Provide Information Regarding Statistics & Econometrics : Introduction to Latin Square (LS) Design lecture

Introduction

Read More: Estimation of Missing Value in LS Design
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Read More: Estimation of Parameters in LSD
Read More: Expectation in LSD
Read More: Graeco-Latin Square Design

lecture - 14

What is a Latin square?

A Latin square is a table filled with p X p different alphabets or symbols in such a way that each alphabet or symbol appears once and only once in each row and exactly once in each column. Here are a few examples.

a	b
b	a

a	b	c
b	c	a
c	a	b

a	b	c	d
b	c	d	a
C	d	a	b
d	a	b	c

Standard Latin Square

A Latin square in which the treatments in the first row and in the first column are arranged in alphabetical or numerical order.

a	b	c
b	c	a
c	a	b

From a standard Latin square of order p, p! (p - 1)! Different Latin squares can be obtained by making p! permutations of columns and (p - 1)! Permutations of rows, which leaves the first row in place.

Orthogonal Latin squares

If two Latin squares of the same order but with different symbols are such that when they are superimposed on each other, every ordered pair of symbols (different) occurs exactly once in the Latin square, then they are called orthogonal.

Introduction to Latin Square Design

(LS Design)

A randomised complete block design is used to control for extraneous sources of variation in the experiment. If there are two sources of variation in the experimental material, there is a chance that the treatments will be influenced by two nuisance factors. The Latin square design is suitable for controlling extraneous variations by blocking the experimental material in two contrast directions.

In Latin square design, the experimental material is divided into rows and columns, each having the same number of experimental units, which is equal to the number of treatments. The treatments are assigned at random to the rows and columns in such a way that each treatment appears once and only once in each row and column.

If there are P treatments, the experimental material is divided into P rows and P columns, resulting in experiment units. Replicates are also included in this design.

Suppose we have four treatments, i.e., A, B, C and D. The standard Latin square is given below:

a	b	c	d
b	c	d	a
c	d	a	b
d	a	b	c

Example:

Suppose a researcher wants to check different brands of petrol consumption per kilometre. He takes 3 different brands of petrol, represented by A, B, and C.

The factors affecting the consumption of petrol are:

· Driver

· Brand of car

The researcher controls these two sources of extraneous variation by LS design as:

Brand of Car	Driver – 1	Driver – 2	Driver – 3
1	A	B	C
2	B	C	A
3	C	A	B

Features of Latin square:

· * Treatments are assigned at random within rows and columns, with each treatment once per row and once per column.

· * There are equal numbers of rows, columns, and treatments.

· * Useful to control variation in two different directions.

Experimental Layout out of LS Design

The Latin square design is used to control two sources of variability in the experimental units. So, both rows and columns are blocking factors. It removes the variation from the measured response variable in both directions.

Let us have 4 treatments, i.e., A, B, C, and D (P = 4). We required experimental units.

Step – 1: Develop a standard Latin square or select any Latin square derived from a standard Latin square.

	Column 1	Column 2	Column 3	Column 4
Row 1	A	B	C	D
Row 2	B	C	D	A
Row 3	C	D	A	B
Row 4	D	A	B	C

Step 2: Row randomisation:

Random Number	0.788	0.943	0.051	0.811
Rank (order)	2	4	1	3
Row	1	2	3	4

Then one should rearrange the order of the rows as follows:

	Column 1	Column 2	Column 3	Column 4
Row 2	B	C	D	A
Row 4	D	A	B	C
Row 1	A	B	C	D
Row 3	C	D	A	B

Step – 3: Column randomisation

Random Number	0.134	0.655	0.210	0.165
Rank (order)	1	4	3	2
Column	1	2	3	4

Then one should rearrange the order of the columns as follows:

	Column 1	Column 4	Column 3	Column 2
Row 2	B	A	D	C
Row 4	D	C	B	A
Row 1	A	D	C	B
Row 2	C	B	A	D

Step – 4: Treatment randomization

Random Number	0.994	0.176	0.441	0.227
Treatment
Rank (order)	A	B	C	D

Assign A as treatment 4, B as treatment 1, C as treatment 3 and D as treatment 2:

	Column 1	Column 4	Column 3	Column 2
Row 2	T1	T4	T2	T3
Row 4	T2	T3	T1	T4
Row 1	T3	T2	T3	T1
Row 2	T4	T1	T4	T2

Statistical Model and Analysis of LS Design

The Latin square design is represented by the following linear statistical model.

Yijk = μ + ρi + τj + ϵijk i =1, 2, ..., P j = 1, 2, ..., P k = 1, 2, ..., P

Where:

Yijk i =1, 2, ..., P j= 1, 2, ..., P k =1, 2, ..., P is the observation for the experimental unit in the ith row block level, the jth column block level and the k^th treatment effect.

Assumptions of the Latin Square Model:

Analysis of Latin Square Design:

Let Yijk be the yield of the pth treatment in the i^th row block and j^th column block in a Latin square. Then it can be tabulated as:

ANOVA Table:

SV	df	SS	MS	F
Row	P – 1	SSR	MSR	F1
Column	P – 1	SSC	MSC	F2
Treatment	P – 1	SST	MST	F3
Error	(P – 1) (P – 2)	SSE	MSE
Total	P^2 – 1	SSTotal

Example:

An experiment to investigate the effects of various dietary starch levels on milk production was conducted on four cows. The four diets, T1, T2, T3, and T4 (in order of increasing starch equivalent), were fed for three weeks to each cow, and the total yield of milk in the third week of each period was recorded (i.e., the third week to minimise carry-over effects due to the use of treatments administered in a previous period). That is, the trial lasted 12 weeks since each cow received each treatment, and each treatment required three weeks. The investigator felt strongly that time period effects might be important (i.e., earlier periods in the experiment might influence milk yields differently compared to later periods). Hence, the investigator wanted to block on both cow and period. However, each cow cannot possibly receive more than one treatment during the same time period; that is, all possible cow-period blocking combinations could not logically be considered.