many possible assignments is in
some sense the best. The randomized block, Latin square
and other complete block types of experiments may be
inefficient for large number of treatments, because of their
Randomized Complete Block Design (RCBD): In a
Completely Randomized Design (CRD), treatments are
assigned to the experimental units in a completely random
manner. The random error component arises because of
all the variables which affect the dependent variable
except the one controlled variable, the treatment (Federer,
1967). Naturally, the experimenter wants to reduce the
errors which account for differences among observations
within each treatment. One of the ways in which this
could be achieved is through blocking. This is done by
identifying supplemental variables that are used to group
experimental subjects that are homogeneous with respect
to that variable. These create differences among the
blocks and make observations within a block similar. The
simplest design that would accomplish this is known as a
Randomized Complete Block Design (RCBD). Each
block is divided into k sub-blocks of equal size. Within
each block the k treatments are assigned at random to the
sub-blocks. The design is complete in the sense that each
block contains all the k treatments. The following layout
shows a RCBD with k treatments and b blocks (Gomez
and Gomez, 1984). There is one observation per treatment
in each block and the treatments are run in a random order
within each block. Hence, the statistical model is:
Corresponding Author: Adam Taiye Mudi, Department of Mathematics, Statistics and Computer Science, Kaduna Polytechnic,
Kaduna
10
Res. J. Math. Stat., 4(1):10-13, 2012
yij = : + Ji + $j + gij
Table 1: Model ANOVA for uniformity trial
Source
df
Blocks
b-1
Within blocks error
b(t-1)
Total
Bt-1
i = 1, … ,t
MS
MSB
MSR
j = 1, … ,b
where, yij
:
Ji
$j
gij
Table 2: Results of farm trial 1
Blocks
----------------------------------------------------Treatment (Fertilizer)
Site A
Site B
Site C
75 kg/hc
10.0
20.0
25.0
100 kg/hc
19.2
14.2
21.0
150 kg/hc
23.5
26.5
24.5
0 kg/hc
7.5
5.5
3.4
= The jth observation on treatment i
= The overall mean
= The ith treatment effect
= The jth block effect
= A random error term associated with the
ijth observation
Table 3: Results of farm trial 2
Blocks
----------------------------------------------------Treatment (Fertilizer)
Site A
Site B
Site C
75 kg/hc
12.0
15.4
22.2
100 kg/hc
16.5
22.4
28.0
150 kg/hc
24.5
27.1
21.7
0 kg/hc
6.7
6.9
5.8
Relative efficiency of the randomized complete block
design: As with CRD, the expected mean squares for
treatment and error are functions of the sample sizes (b,
the number of blocks), the true treatment effects ("1,...,"t)
as well as the variance of the random error terms F2
(Ganju and Lucas, 1999). By assigning all treatments to
units within blocks error variance is smaller for RCBD
than CRD (which combines block variations and random
error into error term). The relative efficiency answers the
question of how many times as many replicates would be
needed for CRD to have as precise of estimates of
treatment means as RCBD does? (Kuehl, 1994).
Moreover, this gives the notion of Relative Efficiency
(RE) in the context of estimation of treatment
comparisons. For two designs, D1 and D2 say, the RE of
D1 to D2 is defined as follows:
RE ( D1 to D2 ) =
or
Table 4: Results of farm trial 3
Blocks
-----------------------------------------------------Treatment (Fertilizer)
Site A
Site B
Site C
75 kg/hc
14.3
12.1
9.2
100 kg/hc
13.4
24.1
17.1
150 kg/hc
16.4
11.2
20.1
0 kg/hc
2.5
12.1
10.2
Table 5: Randomized block descriptives for trial 1
Mean
n
Treatment 1
16.5333
3
Treatment 2
22.3000
3
Treatment 3
24.4333
3
Treatment 4
6.4667
3
Block 1
14.9250
4
Block 2
17.9500
4
Block 3
19.4250
4
Total
17.4333
12
EfficiencyD1
EfficiencyD2
RE ( D1 to D2 ) =
VarD1
VarD2
Table 6: ANOVA table for trial 1
Source
SS
df
MS
Treatments
581.287
3
193.7622
Blocks
42.102
2
21.0508
Error
93.258
6
15.5431
Total
716.647
11
t: 4; b: 3; MSB: 21.0508; MSE: 15.5431
MSE ( D1 )
RE ( D1 to D2 ) =
MSE ( D2 )
where; VarD1 refers to (∑ Ckτ$k ) for design Di (i = 1, 2). In
our case D1 is RCBD with t treatments and r blocks and
D2 is a CRD with r replications for each of the t
treatments. The RE defined depends on the true variances
for the two designs which, of course, are unknown. The
best we can do then is to obtain the estimated RE, which
we shall denote by ERE (Hinkelmann and Kempthorne,
2007).
