Split - Split Plot Design Lecture 50

 

Split - Split Plot Design

(Double Split Plot Design) 

Lecture - 50

Introduction

The split-split plot design is used suite when there three or more factors at several levels where one factor required large experimental unit and other factors can be accommodated in small experimental units and different levels of precision are required for the factors evaluation.

Lay Out of Split - Split Plot Design

Suppose we have three factors A at two levels , B at two levels  and C at three levels . let factor required large experimental units and factor C high precision. Then factor A is assigned to main plot and factor C to sub sub plot. Consider RCBD the step-by-step procedure is given below:

1.  Divide the whole plot in to three blocks (let 3 blocks is taken)

1.     2.  Assign factor A levels (A1, A2) randomly by dividing each block into 2 sub blocks.

Repeat the same procedure of block 2 and block 3.

3.      Divide each sub block into two sub sub blocks and assign randomly factor B levels (i.e., B1, B2). 

Repeat the same procedure of block 2 and block 3.

4. Divide each sub sub block into 3 sub plots and assign randomly factor C levels.

Repeat the same procedure of block 2 and block 3.

Statistical Model

Let  be the response of j^th factor A, k^th level of factor B and l^th level of factor C in i^th block, then it can be represented by the following linear statistical model.

Yijkl = μ + ρi + αj + ϵij + βk + (αβ)jk + ϵijk + γl +(αγ)jl + (βγ)kl + (αβγ)jkl + ϵijkl

i =1, 2, ..., r

 j=1, 2, ...,a 

k = 1, 2, ..., b

Where:

αj : The effect factor A assign randomly to whole plot.

ϵij : The error in the whole plot.

 βk : The effect factor B assign randomly to sub plot.

(αβ)jk : Interaction effect of factor A & factor B.

 ϵijk : The error in the sub plot.

γl : The effect of factor C assign to sub sub plot.

(αγ)jl :Interaction effect of factor A & factor C.

(βγ)kl : Interaction effect of factor B & factor C.

(αβγ)jkl : Interaction effect of factor A, B and factor C.

Statistical Analysis

Step 1: Calculate total sum of squares:

Step 2: Calculate replicate sum of squares:

Step 3: Calculate factor A (assign to whole plot) sum of squares:

Step 4: Calculate Whole plot error sum of squares.

Step 5: Calculate B & (AB) sum of squares:

Step 6: Calculate sub plot error sum of squares:

Step 7: Calculate C sum of squares:

Step 8: Calculate interaction of A with C sum of squares:

Step 9: Calculate interaction of B with C sum of squares:

Step 10: Calculate interaction of ABC sum of squares:

ANOVA Table:

Example

Split-split plot arrangement randomized as an RCBD. Three levels of the whole plot factor, A, two levels of the subplot factor, B, and three levels of the sub-subplot factor, C. Diagram shows the first replicate.

Solution:

Totals for main effects:

Sum of squares for factor A:

Sum of squares for factor B:

Sum of squares of factor C:

ANOVA Table:

Remarks:

All the three factors i.e., A, B, and C are significant.

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