Slide set 6
Stat402 (Spring 2016)
Last update: February 23, 2016
Stat 402 (Spring 2016): slide set 7
Latin Square Design
In this design experimental units (or runs) are grouped into blocks in 2
different ways (usually referred to as row blocks and column blocks)
Example Study effect of 5 different fertilizers A,B,C,D,E (say) on yield of
potatoes. A field is subdivided into a 5 × 5 grid of 25 plots.
Assume that there is a ground moisture variation across the field (say, wet
to dry) and a inherent soil fertility gradient in the other direction. Important
characteristics of a Latin Square Design are:
1. It is required that every treatment appears only once in each row and
once in each column.
2. Every row (column) contains a complete set of the treatments. This
implies that number of treatments = number of row blocks = number
of column blocks
1
Stat 402 (Spring 2016): slide set 7
3. The 2 blocking variables are selected such that the 2 major sources of
variation are eliminated from the estimate of error variance.
4. The major disadvantage of the LSD is that the number of replications of
each treatment equals the number of treatments
5. The above condition places a limit on the number of degrees of freedom
available for Error Mean Square, which is equal to (p-1)(p-2) where p is
the number of treatments.
A possible field plan:
Fertilizer
Gradient
↓
Moisture Gradient −→
Columns
Rows
1
2
3
4
I
A
D
B
E
II
E
C
D
A
III
B
E
A
C
IV
D
A
C
B
V
C
B
E
D
5
C
B
D
E
A
2
Stat 402 (Spring 2016): slide set 7
Randomization Procedures for Latin Squares
1. Randomly select a basic (or standard) latin square (see Table 4.13)
2. Assign latin letters to treatments at random
3. Then randomize rows and columns, respectively.
3
Stat 402 (Spring 2016): slide set 7
Model for Latin Squares
Model: yijk = µ + αi + τj + βk + ijk ,
iid
where ijk ∼ N (0, σ 2) and yijk is an observation in the ith row, k th column
for the j th treatment, αi, τj and βk are the effects of the ith row, j th
treatment, and k th column, respectively.
• Note carefully that observations for all combinations of i, j, and k do
not exist; otherwise there should be p3 observations. We have only a total
of p2 obsevations.
• Thus for any combination of ij (i.e., in any row and column) there is
only one value for k. This is because each appears only once in each row
and a column.
• Thus it is more sensible to define the values of i, j, and k for a
specific experiment as belonging to a set. For example, (i, j, k) ∈ Q where
Q = {(I, A, 1), (I, B, 2), (I, C, 3), . . . , (V, D, 5)}
• The model is an additive model i.e, it is assumed there is no interaction
between row blocking factor, column blocking factor, and the treatments.
4
Stat 402 (Spring 2016): slide set 7
Computations for Latin Squares
•
Means
P
Grand
ȳ...
=
(ijk)∈Q yijk
p2
SST
=
P
=
P
P
Row
ȳi..
=
(j,k)∈Q yijk
P
Trt
ȳ.j.
=
p
(i,k)∈Q yijk
P
•
•
Col.
ANOVA
ȳ..k
=
p
(i,j)∈Q yijk
p
SSRow
i (ȳi..
− ȳ...)2
− ȳ...)2
2
(ȳ
−
ȳ
)
.j.
...
j
P
2
=
(ȳ
−
ȳ
)
..k
...
k
SST rt =
SSCol
i,j,k (yijk
P
SST = SSRow + SSCol + SST rt + SSE
with respective degrees of freedom
p2 − 1 = (p − 1) + (p − 1) + (p − 1) + (p − 2)(p − 1)
Estimates
µ̂ = ȳ...
α̂i = ȳi.. − ȳ...
τ̂j = ȳ.j. − ȳ...
β̂k = ȳ..k − ȳ...
