Stat 512
Spring 2011
Randomized Complete Block Designs
(Ch 15 O&L)
Definition - revisited
The design structure of an experimental design consists of the
structure of the grouping of experimental units into homogeneous
units. This grouping of experimental units in conjunction with the
form of the randomization of treatments to experimental units
defines the design.
Review of Design Types (L 4)
1) Completely Randomized Design (CRD).
2) Randomized Complete Block Design (RCBD).
3) Latin Square Design.
4) Randomized Incomplete Block Designs
Design Advantages
Completely Randomized Design (CRD)
a) Relatively easy to construct.
b) Easy to analyze, even for different sample sizes.
c) Any number of treatments
Randomized Complete Block Design (RCBD)
a) Easily constructed for comparing t treatment means in the
presence of a single source of extraneous variation (blocks).
b) Easy to analyze, but equal sample sizes required.
c) Used for any number of treatments or blocks.
Latin Square Design
a) Relatively easy to construct for comparing t treatment means in
the presence of two extraneous sources of variation (two blocking
factors).
b) Relatively simple analysis.
1
Stat 512
Spring 2011
Design Disadvantages
Completely Randomized Design (CRD)
a) Experimental units should be homogeneous. Any degree of
variation in the experimental units will lower the statistical power.
Randomized Complete Block Design (RCBD)
a) Requires that within block sub-units must be homogeneous, thus
it is best for comparing only a few treatment means.
b) The effect of each treatment on the response must be
approximately the same from block to block.
c) Will have lower statistical power if blocks are homogeneous.
d) The design is less efficient than others in the presence of more
than one source of variation.
e) The efficiency of the design decreases as the number of
treatments and, hence, block size increases.
Latin Square Design
a) Requires that within block sub-units must be homogeneous, thus
it is best for comparing only a few treatment means.
b) The effect of each treatment on the response must be
approximately the same from row to row and column to column.
c) Although a Latin square can be constructed for any value of t, it
require t 2 unit to study t treatments. It is best suited for comparing t
treatments when 5 ≤ t ≤ 10.
d) As t increases the experimental error per unit is likely to
increase.
e) The analysis becomes very complicated if there are missing data
or if treatments are missassigned.
2
Stat 512
Spring 2011
Randomized Complete Block Design
Blocks
Treatments
1
b
Means
1
Y 11
Y1b
2
Y 21
Y2b
.
.
.
.
.
.
.
.
.
.
.
.
t
Y t1
Ytb
Means
2
…
Y12
…
Y22
…
.
…
.
…
.
…
Yt2
…
…
The statistical linear model for a randomized complete block
design with an oneway (fixed) treatment structure is a TWO-WAY
MIXED ANOVA with one fixed and one random factor with no
interactions ie:
Yij = µ + i + bj + eij (effects model)
Yij = i + bj + eij (means model)
i = 1, 2, …, t (treatments)
j = 1, 2, …, r (blocks)
Definitions:
µ grand mean
bj effect of the jth block (random factor) is the average deviation of
the units in block j
(j = dj - µ)
i effect of the ith treatment (fixed factor) (i = i - µ)
eij random error term for the ith treatment and jth block.
3
Stat 512
Spring 2011
Assumptions:

bj are independent for all j follow N(0,sr2)
eij are independent for all i and j follow N(0,se2)
Sufficient conditions for estimation:
∑ 𝜏𝑖 = 0, ∑ 𝑏𝑗 = 0,
Decomposition of the total sum of squares:
Y

Y

t

Y

Y

r

Y

YY


Y

Y

Y











or
t
r
r
2
i
j.
.
i

1
j

1
t
2
.
j.
.
j

1
t
r
2
i
..
.
i

1
2
i
j.
ji
..
.
i

1
j

1
S
S
S
S

S
S

S
S
T
o
t
a
l
B
l
o
c
k
s
T
r
e
a
t
m
e
n
t
s
E
r
r
o
r
Where
t
Sample mean for the jth block
Y. j
Y Sample mean for the ith

i 1
t
r
treatment
Yi. 
Y
j 1
ij
r
t
Sample grand mean
Y.. 
r
Y
i1 j1
ij
rt
4
ij
Stat 512
Spring 2011

