Chapter 6

1. Nested and Split plot designOne consistent feature of the designs discussed in the course so far is that

the size of the experimental units is the same for all the treatment factors.

However, it is not always desirable or possible to use the same size

experimental units for all the treatment factors. The split-plot design is an

example of a design where the experimental units are of different sizes.

Moreover, in some experiments the levels of one factor (e.g. factor B) are

similar but not identical for different levels of another factor (e.g. factor A).

This arrangement is call a nested or “hierarchical” design. Therefore, this

chapter, unlike the pervious chapters, deals with the analyses of split plot

and nested designs.

6.1 Two Stage Nested DesignIn factorial experiment, the assumption is that one factor can have identical

levels to be combined with the levels of the other factor(s) to form the

treatment combination. However, in certain multifactor experiments the

levels of one factor (say, factor B) are similar but not identical for different

levels of another factor (say, A). Such an arrangement is called a nested, or

hierarchical, design, with the levels of factor B nested under the levels of

factor A.

The statistical model for the two-stage nested design is

y ijk=μ+α i+βj (i )+ε ( ij )k

{¿ j=1,2 , . . .b ¿k=1,2 , . . . .k ¿¿¿ i=1,2 , . . . , a

There are a levels of factor A, b levels of factor B nested under each

level of A, and n replicates. The subscript j(i) indicates that the jth level

of factor B is nested under the ith level of factor A.

This is a balanced nested design because there are an equal number of

levels of B within each level of A and an equal number of replicates.

Because every level of factor B does not appear with every level of

factor A, there can be no interaction between A and B.

The total sum of squares can be partitioned into a sum of squares due to

factor A, a sum of squares due to factor B under the levels of A, and a

sum of squares due to error as

With abn−1 degrees of freedom for SST, a−1degrees of freedom for SSA,

a (b−1 ) degrees of freedom for SSB(A), and ab (n−1 ) degrees of freedom for

error.

Computing formulas for the sums of squares are:

If A and B are fixed, the hypotheses of interest areHo :αi=0and this null

hypothesis can be tested byF=

MSA

MSE . The null hypothesis of Ho : β j (i )=0 is

tested by F=

MSB

MSE

Example: consider a company that purchases its raw material from three

different suppliers. The company wishes to determine if the purity of the raw

material is the same from each supplier. There are four batches of raw

material available from each supplier, and three determinations of purity are

to be taken from each batch. This is two stage nested design as given below.

From ANOVA table, we would conclude that there is no significant effect

on purity due to suppliers, but the purity of batches of raw material from

the same supplier does differ significantly.

6.2 The Split-Plot DesignAn important strategy in agricultural and laboratory experiments of studying

two different experimental factors is to employ the so called split-plot design.

For reasons of convenience or necessity, the levels of one factor (the so

called subplot treatments) are applied to the ultimate, single experimental

units and the levels of the other factor (the so called main-plot treatments)

are applied to set of ultimate units.

Notice that the two factors can essentially be applied at different time. As a

result, a split-plot design can be viewed as two experiments combined or

superimposed on each other. One experiment has the whole-plot factor

applied to large experimental units (or it is a factor whose levels are hard

to change) and the other experiment has subplot factor applied to the

smaller experimental units (or it is a factor whose levels are easy to

change).

In other words, the split-plot design is specifically suited for a two factor

experiment that has more treatments that can be accommodated by a

complete block design. In this design, one of the factors is assigned to the

main-plot (whole-plot). The assigned factor is called the main-plot factor.

The main-plot is divided into subplots (split-plots) to which the second factor,

called the subplot factor, is assigned. Thus, each main-plot becomes a block

for the subplot treatments (i.e., the levels of the subplot factor).

With a split-plot design, the precision for the measurement of the effect of

the main-plot factor is sacrificed to improve that of the subplot factors. In

this design, both the procedures for randomization and that for analysis of

variance are accomplished in two stages one of the main-plot level and

another on the subplot level. At each level, the procedure of

randomized complete block design, as described in chapter 4, are

applicable.

