TR

Factorial and split-plot experiments

Enter the levels of both factors and the block values of each combination. Main effects, the interaction, two-way means and letters are calculated. What an interaction means and which design to use when is explained in Chapter 9.

Design
Example:

Three wheat varieties received three nitrogen rates (0, 6, 12 kg/da); each combination was tested in 3 blocks (randomized complete blocks). Yield (kg/da) was measured. Do the varieties respond to nitrogen in the same way?

VarietyNitrogenBlock 1Block 2Block 3
Variety A0 kg
6 kg
12 kg
Variety B0 kg
6 kg
12 kg
Variety C0 kg
6 kg
12 kg
  • VarietysignificantF = 148.2, p < 0.0001
  • NitrogensignificantF = 267.14, p < 0.0001
  • Variety × NitrogensignificantF = 21.96, p < 0.0001
Analysis of variance table
Source of variationdfSSMSF
Block23612.961806.4835.79 **
Variety214963.197481.59148.2 **
Nitrogen226971.6313485.81267.14 **
Variety × Nitrogen44434.811108.721.96 **
Error16807.750.48
Total2650790.3

* significant at 5%, ** at 1%, ns: not significant. CV = 1.61%.

4004505000 kg6 kg12 kgNitrogenVariety A, 0 kg: 397Variety A, 6 kg: 439.67Variety A, 12 kg: 470Variety AVariety B, 0 kg: 414.33Variety B, 6 kg: 473.67Variety B, 12 kg: 530Variety BVariety C, 0 kg: 389Variety C, 6 kg: 428Variety C, 12 kg: 430.67Variety C
Each line is one level of Variety. Parallel lines mean no interaction; if they diverge, converge or cross, the effect of Nitrogen depends on Variety.

What does the interaction say?

The interaction is significant: the effect of Nitrogen depends on the Variety. So “the best Nitrogen” has no single answer; comparisons should be made separately for each Variety. The best Nitrogen level is the same for every Variety (12 kg), but the size of the response differs: the range across levels is 115.7 for Variety B and only 41.7 for Variety C.

Means (letters: Nitrogen levels compared within each Variety, Duncan)
Variety \ Nitrogen0 kg6 kg12 kgVariety mean
Variety A397c439.67b470a435.56
Variety B414.33c473.67b530a472.67
Variety C389b428a430.67a415.89
Nitrogen mean400.11447.11476.89441.37

Because the interaction is significant, the main-effect means (margins) should not be interpreted on their own; the letters in each row compare the Nitrogen levels at that level of Variety. Duncan (5%) critical ranges: R2 = 12.3, R3 = 12.9.

How were the sums of squares calculated?

  1. Correction factor: the square of the grand total divided by the number of plots.

    CF=(∑X)2a⋅b⋅r=1191723⋅3⋅3=5259810.7CF = \dfrac{(\sum X)^2}{a \cdot b \cdot r} = \dfrac{11917^2}{3 \cdot 3 \cdot 3} = 5259810.7

  2. Total sum of squares:

    SST=∑X2−CF=50790.3SS_T = \sum X^2 - CF = 50790.3

  3. Block sum of squares: each block total squared, divided by the plots per block (a·b).

    SSBl=∑Bk29−CF=3612.96SS_{Bl} = \dfrac{\sum B_k^2}{9} - CF = 3612.96

  4. Variety sum of squares: each Variety level total comes from 9 plots.

    SSA=∑Ai2b⋅r=∑Ai29−CF=14963.19SS_A = \dfrac{\sum A_i^2}{b \cdot r} = \dfrac{\sum A_i^2}{9} - CF = 14963.19

  5. Nitrogen sum of squares: each Nitrogen level total comes from 9 plots.

    SSB=∑Bj2a⋅r=∑Bj29−CF=26971.63SS_B = \dfrac{\sum B_j^2}{a \cdot r} = \dfrac{\sum B_j^2}{9} - CF = 26971.63

  6. Interaction: from the variability between combinations (A×B cells), subtract what each factor explains on its own. What remains is how the factors affect each other.

    SSAB=∑Tij2r−CF−SSA−SSB=46369.63−14963.19−26971.63=4434.81SS_{AB} = \dfrac{\sum T_{ij}^2}{r} - CF - SS_A - SS_B = 46369.63 - 14963.19 - 26971.63 = 4434.81

  7. Error: subtract the parts explained by blocks and combinations from the total.

    SSE=SST−SSBl−(SSA+SSB+SSAB)=807.7SS_E = SS_T - SS_{Bl} - (SS_A + SS_B + SS_{AB}) = 807.7

  8. Error degrees of freedom (a·b − 1)(r − 1) = 16. All three F values use the same error:

    FA=148.205,FB=267.144,FAB=21.963F_A = 148.205, \quad F_B = 267.144, \quad F_{AB} = 21.963

  9. Each F value is compared with the F table value for its degrees of freedom; significance levels are marked with stars.

Comparison details: Duncan’s multiple range test

Means are ranked; a larger critical difference is used for means far apart in rank and a smaller one for neighbours. Widely used in agricultural field trials.

  1. Rank the means from largest to smallest. The critical range depends on how many means (p) the pair spans.

    Rp=qαp; p; 16MSE3,αp=1−(1−α)p−1R_p = q_{\alpha_p;\,p;\,16} \sqrt{\dfrac{MS_E}{3}}, \qquad \alpha_p = 1 - (1 - \alpha)^{p-1}

  2. Critical ranges:

    R2=12.3,R3=12.9R_{2} = 12.3,\quad R_{3} = 12.9

  3. If a wider range is not significant, the differences between means inside it are also declared not significant.

  4. Means whose difference is below the critical value share a letter. 3 of 3 pairs differ significantly.