5
Stat 402 (Spring 2016): slide set 7
Computations for Latin Squares(Cont’d 2)
•
Predicted (fitted) Values:
ŷijk
•
Residuals:
= yijk − ŷijk
= yijk − {ȳ... + (ȳi.. − ȳ...) + (ȳ.j. − ȳ...) + (ȳ..k − ȳ...)}
= yijk − ȳi.. − ȳ.j. − ȳ..k + 2ȳ...
rijk
•
= µ̂ + α̂i + τ̂j + β̂k
= ȳ... + (ȳi.. − ȳ...) + (ȳ.j. − ȳ...) + (ȳ..k − ȳ...)
SSE :
2
k rijk ,
P P P
i
j
(i, j, k) ∈ Q
6
Stat 402 (Spring 2016): slide set 7
Computations for Latin Squares(Cont’d 3)
•
•
•
ANOVA TABLE
SV
d.f.
Treatments p − 1
Rows
p−1
Cols
p−1
Error
(p − 1)(p − 2)
Total
p2 − 1
SS
SST rt
SSRows
SSCols
SSE
SST
MS
M ST rt
M SRows
M SCols
M SE = s2E
F
M ST rt/M SE
Hypothesis Tested: H0 : τ1 = τ2 = · · · = τp vs Ha : at least one ineq.
Reject H0 if F0 > Fα,p−1,(p−1)(p−2)
This test is equivalent to testing whether the treatment means µ.j. are
equal.
Confidence Interval: A 100(1 − α)% p
CI for τp − τq is
(ȳ.p. − ȳ.q.) ∓ tα/2,ν · sE 2/p
where s2E = M SE and ν = (p − 1)(p − 2)
7
Stat 402 (Spring 2016): slide set 7
Example (See Section 4.2)
Treatments: 5 methods of Formulation of Rocket Propellant
Row Blocking Variable: Batches of Raw material
Column Blocking Variable: Operators
8
Stat 402 (Spring 2016): slide set 7
JMP Analysis of Rocket Propellent data
•
Model: yijk = µ + αi + τj + βk + ijk (i, j, t) ∈ Q
•
ANOVA Table (extracted from the JMP output)
SV
Formulations
Batches
Operators
Error
Total
•
d.f.
4
4
4
12
24
SS
330.0
68.0
150.0
128.0
676.0
MS
7.7344
1.5938
3.5156
10.6667
F
8.1071
p-value
.0025
Pairwise 95%CI’s for Mean Differences In Burn Rate (from JMP):
µA − µB : (3.89946, 12.9005)∗
µA − µC : (1.69946, 10.7005)∗
9
Stat 402 (Spring 2016): slide set 7
µA − µD : (−5.7005, 3.30054)
µA − µE : (−1.9005, 7.10054)
µB − µC : (−6.7005, 2.30054)
µB − µD : (−14.101, −5.0995)∗
µB − µE : (−10.301, −1.2995)∗
µC − µD : (−11.901, −2.8995)∗
µC − µE : (−8.1005, 0.90054)
µD − µE : (−0.7005, 8.30054)
The CI’s marked with and asterisk do not contain zero; so those pairs of
means are significantly different at α = .05
•
Compute LSD at α = .05:
r
LSD.05 = t.025,12 sE
2
= t.025,12
p
s
2s2E
p
r
= 2.179
2(10.667)
= 4.5
5
10
Stat 402 (Spring 2016): slide set 7
JMP Analysis of Rocket Propellent data) Cont’d
Arrange the sample means in increasing order of magnitude.
underscoring procedure gives:
Formulation
B
C
E
A
D
Means
20.2
22.4
26.0
28.6
29.8
—————–
———————
——————————
•
The
Conclusions:
• The data show that the propellant with the highest mean burning rate is
D but not significantly larger than those of E or A.
• Propellant B and C have similar mean burning rate but significantly lower
than those of E, A, and D.
11
Stat 402 (Spring 2016): slide set 7
Replication of Latin Squares
We can replicate whole latin squares in order to increase the degrees of
freedom for error.There are several ways to do this.
1. Use the same Row Blocks and Column Blocks
2. Use the same Row Blocks but different Column Blocks, or use the same
column blocks but different Row blocks.
3. Use different Row Blocks and different column blocks.
•
Note
It is important to see that when replicating Latin Squares as above the
row and column variables remain the same in each square.
12