F
12.47
1.35
Table 7: Randomized block descriptives for trial 2
Mean
n
Treatment 1
18.3333
3
Treatment 2
18.1333
3
Treatment 3
24.8333
3
Treatment 4
5.4667
3
Block 1
15.0500
4
Block 2
16.5500
4
Block 3
18.4750
4
Total
16.6917
12
Motivation: There is no single blueprint for knowing
whether a particular plant will be more efficient under the
CRD or the RCBD. Hence, the suitable design is
governed by the notion of “fitness for purpose”. The
purpose of this study is to determine the effect of blocking
in tomato crop so as to know the suitability of either the
CRD or the RCBD. More precisely, to determine the
relative efficiency of the CRD to the RCBD to the tomato
crop so as to choose a more suitable design for it.
SD
5.1936
5.7507
2.7006
0.5859
7.5359
8.7964
9.5227
8.0715
p-value
0.0055
0.3270
SD
7.6376
3.5233
1.5275
2.0502
7.5527
8.9168
10.2063
8.2338
METHODOLOGY
Using the concept of uniformity trials, that is pooling
the treatment sums of squares with appropriate error sums
of squares from in the ANOVA table for the respective
designs, we have the respective ANOVAs for uniformity
11
Res. J. Math. Stat., 4(1):10-13, 2012
n
⎤
⎡ p
E ⎢ ∑ ( yi2. / n)⎥ = npµ 2 + n∑ t i2 +
i =1
⎦
⎣ i =1
trial which shall be used to estimate the relative efficiency
of the RCBD (Table 1). However, we need to outline the
variance component of CRD as a basis of comparison as
follows:
p
MSE =
SSE
=
p(n − 1)
∑∑ y
−
2
ij
i =1 j =1
p
1
n
∑y
2
i.
i =1
⎤
⎥
⎦
E ( MSE ) =
⎡ ⎛
⎞
⎛
⎞⎤
1
⎢ E ⎜ ∑ ∑ y 2 ⎟ − n1 E ⎜ ∑ yi2. ⎟ ⎥
P(n − 1) ⎢⎣ ⎝ i =1 j =1 ij ⎠
⎝ i =1 ⎠ ⎥⎦
p
n
( )
i =1
1
[npσ 2 − pσ 2 ] = σ 2
p(n − 1)
Comparison of RCBD and CRD: If the blocks were not
used the estimated error variance would be:
( )
E eij2 = σ 2
E eij = 0
i
(2)
This gives us a simple basis for comparison of RCBD and
CRD as follows:
In every designed experiment, we have:
p
i.
p
p
⎡
⎞⎤
⎛
1
2
2
2
2
2
2
⎢ npµ + n∑ t i + npσ − ⎜ npµ + n∑ t i + pσ ⎟ ⎥
p( n − 1) ⎣
⎠⎦
⎝
i =1
i =1
⇒ E ( MSE ) =
p
∑t =0
i =1
2
i.
i =1
⎡ p n
1
E ⎢ ∑ ∑ y2 −
p(n − 1) ⎣ i =1 j =1 ij
E ( MSE ) =
i =1
2
i.