The statistical linear model for split-plot design is

y ijk=μ+α i+β j+γ k+( αβ )ij+(αγ )ik+ ( βγ ) jk+( αβγ )ijk +εijk{¿ j=1,2 , . . .a ¿k=1,2 , . . .b ¿¿¿ i=1,2 , . . , r

Where α i , β i and (αβ )ij represent the whole-plot and correspond respectively

to blocks (or replicates), main-plot (whole-plot) treatment (factor A),

and whole plot error ( or replicates ( or block) x A); and γ k ,

(αγ )ik , ( βγ ) jk and (αβγ )ijk represent the subplots and respectively to the

subplot treatments (factor B), the replicates (or blocks x B and AB

interactions, and the sub plot error (block x AB). Note that the whole plot

error is replicate (or block) x A interaction and the subplot interaction is the

three interaction block x AB.

The sums of squares for these factors are computed as in the three-

way analysis of variance without interaction. What makes a split-plot

design different form other designs with factorial treatment structure? Here

are three ways to think about what makes the split-plot design different.

First, the split-plot has two sizes of units and two separate randomizations.

Whole-plots act as experimental units for one randomization and other

randomization assigns levels of the split-plot factor variety to split-plots. In

this randomization, split-plots act as experimental units, and whole-plots act

as block for the split-plots. In general, there are two types of separate

randomizations, with two different kinds of units that can be identified

before randomization starts. Second, a split-plot randomization can be

done in one stage, assigning factor-level combinations to split-plots in any

whole-plot get the same level of the whole-plot factor and no two split-plots

in the same whole-plot get the same level of the split-plot factor. Thus, a

split-plot design has a restricted randomization. Third, a split-plot is a

factorial design in incomplete blocks with one main effect confounded

with blocks. The whole-plots are the incomplete blocks, and the whole block

factor is confounded with block.

SSblock x B=1a∑

a

∑b

y2. jk− y

2. . .

rab−SSB−SSblock

Example: A paper manufacturer who is interested in three different pulp

preparation methods and four different cooking temperatures for the pulp

and who wishes to study the effect of these two factors on the tensile

strength of the paper. Each replicate of a factorial experiment requires 12

observations, and the experimenter has decided to run three replicates as

given below.

SStotal=∑i=1

r

∑j=1

a

∑k=1

b

y2

ijk− y2. . .

rab=(302+342+ . . . 392+452 )−1297

3 x 4 x3=822 .97

SSA=1rb

∑ y2. j .− y

2. . .

rab= 13x 4

(4282+4622+ 4072 )−12973 x 4 x3

=128 .39

SSB=1ra

∑ y2. .k− y

2. . .

rab= 13 x 3

(2182+3112+ 3412+3642)−12973 x 4 x 3

=434 .08

SSBlocks=1ab

∑ y2 i. .−y2. . .

rab= 13 x 4

(4172+4232+ 4572)−12973 x 4 x 3

=77 .56

SSError(mainp plot=1b∑

r

∑a

y2ij .− y

2. . .

rab−SSblocks−SS A

=14

(1382+1552+ . . .+1502)−12973 x 4 x 3

−128 .39−77 .56=36 .28

SSAB=1r∑

a

∑b

y2. jk− y

2. . .

rab−SSB−SS A

=13

(892+1002+ . . .+1262+1212 )−12973 x 4 x 3

−434 .08−128 .39=75 .17

SSblock x B=1a∑

a

∑b

y2. jk− y

2. . .

rab−SSB−SSblock

=13

(932+902+ . . .+1212+1292)−12973 x 4 x 3

−434 .08−77 .56=20 .67

SSE (subplot )=SStotal−(sum of all other SS )=50 .83

6.3 The Split Split plot DesignThe concept of split-plot designs can be extended to situations in which

randomization restrictions may occur at any number of levels within the

experiment. If there are two levels of randomization restriction, the layout is

called a split-split-plot design. In other words, what we have split once, we

can split again or split the split-plot is called split-split-plot design.