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Statistical Model & Analysis of Split Plot Design Lecture - 47

Statistical Model

Analysis of Split Plot Design

Lecture - 47

Statistical Model in CRD

Let A and B are two factors. Factor A with “a” levels that’s A1, A2, ....,Aj, ...., Aa and factor B with “b” levels that’s B1, B2, ...., Bk,..., Bb. and factor A is assign to main plot The linear model for the split-plot, with main plots arranged as a CRD is:

Yijk= μ + αj + δij + βk + (αβ)jk + ϵijk

i =1, 2, ... ,r

j=1, 2, ... ,a

k = 1, 2, ..., b

Analysis of Split Design

Let Yijk be the yield of ith replication jth level of factor A (assign to whole plot) and kth level factor B (assign to sub plot).

The factor A total is given by;

The factor B total is given by;

ANOVA Table:

The standard error for the mean of factor A is:

The difference of standard error is:

The standard error for the mean of factor B is:

The standard error for the mean of factor AB is:

Difference between two A means at same or different level of B:

Example:

An agriculture researcher desires to compare two varieties of wheat seeds and 4 levels of nitrogen percentage . The experiment is performed in RCBD split plot design and data obtained given below:

Solution:

a = 2 , b = 4, r = 3

Now consider factor A & B:

ANOVA Table:

Remarks: The factor A is significant, factor B and AB interaction is insignificant.

Read More: Post Hock test in Split Plot Design
Read More: Solved Example on Split Plot Design

Split Plot Design Lecture - 46

Split Plot Design

(SPD)

Lecture - 46

Introduction

Factorial experiment is used to compare two or more factors each at several levels, now if one factor required large experimental units and other factor can be accommodate in small experimental units, and then factorial experiment is replaced by split plot design.

Suppose we want to study two factors, say method of cultivation at levels (Broadcasting, Behind Local Plough, Drilling) and variety of wheat seed (Pak 2013, Markaz 2019, Zincol 2016). The first factor ploughing requires relatively large plots of land. This will require higher cost and puts a restriction on the number of plots to be used. The second factor can be accommodated in much smaller plots. To achieve this, the large plots are split into smaller plots at the planting stage. . This suggests that the experiment can be conducted with two strata.

1. The whole-plot stratum consists of large plots in which the plots can be assigned as per any standard design, e.g. CRD, RBD, or Latin square design.

2. The stratum is the split-plot stratum which consists of the split-plots. There are the smaller plots that are obtained by splitting each of the large plots into three parts.

The treatments assigned to the large whole-plots are replicated r times, and treatments assigned to the split-plots are replicated rt times. The smaller variation is expected in sub plots. The interaction contrasts between whole- and split-plot treatment also fall into the split-plot stratum and benefit due to smaller variance.

There are two distinct randomizations in the split plot designs:

i. The first randomization takes place in stratum 1, when the levels of the whole-plot treatment are randomly assigned to the whole–plots.

ii. The second randomization takes place in stratum 2 where the levels in the split-plot treatment are randomly assigned in the split-plot.

Determining Which Factor to Use as the Whole and Subplot Factors

With the split plot arrangement, plot size and precision of measurement of the effects are not the same for whole and subplot factors. Thus, assignment of a particular factor to either the whole or subplot is extremely important. To make a choice, the following guidelines are suggested:

1. Degree of Precision: for a greater deal of precision for factor B than factor A, assign factor B to the subplots and factor A to the whole plots.

2. Relative Size of the Main Effects: If the main effect of one factor (.e.g., factor A) is expected to be much larger and easier to detect than that of the other factor (e.g., factor B), factor A should be assigned to the whole plots and factor B to the subplots. This may increase the chances of detecting differences among levels of factor B.

3. Management Practices: Cultural practices required by a factor may dictate use of large plots. In such a case, such factors should be assigned to whole plots.

Split Plot Design

The split plot design involve assigning the levels of one factor to the main plots arranged in a CRD, RCBD, or Latin square and the level of the second factor to split or sub plot with in each main plot. There will be two stage randomization procedures. First, levels of factor A are randomized over the main plots and then levels of factor B are randomized over the subplots within each main plot.

The split plot design was developed in 1925 by mathematician Ronald Fisher for use in those agricultural experiments, where

1. Two or more factors each at several levels are involved and one factor required large experimental units, or

2. One factor required greatly accuracy as compare to other factor(s), or

3. A new factor can wants to introduce during the experiment is in progress.

Lay Out of Split Plot Design

Let we have two factors A and B. Factor A have two levels say A1, A2 and factor B have three levels say B1, B2, B3. Further it is assumed that factor A required large plot and factor B can be accommodated in sub plots.

The step by step procedure is given below:

1. Set number of replications / Blocking

Divide the experiment material into r plots. Let r = 3

Replications

III

2. Number of Main Plots

Divide each main plot into sub plot on the basis of factor level assign to large plot. Here we have two levels of factor A. Divide each plot in to two sub plots.

Repeat the same procedure for II and III.

3. Assign randomly factor A levels to sub plot of I using the concepts of basic designs.

Repeat the same procedure for II and III.

4. Splitting Procedure of Each Plot

Split each main sub plot into a number of split plots according to levels of factor B. here we have three levels for factor B. so, divide each sub plot in to 3 split plots and assign factor B levels to split plots randomly.

Repeat the same procedure for II and III.

Read More: Analysis Method of Split Block Design

MCQ's on BIBD

MCQ's on BIBD

1. If t = 5, b = 5 and each treatment is replicated 3 times, the total number of observations in incomplete block design is

i. 5

ii. 10

iii. 15

iv. 25

2. If t = 5, b = 5 and each treatment is replicated 3 times, then the number of experiments per block in incomplete block design is

i. 3

ii. 5

iii. 10

iv. 15

3. IBD is used when

i. t< b

ii. t > k

iii. t < k

iv. t > k

4. IBD is applied to

i. secure accurate estimates

ii. Reduce expenses

iii. A block can not accommodate all treatments.

Both ii) & iii)

5. An IBD is said to be BIBD, if

i. r is constant.

ii.λ is constant.

iii. K is constant.

iv. Both i) & ii).

6. The value in BIBD (4, 4, 3, 3, ?)

i. 1

ii. 2

iii. 3

iv. 4

7. A BIBD (3, 4, ?, 3, ?)

i. (4, 5)

ii. (5, 3)

iii. (5, 5)

iv. (4, 6)

8. A bock design with t treatments and b blocks is connected, the rank will be:

i. A BIBD with # blocks / r is an integer is called

ii. Symmetrical BIBD

iii. Resolvable BIBD

iv. Affine Resolvable BIBD

9. Which of the following statement is true for BIBD (t = 4, b = 6, r = 3, k = 2, λ = 1)

i. Symmetrical BIBD

ii. Resolvable BIBD

iii. Affine Resolvable BIBD

10. A BIBD is always exist, when

i. K = t

ii. K < t

iii. K > t

iv. K = t -1

11. The total number of observations in a un reduced BIBD with t = 5, k = 2 and λ = 1 is

i. 5

ii. 15

iii. 20

iv. 25

12. In IBD, if blocks are grouped into larger blocks and form a complete block design is called

i. Symmetrical BIBD

ii. Resolvable BIBD

iii. Affine Resolvable BIBD

iv. Complementary BIBD

13. A special case of IBD, if k^2 = t, the design is called

i. Symmetrical BIBD

ii. Resolvable BIBD

iii. Affine Resolvable BIBD

iii. Lattice BIBD

Read More: Split Plot Design

Youden Square Design Lecture - 45

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: