efficiency of lattice design in relation to randomized complete ... - PJAR

Pakistan J. Agric. Res. Vol 22 No. 3-4, 2009.
EFFICIENCY OF LATTICE DESIGN IN RELATION TO RANDOMIZED
COMPLETE BLOCK DESIGN IN AGRICULTURAL FIELD
EXPERIMENTS
Irum Raza and M. Asif Masood*
ABSTRACT: This study was conducted to compare the relative efficiency of
two statistical experimental designs based on mean square errors. For this purpose,
three datasets were analyzed with Lattice design and randomized complete
block design (RCBD). The results of the first dataset show that 26% precision
increased with Lattice design over RCBD. Coefficient of variation of lattice design
is 19% while that of RCBD is 21%, which proves the efficiency of Lattice
design. In addition, larger F- value of Lattice design indicates greater variability
among the treatments as compared to RCBD. Results of the second dataset signify
that Lattice design increases the precision of experiment by 17% and also shows
less coefficient of variation than RCBD which implies that lattice design is again
more efficient. In third dataset, Lattice design is again more efficient than RCBD
in terms of relative efficiency (55%) and c.v (7%). The relative efficiencies of
three datasets were 26%, 17% and 55%, respectively, and specify that the precision
of experiment increased significantly using Lattice design.
Key Words: Complete Block Design; Incomplete Block Design; Randomized Complete
Block Design; Lattice Design; Relative Efficiency; Coefficient of Variation; Pakistan.
150
INTRODUCTION
Experimentation plays a momentous
role in the field of agriculture. A good experiment
is the one which involves good
planning, accurate data collection, proper
data analysis and precise interpretation of
the data. A statistician is supportive in
drawing inferences and conclusions from
the experiment however, before that the
researcher must properly define the objectives
of the experiment.
Experimental designs are basically divided
into two categories: complete block
designs and incomplete block designs.
Complete block designs include completely
randomized design (CRD), RCBD, Latin
squares etc. Among these designs, RCBD
is one of the most extensively used designs
in agriculture.
In RCBD blocks size should be homogeneous
and each block must contain a
complete set of treatments. It also reduces
experimental error through proper blocking
though blocking becomes ineffective
when the block size increases and cannot
be used for a large number of treatments.
Therefore in such a situation, RCBD becomes
less powerful in controlling experimental
error due to soil heterogeneity in
experimental site.
Patterson et al. (1978) suggested that
if blocking is not done properly then lattice
designs can be analyzed as RCBD by considering
super blocks as ordinary blocks.
These designs do not necessitate the number
of blocks to be equal to the number of
replications although the replication is further
divided into smaller blocks and then
treatments are assigned to these blocks.
Another advantage of incomplete block design
is that it can be used for any number
of treatments and replications.
Yates (1936 b) launched an important
type of incomplete block design named ‘Lattice
designs’. These designs were initially
termed as pseudo-factor designs and were
capable of providing efficient incomplete
block designs for unstructured treatments
by inflicting artificial pseudo-factor on the
treatments and then confounding the contrasts
between blocks.
Lattice designs are now frequently
used in the field of agriculture to test the
yield of annual crops. A condition required
*Social Sceinces Institute, National Agricultural Research Centre, Islamabad, Pakistan.

in these designs is that the number of treatments
used must be a perfect square such
as 5 2 or 25, 6 2 or 36, 7 2 or 49, 8 2 or 64, 9 2 or
81, 10 2 or 100. The two most commonly used
lattice designs include balanced lattice and
partially balanced lattice. The discrepancy
between these two designs occurs on the
use of the number of replications. In balanced
lattice design, block size (k) is equal
to the square root of the total number of
treatments and the number of replications
required is one more than the block size
i.e., k+1. However in partially balanced lattice
design any number of replications can
be used so with two replications the design
is called a simple lattice; with three replications,
a triple lattice; with four replications,
a quadruple lattice, etc.
Sarker et al. (2001) performed an incomplete
block analysis of 53 lentil yield
trials in lattice block design. In field experiment
spatial variability exists in row
and column direction and can be modeled
using covariance structure for plot errors.
First order auto correlation error structure
(AR1) was fitted in both the column and row
direction and the best model was selected
on the basis of residual deviance of 53 trials.
The efficiency for the pair wise comparison
of genotypes over RCB was obtained
for the lattice blocks and the selected models.
Spatial models were found efficient for
the 74% of the trials as compared to the
lattice blocks. The average efficiency of
spatial models over RCB was 50% at the
analysis stage. The results emphasized
IRUM RAZA AND M. ASIF MASOOD
151
that a combined lattice and neighbor analysis
increase the efficiency of experiment.
MATERIALS AND METHODS
To compare the relative efficiency of
RCBD and Lattice design, three datasets
containing 3x3 and 4x4 as a balanced lattice
design with 4 and 5 replications respectively
and a 5x5 partially balanced lattice
design with 2 replications were taken
from Gomez and Gomez (1976). Layout and
analysis were obtained using the MSTAT-
C software.
The efficiency of Lattice design relative
to RCBD was based on Intra block error
MS, Effective error MS, F-value and
Coefficient of variation (c.v) .The relative
efficiency (R.E) was calculated as
R.E = 100 * block (adj.)SS + intrablock
error SS / k (k 2 – 1) (effective error MS)
For each dataset, c.v, mean square
errors (MSE) and F-values were studied.
RESULTS AND DISCUSSION
First dataset consisted of 9 treatments
and 4 replications and the results revealed
that the relative efficiency of Lattice compared
with randomized complete block was
26% (Table 1). Coefficient of variation of
lattice design was 19% while that of RCBD
was 21% which also showed the efficiency
of lattice design. In addition larger F- value
of Lattice indicated greater variability
among the treatments as compared to
RCBD.
Table 1. Analysis of Variance for 3x3 Square Lattice Design
Source of Degrees of Sum of
variance freedom squares Mean square F-value Prob
Replications 3 0.077 0.026
Treatments
-Unadjusted 8 3.226 0.403 3.64 0.013
-Adjusted 8 3.172 0.396 4.32 0.006
Blocks within
Reps (adj.) 8 1.421 0.178
Error
-Effective 16 1.470 0.092
-RCB Design 24 2.657 0.111
-Intrablock 16 1.237 0.077
Total 35 5.961

