2 X 2 X 3 Factorial Design with Basic Design: RCBD
The following is a description of the 2 X 2 X 3 Factorial Design Data Tabulation, Linear Models, Assumptions, Variety Analysis Formulas and hypotheses. The values of the observational data from the experiment can be tabulated as follows:
Analyse data using SmartstatXL Add-In can be learned on the following link: How to Analyze RCBD Factorial Experiments (3 Factors)
The values of the experiment can be tabulated as follows :
Table 29. Tabulation of Factorial Design Data 2 X 2 X 3 In Complete Block Random Design
Block | Varieties | P1 | P2 | ||||
|
| T1 | T2 | T3 | T1 | T2 | T3 |
1 | V1 | Y1111 | Y1112 | Y1113 | Y1121 | Y1122 | Y1123 |
| V2 | Y1211 | Y1212 | Y1213 | Y1221 | Y1222 | Y1223 |
2 | V1 | Y2111 | Y2112 | Y2113 | Y2121 | Y2122 | Y2123 |
| V2 | Y2211 | Y2212 | Y2213 | Y2221 | Y2222 | Y2223 |
3 | V1 | Y3111 | Y3112 | Y3113 | Y3121 | Y3122 | Y3123 |
| V2 | Y3211 | Y3212 | Y3213 | Y3221 | Y3222 | Y3223 |
Linear Model and Anova of Factorial Experiments with Three Factor In RCBD
The linear model of the 3-factor factorial experiment in the Randomized Complete Block Design is as follows:
$$Y_{ijkl}=\mu+\kappa_i+\alpha_j+\beta_k+\gamma_l+(\alpha\beta)_{jk}+(\alpha\gamma)_{jl}+(\beta\gamma)_{kl}+(\alpha\beta\gamma)_{jkl}+\varepsilon_{ijkl}$$
i = 1.2,... r j = 1, 2, ..., a k = 1, 2, ..., b l = 1.2,... c
with Yijkl= the observation value of the i-th block obtaining the j-th level of factor A, the k-th level of factor B and the level of –l of factor c.
μ = population mean
κi = effect of additives of the i-th block
αj = effect of additives of j-rated factor A
βk = effect of additives of the k-th level of factor B
γl = effect of additives of the l-th level of factor C
(αβ) jk = effect of the interaction of the j-th level of factor A and the k-th level of factor B
(αγ) jl = effect of the interaction of the j-th level of factor A and the l-th level of factor C
(βγ) kl = effect of the interaction of the k-th level of factor B and the l-th level of factor C
(αβγ) jkl = effect of the interaction of the j-th level of factor A, the k-th level of factor B and the l-th level of factor C
εijkl = random effect of the i-th block obtaining the j-th level of factor A, the k-th level of factor B and the lth level of factor C.
εijkl ~N(0, σ2)
Assumption
Assumptions if all factors (factors A, B and C) are fixed :
$$\begin{matrix}\sum_{j}{\alpha_j=\sum_{k}\beta_k=\sum_{l}\gamma_l=\sum_{j}\left(\alpha\beta\right)_{jk}}=\sum_{k}\left(\alpha\beta\right)_{jk}=\sum_{j}\left(\alpha\gamma\right)_{jl}=\sum_{l}\left(\alpha\gamma\right)_{jl}=\sum_{k}\left(\beta\gamma\right)_{kl}=\\\sum_{l}\left(\beta\gamma\right)_{kl}=\sum_{j}\left(\alpha\beta\gamma\right)_{jkl}=\sum_{k}\left(\alpha\beta\gamma\right)_{jkl}=\sum_{l}\left(\alpha\beta\gamma\right)_{jkl}=0\\\end{matrix}$$
Analysis of variance
