Effectiveness of split plot design over randomized complete

Journal of Biology, Agriculture and Healthcare www.iiste.org
ISSN 2224-3208 (Paper) ISSN 2225-093X (Online)
Vol.4, No.19, 2014
Effectiveness of Split-Plot Design over Randomized Complete
Block Design in Some Experiments
1David, I. J. and 2Adehi, M. U.
1Department of Mathematics, Ahmadu Bello University, Zaria-Nigeria
2Department of Mathematics, Nassarawa State University, Nasarawa-Nigeria
Correspondence: David, I. J. Email: ikwuoche8@gmail.com
ABSTRACT
One of the main features that distinguish split-plot experiments from other experiments is that they involve two
types of experimental errors; the whole plot (WP) error and the split-plot (SP) error. This research paper
compared the effectiveness of split-plot design (SPD) over randomized complete block design (RCBD). The data
used for comparison is a 21 x 52 split-plot experiment with three replicates. It is been carried out to evaluate the
threshing efficiency of an improved sorghum thresher; the three factors considered in the experiment are the feed
at two different rates, moisture content at five different levels and speed at five different rates. The data is
analyzed as split-plot and as randomized complete block design and their ANOVA results and relative efficiency
(RE) statistic values were compared. The result reveals the effectiveness of split-plot design over randomized
complete block design.
Key words: SPD, SP error, WP error, RCBD, Relative Efficiency
75
1. INTRODUCTION
Experiments performed by investigators in virtually all field of inquiry, are usually, to discover
something about a particular process or system. Literally, an experiment is a test (Montgomery, 2005). The
choice of experimental unit has been one of the signifying problems for various types of experiments. In field
experiments the size and shape of plot and the size and shape of block are of great influence and the choice on
these matters are of two types: statistical and others. Under statistical considerations we include topics such as
effect of size and shape of plot on error variance and accuracy of estimation while the non-statistical
considerations include such matters as the feasibility of particular sizes and shapes of plot from the point of view
of experimentation.
Most field (agronomic) experiments are large and the choice of a suitable experimental unit may have
some scope of ingenuity. The experimenter have to make the division of the experimental material into blocks in
such a way that the plots within blocks are as homogeneous as possible, that is, block should remove as much
trends in the material as possible. Kempthorne (1952) stated that if an inappropriate design for an experiment is
used it faces considerable defects and it may not be easy to find a reasonably homogeneous area of dimensions
of the experiment. In addition, column effects at one end of the experiment may be very different from column
effects at the other end and such an effect would result in lower efficiency.
Cox (1958) stated that in a randomized complete block design (RCBD) the effects of certain sources of
variation reduced by grouping the experimental units or by the use of adjustments based on a concomitant
variable, the remaining variation convert into effective random variation by randomization. He further discussed,
an agricultural field trials in the form of factorial experiments arranged in randomized complete block design the
treatments must not be too large if these design is to be effective: often the number of treatments needs to be less
than 20 for randomized complete block design, and sometimes the limit is lower than this. When the numbers of
treatments exceed this limit, the blocks or rows and columns tend to become too heterogeneous resulting in a
high residual standard deviation. He concluded that the grouping of the material into blocks eliminates the effect
of constant differences between blocks and the randomization allows us to treat the remaining variation between
units as random variation, so far as assessing treatment comparisons is concerned. The success of the
randomized complete block design depends on a good grouping of the units into blocks, and the general idea of
grouping into blocks is of fundamental importance and is not only frequently used in simple experiments but also
forms the basis for most of the more complicated designs.
Cox (1958) again argued that the use of split-plot design is necessarily not when the whole plot
treatments are likely to be compared with less precision than the subplot treatments because the whole plot
treatments need not be restricted to a single factor but may consist of all combinations of the levels of several
factors. He stated that the split-plot principle is applied to experiments on processes in which there are several
stages and it may then be convenient to work with large batches of material at the first stage, dividing into
smaller batches for the application of the treatment at the second stage.
Cochran and Cox (1956) stated that the chief practical advantage of the split-plot arrangement is, it
enables factors that require relatively large amount of materials and factors that require only small amount of
materials to be used. In addition, factors of second type can often be included at very little extra cost, and some
additional information obtained very cheaply. They summarized that , the split-plot design is advantageous if the

Vol.4, No.19, 2014
sub-plot (B) treatment and the interaction of the whole plot (A) and sub-plot (B) treatment effects are of greater
interest than the whole plot treatment (A) effect, or if the whole plot (A) effect cannot be tested on small amount
of materials. They gave two disadvantages mentioned by experimenters, which are; sometimes the whole-unit
error is much larger than the sub-unit error. It may occur at times that the effect of whole-plot (A), though large
and exciting, is not significant, whereas that of sub-plot (B), which is too small to be of practical interest, is
statistically significant. The experimenter tends to be uncomfortable in reporting results of this type. Secondly,
the fact that different treatment comparisons have different basic error variances makes the analysis more
complex than with randomized complete block design, especially if some unusual type of comparison is being
made. They commended that the split-plot Latin square eliminates error variation, which arise from two types of
grouping, and is preferable to randomized complete block design (RCBD). They cited an example presented by
Yates (1935) where he summarized 22 field experiments in Latin squares where the plots were split into halves,
he found a substantial net increase in precision over randomized complete block design, and the superiority was
so pronounced that even the whole-plot comparisons would have been less precisely determined in randomized
complete block design.
Montgomery (2005) stated that the incorrect testing of effects for running an experiment in a
completely randomized fashion instead of the split-plot design is because the error associated with the hard-to-change
factor inflates the variance of the regression coefficient for the easily changed factor. Jones and
Nachtcheim (2009) stated that when an industrial experiment, is prescribed wrongly it lead to incorrect analysis,
which inflates the Type I, error rate for whole plot sub factors, as does the Type II error rate for split plot sub
factors and whole plot by split-plot interactions. Therefore, one way to avoid these mistakes is to plan the
experiment as a split-plot design in the first place. Not only does this avoid mistakes, it also leverages the
economic and statistical efficiencies. Besides the less expensiveness of running the split-plot design, it is often a
more statistically efficient design compared to other experimental designs because it handles large data with less
error. Bisgaard et al. (1996) discussed this issue and stated that if a split-plot experiment is wrongly analyzed as
a completely randomized experiment, some factors may be declared significant when they are not and vice versa.
Kowalski and Potcner (2003) stated that when a designed experiment uses blocks such as days or
batches, the analysis of the experiment includes a term for these blocks. When a designed experiment is
performed by fixing a factor and then running the combinations of the other factors, using different sized
experimental units or using a different randomization for the factors (a split-plot design), the analysis should
incorporate these features. They analyzed 23 factorial treatments design with two replicates using the split-plot
approach. The responses were first analyzed incorrectly as if they came from a completely randomized design
and then ran correctly as a split-plot design. It was observed that some treatment effects were declared
significant for the completely randomized design and not for the split-plot design and vice versa. They stated that
the cause is, in the completely randomized design, all factors effect use the mean square error as the estimate of
experimental error. In a split-plot experiment, however, there are two different experimental error structures: one
for the WP factor and one for the SP factor. They concluded that it is because of the two separate randomizations
that occur when running the experiment.
2. MATERIALS AND METHODS
A 21 x 52 split-plot experiment with three replicates was carried out to evaluate the threshing efficiency
of an improved sorghum thresher. The three factors considered in the experiment are the feed (whole plot factor)
at two different rates, moisture content (first sub-plot factor) at five different levels and speed (second sub-plot
factor) at five different rates. The data will be analyzed as split-plot design and as randomized complete block
design, this is to check the effectiveness of the split-plot design over randomized complete block design. Their
respective models are as follows;
76
2.1 Split-Plot Design Model
The model is a linear additive model, it is given as;
ijkl i k ik j l ij il jl ijkl Y = μ +a +g + ℓ +b +h + (ab ) + (ah ) + (bh ) + ℓ …………………… (1)
where: Yijkl is the response; μ is a constant; i a is the WP factor; k g is the block effect; ik
ℓ is the WP error, NID
2 s ); j b is the first SP factor; l h
~ (0,
is the second SP factor; ij (ab ) , il (ah ) , jl (bh ) are the interaction
ℓ is the SP error, NID ~ (0,
factors; ijkl
2 s ); i = 1, 2, 3, …, a (levels of WP factor), j = 1, 2, 3, …, b (levels of
SP factor), k = 1, 2, 3, …, r (number of replicates or blocks). Note, the model did not include the three-factor
interaction.

Vol.4, No.19, 2014
Table I: Sketch of the ANOVA Table for Split-Plot Design Model
Source Df Sum of Square Mean Square Fcal
Block r – 1 SSBLOCK MSBLOCK MSBlock/MSMPE
A a – 1 SSA MSA MSA / MSMPE
Main plot error (r – 1)( a – 1) SSMPE MSMPE
B b – 1 SSB MSB MSB / MSSPE
C c – 1 SSC MSC MSC/MSSPE
AB (a – 1)( b – 1) SSAB MSAB MSAB / MSSPE
AC (a – 1)( c – 1) SSAC MSAC MSAC / MSSPE
BC (b – 1)( c – 1) SSBC MSBC MSBC / MSSPE
Split-plot error a(r-1)( b – 1)(c – 1 ) SSSPE MSSPE
TOTAL abcr-1 SSTOTAL
Source: Jones and Nachtcheim (2009)
where,
2
... , SSA =Σ
2
.. , SSB =Σ
U = μ +a + b + ℓ (2)
U = μ +a +d + b +h + (ab ) + (ah ) + (bh ) + ℓ ……………………….. (4)
77
SSTOTAL =
abcr
a
i
b
j
c
l
r
k
ijkl
2
2 .... U
ΣΣΣΣU − , SSBLOCK =Σ
2
....
U
−
r U
k
k
abc abcr
2
U
2
... ...
,
−
a U
i
i
bcr abcr
SSMPE =ΣΣ Σ Σ
U
i k i k
+
bc bcr abc abcr
U
−
U
−
a U
i
r
k
a
i
r
k
2
....
2
...
2
...
2
....
U
−
2
. ..
b U
j
j
acr abcr
, SSC = Σ
2
....
U
−
2
.. .
c U
l
l
abr abcr
a
Y Y
2
Y
ij i j
SSAB = ΣΣ −Σ −Σ +
i
b
j
a
i
b
j
Y
abcr
acr
bcr
cr
....
2
. ..
2
...
2
..
a
Y Y
Y
2
, SSAC = ΣΣ −Σ −Σ +
i
c
l
a
i l i l
i
c
l
Y
abcr
abr
bcr
br
....
2
.. .
2
...
2
. .
and
b
Y 2
Y
Y
jl j l
SSBC = ΣΣ −Σ −Σ +
j
c
l
b
j
c
l
Y
abcr
abr
acr
ar
....
2
.. .
2
. ..
2
. .
hence, the split-plot error will be,
SSSPE = SSTOTAL – SSBLOCK – SSA – SSMPE – SSB – SSC – SSAB – SSAC – SSBC
2.2 Randomized Complete Block Design Model (RCBD)
Three models can be identified from a randomized complete block design, they are;
1. RCBD model without replication within blocks and one observation per cell.
ij i j ij
where;
Yij is the response yield; μ is the overall mean; αi is the effect of the ith row (treatment); βi is the effect of the jth
2 column (block) and ℓ is the experimental error which is NID ~ (0,
s ); i = 1, 2, 3,…, p j = 1, 2, 3,…, b.
ij
2. RCBD model with replication within blocks but no interaction between treatments.
U = μ +a + b +d + ℓ
ijkl i j (i )k (ijk ) l
(3)
where;
Yijkl is the response from the lth experimental unit in block i, the kth randomization and given the jth treatment; μ
is the overall mean; i a
is the ith block effect; j b is the jth treatment effect; (i )k d is the kth restriction error within
the ith block (i.e. Df for (i )k d is zero) and ijk
2 s ).