EFFICIENCY OF LATTICE DESIGN
Table 2. Analysis of Variance for 4x4 Square Lattice Design
Source of Degrees of Sum of
variance freedom squares Mean square F-value Prob
Replications 4 5946 1486
Treatments
-Unadjusted 15 26994 1799 4.17 0.000
-Adjusted 15 24001 1600 4.33 0.000
Blocks within
Reps (adj.) 15 11382 759
Error
-Effective 45 16620 369
-RCB Design 60 25915 431
-Intrablock 45 14533 323
Total 79 58855
Table 3. Analysis of Variance for 5x5 Square Lattice Design
Source of Degrees of Sum of
variance freedom squares Mean square F-value Prob
Replications 3 1113 371.0
Treatments
-Unadjusted 24 420 17.5 2.21 0.008
-Adjusted 24 406 16.9 3.32 0.000
Blocks within
Reps (adj.) 16 327 20.4
Error
-Effective 56 285 5.0
-RCB Design 72 570 7.9
-Intrablock 56 242 4.3
Total 99 2104
If E e
is greater than E b
, than mean is taken to be zero and no adjustments are made for the treatments and
therefore the design is treated as randomized complete blocks.
Where E e
and E b
are respectively, the mean squares for intra block and blocks error.
Results for the second dataset containing
16 treatments and 5 replications depict
that Lattice design increased the precision
of experiment by 17% and also
showed less coefficient of variation than
RCBD which implied that lattice design was
again more efficient (Table 2).
Third dataset used 25 treatments and
4 replications and showed that lattice design
was again more efficient than RCBD
in terms of relative efficiency (55%), c.v
(7%), F-value and Mean square errors
(Table 3). Lin et al. (1993) studied the performance
of randomized complete block design
in field experiment and found similar
results. They used 60 sets of historical data
of soybean yield to estimate soil variation
in one and two directions. The total sum of
152
squares of soil variation in one direction
was 39.3% and in two directions was 52.7%.
The significant difference of 13.4% depicted
that blocking was not done properly and in
this situation randomized complete block
design became less efficient. Heterogeneity
within the blocks was tested using two
datasets of soybean data and statistical
techniques were compared such as lattice
design and analysis, neighbor analysis. The
results showed that heterogeneity within
the blocks was controlled using lattice design
Use of Lattice design and analysis is
advocated for large number of treatments
to control soil heterogeneity in the experimental
environment.

IRUM RAZA AND M. ASIF MASOOD
LITERATURE CITED
Patterson, H.D. William, E.R. Hunter, E.A.
Edmondson R.N. 2004. Past developments
1978. Block designs for variety trials. J.
Agric. Sci. Cambridge, 90: 398-40.
and future opportunities in the design
Snyder, E. B. Lattice and compact family
and analysis of crop experiments. J.
block designs in forest genetics. U.S
Agric. Sci. 143: 27-33.
Department of Agriculture, Forest service,
North central Forest Experiment
Erwin, L.L. Leonard, W. H. and Clark, A.G.
1972. Field plot technique. 2nd edn. station. p. 12-17.
USA.
Sarker, A. Singh, M. and Ergkinae, W.
Gomez, A.K. and Gomez, A.A. 1984. Statistical
procedure for agricultural research yield trials in lentil. (Lens culinaris ssp.)
2001. Efficiency of spatial methods in
2nd edn, John Wiley and Sons, Inc. J. Agric. Sci. 137: 427-438.
Lin C.S. Binns, M.R. Voldeng, H.D. and Yates, F. 1936b. A new method of arranging
variety trials involving a large num-
Guillenette, R. 1993. Performance of
randomized complete block designs in ber of varieties. J. Agric. Sci. Cambridge,
field experiments. Agron. J. 85:168-171. 26: 242-255.
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