Pros & Cons of Latin Square Design lecture - 16

 Pros & Cons

of 

Latin Square Design

lecture - 16 

Pros and cons of Latin square design

The advantages of Latin square designs are:

i.                     The LS design reduces the experimental error by controlling two sources of variations.

ii.                   LS design allows experiments with a relatively small number of runs.

iii.                 The LS design is more efficient than CRD and RCBD, when the heterogeneity appearing in two directions.

iv.                The LS design is flexible from 5 to 10 treatments.

The disadvantages are:

i.                    LS design is not possible for two treatments i.e. P = 2.

ii.                  Replication in LS design is costly.

iii.                LS design manually laborious and time consuming for large of treatments. 

Relative Efficiency of LSD vs. RCBD

The relative efficiency measures the estimation power or capacity of a design relative to other designs. If we have P  Latin square design, then the relative efficiency of LS design to RCB design is estimated as:

1.     1.  To compare with an RCBD using columns as blocks:

Consider columns as blocks and rows omitted.

1.      2. To compare with an RCBD using rows as blocks:

Consider rows as blocks and columns omitted.

In all cases RE > 1, the blocking has improved the efficiency.

Relative Efficiency of LSD vs. CRD

The relative efficiency of LS design to CR design is estimated by the following formula:

Example: 

An investigator wants to test the significance of four manufacturing methods to control two sources of variations corresponding to the operators and corresponding to machines. The data on the response variable is given below:

Operator	Machines
	1	2	3	4
1	4 (A)	2 (B)	5 (C)	7 (D)
2	1 (B)	2 (C)	6 (D)	5 (A)
3	8 (C)	9 (D)	6 (A)	3 (B)
4	11 (D)	3 (A)	7 (B)	8 (C)

Test the hypothesis about the significance of four manufacturing methods, also explain whether row and columns blocking are effective in this experiment.

Solution: We setup our hypothesis as:

$ii. \; \; The\; significance\; level;$α = 0.05

iii.    The test statistic: LS design is used to control two sources of variation between operators and machines.

F = MST / MSE  ~ F(3, 6)

iv.     Reject  H0, when F > 4.53

v.      Computation:

The treatment total can be arranged as:

 

ANOVA Table:

SV	df	SS	MS	F
Treatment	3	54.6875	18.23	6.11
Operator	3	36.1875	12.0625	4.05
Machine	3	11.1875	3.73	1.25
Error	6	17.8751	2.98
Total	15	119.9375

vi.                    From ANOVA table, it is observed that the four treatments are significant.

Now it is desired to compute the relative efficiency

RE of LSD TO RCBD

To compare with an RCBD using columns as blocks:

As RE > 1

The situation is improved by taking columns ( machines) as blocks.

To compare with an RCBD using rows as blocks:

As RE > 1

The situation is improved by taking rows (operators)as blocks.

The relative efficiency of LS design to CR design is estimated by the following formula:

It means that 65 % more observations per treatment would be required in CRD to obtain the same precision for estimating the treatment means as with this LSD with P =4.

Example: 

The ANOVA Table for LS Design is given below:

SV	df	SS	MS	F
Row	7	6.17	0.88	5.87
Column	7	0.79	0.11	0.65
Treatment	7	3.77	0.54	3.18
Error	42	7.01	0.17
Total	63	17.74

a.       Are treatments means are significant?

b.      Are columns means are significant?

c.       Are rows means are significant?

d.      Discuss the relative efficiency to RCBD & CRD.

       Solution:

SV	df	SS	SS	F	F 0.05(7, 42)
Row	7	6.17	0.88	5.87	2.32
Column	7	0.79	0.11	0.65	2.32
Treatment	7	3.77	0.54	3.18	2.32
Error	42	7.01	0.17
Total	63	17.74

From the above ANOVA, it is conclude that the treatment means and row means are significant, where the column means are insignificant.

RE of LSD TO RCBD

To compare with an RCBD using columns as blocks:

As RE > 1

The situation is improved by taking columns as blocks.

To compare with an RCBD using rows as blocks:

As RE < 1

The situation is not improved by taking rows as blocks.

 The relative efficiency of LS design to CR design is estimated by the following formula:

It means that 42 % more observations per treatment would be required in CRD to obtain the same precision for estimating the treatment means as with this LSD with P = 8.