Pot Experiments Were Conducted to Screen Fifteen Sweet Potato Cultivars for Drought Tolerance

ST 524NCSU - Fall 2008

Factorial Experiments

Example: Study of the effects of five cowpea varieties and method of cultivation on yield. Yield, in lb. per plot of morgen[1], Cowpea varieties: A, B, C, D; Method of cultivation: 1, 2, 3.

Methods of Cultivation correspond to the factor “spacing in the rows” (S) with three levels: 4” (Method 1), 8” (Method 3), and 12” (Method 2).

1. Analysis of Variance: Factorial Experiment RCBD, 5 Variety × 3 Row Spacing × 4 Blk

• Fixed effects model,
• Model, RCBD:

, , , ,

------

original scale yield in lb/plot 1

The GLM Procedure

Class Level Information

Class Levels Values

blk 4 1 2 3 4

Variety 5 A B C D E

spacing 3 4 8 12

Number of Observations Read 60

Number of Observations Used 60

------

original scale yield in lb/plot 2

The GLM Procedure

Dependent Variable: yield

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 17 2711.900000 159.523529 12.59 <.0001

Error 42 532.100000 12.669048

Corrected Total 59 3244.000000

R-Square Coeff Var Root MSE yield Mean

0.835974 6.244492 3.559361 57.00000

Source DF Type I SS Mean Square F Value Pr > F

blk 3 638.400000 212.800000 16.80 <.0001

Variety 4 1089.166667 272.291667 21.49 <.0001

spacing 2 109.200000 54.600000 4.31 0.0198

Variety*spacing 8 875.133333 109.391667 8.63 <.0001

Source DF Type III SS Mean Square F Value Pr > F

blk 3 638.400000 212.800000 16.80 <.0001

Variety 4 1089.166667 272.291667 21.49 <.0001

spacing 2 109.200000 54.600000 4.31 0.0198

Variety*spacing 8 875.133333 109.391667 8.63 <.0001

1. Hypothesis for main effect of Variety
2. Hypothesis for main effect of Row Spacing
3. Hypothesis for interaction effects Variety* Row Spacing

1. Since all P values (Pr > F)are lower than the significance level α = 0.05, we can reject each null hypothesis and conclude that the effects of variety on yield is dependent on the row spacing selected (Variety*Spacing is significant, P <0.0001).
2. Main effect of Variety is significant, (P < 0.0001 ). There are significant differences on the response (yield) to the distinct Varieties on average of Row Spacing.
3. Main effect of Row Spacing is significant, (P = 0.0198 < 0.05 = There are significant differences on the response (yield) to the distinct Row Spacing on average of Varieties.
4. Main effects of Variety and Row Spacing are less important since their interaction is significant.

Means

original scale yield in lb/plot 4

var_ stderr_

Obs Variety spacing mn_yield yield yield

1 . 57.0000 54.9831 0.95728

2 4 55.3000 44.1158 1.48519

3 8 57.1000 39.5684 1.40656

4 12 58.6000 81.3053 2.01625

5 A . 51.3333 48.9697 2.02010

6 B . 56.1667 10.3333 0.92796

7 C . 55.4167 40.0833 1.82764

8 D . 57.6667 23.3333 1.39443

9 E . 64.4167 73.1742 2.46938

10 A 4 47.5000 33.6667 2.90115

11 A 8 50.7500 42.2500 3.25000

12 A 12 55.7500 57.5833 3.79418

13 B 4 57.5000 7.0000 1.32288

14 B 8 56.7500 10.2500 1.60078

15 B 12 54.2500 12.9167 1.79699

16 C 4 53.2500 46.9167 3.42479

17 C 8 55.2500 50.2500 3.54436

18 C 12 57.7500 36.2500 3.01040

19 D 4 62.2500 4.9167 1.10868

20 D 8 58.5000 3.6667 0.95743

21 D 12 52.2500 8.9167 1.49304

22 E 4 56.0000 28.6667 2.67706

23 E 8 64.2500 14.9167 1.93111

24 E 12 73.0000 32.0000 2.82843

• Analysis of Variety Main Effect

• Analysis of Row Spacing Main Effect and Row Spacing*Variety Interaction Effect

Graphical representation of the mean response (yield)

Orthogonal Contrasts – Orthogonal Polynomial Coefficients

• Equally spaced quantitative treatments or levels of a quantitative factor.
• Their use allows for the analysis of the independent computation of the contribution of a given power of the independent variable (factor), X, X2,X3, . . .
• For a quadratic curve, orthogonal polynomial curve, , can be analyzed using the coefficients in table, where are the orthogonal transformation of X, X2.
• Main effect is partitioned in a set of mutually orthogonal effects, each with one degree of freedom, and associated test for the null hypothesis that the polynomial term equal 0,
• Sequentially, each sum of squares is the additional contributiondue tofitting a curve one degree higher.
• Similar decomposition may be used to study significant interactions between factors.

Table of coefficients – orthogonal polynomial contrasts – 1 degree of freedom

Analysis of Spacing Main Effects - Orthogonal polynomial

In example, factor “Spacing in the rows” (S) is a quantitative factor with three levels: 4”, 8” and 12”.

