Provide Information Regarding Statistics & Econometrics : Youden Square Design Lecture

Youden Square Design

Lecture - 45

In conventional designs the Latin square design is used to control two sources of extraneous variation by performing blocking into two mutual perpendicular directions. The same role is performed by Youden square design in case of incomplete block. The Youden square design is used to control two sources of extraneous sources of variation. A symmetric BIBD with t = b and r = k forms a Youden square design.

Example:

let four treatments is distributed in to 3 blocks to control to extraneous sources of variation by using Youden square design.

Block 1	A	B	C	D
Block 2	B	A	D	C
Block 3	C	D	A	B

Construction of Youden Square Design

1. The Youden square design is obtained from a Latin square design by deleting

i. One of its rows or

ii. One of its columns or

iii. Diagonal entries

2. A BIBD with t = b forms a Youden square design.

Properties of Youden Square Design

· There are t treatments and each treatment is replicated r times.

· There are b blocks each of size k. where k < t.

· r t = b k = n

· No treatment appears more than once in a block

· There are m associates, i^th associates appear in λi blocks.

· λi’s are usually arranged in decreasing order of magnitude

Statistical Model

The Youden square design is represented by the following linear model

Yijm = μ + βi + τj + ϵijm

Where:

βi: i^th block effect

τj: j^th treatment effect

ϵijm: m^th position effect

Analysis

Consider a BIBD with t = b, r = k

Let Yijm be the effect of jth treatment in ith block and appear in mth position. Then it can be arranged as:

Where:

Now to obtain treatment totals

The adjusted treatment can be obtained as:

The block adjusted can be obtained as:

ANOVA Table:

SV	df	SS	MS	F
Treatment (Adjusted)	t-1	SST	MST adjusted
Rows	r-1	SSR	MSR
Columns	c-1	SSC	MSC
Error	diff	SSE	MSE
Total	bk -1

Example:

There are twenty experimental units available, and the researcher is interested in comparing five treatments. With one observation per cell, these 20 experimental units are set up in a Youden square pattern with 5 rows and 4 columns. The design parameters were as follows: each pair of treatments was duplicated three times, with t (treatments) = 5, b (blocks) = 5, k (number treatment) = 4, and r (replications) = 4. The following is the design arrangement along with the observations: