Transcript Design of Engineering Experiments Part 7 – The 2k

Nested and Split
Plot Designs
Nested and Split-Plot Designs
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These are multifactor experiments that address
common economic and practical constraints
encountered in experimentation with real
systems.
Nested and split-plot designs frequently involve
one or more random factors.
There are many variations of these designs.
Agricultural Field Trial
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Investigate the yield of a new variety of crop
Factors
• Insecticides
• Fertilizers
Experimental Units
Experimental Design ?
• Farms
• Fields within farms
Fertilizers can be applied to individual fields;
Insecticides must be applied to an entire farm
from an airplane
Agricultural Field Trial
Farms
 Insecticides applied to farms
 One-factor ANOVA
 Main effect: Insecticides
 MSE: Farm-to-farm
variability
Agricultural Field Trial
Fertilizers applied to
fields
One-factor ANOVA
Main Effect: Fertilizers
MSE: Field-to-field
variability
Fields
Agricultural Field Trial
Farms
Fields
 Insecticides applied to
farms, fertilizers to fields
 Two sources of variability
 Insecticides subject to
farm-to-farm variability
 Fertilizers and insecticides
x fertilizers subject to
field-to-field variability
Two-Stage Nested Design
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Nested design
the levels of one factor (B) are similar to, but not
identical to each other at different levels of another
factor (A).
Consider a company that purchases material from
three suppliers
• The material comes in batches.
• Is the purity of the material uniform?
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Experimental design
• Select four batches at random from each supplier.
• Make three purity determinations from each batch.
Two-Stage Nested Design
Nested Design
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A factor B is considered nested in another
factor, A if the levels of factor B differ for
different levels of factor A.
The levels of B are different for different levels
of A.
Synonyms indicating nesting:
• Depends on, different for, within, in, each
Examples - Nested
Examples - Nested
Examples - Crossed
5-1 The yield of a chemical process is being studied. The two most important variables are thought to be
the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with
two replicates is performed. The yield data follow:
Temperature 200
150
90.4
90.2
160
90.1
90.3
170
90.5
90.7
Pressure
215
90.7
90.6
90.5
90.6
90.8
90.9
230
90.2
90.4
89.9
90.1
90.4
90.1
Examples - Crossed
5-2 An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the
depth of cut. She selects three feed rates and four depths of cut. She then conducts a factorial experiment
and obtains the following data:
Feed Rate (in/min)
0.20
0.15
74
64
60
Depth of Cut (in)
0.18
0.20
79
82
68
88
73
92
0.25
99
104
96
0.25
92
86
88
98
104
88
99
108
95
104
110
99
0.30
99
98
102
104
99
95
108
110
99
114
111
107
Examples - Nested
Two-Stage Nested Design
Statistical Model and ANOVA
 i  1, 2,..., a

yijk     i   j (i )   (ij ) k  j  1, 2,..., b
k  1, 2,..., n

SST  SS A  SS B ( A)  SS E
df : abn  1  a  1  a(b  1)  ab(n  1)
1 a 2 y...2
SS A 
yi.. 

