ANOVAPresentation Title Goes Here PARTITIONING SUMS OF
SQUARESpresentation subtitle. Tells us only if treatment effect is
significant or not If treatment effect is significant, does not
tell us the nature of the significance
Violeta Bartolome Senior Associate Scientist-Biometrics Crop
Research Informatics Laboratory International Rice Research
Institute
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What to do after ANOVA if treatment effect is significant Mean
comparison if treatments have no known structure Partition sum of
squares if treatments have a known structure o Treatments can be
grouped o Treatment levels are quantitative o Treatments are a
combination of 2 or more factors and at least one factor is
partitioned
Partitioning Sum of Squares (PSS)PSS breaks the treatment sums
of squares into components. The maximum number of components is
equal to the Treatment df, i.e. number of treatment levels minus
one.
BlockSS ErrorSS TreatmentSS
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Three ways of partitioning sums of squares Group Comparison
Orthogonal Polynomials Factorial Comparison
Group ComparisonExample 1: 3 varieties tested: Traditional
variety A New varieties B and CSV Blocks Varieties Error Total df 3
2 6 11 SS 0.587 1.040 0.533 2.16 MS 0.196 0.520 0.089 5.84*
F-value
Objective is to compare yield of traditional variety with
average yield of new varieties H0: A = ( B + C)
Pairwise comparison will not answer this objective.5 :: color,
composition, and layout 6 :: color, composition, and layout
Group ComparisonTo answer objective, we have to do between group
and within group comparisons.SV Blocks Varieties A vs (B and C) B
vs C Error Total 6 11 df 3 2 1 1 SS 0.587 MS 0.196 F-value
Group ComparisonTo compute sum of squares for each component we
have to construct single df contrasts.Contrast coefficients should
be assigned to each level of the treatment factor. Sum of contrast
coefficients should be zero.Group 1 Group 2
Between 1.040 0.520
group comparison 5.84*Components A vs (B & C) B vs C A B C
Sum
Within group comparison0.533 2.16 0.089
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Group ComparisonComponents A vs (B & C) B vs CGroup 1 Group
2
Group ComparisonC Sum
A
B
If Treatment SS = Component 1 SS + Component 2 SS Then contrasts
are orthogonal or independent Contrasts are orthogonal if the sum
of the products of the corresponding coefficients of any two
comparisons is zero.Components A vs (B & C) B vs C
Product10
-2
1-1
1 1
0
0 0
Contrast coefficients to be assigned to one group should be
equal to the number of members of the other group. Assign a
negative sign to either of the groups. Treatment levels not
included in the comparison should be assigned a 0 coefficient.
Group 1
Group 2
A
B
C
Sum
-2 0 0
1 -1 -1
1 1 1
0 0 0
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Group Comparison Group ComparisonTo visually check for
orthogonality, treatments should be compared only once. Example:A
A,B vs (C,D) B vs C Product B C D SV Blocks Varieties A vs B&C
B vs C ANOVA table df 3 2 1 1 6 11 SS Total equals Variety SS MS
F-value 5.84 * 10.70 ** 0.89 0.587 0.196 1.040 0.520 0.960 0.960
0.080 0.080 0.533 0.089 2.16
-1 0 0
-1 -1 1
1 1 1
1 0 0
0 0 2
Error Total
Results indicate that the significant variety effect is mainly
due to the difference in yield of the traditional and the new
varieties.12 :: color, composition, and layout
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Group Comparison Presentation of Results5.0
Example 2: Consider a variety trial involving 5 varieties and 4
replications laid out in RCB.a
Variety Traditional New Difference
Mean 4.0 4.6 0.6**Yield (t/ha)
Varieties:Japonica group V1 V2 V3 V4 V5 Indica group
4.5
4.0
b
3.5
The following comparisons are of interest :Japonica group vs
Indica groupTraditional New
3.0
(V1, V2) vs (V3, V4, V5) V1 vs V2 V3 vs V4 vs V5:: color,
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Within Japonica group Within Indica group13 :: color,
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Group ComparisonANOVA Table Outline SV Block Variety (V1, V2) vs
(V3, V4, V5) V1 vs V2 V3 vs V4 vs V5 Error Total df 3 4 (1) (1) (2)
12 19
Group ComparisonComparison Japonica vs Indica Within Japonica
group Contrast V1,V2 vs V3,V4,V5 V1 vs V2 V3 vs V4,V5 Within Indica
group V4 vs V5 0 0 0 -1 1 V1 3 -1 0 V2 3 1 0 V3 -2 0 -2 V4 -2 0 1
V5 -2 0 1
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Table of means for grain yield (t/ha) (Ave. of 3 reps)N-rate
(kg/ha) 0 60 90 120 Mean 4.306 5.982 6.259 6.895
When treatments are quantitative Interest usually is not to
compare treatment means Most often the interest is to estimate an
optimum rate To estimate an optimum, a response curve between
dependent variable and the treatment should be estimated
Partitioning the treatment sums of squares into orthogonal
polynomials will guide us on the appropriate relationship to use to
estimate the response curve
If the interest is to estimate the optimum N-rate, would a pair
wise comparison answer this objective?17 :: color, composition, and
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Orthogonal Polynomials Seeks the lowest degree polynomial that
can adequately represent the relationship between crop response and
treatment The coefficients for equally spaced factors can be found
in common statistical tables They are used exactly the same way as
in group comparison
Orthogonal PolynomialsTable of coefficients for equally spaced
factorsNo. of Levels 3 4 Trend Linear Quadratic Linear Quadratic
Cubic 5 Linear Quadratic Cubic Quartic T1 -1 +1 -3 +1 -1 -2 +2 -1
+1 T2 0 -2 -1 -1 +3 -1 -1 +2 -4 T3 +1 +1 +1 -1 -3 0 -2 0 +6 +3 +1
+1 +1 -1 -2 -4 +2 +2 +1 +1 T4 T5
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Orthogonal Polynomials Orthogonal PolynomialsANOVA for grain
yield (t/ha) SVTrend 0 kg N/ha 60 kg N/ha 90 kg N/ha 120 kg N/ha
Sum
df 2 3 (1) (1) (1) 6
MS 0.2134 3.6602 10.6619 0 .1869 0.1315 0 .3307
F