 
S
S
2 2
2
B
l
o
c
k
s
M
S

E
M
S


r


t

r



B
l
o
c
k
s
B
l
o
c
k
s
B
l
o
c
k
B
l
o
c
k
R
e
p


r

1

t
2
r


i
S
S
2
T
r
e
a
t
m
e
n
t
s
i

1
M
S

E
M
S




T
r
e
a
t
m
e
n
t
s
T
r
e
a
t
m
e
n
t
s
t

1
t

1


S
S
2
M
S
 E
E
M
S



E
E
t

1
r
1




ANOVA table for the RCBD
Source
Blocks
Treatments
Error
Total
df
r-1
t-1
(r-1)(t-1)
rt - 1
SS
SSB
SST
SS-SST-SSB
SS
MS
F
SSB/(r-1)
SST/(t-1)
MST/MSE
SSE/(r-1)(t-1)
ANOVA table for the CRD
Source
Treatments
Error
Total
df
t-1
t(r - 1)
rt – 1
SS
SST
SS - SST
SS
MS
SST/(t-1)
SSE/r(t-1)
F
MST/MSE
Notice that there are fewer degrees of freedom for error in the RCBD design
than in the CRD design, (r-1)(t-1) vs. t(r-1), or (r - 1) fewer degrees of
freedom. In the RCBD, these r - 1 degrees of freedom have been partitioned
from the error and assigned to the blocks. However, the SSE in RCBD is
generally smaller than that of CRD (which supposedly contains block
effects).
Obviously one should only use the RCBD when the variation explained by
the blocks more than offsets the degrees of freedom they consume. So how
can one determined when an RCBD is appropriate?
The concept of efficiency.
5
Stat 512
Spring 2011
Variability in the completely randomized design (CRD)
In the CRD it is assumed that the experimental units are uniform.
This is not always true in practice and it is necessary to develop
methods to deal with variability.
 If in comparing two methods of fertilization one region of the
field has much greater fertility than the others, then a
treatment effect might be incorrectly ascribed to the
treatment applied to this part of the field, making a Type I
error.
 For this reason in CRD it is always advocated to include as
much of the native variability of the experiment as possible
within each plot, making each plot as representative of the
whole experiment, and the whole experiment as uniform, as
possible.
 In actual field studies plots are designed long and narrow to
achieve this effect.
 However, if the plots are more variable, experimental error
(MSE) is larger, F (MST/MSE) is smaller, and the
experiment is less sensitive.
 Finally, if the experiment is replicated in a variety of
situations to increase the scope of the experiment, this
additional variability needs to be removed from the analysis
to focus on the treatment effect.
This is the purpose of blocking.
6
Stat 512
Spring 2011
No difference among blocks: if the RCBD design were applied to
an experiment in which the blocks were really no different (i.e. no
significant block effect), the MSE for the CRD would be smaller
than the MSE for the RCBD simply due to degrees of freedom.
For example, if t=3 and r=4, MSECRD = SSE/9, and MSERCBD =
SSE/6. Therefore, the F statistic for the CRD would be larger.
Consider a confidence interval for the differences between two
means
2
Y