bn i 1
abn
SSB ( A)
1 a b 2 1 a 2
   yij.   yi..
n i 1 j 1
bn i 1
a
b
n
1 a b 2
SSE    y   yij.
n i 1 j 1
i 1 j 1 k 1
2
ijk
2
y
2
SST   yijk
 ...
abn
i 1 j 1 k 1
a
b
n
Two-Stage Nested Design
Statistical Model and ANOVA
Residual Analysis
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Calculation of residuals.
yijk     i   j ( i )   ( ij ) k
ˆ  y... , ˆi  yi..  y... , ˆ j (i )  yij.  yi..
 yˆ ijk  y...  ( yi..  y...)  ( yij.  yi.. )  yij.
 eijk  yijk  yij.
m-Stage Nested Design
yijkl     i   j (i )   k (ij)   l (ijk)
m-Stage Nested Design
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Test statistics depend on the type of factors and
the expected mean squares.
• Random.
• Fixed.
Expected Mean Squares
Assume that fixtures and layouts are fixed, operators are
random – gives a mixed model (use restricted form).
Alternative Analysis
If the need detailed analysis is not available, start
with multi-factor ANOVA and then combine
sum of squares and degrees of freedom.
Applicable to experiments with only nested
factors as well as experiments with crossed and
nested factors.
Sum of squares from interactions are combined with
the sum of squares for a nested factor – no
interaction can be determined from the nested
factor.
Alternative Analysis
An Experiment Can Have Either of these Features
Two hierarchically nested factors,
with additional crossed factors occurring within
levels of the nested factor
Two sizes of experimental units,
one nested within the other, with crossed
factors applied to the smaller units
Split-Plot Design
Split-Plot Design
Whole-Plot Experiment : Whole-Plot Factor = A
Level A1
Level A2
Level A2
Level A1
Split Plot Designs
Analysis of Variance Table
Source
Whole-Plot Analysis
Factor A
Whole-Plot Error
df
a-1
a(r-1)
Split-Plot Design
Split-Plot Experiment : Split-Plot Factor = B
B2
B1
B1
B2
B1
B1
B2
B1
B2
B2
B2
B1
B1
B2
B1
B2
Level A1
Level A2
Level A2
Level A1
Split Plot Designs
Analysis of Variance Table
Source
Whole-Plot Analysis
Factor A
Whole-Plot Error
Split-Plot Analysis
Factor B
A x B
Split-Plot Error
Total
df
a-1
a(r-1)
b-1
(a-1)(b-1)
a(b-1)(r-1)
abr-1
Agricultural Field Trial
Agricultural Field Trial
Insecticide 2
Insecticide 1
Insecticide 2
Insecticide 1
Insecticide 1
Insecticide 2
Agricultural Field Trial
Insecticide 2
Insecticide 1
Fert A
Fert B
Fert B
Fert A
Fert B
Fert A
Fert B
Fert A
Fert A
Fert B
Fert B
Fert A
Fert A
Fert B
Insecticide 2
Fert B
Fert B
Fert B
Fert B
Insecticide 1
Fert A
Fert A
Fert A
Fert A
Fert B
Fert A
Insecticide 2
Fert B
Fert A
Fert A
Fert B
Fert B
Fert A
Fert A
Fert A
Fert B
Fert B
Fert A
Insecticide 1
Fert B
Fert A
Fert A
Fert B
Fert B
Agricultural Field Trial
Whole Plots = Farms
Large Experimental Units
Split Plots = Fields
Small Experimental Units
Agricultural Field Trial
Whole Plots = Farms
Large Experimental Units
Whole-Plot Factor = Insecticide
Whole-Plot Error = Whole-Plot Replicates
Split Plots = Fields
Small Experimental Units
Split-Plot Factor = Fertilizer
Split-Plot Error = Split-Plot Replicates
The Split-Plot Design
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The split-plot is a multifactor experiment
where it is not practical to completely
randomize the order of the runs.
Example – paper manufacturing
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Three pulp preparation methods.
Four different temperatures.
The experimenters want to use three replicates.
How many batches of pulp are required?
The Split-Plot Design
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Pulp preparation method 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.
The Split-Plot Design
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Each replicate has been divided into three parts,
called the whole plots.
• Pulp preparation methods is the whole plot treatment.
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Each whole plot has been divided into four subplots
or split-plots.
• Temperature is the subplot treatment.
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Generally, the hard-to-change factor is assigned to
the whole plots.
This design requires 9 batches of pulp (assuming
three replicates).
The Split-Plot Design
Tensile Strength
Rep (Day) 1
Pulp Prep Method
Temperature
200
225
250
275
Rep (Day) 2
Rep (Day) 3
1
2
3
1
2
3
1
2
3
30
34
29
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31
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31
35
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35
41
26
32
36
30
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40
34
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32
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36
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The Split-Plot Design
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There are two levels of randomization
restriction.
• Two levels of experimentation
Tensile Strength
Rep (Day) 1
Pulp Prep Method
Temperature
200
225
250
275
Rep (Day) 2
Rep (Day) 3
1
2
3
1
2
3
1
2
3
30
34
29
28
31
31
31
35
32
35
41
26
32
36
30
37
40
34
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Experimental Units in Split
Plot Designs
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Possibilities for executing the example split plot design.
• Run separate replicates. Each pulp prep method (randomly selected) is tested at
four temperatures (randomly selected).
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Large experimental unit is four pulp samples.
Smaller experimental unit is a an individual sample.
• If temperature is hard to vary select a temperature at random and then run (in
random order) tests with the three pulp preparation methods.
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Large experimental unit is three pulp samples.
Smaller experimental unit is a an individual sample.
Tensile Strength
Rep (Day) 1
Pulp Prep Method
Temperature
200
225
250
275
Rep (Day) 2
Rep (Day) 3
1
2
3
1
2
3
1
2
3
30
34
29
28
31
31
31
35
32
35
41
26
32
36
30
37
40
34
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38
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40
42
32
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36
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The Split-Plot Design
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Another way to view a split-plot design is a
RCBD with replication.
• Inferences on the blocking factor can be made with
data from replications.
The Split-Plot Design Model
and Statistical Analysis
yijk     i   j  ( )ij   k  ( )ik  (  ) jk
 i  1, 2,..., r
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( )ijk   ijk  j  1, 2,..., a
 k  1, 2,..., b
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Sum of squares are computed as for a three factor factorial
design without replication.
RCBD Model
Statistical model(effectsmodel):
yij     i   j   ij for i  1,2,, a; j  1,2,, b.
or thesum of squares can also be obtainedfrom
(two factorfactorialmodel with no replication)
yij     i   j  ij   ij for i  1,2,, a; j  1,2,, b.
The Split-Plot Design Model and
Statistical Analysis
There are two error structures; the whole-plot error and the subplot
error