ℓ is the experimental error, NID~ (0,
3. RCBD model with replication within blocks and interaction between treatments.
ijkl i k j l ij il jl ijkl
l where;
Yijk, h
is the kth response from the ith and jth effects; μ , is the overall mean constant; a , is the ith effect of
i treatment A; d , is the block effect; b , is the jth effect of treatment B; , is the lth effect of treatment C;
k j ij (ab ) , il (ah ) , jl (bh ) are the interaction of the ith, jth and lth effect of treatment A and B, A and C, B and C;

Vol.4, No.19, 2014
ℓ , is the random error caused by the kth response from the jth effect of B and lth effect of C in the ith effect of
2
... , SSA =Σ
a
a MS a b MSWPE SPE …………………………………………………… 5
78
ijk
A, NID ~ (0,
2 s ); i = 1, …, a, j = 1, …, b, l = 1, …, c, k = 1, …, n.
The third model is the adopted model for this research in comparison with the split-plot design model
also; the three-factor interaction was not included in the model just to obtain adequate degree of freedom for
error to estimate the interaction effect adequately.
Table II: Sketch of the ANOVA Table for the Third RCBD Model
Source Df Sum of Square Mean Square Fcal
Block r-1 SSBLOCK MSBLOCK MSBlock/MSE
A a-1 SSA MSA MSA / MSE
B b-1 SSB MSB MSB / MSE
C c-1
AB (a-1)(b-1) SSAB MSAB MSAB / MSE
AC (a-1)(c-1)
BC (b-1)(c-1)
Error (abc-1)(r-1) SSE MSE
TOTAL abcr-1 SSTOTAL
Source: Montgomery (2005).
where,
SSTOTAL =
abcr
a
i
b
j
c
l
r
k
ijkl
2
2 .... U
ΣΣΣΣU − , SSBLOCK =Σ
2
....
U
−
r U
k
k
abc abcr
2
U
2
... −
....
, SSB
a U
i
i
bcr abcr
=Σ
2
....
U
−
2
. ..
b U
j
j
acr abcr
, SSC =Σ
2
....
U
−
2
.. .
c U
l
l
abr abcr
Y Y
2
Y
ij i j
, SSAB = ΣΣ −Σ −Σ +
i
b
j
a
i
b
j
Y
abcr
acr
bcr
cr
....
2
. ..
2
...
2
..
,
a
Y Y
Y
2
SSAC = ΣΣ −Σ −Σ +
i
c
l
a
i l i l
i
c
l
Y
abcr
abr
bcr
br
....
2
.. .
2
...
2
. .
b
Y 2
Y
Y
jl j l
and SSBC = ΣΣ −Σ −Σ +
j
c
l
b
j
c
l
Y
abcr
abr
acr
ar
....
2
.. .
2
. ..
2
. .
hence, the sum of square error will be;
SSError = SSTOTAL – SSBLOCK – SSA – SSB – SSC – SSAB – SSAC – SSBC
2.3 Efficiency of Split-Plot Design Relative to RCBD
According to Hinkelmann and Kempthorne (2008), experimenters utilize the split-plot design in many
instances and circumstances for technical reasons. Under most circumstances, the split-plot design is been used
for purely technical and practical reasons, as the levels of some factor can be applied only to large experimental
units, which can then be “split” into smaller experimental units for application of the levels of the other factor.