∑y
1
n
p(n − 1)
∴
E ( MSE ) =
p
∑ E (e ) + 2 µ ∑ E (e )
n
⎡ p
⎤
⇒ E ⎢ ∑ ( yi2. / n)⎥ = npµ 2 + n∑ ti2 + pσ 2
i =1
⎣ i =1
⎦
p
n
p
1
n
SSB + SSR (b − 1) MSB + b(t − 1) MSR
=
bt − 1
bt − 1
Now;
yij = : + Ji +eij
∑ y = ∑ (µ + t
n
n
ij
j =1
i
j =1
+ eij
The estimated error variance with blocks is, of course,
MS(R), so that:
)
ERE ( RCBD to CRD) =
yi. = n: + nti + ei.
(
yij2 µ + ti + eij
p
i =1 j =1
2
= µ 2 + ti2 + eij2 + 2 µti + 2 µeij + 2ti eij
p
n
∑∑y
)
2
ij
= npµ 2 + n∑ t i2 +
i =1
p
p
p
n
∑t
i =1 i =1
i
+ 2µ∑
i =1
n
∑e
i =1 j =1
ij
p
+ 2∑
n
∑te
i ij
i = 1 j =1
∑ ∑ E (e ) + 2nµ ∑ t
p
p
n
2
ij
i =1 j =1
p
n
ij
i =1
ERE ( RCBD to CRD) =
i
ij
p
p
n
2
ij
i =1 j =1
i
ij
p
n
i =1 j =1
where; MSE and MSB are obtained from the ANOVA for
the completed experiment using an RCBD.
ij
n
i =1 j =1
i
ij
Data presentation and analysis: The following data
were obtained from Kwara State Agricultural
Development Project for three farm trials. The experiment
is for testing the effect of fertilizer levels on tomato yield
in kilogram/hectare using sites as the blocking factor. The
result is displayed in the Table 2, 3, 4 for the three farm
trials.
The Microsoft Excel 2007 spreadsheet was used and
produced the following ANOVA tables for the respective
analyses (Table 5 to10).
∑ ∑ E (e )
p
n
i =1 j =1
( b − 1) MSB + b( t − 1) MSE
( bt − 1) MSE
n
i =1 j =1
∑ ∑ E (e ) + 2 µ ∑ ∑ E (e ) + 2∑ ∑ t E (e )
p
n
p
n
i = 1 j =1
⎡
⎤
E ⎢ ∑ ∑ yij2 ⎥ = npµ 2 + n∑ ti2 +
i =1
⎣ i =1 j =1 ⎦
p
p
+ 2nµ ∑ t i + 2 µ ∑
Since we have carried out an experiment with real
treatments and not dummy treatments we do not know
MSR. Instead we only know MSE. Hence substituting
MSE for MSR in the equation above we have:
∑ E (e ) + 2 µ ∑ ∑ E (e ) + 2 ∑ ∑ t E ( e )
i = 1 j =1
p
⎡ p n
⎤
⇒ E ⎢ ∑ ∑ yij2 ⎥ = npµ 2 + n∑ ti2 +
i =1
⎣ i =1 j =1 ⎦
⇒
2
ij
i =1 j =1
p
⎡ p n
⎤
E ⎢ ∑ ∑ yij2 ⎥ = npµ 2 + n∑ ti2 +
i =1
⎣ i =1 j = 1 ⎦
+ 2µ∑
p
n
∑∑e
(b − 1) MSB + b( t − 1) MSR
( bt − 1) MSR
2
ij
p
⎡ p n
⎤
⇒ E ⎢ ∑ ∑ yij2 ⎥ = npµ 2 + n∑ ti2 + npσ 2
i
j
i
1
1
1
=
=
=
⎣
⎦
(1)
yi2. = ( µ + t i + ei . ) = n 2 µ 2 + n 2 t i2 + ei2. + 2n 2 µt i + 2nµei . + 2nt i ei
2
⇒ yi2. / n = ( µ + t i + ei . ) = nµ 2 + nt i2 + ei2. / n + 2nµt i . + 2nµei . + 2t i ei
2
⇒ yi2. / nµ 2 + nt i2 + ei2. / n + 2nµt i + 2 µei . + 2t i . ei .
Analysis 1 (Farm trial 1):
p
⇒
∑(y
i =1
2
i.
