Stat 402 (Spring 2016): slide set 7
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
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.
Last update: February 23, 2016
Stat402 (Spring 2016)
Slide set 6
Stat 402 (Spring 2016): slide set 7
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.
Randomization Procedures for Latin Squares
3
Stat 402 (Spring 2016): slide set 7
1
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. It is required that every treatment appears only once in each row and
once in each column.
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:
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.
In this design experimental units (or runs) are grouped into blocks in 2
different ways (usually referred to as row blocks and column blocks)
Latin Square Design
Stat 402 (Spring 2016): slide set 7
•
•
•
SSE :
= yijk − ŷijk
= yijk − {ȳ... + (ȳi.. − ȳ...) + (ȳ.j. − ȳ...) + (ȳ..k − ȳ...)}
= yijk − ȳi.. − ȳ.j. − ȳ..k + 2ȳ...
i
j
2
k rijk , (i, j, k) ∈ Q
=
=
SSCol =
SST rt =
SSRow
SST
− ȳ...)2
− ȳ...)2
− ȳ...)2
− ȳ...)2
k (ȳ..k
j (ȳ.j.
i (ȳi..
i,j,k (yijk
6
•
F
M ST rt/M SE
7
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)% 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)
MS
M ST rt
M SRows
M SCols
M SE = s2E
•
SS
SST rt
SSRows
SSCols
SSE
SST
ANOVA TABLE
SV
d.f.
Treatments p − 1
Rows
p−1
Cols
p−1
Error
(p − 1)(p − 2)
Total
p2 − 1
•
Computations for Latin Squares(Cont’d 3)
Stat 402 (Spring 2016): slide set 7
= μ̂ + α̂i + τ̂j + β̂k
= ȳ... + (ȳi.. − ȳ...) + (ȳ.j. − ȳ...) + (ȳ..k − ȳ...)
rijk
Residuals:
ŷijk
(j,k)∈Q yijk
p
(i,k)∈Q yijk
p
(i,j)∈Q yijk
p
(ijk)∈Q yijk
p2
SST = SSRow + SSCol + SST rt + SSE
with respective degrees of freedom
p2 − 1 = (p − 1) + (p − 1) + (p − 1) + (p − 2)(p − 1)
Estimates
α̂i = ȳi.. − ȳ...
μ̂ = ȳ...
τ̂j = ȳ.j. − ȳ...
β̂k = ȳ..k − ȳ...
=
=
=
=
Stat 402 (Spring 2016): slide set 7
Computations for Latin Squares(Cont’d 2)
Predicted (fitted) Values:
ȳ.j.
Trt
ȳ..k
ȳi..
Row
Col.
ANOVA
ȳ...
Grand
Means
5
•
•
•
Stat 402 (Spring 2016): slide set 7
Computations for Latin Squares
4
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.
iid
Model: yijk = μ + αi + τj + βk + ijk ,
Model for Latin Squares
Stat 402 (Spring 2016): slide set 7
•
LSD.05 = t.025,12 sE
2
= t.025,12
p
Compute LSD at α = .05:
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
2s2E
= 2.179
p
10
2(10.667)
= 4.5
5
The
11
• 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.
Conclusions:
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
—————–
———————
——————————
JMP Analysis of Rocket Propellent data) Cont’d
Stat 402 (Spring 2016): slide set 7
μA − μB : (3.89946, 12.9005)∗
μA − μC : (1.69946, 10.7005)∗
Stat 402 (Spring 2016): slide set 7
•
d.f.
4
4
4
12
24
Pairwise 95%CI’s for Mean Differences In Burn Rate (from JMP):
SV
Formulations
Batches
Operators
Error
Total
ANOVA Table (extracted from the JMP output)
Model: yijk = μ + αi + τj + βk + ijk (i, j, t) ∈ Q
9
•
•
•
Stat 402 (Spring 2016): slide set 7
JMP Analysis of Rocket Propellent data
8
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
Column Blocking Variable: Operators
Row Blocking Variable: Batches of Raw material
Treatments: 5 methods of Formulation of Rocket Propellant
Example (See Section 4.2)
Stat 402 (Spring 2016): slide set 7
•
12
Note
It is important to see that when replicating Latin Squares as above the
row and column variables remain the same in each square.
3. Use different Row Blocks and different column blocks.
2. Use the same Row Blocks but different Column Blocks, or use the same
column blocks but different Row blocks.
1. Use the same Row Blocks and Column Blocks
We can replicate whole latin squares in order to increase the degrees of
freedom for error.There are several ways to do this.
Replication of Latin Squares