Y

CriticalF
*
MSE
A
B
(
1
,
MSEdf
),

r
Under H0 = Y A - Y B =0
 The CRD has a smaller critical F value than the RCBD
because of its larger df. In addition if there are no differences
among blocks then MSECRD=MSERCBD.
 Therefore, the larger critical F value in the RCBD moves the
threshold of the rejection further from the mean (0) than in
the CRD.
 This change in the position of the rejection threshold affects
the Type II error () and the power of the test (1-).
 Under this scenario, the probability of failing to reject a false
null hypothesis () will be smaller in the CRD than in the
RCBD.
 In other words, the CRD would in this situation be more
powerful (larger 1- ).
Significant difference among blocks: On the other hand, suppose
that there really were a substantial difference among blocks as well
7
Stat 512
Spring 2011
as among the treatments (H0 false). If the CRD were used, this
difference among blocks would be allocated to the error, so the F
statistic for the CRD would be smaller than the F statistic of the
RCBD.
Under this scenario,
the RCBD would still
have a larger critical
F value because of
the lost degrees of
freedom, but this may
be more than
compensated by the
smaller MSE.
If the effect of the
reduced MSE
(threshold closer to 0)
is larger than effect of the larger critical value (threshold further
from 0) the net result will be a smaller , and a larger power (1-)
in the RCBD relative to the CRD.
Summary:
The MSError is the estimator of the variance used for assessing
hypotheses, whether for a CRD or an RCBD. If experimental units
are relatively homogeneous, then the CRD is preferred. This is
because the relatively larger dfError reduces the MSError.
If the experimental units are heterogeneous, then the RCBD is
preferred. This is because the SSBlock is large and subsequently
the SSError is small, and the relative decrease in SSError is larger
than the relative decrease in dfError.
Example 1:
8
Stat 512
Spring 2011
A greenhouse consisting of six benches was to be used for an
experiment assessing growth among four varieties of house plants.
Because light intensity, humidity and temperature varied
throughout the greenhouse it was decided that each bench should
contain a complete replication of the experiment. Thus, each bench
received each variety of potted plant.
The change in plant height (cm) after 2 weeks was recorded:
Bench
1
2
3
4
5
6
1
19.8
16.7
17.7
18.2
20.3
15.5
Varieties
3
16.4
15.4
14.8
15.6
16.4
14.6
2
21.9
19.8
21.0
21.4
22.1
20.8
4
14.7
13.5
12.8
13.7
14.6
12.9
Variety Means:
Block Means:
Grand Mean:
ANOVA Table
Source
Sum
of Degrees of
Squares
Freedom
19.793
5
Bench
3
Varieties 188.538
6.527
15
Error
214.858
23
Total
H0:
Ha: not all µi are equal.
F=144.44
9
Mean
Squares
3.959
62.846
0.435
F0
144.44
Stat 512
Spring 2011
Reject H0 if F > F (0.05, 3, 15) = 3.287
Conclusion: Reject H0 and assume the varieties do not have the
same mean growth.
P-value = 2.741011
Standard Error for the Treatment Mean:
which is estimated by:
S
E
Y
i.
SEY
 i. 

r
M
S
E
r
Estimated Standard Error for the Difference in Two Treatment
Means:
1
1
M
S
E

2
S
E
Y

Y

M
S
E





i
.
i
.


rr

 r
Multiple Comparisons - Fisher’s LSD:
1
1
1
1




L
S
D

t

M
S
E



2
.
1
3
1

0
.
4
3
5



0
.
8
1
1
5




2
,
t

1
r

1






r
r
6
6





Y

Y

3
.
1
3
4
*
Y

Y

2
.
5
0
0
*
Y

Y

4
.
3
3
3
*
1
.
2
.
1
.
3
.
1
.
4
.
Y

Y

5
.
6
3
4
*
Y

Y

7
.
4
6
7
*
Y

Y

1
.
8
3
3
*
2
.
3
.
2
.
4
.
3
.
4
.
* significant at the 0.05 level.
Equivalently you could perform t - tests:
H0:
VS. Ha:
t
Yi. Yi.
1 1
M
SE  
r r
Reject if | t | > t (α/2, (t-1)(r-1)), or P-value ≤ α
Conclusion:
Example (Greenhouse example cont)
MSE = 0.435, df = 15 and t(0.025, 15) = 2.131
:
t = -8.230
P-value < 0.0001*
:
t = 6.565
P-value < 0.0001*
10
Stat 512
Spring 2011
:
t = 11.379
P-value < 0.0001*
:
t = 14.796
P-value < 0.0001*
:
t = 19.609
P-value < 0.0001*
:
t = 4.814
P-value = 0.0002*
* significant at the 0.05 level.
Group
Mean
Treatment
A
21.167
2
B
18.033
1
C
15.533
3
D
13.700
4
Note: In addition to the LSD procedure (or t-tests), Scheffe’s
procedure, Bonferroni’s procedure and any of the other multiple
comparison procedures or contrasts discussed in class can be used
Design Efficiency
One cannot say that a block design is more efficient than a
completely randomized design, except when viewed in the context
of the variability of the response among experimental units. The
reverse assertion also cannot be made.
Suppose it were possible to analyze the same data set under a
randomized complete block design and a completely randomized
design.
11
Stat 512
Spring 2011
Example 2: This example involves the response of sheep to
estrogen. The sheep are blocked by ranch, with four treatments per
block. The treatments are combinations of sex of the sheep (M or
F) and level of estrogen treatment (S0 or S3). Although these data
could be analyzed as a factorial experiment, in this example they
are treated as four separate treatments.
RCBD. Effect of estrogen on weight gains. Blocks are 4 different
ranches.
Block
Treatment
F-S0
M-S0
F-S3
M-S3
Block Total
Block Mean
I
47
50
57
54
208
52
II
III
52
54
53
65
224
56
62
67
69
74
272
68
IV
51
57
57
59
224
56
Treatment
Total Mean
212
53
228
57
236
59
252
63
928
58
Table 8.2 RCBD ANOVA
Source of Variation
Totals
Blocks
Treatments
Error
df
15
3
3
9
SS
854
576
208
70
MS
df
15
3
12
SS
854
208
646
MS
192.00
69.33
7.78
F
24.69**
8.91**
Table 8.3 CRD ANOVA
Source of Variation
Totals
Treatments
Error
12
69.33
53.83
F
1.29 NS
Stat 512
Spring 2011
Since each treatment occurs the same number of times in each
block, differences among blocks do not result from treatments but
from other differences associated with the blocks. This component
of the total sum of squares can be removed and the experimental
error reduced accordingly. Compare the SSerror in Tables 8.2 and
8.3
Checking Model Assumptions
ˆ
ˆ
eY
Y


Y
Y

Residuals:
i
j
i
j
i
j Y
i
j
.
j
i
.
Normality:
Wilks-Shapiro test
Normal Probability Plot
Equality of Variances:
Levene's Test
Likelihood Ratio Test
Plots: Plot Residuals vs. Factor Levels
Plot Residuals vs. Block Levels
Plot Residuals vs. Predicted Values
Other Problems:
A block by treatment interaction can result from a poorly
controlled experiment. If treatments are not manipulated in a
consistent manner an interaction may result.
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Stat 512
Spring 2011
Relative Efficiency of Blocking
We saw earlier that if the variation among blocks is large then we
can expect the RCBD method to work better than the CRD while if
this variation is small it may not. There is no F-test for assessing
whether blocks are significant and therefore effectively reduce the
mean square error. The concept of relative efficiency formalizes
the comparison between two experimental methods. Recall that
the F statistic is defined by the formula F = MST/MSE. The
experimental design affects primarily the MSE since the degrees of
freedom for treatments is always t - 1. The information in the
design is 1/MSE, so the relative efficiency of design to design to is
(1/MSE1)/ (1/MSE2) = MSE2/MSE1.
A relative efficiency measure is:
r

1
M
S

r

t

1

M
S
EM



S
E
l
o
c
k
s
B
l
o
c
k
s
R
E
 B
H

r

t

1

M
S
E
M
S
E


RE < 1  H < 1 => Ineffective Blocking
RE = 1  H = 1 =>
RE > 1  H > 1 => Effective Blocking
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Stat 512
Spring 2011
The Latin Square Design
The randomized complete block design reduces the variation
associated with each treatment mean by controlling for variation
due to known nuisance variables (blocks). The concept can be
extended to two levels of control. That is, two separate blocking
factors (rows and columns) can be used to control for some forms
of variation.
In agricultural yield experiments one might find a moisture
gradient running East - West and a fertility gradient running North
- South. Neither of these components is comparatively of interest
to the researcher, but it would be hove the researcher to control for
these sources of variation when he or she desires to compare the
treatment means of interest.
Linear Statistical Model for the Latin Square (3-way ANOVA
with 2 random effects and no interactions)
Yij = µ + ri + cj +k + eij (effects model)
Yij = Ri + Cj +k +eij (cell means model)
i = 1, 2, , t
j = 1, 2, , t
k = 1, 2, , t
Yij response for ith row, jth column.
µ grand mean
ri effect of the ith row effect (random) (ri = Ri - µ, where ri is the ith
row mean)
cj effect of the jth column effect (random) (cj = Cj - µ, where cj is
the jth column mean)
k effect of the kth treatment effect (fixed) (k = k - µ, where k is
the kth treatment mean)
eij random error component for the ith row, jth column.
15
Stat 512
Spring 2011
Assumptions:
The random effects ri, cj and eij are normally distributed and
mutually independent.
Define:
t
Observed row mean:
Yi. 
Y
ij
j 1
t
t
Observed column mean:
Y. j 
Y
ij
i 1
t
t
Observed treatment mean
Yk 
Y
k 1
ij
t
This last summed over the response for the kth treatment.
Decomposition of the Sum of Squares
Y
Y
tY
Y
tY
Y
tY
Y
i.

k



.j




i
j
.
.
.
.
.
.
.
.
t t
2
i j
t
t
2
i

1
2
j

1
t
2
k

1

Y
Y
Y
Y

2

Y




i
j
i
.
.
j
k
.
.
t t
2
i j
or
S
S
=
S
S
+
S
S
+
S
S
+
S
S
T
R
o
w
s
C
o
l
u
m
n
s
T
r
e
a
t
m
e
n
t
s
E
ANOVA Table
Source
Sum of
Squares
Mean
Squares
SSTreatments
Degrees of
Freedom
t-1
t-1
t-1
Rows
Columns
Treatment
s
Error
SSRows
SSE
(t - 1)(t - 2)
MSE
Total
SST
t2 1
SSColumns
16
F0
MSRows
MSColumns
MSTreatments
MSTreatments
MSE
Stat 512
Spring 2011
Hypothesis Testing
H
:
μ
=
μ
=
=
μ
s
.
H
:
N
o
t
a
l
l
μ
a
r
e
e
q
u
a
l
0
1
2
tv
a
k
M
S
T
rea
tm
en
ts
F
0
M
SE
Reject Ho if Fo > F( ,t - 1,(t - 1)(t - 2))
Conclusion:
A Latin Square Example
A latin square design was used to investigate the effect of shelf
space on food sales. The experiment was carried out over a sixweek period using six different stores. The resulting sales of coffee
creamer are presented in the following table (with shelf space
index in parentheses).
Store
1
2
3
4
5
6
1
27 (5)
34 (6)
39 (2)
40 (3)
15 (4)
16 (1)
2
14 (4)
31 (5)
67 (6)
57 (1)
15 (3)
15 (2)
3
18 (3)
34 (4)
31 (5)
39 (2)
11 (1)
14 (6)
4
35 (1)
46 (3)
49 (4)
70 (6)
9 (2)
12 (5)
Weeks
5
26 (6)
37 (2)
38 (1)
37 (4)
18 (5)
19 (3)
6
22 (2)
23 (1)
48 (3)
50 (5)
17 (6)
22 (4)
Analysis of Variance Table
Source
Store
(Rows)
Week
(Column)
Shelf
Error
Total
SS
6502.25
df
5
MS
1300.45
533.92
5
106.78
477.58
1291.00
8804.75
5
20
95.52
64.55
35
17
F0
1.48
Stat 512
Spring 2011
Estimated Standard Error for
Yk and Yk  Yk 
M
S
E
2

M
S
E
S
E
Y

S
E
Y

Y






k
k
k
t
t
Multiple Comparisons - Fisher’ s LSD:
1
1


L
S
D

t
S
E


Reject



2
,
t

12
t






M
tt


equality of
k and k  if
Y
Y
SD
k
k L
Equivalently you could perform t - tests:


H
:k

0
v
s
.H
:k

0


0
k
a
k
t0 
Y
Y
k
k
1 1
M
SE  
t t
Reject Ho if |to| >
t 2,t1t1 or
Pvalue < 0.05
Conclusion
Note: In addition to the LSD procedure (or t-tests), Scheffe’ s
procedure, Bonferroni’ s procedure and any of the other multiple
comparison procedures or contrasts discussed in class can be used.
Replicated Latin Squares
Latin Squares can be replicated to increase the precision of an
experiment. Replication can be approached in two manners:
multiple independent tables; or multiple tables with a common set
of rows or columns. For at four (t = 4) treatment Latin square
design with two replicates we have:
18
Stat 512
Spring 2011
Independent Latin Squares
Row
1
2
3
4
5
6
7
8
1
A
B
C
D
2
B
C
D
A
3
C
D
A
B
4
D
A
B
C
5
A
B
C
D
Column
6
7
B
C
D
A
Replicated Latin Squares with Common Rows
Column
Row
1
2
3
4
5
6
1
A
B
C
D
A
B
2
B
C
D
A
B
C
3
C
D
A
B
C
D
4
D
A
B
C
D
A
8
C
D
A
B
D
A
B
C
7
C
D
A
B
8
D
A
B
C
The analysis of a Latin square design with r independently
replicated tables is similar to the analysis for a single Latin square
design, except the model for analysis must account for variation
among replicates. Thus, the model is of the form:
Y
=
μ
+
κ
ρ
γ+
+
e
i
j
k
l+
k
i
j
k
i
l
j
l

+

τ
Yijk = µ + tk + sl +ri(l) + cj(l) + eijk
Source
df
Table
s-1
Row
s(t - 1)
Column
s(t - 1)
Treatment
t-1
Error
(st - s - 1)(t - 1)
19
Stat 512
Spring 2011
The analysis of a Latin square design with r replicated tables
having common rows is adjusted for independent columns, but
uses the same rows in each table. Thus, the model is of the form:
Yijk = µ + rk + sl +i + cj(l) + eijk
Source
Table
Row
Column
Treatment
Error
df
s-1
t-1
s(t - 1)
t-1
(st - 2)(t - 1)
An Experiment with six yearling dairy heifers was conducted as
two latin squares. Treatments were three rations selected on the
basis of diverse quality and physical characteristic and fed ad
libitum. Each animal ate the three rations sequentially, one week
on each. The response, Y, is pounds of dry matter consumed per
100 lb of body weight. The three treatments were (1) alfalfa hay,
(2) corn silage, (3) blue-grass straw pellets.
Square
1
2
Heifer
Heifer
Week
1
2
3
1
2.7 (1)
2.2 (2)
1.9 (3)
2
2.6 (2)
0.2 (3)
2.1 (1)
3
1.9 (3)
2.3 (1)
2.4 (2)
20
4
3.3 (1)
1.7 (3)
2.1 (2)
5
2.3 (2)
2.8 (1)
1.7 (3)
6
0.1 (3)
1.8 (2)
2.7 (1)
Stat 512
Spring 2011
Source
Sum of
Squares
Mean
Squares
0.0050
0.4444
1.9244
Degrees
of
Freedom
1
2
4
Square
Weeks (Rows)
Heifers(Square)
(Columns)
Feed
(Treatments)
Error
Total
6.1644
2
3.0822
2.5644
11.2028
8
17
0.3206
F0
0.0500
2.2222
0.4811
9.62
F(0.05, 2, 8) = 4.4590
For the above example we would reject the null hypothesis of
equal means for the three feed treatments.
21
Stat 512
Spring 2011
Randomized Complete Block Design with a Twoway
Treatment Structure
Example
A computer company, to test the efficiency of its new
programmable calculator, selected size engineers who were
proficient in the use of both this calculator and an earlier model
and asked them to work out two problems on both calculators. One
of the problems was statistical in nature, the other was an
engineering problem. The order of the four calculations was
randomized independently for each engineer. The length of time
(in minutes) required to solve each problem was observed and is
presented in the following table:
Data for the RCBD with Two-way Treatment Structure
Problem
Statistical
Engineering
Model
Model
Engineer
1
2
3
4
5
6
Cell Means:
New
3.1
3.8
3.0
3.4
3.3
3.6
Earlier
7.5
8.1
7.6
7.8
6.9
7.8
New
2.5
2.8
2.0
2.7
2.5
2.4
Y

3
.
3
6
7
Y

7
.
6
1
7
S
t
a
t
i
s
t
i
c
a
l
,
N
e
w
S
t
a
t
i
s
t
i
c
a
l
,
E
a
r
l
i
e
r
Y

2
.
4
8
3
Y

5
.
1
6
7
E
n
g
i
n
e
e
r
i
n
g
,
N
e
w
E
n
g
i
n
e
e
r
i
n
g
,
E
a
r
l
i
e
r
Marginal Means:
Y

2
.
9
2
5
N
e
w
Y

6
.
3
9
2
E
a
r
l
i
e
r
Y
5
.
4
9
2Y

3
.
8
2
5
s
t
a
t
i
s
t
i
c
a
l
E
n
g
i
n
e
e
r
i
n
g
Model:
22
Earlier
5.1
5.3
4.9
5.5
5.4
4.8
Stat 512
Spring 2011
Yijk = µ + rk + i + j + ()ij+ eijk
(effects model)
i = 1, 2, , a
j = 1, 2, , b
k = 1, 2, , r
µ - grand mean
i - treatment effect for the ith level of factor A (i = i.. - ...)
j - treatment effect for the jth level of factor B (j = .j. - ...)
()ij - interaction effect between the ith level of factor A and the jth
level of factor B
(()ij = ij. -i.. -.j. + ...)
rk - block effect (random factor) (rk = ..k - ...)
eijk - error for the kth block within the ijth treatment combination
Decomposition of the total sum of squares:
Y

Y

a

b

Y

Y

b

r

Y

Y

a

r

Y

Y














i
j
k
.
.
.
k
.
.
.
.
.
i
.
.
.
.
.
.
j
.
.
.
.
a
b
r
i

1
j

1
k

1
2
r
2
k

1
a
2
i

1
b
2
j

1

r

Y

Y

Y

Y

Y

Y

Y

Y









i
j
.
i
.
.
.
j
.
.
.
.
i
j
k
i
j
.
.
.
k
.
.
.
a
b
2
i

1
j

1
a
b
r
i

1
j

1
k

1
or
S
S

S
S

S
S

S
S

S
S
S
S
T
o
t
a
l
B
l
o
c
k
s
A
B
A
B
E
r
r
o
r

 
S
S
2 2
2
B
l
o
c
k
s
M
S

,E
M
S


r


a

r



B
l
o
c
k
s
B
l
o
c
k
s
B
l
o
c
k
B
l
o
c
k
R
e
p


r

1
S
S
2
2
A
M
S

, E
M
S


r

b



A
A
a

1
S
S
2
2
B
M
S

, E
M
S


r

a



B
B
b

1
S
S
2
2
B
M
S
A
, E
M
S


r



A
B
A
B


a

1
b

1


 
 

23
2
Stat 512
Spring 2011

S
S
2
M
S
 E , E
M
S



E
a
b

11
r





ANOVA Table
Source
Sum of
Squares
Mean
Squares
SSA
Degrees of
Freedom
r-1
a-1
Blocks
Factor A
SSBlocks
MSA
MS A
MS E
Factor B
SSB
b- 1
MSB
MS B
MS E
Interaction
SSAB
MSAB
MS AB
MS E
Error
Total
SSE
(a - 1)(b 1)
(ab-1)(r-1)
abr – 1
SST
F0
MSBlocks
MSE
Data for the RCBD with Two-way Treatment Structure
Problem
Statistical
Engineering
Model
Model
Engineer
New
Earlier
New
Earlier
1
3.1
7.5
2.5
5.1
2
3.8
8.1
2.8
5.3
3
3.0
7.6
2.0
4.9
4
3.4
7.8
2.7
5.5
5
3.3
6.9
2.5
5.4
6
3.6
7.8
2.4
4.8
24
Stat 512
Spring 2011
ANOVA Table
Source
Blocks
(Engineers)
Problem
Model
Problem*M
odel
Error
Total
Sum of
Squares
1.0533
Degrees of
Freedom
5
Mean
Squares
0.2107
F0
72.1067
16.6667
3.6817
1
1
1
72.1067
16.6667
3.6817
1070.89
247.52
54.68
1.0100
15
0.0673
F(0.05, 1, 15) = 4.5431
For the above example we would reject the null hypothesis of no
interaction and therefore have to compare cell means, not marginal
means.
25