Chapter 17 : Crossed versus Nested Designs:
Definitions:
A factor A is said to be crossed with respect to a second factor, B, if
each level of factor A is exactly the same for each level of factor B.
Otherwise, the factor is said to be nested.
Example 1:
A PhD candidate in the School of Biological Sciences at WSU designed
an experiment in which mice were either selectively bred for wheel
running or not selectively bred for the trait. Within each selection group,
were four randomly initiated closed lines of mice. Within each line were
randomly selected eight families, and within each family wheel running
measurements were taken on two randomly selected offspring.
In this example, the selection group is a fixed effect, while line, family
and offspring are random effects.
Furthermore, lines are nested within selection group, family is nested
within line and offspring is nested within family.
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Example 2:
Clinical laboratories perform patient serum assays critical for correct
medical diagnoses. The laboratories maintain quality control programs
to monitor the performance of assays and insure the physician is
receiving accurate information. Quality control is maintained if variation
across several days of testing and two different, but equivalent, glucose
standard serums cannot be detected. The experimental protocol requires
three days of observation. Within each day, two runs of the glucose
standard are constructed. Within each run, three separate preparations
(replicates) are run through a spectrophotometer, and the glucose
(mg/dl) measured In this experiment the researcher wanted to treat days
as a random sample from a large population of days.
In addition, run is also considered a random sample from a large
population of potential runs. Lastly, the replication is, as usual, a
random sample from a large population of potential replicates.
Days and run are crossed effects, while replication is nested within
both days and run.
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The data for this experiment are shown in the table below. The response
is glucose (mg/dl) in quality control standards (Source: Dr. J. Anderson,
Beckman Instruments, Inc.)
Day
------------------------------------------------------------Run
Rep
1
2
3
--------------------------------------------------------------
1
2
1
42.5
48.0
41.7
2
43.3
44.6
43.4
3
42.9
43.7
42.5
1
42.2
42.0
40.6
2
41.4
42.8
41.8
3
41.8
42.8
41.8
--------------------------------------------------------------
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Researchers conducted an experiment to determine the content of a
cardio-vascular drug. They obtained a random sample of three batches
and from each of two blending sites, and from each batch they selected
5 tablets each.
Data:
Site
Batch
Tablet
1
2
3
4
5
1
1
5.03
2
4.64
5.10
5.25
4.98
5.05
4.73
4.95
4.82
5.06
3
5.10
2
1
5.05
2
5.46
3
4.90
5.15
5.20
5.08
5.14
4.96
5.12
5.12
5.05
5.15
5.18
5.18
5.11
4.95
4.86
4.86
5.07
Here Site A is fixed with 2 levels and Batch is nested within Site and is a
random Factor.
Model:
Yijk =  + i + bj(i) + ijk
We write the Factors as A and B(A).
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Now we will discuss the ANOVA Tables for Fixed, Random, Mixed with
Crossed and Random Factors
Model 1:
Mixed effects model - A fixed and B random, and crossed
ANOVA Table of Expected Mean Squares
Source
df
Expected Mean
Square
A
a-1
2
2
2

r
br
e 
a
b
a
B
b-1
e2 arb2+rab
AB
(a - 1)(b - 1)
e2 ra2b
Error
ab (r - 1)
 e2
Total
abr – 1
Use the MSError as the denominator term of the F statistics for assessing
AB effects.
Use the MSAB as the denominator term of the F statistics for assessing
the A and B effects.
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Model 2.
For Crossed A and B either Fixed or Random
Expected Mean Square (EMS)
Sour
ce
df
Fixed
Random
A Fixed, B
Random
A
a-1
e2 bra2
2
2
2
2
2
2

r
br

r
br
e 
a
b
a
e
a
b
a
B
b-1
e2 arb2
2
2
2
2
2
2

r
ar

r
ar
e
a
b
b
e
a
b
b
AB
(a - 1)(b 1)
2
e2 r ab
e2 ra2b
e2 ra2b
Error
ab (r - 1)  e2
 e2
 e2
Total
abr – 1
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3. Nested effects model – A, B fixed with B factor nested in factor A
ANOVA Table of Expected Mean Squares
Source
df
Expected Mean
Square
A
a-1
e2 bra2
B(A)
a (b – 1)
e2  r b2
Error
ab (r - 1)
 e2
Total
abr – 1
Use the MSError as the denominator term of the F statistics for assessing
the A and B(A) effects.
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4. Nested effects model - A and B random with B factor nested in
factor A
ANOVA Table of Expected Mean Squares
Source
df
Expected Mean
Square
A
a-1
2
2
2

r
br
e 


B(A)
a (b – 1)
e2 r 2
Error
ab (r - 1)
 e2
Total
abr – 1
Use the MSError as the denominator term of the F statistics for assessing
the B(A) effect.
Use the MSB(A) as the denominator term of the F statistics for assessing
the A effects.
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Nested effects model - A fixed and B random with B factor nested in
factor A
Source
df
Expected Mean
Square
A
a-1
2
2
2

r
br
e 

a
B(A)
a (b – 1)
e2 r 2
Error
ab (r - 1)
 e2
Total
abr – 1
(similar to our medicine data)
Summary of Nested ANOVA Table
Expected Mean Square (EMS)
Sour
ce
df
Fixed
Random
A Fixed, B
Random
A
a-1
e2 bra2
2
2
2
2
2
2

r
br
r
br
e 

 
e 

a
B(A)
a (b – 1)
e2  r b2
e2 r 2
e2 r 2
Error
ab (r - 1)  e2
 e2
 e2
Total
abr – 1
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Comparing the CRD and Nested Design for Two Factors A and B
A and B are crossed:
Y
=
μ
+
a
+
b
+
a
b
+
e
w
i
t
h
i
=
1
,
2
,
,
a
;
j
=
1
,
2
,
,
b
;
a
n
d
k
=
1
,
2
,
,
r


i
j
k
i
j
i
j
k

i
j
B is nested within A:
Y
=
μ
+
a
+
b
+
e
w
i
t
h
i
=
1
,
2
,
,
a
;
j
=
1
,
2
,
,
b
;
a
n
d
k
=
1
,
2
,
,
r
i
j
ki
i
j i
j
k



ANOVA Table
Crossed
Source
Nested
df
Source
df
--------------------------------------------------------------------------------------Ai
a-1
Ai
a-1
Bj
b-1
B i  j
a(b-1)
ABij
Errorijk
(a-1)(b-1)
ab(r-1)
Errorijk
ab(r-1)
--------------------------------------------------------------------------------------Total
abr - 1
Total
abr - 1
In the nested design, B(i)j absorbs the degrees of freedom for Bj and
ABij from the crossed effects.
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