This includes also the distinction between hard-to-change and easy-to-change factors in industrial
experimentation. It is, however, of interest to evaluate the efficiency of the split-plot design relative to the RCBD
with r blocks. The question then is given that we have carried out a split-plot experiment, what would have been
MSE for the RCBD? This, of course, determines how much information would have been available for all
treatment comparisons. Using the pooled treatment sums of squares with appropriate error sum of squares, that
r (ab – 1) MSE = r (a – 1) MSWPE + ra(b – 1)MSSPE
divide through by r we have
E΄ =
− + −
( 1) ( 1)
1
−
ab
where MSE = E΄ = is a weighted average of mean square WP error and mean square SP error; MSWPE, is the
mean square WP error; MSSPE, is the mean square split-plot error; a, is the number of levels of WP factor; b, is
the number of levels of SP factor.
From equation (1) and equation (4) the information on all treatment comparisons would then have been
proportional to 1/ E΄. The information on WP treatments from the split-plot experiment relative to randomized
complete block design is then E΄/MSMPE that is less than 1. For SP treatments and interaction effects from the
split-plot experiment relative to the randomized complete block, design is E΄/MSSPE that is greater than 1. These
results express the obvious: that the arrangement of split-plot treatments together within a whole plot results in a
lower accuracy on whole plot treatment comparisons and an increased accuracy on other treatment comparisons,
the formulas enable a quantitative evaluation of these effects. Recourse should be taken to a split-plot design

Vol.4, No.19, 2014
when experimental conditions necessitates the special arrangement, or when the experimenter is more interested
in one factor, which he/she arranges within whole plots, than in the other (Hinkelmann and Kempthorne, 2008).
Equation (5) above as presented by Hinkelmann and Kempthorne (2008) is inadequate for our designed
models since its applicability is for two factors only. Hence, for a three-factor design the relative efficiency of
the split-plot design over RCBD will be obtained from Table I and II, using the pooled treatment sums of squares
with appropriate error sums of squares, that
r(abc – 1)MSE = r(a – 1)MSWPE + ra(b – 1)(c – 1)MSSPE
divide through by r we have
a MS a b c MSWPE SPE
79
E΄ =
− + − −
( 1) ( 1)( 1)
1
−
abc
………………………………………………. 6
The interpretations for the two factors as given by Hinkelmann and Kempthorne (2008) and their respective
conclusions hold for the three factors too.
3. RESULT
Table III: ANOVA Table for the Split-Plot Model
Source Df SS MS FCAL P- Values
Blocks 2 8.972 4.486 30.73 <.0001
FR 1 75.69 75.69 22.15 0.0423
Mp error 2 6.834 3.42 23.41
GM 4 52.68 13.17 90.22 <.0001
BFS 4 6.905 1.726 11.83 <.0001
FR*GM 4 15.12 3.78 25.90 <.0001
FR*BFS 4 1.62 0.40 2.77 0.0305
GM*BFS 16 0.224 0.014 0.10 1.000
SP Error 112 16.35 0.146
Total 149 184.394
Source: Author’s computation
Table IV: ANOVA Table for the RCBD Model
Source Df SS MS FCAL P- Values
Blocks 2 8.97 4.49 22.06 <.0001
FR 1 75.69 75.69 372.16 <.0001
GM 4 52.68 13.17 64.76 <.0001
BFS 4 6.90 1.73 8.49 <.0001
FR*GM 4 15.12 3.78 18.59 <.0001
FR*BFS 4 1.62 0.405 1.99 0.1005
GM*BFS 16 0.224 0.014 0.07 1.0000
Error 114 23.18 0.203
Total 149 184.39
Source: Author’s computation
4. DISCUSSION
From equation (5) the weighted mean square error, E΄ is estimated as,
E΄ = 0.165142857.
