Powerpoint presentation - Partitioning Sum of Squares

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

2

:: color, composition, and layout

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

3


4


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

0 0:: color, composition, and layout

7


8

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

9



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

11


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, composition, and layout

Within Japonica group Within Indica group13 :: color, composition, and layout 14

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

15


16


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 layout 18


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

19


20


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

Documents

Powerpoint presentation - Partitioning Sum of Squares