• Slinearis used to analyzed whether response (yield) to increasing levels of spacing presents a linear trend.
• Sdev. linearallows us to test whether response (yield) to increasing levels of spacing is not simply linear, but may require a higher degree polynomial.
• Table of Means and Totals for each Method (Spacing) with associated coefficients for orthogonal polynomial contrasts

• Estimated value of the linear combination of means related to Slinear and Sdev. linear

Decomposition of the Sum of Squares for Spacing in SS(Slinear) and SS(Sdev. linear)

SS(Spacing) = SS(due to linear effect of Spacing) + SS(deviation from linear effect of Spacing)

SS(deviation from linear effect of Spacing) is what is remaining after fitting the linear effect of Spacing if this term is significant, then the effect of Spacing on the response (yield) is not just linear, it may be necessary to run a new experiment with more levels of spacing to get a better idea of the trend of the response to increasing levels of spacing.

Working with Totals for each spacing level:

Contrast DF Contrast SS Mean Square F Value Pr > F

Row Spacing linear 1 108.9000000 108.9000000 8.60 0.0054

Row Spacing dev linear 1 0.3000000 0.3000000 0.02 0.8784

Estimate

Dependent Variable: yield

Standard

Parameter Estimate Error t Value Pr > |t|

Row Spacing linear 1.65000000 0.56278432 2.93 0.0054

Row Spacing dev linear -0.07500000 0.48738552 -0.15 0.8784

Conclusion: Linear Main effect of Row Spacing is highly significant, while Dev. From Linear is not significant. A linear trend is adequate to represent the trend on yield response to increasing levels of Row Spacing.

Analysis Interaction effects of Variety*Row Spacing

Contrasts Sum of Squares within each cultivar

Contrast DF Contrast SS Mean Square F Value Pr > F

Row Spacing linear in A 1 136.1250000 136.1250000 10.74 0.0021 *

Row Spacing linear in B 1 21.1250000 21.1250000 1.67 0.2037 ns

Row Spacing linear in C 1 40.5000000 40.5000000 3.20 0.0810 ns

Row Spacing linear in D 1 200.0000000 200.0000000 15.79 0.0003 *

Row Spacing linear in E 1 578.0000000 578.0000000 45.62 <.0001 *

Row Spacing dev linear in A 1 2.0416667 2.0416667 0.16 0.6901 ns

Row Spacing dev linear in B 1 2.0416667 2.0416667 0.16 0.6901 ns

Row Spacing dev linear in C 1 0.1666667 0.1666667 0.01 0.9092 ns

Row Spacing dev linear in D 1 4.1666667 4.1666667 0.33 0.5694 ns

Row Spacing dev linear in E 1 0.1666667 0.1666667 0.01 0.9092 ns

Sum = 984.33 = (109.2+875.13) = [SS (S) + SS (V*S)]

Estimated coefficients – orthogonal polynomial contrasts

Dependent Variable: yield

Standard

Parameter Estimate Error t Value Pr > |t|

Row Spacing linear in A 4.12500000 1.25842400 3.28 0.0021

Row Spacing linear in B -1.62500000 1.25842400 -1.29 0.2037

Row Spacing linear in C 2.25000000 1.25842400 1.79 0.0810

Row Spacing linear in D -5.00000000 1.25842400 -3.97 0.0003

Row Spacing linear in E 8.50000000 1.25842400 6.75 <.0001

Row Spacing linear in A =

Row Spacing linear in B

The GLM Procedure

Dependent Variable: yield

Standard

Parameter Estimate Error t Value Pr > |t|

Row Spacing linear dev in A 0.43750000 1.08982715 0.40 0.6901

Row Spacing linear dev in B -0.43750000 1.08982715 -0.40 0.6901

Row Spacing linear dev in C 0.12500000 1.08982715 0.11 0.9092

Row Spacing linear dev in D -0.62500000 1.08982715 -0.57 0.5694

Row Spacing linear dev in E 0.12500000 1.08982715 0.11 0.9092

Row Spacing Dev. from linear in A =

Row Spacing Dev. from linear in B =

Analysis of Simple effects

• Analyze Variety effect within eah spacing level

The GLM Procedure

Least Squares Means

Variety*spacing Effect Sliced by spacing for yield

Sum of

spacing DF Squares Mean Square F Value Pr > F

4 4 474.700000 118.675000 9.37 <.0001

8 4 387.800000 96.950000 7.65 0.0001

12 4 1101.800000 275.450000 21.74 <.0001

• Analyze Row Spacing Effect within each Variety

The GLM Procedure

Least Squares Means

Variety*spacing Effect Sliced by Variety for yield

Sum of

Variety DF Squares Mean Square F Value Pr > F

A 2 138.166667 69.083333 5.45 0.0078

B 2 23.166667 11.583333 0.91 0.4086

C 2 40.666667 20.333333 1.60 0.2130

D 2 204.166667 102.083333 8.06 0.0011

E 2 578.166667 289.083333 22.82 <.0001

Tuesday October 21, 2008 Orthogonal Polynomial contrasts1

[1] South African unit of measure equal to about 2 acres.