Detail explanation of RBD

1. 1
Chapter 4 Randomized Blocks, Latin
Squares, and Related Designs

2. 2
4.1 The Randomized Complete
Block Design
• Nuisance factor: a design factor that probably has
an effect on the response, but we are not interested
in that factor.
• If the nuisance variable is known and
controllable, we use blocking
• If the nuisance factor is known and
uncontrollable, sometimes we can use the
analysis of covariance (see Chapter 14) to
remove the effect of the nuisance factor from the
analysis

3. 3
• If the nuisance factor is unknown and
uncontrollable (a “lurking” variable), we hope
that randomization balances out its impact across
the experiment
• Sometimes several sources of variability are
combined in a block, so the block becomes an
aggregate variable

4. 4
• We wish to determine whether 4 different tips
produce different (mean) hardness reading on a
Rockwell hardness tester
• Assignment of the tips to an experimental unit;
that is, a test coupon
• Structure of a completely randomized experiment
• The test coupons are a source of nuisance
variability
• Alternatively, the experimenter may want to test
the tips across coupons of various hardness levels
• The need for blocking

5. 5
• To conduct this experiment as a RCBD, assign all
4 tips to each coupon
• Each coupon is called a “block”; that is, it’s a
more homogenous experimental unit on which to
test the tips
• Variability between blocks can be large,
variability within a block should be relatively
small
• In general, a block is a specific level of the
nuisance factor
• A complete replicate of the basic experiment is
conducted in each block
• A block represents a restriction on
randomization
• All runs within a block are randomized

6. 6
• Suppose that we use b = 4 blocks:
• Once again, we are interested in testing the
equality of treatment means, but now we have to
remove the variability associated with the
nuisance factor (the blocks)

7. 7
Statistical Analysis of the RCBD
• Suppose that there are a treatments (factor levels)
and b blocks
• A statistical model (effects model) for the RCBD
is
–  is an overall mean, i is the effect of the ith
treatment, and j is the effect of the jth block
– ij ~ NID(0,2)
–
1,2,...,
1,2,...,
ij i j ij
i a
y
j b
   


    


0
,
0
1
1

 
 

k
j
j
a
i
i 


8. 8
• Means model for the RCBD
• The relevant (fixed effects) hypotheses are
• An equivalent way for the above hypothesis
• Notations:
j
i
ij
ij
ij
ij
y 




 



 ,
0 1 2 1
: where (1/ ) ( )
b
a i i j i
j
H b
        

       

0
: 2
1
0 


 a
H 

 
N
y
y
a
y
y
b
y
y
y
y
y
y
b
j
y
y
a
i
y
y
j
j
i
i
a
i
i
b
j
j
a
i
b
j
ij
a
i
ij
j
b
j
ij
i
/
,
/
,
/
,...,
1
,
,...,
1
,
1
1
1 1
1
1












 






















9. 9
• ANOVA partitioning of total variability:
2
.. . .. . ..
1 1 1 1
2
. . ..
2 2
. .. . ..
1 1
2
. . ..
1 1
( ) [( ) ( )
( )]
( ) ( )
( )
a b a b
ij i j
i j i j
ij i j
a b
i j
i j
a b
ij i j
i j
T Treatments Blocks E
y y y y y y
y y y y
b y y a y y
y y y y
SS SS SS SS
   
 
 
    
   
   
   
  
 
 


10. 10
SST = SSTreatment + SSBlocks + SSE
• Total N = ab observations, SST has N – 1 degrees
of freedom.
• a treatments and b blocks, SSTreatment and SSBlocks
have a – 1 and b – 1 degrees of freedom.
• SSE has ab – 1 – (a – 1) – (b – 1) = (a – 1)(b – 1)
degrees of freedom.
• From Theorem 3.1, SSTreatment /2, SSBlocks / 2 and
SSE / 2 are independently chi-square
distributions.

11. 11
• The expected values of mean squares:
• For testing the equality of treatment means,
2
1
2
2
1
2
2
)
(
1
)
(
1
)
(
















E
b
j
j
Blocks
a
i
i
Treatment
MS
E
b
a
MS
E
a
b
MS
E
)
1
)(
1
(
,
1
0 ~ 


 b
a
a
E
Treatments
F
MS
MS
F

12. 12
• The ANOVA table
• Another computing formulas:
Blocks
Treatments
T
E
b
j
j
Blocks
a
i
i
Treatments
a
i
b
j
ij
T
SS
SS
SS
SS
N
y
y
a
SS
N
y
y
b
SS
N
y
y
SS



















 



,
1
1
,
2
1
2
2
1
2
2
1 1
2

13. 13
• Example 4.1
4.1.2 Model Adequacy Checking
• Residual Analysis
• Residual:
• Basic residual plots indicate that normality,
constant variance assumptions are satisfied
• No obvious problems with randomization



 




 y
y
y
y
y
y
e j
i
ij
ij
ij
ij
ˆ

14. 14
ESIGN-EXPERT Plot
ardness
Residual
Norm
al
%
probability
Normal plot of residuals
-1 -0.375 0.25 0.875 1.5
1
5
10
20
30
50
70
80
90
95
99

15. 15
EXPERT Plot
2
2
2
2
Predicted
Res
iduals
Residuals vs. Predicted
-1
-0.375
0.25
0.875
1.5
-2.75 -0.31 2.13 4.56 7.00
DESIGN-EXPERT Plot
Hardness
Run Number
Res
iduals
Residuals vs. Run
-1
-0.375
0.25
0.875
1.5
1 4 7 10 13 16

16. 16
• Can also plot residuals versus the type of tip
(residuals by factor) and versus the blocks. Also
plot residuals v.s. the fitted values. Figure 4.5 and
4.6 in Page 137
• These plots provide more information about the
constant variance assumption, possible outliers
4.1.3 Some Other Aspects of the Randomized
Complete Block Design
• The model for RCBD is complete additive.
1,2,...,
1,2,...,
ij i j ij
i a
y
j b
   


    



17. 17
• Interactions?
• For example:
• The treatments and blocks are random.
• Choice of sample size:
– Number of blocks , the number of replicates
and the number of error degrees of freedom 
–
j
i
ij
j
i
ij y
E
y
E 



 ln
ln
ln
)
(
ln
)
( 




2
1
2
2


a
b
a
i
i





18. 18
• Estimating miss values:
– Approximate analysis: estimate the missing
values and then do ANOVA.
– Assume the missing value is x. Minimize SSE to
find x
– Table 4.8
– Exact analysis
)
1
)(
1
(
'
'
'









b
a
y
by
ay
x
j
i

19. 19
4.1.4 Estimating Model Parameters and the General
Regression Significance Test
• The linear statistical model
• The normal equations
1,2,...,
1,2,...,
ij i j ij
i a
y
j b
   


    


b
b
a
a
a
b
a
b
b
a
y
a
a
y
a
a
y
b
b
y
b
b
y
a
a
b
b
ab






















































ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
1
1
1
1
1
1
1
1
1
1

















20. 20
• Under the constraints,
the solution is
and the fitted values,
• The sum of squares for fitting the full model:
• The error sum of squares
0
ˆ
,
0
ˆ
1
1

 
 

b
j
j
a
i
i 








 



 y
y
y
y
y j
j
i
i 

 ˆ
,
ˆ
,
ˆ



 




 y
y
y
y j
i
j
i
ij 

 ˆ
ˆ
ˆ
ˆ
ab
y
a
y
b
y
y
y
y
R
b
j
j
a
i
i
b
j
j
j
a
i
i
i
2
1
2
1
2
1
1
ˆ
ˆ
ˆ
)
,
,
( 










 




 


 





 
2
1 1
1 1
2
)
,
,
( 
  




 






a
i
b
j
j
i
ij
a
i
b
j
ij
E y
y
y
y
R
y
SS 



21. 21
• The sum of squares due to treatments:
ab
y
b
y
R
a
i
i
2
1
2
)
,
|
( 




 




22. 22
4.2 The Latin Square Design
• RCBD removes a known and controllable
nuisance variable.
• Example: the effects of five different formulations
of a rocket propellant used in aircrew escape
systems on the observed burning rate.
– Remove two nuisance factors: batches of raw
material and operators
• Latin square design: rows and columns are
orthogonal to treatments.

23. 23
• The Latin square design is used to eliminate two
nuisance sources, and allows blocking in two
directions (rows and columns)
• Usually Latin Square is a p  p squares, and each
cell contains one of the p letters that corresponds
to the treatments, and each letter occurs once and
only once in each row and column.
• See Page 145

24. 24
• The statistical (effects) model is
– yijk is the observation in the ith row and kth
column for the jth treatment,  is the overall
mean, i is the ith row effect, j is the jth
treatment effect, k is the kth column effect and
ijk is the random error.
– This model is completely additive.
– Only two of three subscripts are needed to
denote a particular observation.
1,2,...,
1,2,...,
1,2,...,
ijk i j k ijk
i p
y j p
k p
    



     

 


25. 25
• Sum of squares:
SST = SSRows + SSColumns + SSTreatments + SSE
• The degrees of freedom:
p2 – 1 = p – 1 + p – 1 + p – 1 + (p – 2)(p – 1)
• The appropriate statistic for testing for no
differences in treatment means is
• ANOVA table (Table 4-10) (Page 146)
• Example 4.3
)
1
)(
2
(
,
1
0 ~ 


 p
p
p
E
Treatments
F
MS
MS
F

26. 26
• The residuals
• Table 4.13
• If one observation is missing,
• Replication of Latin Squares:
– Three different cases
– See Table 4.14, 4.15 and 4.16
• Crossover design: Pages 150 and 151








 





 y
y
y
y
y
y
y
e k
j
i
ijk
ijk
ijk
ijk 2
ˆ
)
1
)(
2
(
2
)
( '
'
'
'















p
p
y
y
y
y
p
y
k
j
i
ijk

27. 27
4.3 The Graeco-Latin Square Design
• Graeco-Latin square:
– Two Latin Squares
– One is Greek letter and the other is Latin letter.
– Two Latin Squares are orthogonal
– Table 4.18
– Block in three directions
– Four factors (row, column, Latin letter and
Greek letter)
– Each factor has p levels. Total p2 runs

28. 28
• The statistical model:
– yijkl is the observation in the ith row and lth
column for Latin letter j, and Greek letter k
–  is the overall mean, i is the ith row effect, j
is the effect of Latin letter treatment j , k is the
effect of Greek letter treatment k, l is the
effect of column l.
– ANOVA table (Table 4.19)
– Under H0, the testing statistic is Fp-1,(p-3)(p-1)
distribution.
• Example 4.4
p
l
k
j
i
y ijkl
l
k
j
i
ijkl ,
,
1
,
,
,
, 







 





29. 29
4.4 Balance Incomplete Block
Designs
• May not run all the treatment combinations in
each block.
• Randomized incomplete block design (BIBD)
• Any two treatments appear together an equal
number of times.
• There are a treatments and each block can hold
exactly k (k < a) treatments.
• For example: A chemical process is a function of
the type of catalyst employed. See Table 4.22

30. 30
4.4.1 Statistical Analysis of the BIBD
• a treatments and b blocks. Each block contains k
treatments, and each treatment occurs r times.
There are N = ar = bk total observations. The
number of times each pairs of treatments appears
in the same block is
• The statistical model for the BIBD is
)
1
(
)
1
(



a
k
r

ij
j
i
ij
y 


 




31. 31
• The sum of squares
Blocks
adjusted
Treatments
T
E
b
j
j
ij
i
i
a
i
i
adjusted
Treatments
b
j
j
Blocks
i j
ij
T
E
Blocks
Treatments
T
SS
SS
SS
SS
y
n
k
y
Q
a
Q
k
SS
N
y
y
k
SS
N
y
y
SS
SS
SS
SS
SS



























)
(
1
1
2
)
(
2
1
2
2
2
1
,
/
1
/


32. 32
• The degree of freedom:
– Treatments(adjusted): a – 1
– Error: N – a – b – 1
• The testing statistic for testing equality of the
treatment effects:
• ANOVA table (see Table 4.23)
• Example 4.5
E
adjusted
Treatments
MS
MS
F
)
(
0 

33. 33
4.4.2 Least Squares Estimation of the Parameters
• The least squares normal equations:
• Under the constrains,
we have
j
j
a
i
i
ij
j
i
b
j
j
ij
i
i
b
j
j
a
i
i
y
k
n
k
y
n
r
r
y
k
r
N


































ˆ
ˆ
ˆ
:
ˆ
ˆ
ˆ
:
ˆ
ˆ
ˆ
:
1
1
1
1
0
ˆ
,
0
ˆ
1
1

 
 

b
j
j
a
i
i 



 y
̂

34. 34
• For the treatment effects,
a
i
a
kQ
kQ
k
r
i
i
i
a
i
p
p
p
i
,
,
2
,
1
,
ˆ
ˆ
ˆ
)
1
(
,
1





 





