# Lecture 26

```Experiments with both nested and
“crossed” or factorial factors
•Example 14-2 Nested and Factorial Factors (Hand insertion
of electronic components)
•Three assembly fixtures
•Two workplace layouts
•Four operators for each fixture-layout combination
•It’s difficult to use the same (4) operators for each layout
because of location
• Four operators for layout 1
•
Four operators for layout 2
•Assume that fixtures and layouts are fixed, operators are
random – gives a mixed model (use restricted form)
•Fixtures and layouts: factorial
•Operators: nested within layouts
1
Table 14-9 Assembly Time Data for Example 14-2
yijkl
 i  1,2,3
 j  1,2

    i   j   k ( j )  ( )ij  ( )ik ( j )   (ijk )l 
k  1,2,3,4
 l  1,2
2
Example 14-2 – Expected Mean Squares
3
Example 14-2 – Minitab Analysis
Table 14-12 Minitab Balanced ANOVA Analysis of Example 14-2 Using the Restricted Model
4
The Split-Plot Design
• Text reference, Section 14-4 page 540
• The split-plot is a multifactor experiment where it
is not possible to completely randomize the order
of the runs
• Example – paper manufacturing
–
–
–
–
–
Three pulp preparation methods
Four different temperatures
Each replicate requires 12 runs
The experimenters want to use three replicates
How many batches of pulp are required?
5
The Split-Plot Design
• Pulp preparation methods is a hard-to-change
factor
• Consider an alternate experimental design:
– In replicate 1, select a pulp preparation method,
prepare a batch
– Divide the batch into four sections or samples, and
assign one of the temperature levels to each
– Repeat for each pulp preparation method
– Conduct replicates 2 and 3 similarly
6
The Split-Plot Design
• Each replicate (sometimes called blocks) has been
divided into three parts, called the whole plots
• Pulp preparation methods is the whole plot
treatment
• Each whole plot has been divided into four
subplots or split-plots
• Temperature is the subplot treatment
• Generally, the hard-to-change factor is assigned to
the whole plots
• This design requires only 9 batches of pulp
(assuming three replicates)
7
The Split-Plot Design
Model and Statistical Analysis
yijk     i   j  ( )ij   k  ( )ik  (  ) jk
 ( )ijk   ijk
 i  1, 2,..., r

 j  1, 2,..., a
 k  1, 2,..., b

Table 14-15 Expected Mean Square Derivation for Split-Plot Design
8
Split-Plot ANOVA
9
Split-Plot ANOVA
Table 14-16 Analysis of Variance for the Split-Plot Design Using Tensile Strength Data from
Table 14-14
Calculations follow a three-factor ANOVA with one replicate
Note the two different error structures; whole plot and subplot
10
Alternate Model for the Split-Plot
yijk     i   j  ( )ij   k  (  ) jk   ijk
 i  1, 2,..., r

 j  1, 2,..., a
 k  1, 2,..., b

11
The Agriculture Heritage of Split-Plot
Design
• Whole plots: large areas of land
• Subplots: smaller areas of land within large areas
• Example: Effects of variety, field, and fertilizer on
the growth of a crop
• One variety is planted in a field (a whole plot)
• Each field is divided into subplots with each
subplot is treated with one type of fertilizer
• Crop varieties: main treatments
• Fertilizers: subtreatments
12
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