Introduction to Nested Designs
Mark S. Rusco
for
ASQ – Grand Rapids Section
Definition
A type of Designed Experiment
 You identify possible sources of variation
 Assigns values to sources of variation
 You work on the biggest source

Example 1

(Barker) Coal is loaded from a strip mine
into train cars. If the sulfur content is
<5% it can be burned. Three hopper cars
are sampled four times each and each
sample is tested twice. How much
variability is from car-to-car variation?
Sample to sample? Test-to-test?
Coal Car Schematic
Power
Plant
Coal
Car 1
Coal
Car 2
Coal
Car 3
Sample 1
Sample 2
Sample 3
Sample 4
Sample 1
Sample
2…
Test 1
Test 1
Test 1
Test 1
Test 1
Test 1…
Test 2
Test 2
Test 2
Test 2
Test 2
Sample
1…
Example 2

(Montgomery) Raw materials are
purchased from three suppliers and there
is too much variability in the finished
product. Each supplier is sampled four
times and each sample is tested three
times. Is the variability between suppliers?
Between batches? Between tests?
Raw Material Schematic
Plant
Process
Supplier
1
Supplier
2
Supplier
3
Samp 1
Samp 2
Samp 3
Samp 4
Samp 1
Samp
2…
Test 1
Test 1
Test 1
Test 1
Test 1
Test 1…
Test 2
Test 2
Test 2
Test 2
Test 2…
Test 3
Test 3
Test 3
Test 3
Example 3

(Box, Hunter, Hunter) Paint pigment paste
is produced in 80 drum batches.
Moisture content is important, but varies
widely. Fifteen batches were sampled
twice each and each sample was tested
twice. Is the variability between batches?
Between samples? Between tests?
Paint Paste Example Illustration

Standard deviation for each curve?
Total Variation
about the
Process Mean
Sampling
Variation
Measurement
Error
The measurement
value that you write
down.
Process Mean
Batch
Mean
Variation of all of the batch
means around the long
term process mean.
Variation of the drums
means around the batch
mean.
Variation of the tests
around the mean for that
sample of material.
Nested vs. Crossed Designs






This is a choice for many GRR software
packages.
Which do you choose?
In a Crossed Design it is possible for the
different levels to participate in all of the other
levels.
In a Nested Design each level cannot participate
in all of the other levels.
Crossed Designs can test for interactions.
Nested Designs cannot test for interactions.
Example 1
A typical Gauge R&R is performed using a
1” micrometer. Ten parts are selected,
and 3 well-trained operators measure
each part 3 times. The GRR is performed
at a single location and all operators are
available on the same shift.
 Crossed or Nested? (crossed)

Example 2
Two locations of a company produce
mating parts that come together to
provide a sliding fit in use. Each location
sends 10 of their parts to the other
location where they are matched to parts
and tested for effort of sliding fit. Each
location uses their tensile tester (same
brand) with 2 operators, 10 parts, 3 trials.
 Crossed or Nested? (nested)

Example 3
You produce a part made from platinum
which is supplied to you in very small
batches. You have two machines set up to
run the part, you clean one machine while
you run the other so a batch is run
completely on one machine. You want to
test for variation from machine, batch,
operator, and measurement.
 Crossed or Nested? (nested)

Overall Steps
1.
2.
3.
4.
5.
6.
Recognize a situation as a possible Nested
Design.
Collect and organize the data.
Calculate the Sum of Squares (SS) for each
level in the hierarchy.
Create an ANOVA table to calculate Mean
Square (i.e.,Variance) for each source.
Create an Estimated Mean Square table and
the set of equations to reflect the nesting.
Solve 4 and 5 for the Variance for each source.
Master Equation for SS
∑(x2)/n in x – (∑x)2/n in x
 x is the measured values.

◦ At the lowest level it is the individual value
◦ At higher levels it is a sum.
n is the number of values that went into
each x and usually differs.
 Example on next page.

Example – Short Shots
Date
03/01 (W)
03/01 (W)
03/02 (Th)
03/05 (Th)
03/06 (M)
03/06 (M)
03/07 (T)
03/07 (T)
03/08 (W)
03/13 (M)
03/13 (M)
03/14 (T)
03/15 (W)
03/15 (W)
03/20 (M)
03/21 (T)
Shift
st
1
nd
2
st
1
nd
2
st
1
nd
2
st
1
nd
2
st
1
st
1
nd
2
nd
2
st
1
nd
2
nd
2
st
1
Lot
A
A
B
E
F
F
E
G
G
C
C
B
D
D
H
H
Machine
1
1
1
2
2
2
2
2
2
1
1
1
1
1
2
2
Short Shots
4
3
4
5
7
3
8
8
9
3
5
3
7
8
15
17
Short Shot Data Re-Aligned
1
Machine
2
Lot
A
B
C
D
E
F
G
H
1st Shift
4
4
3
7
8
7
9
17
2nd Shift
3
3
5
8
5
3
8
15
Sum of Squares (SS) for Shift
1st Shift ^2
2nd Shift ^2
Sum of
Shifts
Sum^2
SS w/in Lot
SS for Shift
16
16
9
49
64
49
81
289
9
9
25
64
25
9
64
225
7
7
8
15
13
10
17
32
49
49
64
225
169
100
289
1024
0.5
0.5
2
0.5
4.5
8
0.5
2
18.5
SS for Lot
Lot tot
7
7
8
15
13
10
17
32
Lot Tot)^2
49
49
64
225
169
100
289
1024
Tot of Lots
37
72
SS w/in
Mach
22.375
143
SS Lot
165.375
SS for Machine
Mach
Total
Total^2
Total
SS Machine
109
76.56
1
2
37
72
1369
5184
Co-efficients for EMS
R
R
R
i:2
j:4
k:2
Mach i
1
4
2
=8*σ2mach + 2*σ2lot + σ2shift
Op(i)j
1
1
2
=2*σ2lot + σ2shift
Samp(ij)k
1
1
1
=σ2shift
EMS
Short Shots ANOVA Table
Source
SS
Machine 76.5625
dF
MS
1
76.5625 =8*σ2mach + 2*σ2lot + σ2shift
Lot
165.375
6
27.5625 =2*σ2lot + σ2shift
Shift
18.5
8
2.3125 =σ2shift
EMS and % Contribution
EMS
%Contrib.
σ2mach =
6.125
29.1
σ2lot =
12.625
59.9
σ2shift =
2.3125
11.0
Total
21.0625
Interpret
Batch to batch variation is the biggest
contributor. Work with the supplier to
minimize variation.
 Machine to machine variation is next largest.
Settle on one machine to run the parts.
 No need to write up any operators.

Questions?