Hence, the information on WP treatment from the split-plot experiment relative to randomized complete block
design is then,
(E΄/MSMPE ) = 0.165142857/3.42
= 0.0483 < 1
While the information on SP treatments from the split-plot experiment relative to randomized complete block
design is then
(E΄/MSSPE ) = 0.165142857/0.146

Vol.4, No.19, 2014
80
= 1.131 > 1
Table III and IV above shows the ANOVA result for the split-plot design and the randomized complete block
design respectively. From observation on table III, at α = 5% significance level, all the main effects FR, GM and
BFS are significant since, their p-values are less than α = 5% significance level. While for the interaction effects
FR*GM, FR*BFS and GM*BFS it is observed that GM*BFS is not significant since its p-value of 1.000 is
greater than α = 5% significance level. Likewise, from table IV all main effects are significant while for the
interaction effects it is observed that FR*BFS and GM*BFS are not significant since their p-values of 0.1005
and 1.000 respectively are greater than α = 5% significant level. This is an evidence of the split-plot design
superiority over the randomized complete block design. Hence, for a complete proof of the ANOVA results the
relative efficiency of the split-plot design over the RCBD was been computed. It was clear when the efficiency
of the WP treatment relative to the RCBD computed using equation (6) the value (0.0483) obtained is less than
one and that of the sub-plot treatments relative to the RCBD show that the value (1.131) obtained is greater than
one. Hence, we can agree on the superiority of the SPD over the RCBD. This result obtained is not far from that
obtained by Kowalski and Potcner (2003) though they observed the efficiency of SPD relative to complete
randomized design (CRD) using ANOVA results only. The difference between the RCBD and CRD is the block
factor besides; the issue of blocking is to improve the design and to remove every element of heterogeneity as
much as possible. Hence, the SPD is not an exception, its form of design is to develop an adequate and precise
design that can reduce cost, remove heterogeneity and study factors of less and important interest together.
5. CONCLUSION
This research is an attempt to compare the effectiveness of SPD over RCBD. The result obtained clearly
put the SPD ahead of the RCBD since from the ANOVA table result for the interaction of the feed rate by the
machine blowing fan speed (FR*BFS) is significant from the SPD ANOVA model but not significant from the
RCBD ANOVA model at 5% significance level. Likewise, the relative efficiency statistic computed was to
compare the two designs and the values obtained also reveal that the SPD model is more efficient relative to the
RCBD model. Before embarking on the plan and design of experiment, experimenters should adequately study
the type of factors for their experiments to know how important they are for achieving their experimental goals.
Because the results from this research shows that the SPD will be more efficient than the RCBD especially when
one or some of the factors to be studied are of less importance or is hard-to-change as in the industrial
experiments. Instead of avoiding the SPD due to computational complexities as viewed by some experimenters
and scholars, a professional should be contacted it will go a long way in reducing biased results in estimating
factors significant contribution to the experimental response.
REFERENCES
1. Bisgaard, S., Fuller, H. and Barrios. E. 1996. “Two-level Factorials run as Split–plot Experiments”.
Quality engineering 8: 705-708.
2. Cochran, W.G. and Cox, G. M. 1956. Experimental Designs, John Wiley and Sons New York, pp. 294
– 305, 106 – 107.
3. Cox, D. R. 1958. Planning of Experiments, John Wiley and Sons New York, pp. 26 – 30, 34 and 142 –
151.
4. Hinkelman, K. and Kempthorne, O. (2008). Design and Analysis of Experiments, vol. I: Introduction to
Experimental Design, 2nd ed. Wiley-Interscience, New York.
5. Jones, B. and Nachtsheim, C. J. 2009. Split-plot Designs: What, why, and How. Journal of Quality
Technology, 41(4): 340-361.
6. Kempthorne, O. 1952. Design and analysis of Experiments, John Wiley and Sons New York, pp. 163 –
183, 207 – 213 and 371 – 389.
7. Kowalski, S. M. and Potcner, K. J. 2003. How to Analyze a Split-Plot Experiment, Quality Progress,
37(12): 67-74.
8. Montgomery, D. C. 2005. Design and Analysis of Experiments, 5th ed. John Wiley and Sons New York,
PP. 76-104 and 392 – 422.
9. Yates, F. 1935. Complex Experiments. Journal of Royal Statistics Society Supplementary: 181-247.

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