/ n) = npµ 2 + n∑ ti2 +
n
i =1
p
∑ (e
i =1
2
i.
p
p
p
i =1
i =1
i =1
/ n) + 2nµ ∑ ti + 2 µ ∑ ei . + 2∑ ti ei .
ERE ( RCBD to CRD) =
12
.
) + (3)(3)(155431
.
)
2(210508
= 10644
.
(11)(155431
.
)
Res. J. Math. Stat., 4(1):10-13, 2012
Table 8: ANOVA table for trial 2
Source
SS
df
MS
Treatments
591.183
3
197.0608
Blocks
23.582
2
11.7908
Error
130.985
6
21.8308
Total
745.749
11
t: 4; b: 3; MSB: 11.7908; MSE: 21.8308
F
9.03
0.54
CONCLUSION AND RECOMMENDATIONS
Table 9: Randomized block descriptives for trial 3
Mean
n
Treatment 1
11.8667
3
Treatment 2
18.2000
3
Treatment 3
15.9000
3
Treatment 4
8.2667
3
Block 1
11.6500
4
Block 2
14.8750
4
Block 3
14.1500
4
Total
13.5583
12
Table 10: ANOVA table for trial 3
Source
SS
df
MS
Treatments
173.676
3
57.8919
Blocks
22.902
2
11.4508
Error
140.912
6
23.4853
Total
337.489
11
t: 4; b: 3; MSB: 11.4508; MSE: 23.4853
depicted in Table 9 and the corresponding ANOVA in
Table 10. Hence, in particular in the area under study, for
tomato crop, the CRD has a better precision.
p-value
0.0121
0.6086
Hence future study on the effect of fertilizer levels,
(same levels) on tomato yield using the same sites as the
blocking factors may not be blocked. In a general note,
this case of tomato yield has provided us with evidence
that not in all cases that blocking could minimize
experimental error. It is therefore important to investigate
at the design stage whether blocking should be introduced
or not. It is therefore recommended that experimenters
and agronomist should subject individual crops to field
experiments before choosing a suitable design for it.
SD
2.5580
5.4342
4.4710
5.0836
6.2282
6.1646
5.2981
5.5390
F
2.47
0.49
p-value
0.1599
0.6365
REFERENCES
Anbari, F.T. and J.M. Lucas, 1994. Super-efficient
designs: How to run your experiment for higher
efficiency and lower cost. ASQC Technical
Conference Trans., 14: 852-863.
Bisgaard, S. and D.M. Steinberg, 2007. The design and
analysis of 2k- p X s prototype experiments,
Technometrics, 39: 52-62.
Federer, W.T., 1967. Experimental Design: Theory and
Application. Oxford Press, New Delhi, pp: 196-197.
Ganju, J. and J.M. Lucas, 1999. Detecting randomization
restrictions caused by factors. J. Stat. Planning
Inference, 81: 129-140.
Gomez, K.A. and A.A. Gomez, 1984. A Statistical
Procedure for Agricultural Research. John Wiley and
Sons, New York.
Hinkelmann, K. and O. Kempthorne, 2007. Design and
Analysis of Experiment. John Wiley and Sons. New
York.
Kuehl, R.O., 1994. Statistical Principles of Research
Design and Analysis. Duxbury Press, New York.
Montgomery, D.C., 2009. Design and Analysis of
Experiment, 4th Edn., John Wiley and Sons, New
Delhi.
Analysis 2 (Farm trial 2):
ERE ( RCBD to CRD) =
.
) + ( 3)(3)(218308
.
)
2(117908
= 0.9164
(11)(218308
.
)
Analysis 3 (Farm trial 3):
ERE ( RCBD to CRD) =
2(114508
.
) + (3)(3)(23.4853)
= 0.9068
(11)(23.4853)
DISCUSSION
The result showed that ERE (RCBD to CRD) are
(1.064, 0.9164 and 0.9068) respectively for the three trials
in Table 6 through Table 8 while the respective
descriptive statistics were depicted in Table 2 through
Table 4. Since the values are very close to 1, Completely
Randomized Designs (CRD) may have had better
precision with which to estimate treatment effects
compared with RCBD. Combined descriptive statistics
13