The sum of the squares of the 3-factor factorial experiment in the Randomized Complete Block Design is as follows:
CF= $\frac{Y^2....}{rabc}$
SSTOT=- CF $\sum_{i,j,k,l}{Y^2}_{ijkl}$
SSB= $\frac{\sum_{i} Y_{i...}}{abc}-CF$
SST= $\frac{\sum_{j,k,l}{Y^2}_{jkl}}{r}-CF$
SSE=SSTOT – SSB –SST
SS(A)= $\frac{\sum_{j}\left(\alpha_j\right)^2}{rbc}-CF=\frac{\sum_{j}\left(total\ level\ factor\ \ A\right)^2}{rbc}\ \ -\ CF$
SS(B)= $\frac{\sum_{k}\left(\beta_k\right)^2}{rac}-CF=\frac{\sum_{k}\left(total\ level\ factor\ \ B\right)^2}{rac}\ \ -\ CF$
SS(C)= $\frac{\sum_{l}\left(\gamma_l\right)^2}{rab}-CF=\frac{\sum_{l}\left(total\ level\ factor\ \ C\right)^2}{rab}\ \ -\ CF$
SS(AB)= $\frac{\sum_{j,k}\left(\alpha_j\beta_k\right)^2}{rc}-CF-SS(A)-SS(B)$
SS(AC)= $\frac{\sum_{j,k}\left(\alpha_j\gamma_l\right)^2}{rb}-CF-SS(A)-SS(C)$
SS(BC)= $\frac{\sum_{k,l}\left(\beta_k\gamma_l\right)^2}{ra}-CF-SS(B)-SS(C)$
SS(ABC)=SST-SS(A) – SS(B) – SS(C) – SS(AB) – SS(AC) – SS(BC)
The variance analysis table of the calculations above is as follows:
Table 30. Anova Table of Three-Factorial Design in a Randomized Complete Block Design
Sources of Variance (SK) | Degree of freedom (df) | Sum of squares (SS) | Mean Square (MS) | F-stat | E(MS) |
Block | r-1 | SSB | MSB | $$\frac{MSB}{MSE}$$ | $$\sigma^2+abc\frac{\sum_{i}{\kappa^2}_i}{(r-1)}$$ |
Treatment | abc-1 | SST | MST |
|
|
A | a-1 | SS(A) | MS(A) | $$\frac{MS\left(A\right)}{MSE}$$ | $$\sigma^2+rbc\frac{\sum_{j}{\alpha^2}_j}{(a-1)}$$ |
B | b-1 | SS(B) | MS(B) | $$\frac{MS\left(B\right)}{MSE}$$ | $$\sigma^2+rac\frac{\sum_{k}{\beta^2}_k}{(b-1)}$$ |
C | c-1 | SS(C) | MS (C) | $$\frac{MS\left(C\right)}{MSE}$$ | $$\sigma^2+rab\frac{\sum_{l}{\gamma^2}_l}{(c-1)}$$ |
AB | (a-1) (b-1) | SS(AB) | MS(AB) | $$\frac{MS\left(AB\right)}{MSE}$$ | $$\sigma^2+rc\frac{\sum_{j,k}\left(\alpha_j\beta_k\right)^2}{(a-1)(b-1)}$$ |
AC | (a-1) (c-1) | SS(AC) | MS(AC) | $$\frac{MS\left(AC\right)}{MSE}$$ | $$\sigma^2+rb\frac{\sum_{j,l}\left(\alpha_j\gamma_/\right)^2}{(a-1)(c-1)}$$ |
BC | (b-1) (c-1) | SS(BC) | MS(BC) | $$\frac{MS\left(BC\right)}{MSE}$$ | $$\sigma^2+ra\frac{\sum_{k,l}\left(\beta_k\gamma_/\right)^2}{(b-1)(c-1)}$$ |
ABC | (a-1) (b-1) (c-1) | SS(ABC) | MS(ABC) | $$\frac{MS\left(ABC\right)}{MSE}$$ | $$\sigma^2+r\frac{\sum_{j,k,l}\left(\alpha_j\beta_k\gamma_/\right)^2}{(a-1)(b-1)(c-1)}$$ |
Error | (r-1) (abc-1) | SSE | MSE |
| $$\sigma^2$$ |
Total | rabc-1 | SSTOT |
|
|
|
Hypothesis
Hypothesis that needs to be tested if all factors are fixed:
1.Ho: (αβγ)jkl = 0
H1:there is at least one (αβγ)jkl ≠ 0
2.Ho: (αβ)jk = 0
H1:there is at least one (αβ)jk ≠ 0
3.Ho: (αγ)jl = 0
H1:there is at least one (αγ)jl ≠ 0
4.Ho: (βγ)kl = 0
H1:there is at least one (βγ)kl ≠ 0
5.Ho: αj = 0
H1:there is at least one αj ≠ 0
6.Ho: βk = 0
H1:there is a minimum of one βk ≠ 0
7.Ho: γl = 0
H1:there is a minimum of one γl ≠ 0
Example 1. Calculation By SmartstatXL Excel Add-In
Sample Data
Analysis Output:
Interaction (Not Significant)
