Factorial experiment with a basic design of Randomized Block Design (RCBD) is an experiment in which more than one factor is tested and using RCBD as the experimental design. This design was chosen if the experimental unit used was not uniform, so it was necessary to group it, while in Completely Randomized Design (CRD) Factorial , the experimental unit was relatively uniform so there was no need for grouping. Basically, the RCBD Factorial experiment is the same as the Randomized Complete Block Design (RCBD) experiment previously discussed, but in this experiment it consists of two or more factors.
Sub-discussion:
- Factorial Experiment in Randomized Complete Block Design
- Randomization and Trial Layout
- Linear Model for RCBD Factorial
- Example of RCBD Factorial
- Example 1: RCBD Factorial (Interaction not significant)
- Example 2: RCBD Factorial
The full discussion is in the document below.
Factorial Experiments In Randomized Complete Block Design
A Factorial experiment with a basic RCBD design is an experiment in which more than one factor is tried and uses RCBD as the experimental design. This design was chosen because the experimental units used were not uniform, so they needed grouping. In principle this experiment is the same as the single RCBD experiment discussed earlier but, in this experiment, it consists of two or more factors.
Randomization and Experimental Plan
The consideration of determining the Factorial Experiment with the basic design of the RCBD is almost the same as the consideration of the RCBD one factor chosen if the environmental conditions are not uniform. The proper ways of grouping can be seen again in the RCBD discussion. The placement of treatments that are a combination of the level of factors to be tried is carried out in the same way as RCBD. Consider the following case example. An experiment wanted to study the effect of Nitrogen and Variety fertilization on the results of production carried out in the field. Environmental conditions are estimated to be heterogeneous. The fertilization factor consists of 2 levels, namely 0 kg N / ha (n0) and 60 kg N / ha (n1). The Variety Factor consists of two levels, namely the IR-64 Variety (v1) and the S-969 Variety (v2). The experiment was designed using the basic CRD design which was repeated 3 times. The experiment was a 2x2 factorial experiment so there were 4 treatment combinations: n0v1; n0v2; n1v1; and n1v2. Because it is repeated 3 times, the unit of experiment consists of 4x3 = 12 units of experiments. The experimental unit was divided into three groups. The placement of treatment combinations was carried out randomly for each block separately. This is different from randomization in CRD, where randomization is carried out thoroughly, while in RCBD randomization is carried out separately.
Randomization can be by using a Random Number List, a Sweepstakes, or by a computer device (can be seen again in the one-factor RCBD discussion). In this case, the randomization process is carried out using MS Excel. Create 12 plots (experimental units) and the units of experiments are numbered from 1 to 12. Although in RCBD randomization for each block must be done separately, but using MS Excel, the randomization process can be done at once, as long as the randomization is grouped by block.
- Create a Table as below, in each block there are 4 treatment combinations: n0v1; n0v2; n1v1; and n1v2. A Random Number is generated by using the = RAND() function.
- Sort by pressing the Sort Toolbar (located in the Data Tabs block, Office 2007). Notice how it's sorted: Highlight (select) Range B1:D13. Sorting was only performed on Three Variables (Treatment, Block, and Random Number). Multi sort by exact order as in the example below, by Block, then Random Number. First, MS Excel will sort by block, then the next sorting by Random Numbers, so with this technique, the sorting of random numbers will be done per block.
- The result of the sorting looks like in the following image. Place a combination of treatments for each block on the experimental unit according to its sequence number.
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Block | ||
I | II | III |
n1v1 | n1v1 | n0v2 |
n0v2 | n0v1 | 1n0v1 |
n1v2 | n1v2 | 1n1v1 |
n0v1 | n0v2 | 1n1v2 |
Figure 11. Layout of 2 x 2 Factorial Experiment with RCBD Environmental Design
Linear Model of Factorial Design In RCBD
The linear model of additives for a two-factor factorial design with its environmental design random design block is as follows :
Yijk = μ + αi + βj + (αβ)ij + ρk + εijk
with i =1.2...,r; j = 1, 2, ..., a; k = 1, 2, ..., b
Yijk = observations on the i-th experimental unit that obtained a combination of j-th level treatment of factor A and k-th level of factor B
μ = population mean
ρk = k-th degree effect of Block factors
αi = i-th degree effect of factor A
βj = j-th degree effect of factor B
(αβ) ij = effect of the i-th level of factor A and the j-th level of factor B
εijk = random effect of the kth experimental unit that obtained a combination of ij treatments. εijk ~ N(0.σ2).
Assumption:
If all factors (factors A and B) are fixed | If all factors (factors A and B) are random |
$\begin{matrix}\sum{{\alpha}_{i}\ \ =\ \mathbf{0}\ ;\ \ \ \ \ \sum{\beta}_{j}}\ =\ \mathbf{0}\ ;\ \ \ \ \ \\\sum_{{i}}{({\alpha\beta})_{{ij}}=\sum_{{j}}{({\alpha\beta})_{{ij}}=}}\mathbf{0}\ ;\ \ \ \ \ {\varepsilon}_{{ijk}}\buildrel~\over~{bsi}{N}(\mathbf{0},{\sigma}^\mathbf{2})\\\end{matrix}$ | $\begin{matrix}\ \ \alpha_i\buildrel~\over~N(0,{\sigma_\alpha}^2)\ \ ;\ \ \ \ \ \beta_j\buildrel~\over~N(0,{\sigma_\beta}^2)\ ;\ \ \ \ \\\ (\alpha\beta)_{ij}\buildrel~\over~N(0,{\sigma_{\alpha\beta}}^2)\ \ ;\ \ \ \ \ \ \varepsilon_{ijk}\buildrel~\over~bsiN(0,\sigma^2)\\\end{matrix}$ |
Hypothesis:
The hypotheses tested in a factorial design consisting of two factors with a Completely Randomized Design environment design are:
Hypotheses to Be Tested: | Fixed Model (Model I) | Random Model (Model II) |
Effect of AxB Interactions | ||
H0 | (αβ) ij =0 (no effect of interaction on the observed response) | σ2αβ=0 (no variance in the population of combination treatments) |
H1 | there is at least a pair (i,j) so that (αβ)ij ≠0 (there is an effect of the interaction on the observed response) | σ2αβ>0 (there is variance in the combined treatment population) |
Main effect of Factor A | ||
H0 | α1 =α2 =...=αa=0 (no response difference between the levels of factor A attempted) | σ2α=0 (no variance in the population of factor A level) |
H1 | there is at least one i so that αi ≠0 (there is a difference in response among the level of factor A tried) | σ2α>0 (there is variance in the population of factor A level) |
Main effect of Factor B | ||
H0 | β1 =β2 =...=βb=0 (no response difference between the B factor levels attempted) | σ2β=0 (no variance in the population of factor B level) |
H1 | there is at least one j so that βj ≠0 (there is a difference in response between the level of factor B that is tried) | σ2β>0 (there is variance in the population of factor B level)
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Analysis of variance:
The data representation of the linear model Yijk = μ + αi + βj + (αβ)ij + ρk + εijk is as follows:
$$Y_{ijk}={\overline{Y}}_{...}+({\overline{Y}}_{i..}-{\overline{Y}}_{...})+({\overline{Y}}_{.j.}-{\overline{Y}}_{...})+({\overline{Y}}_{ij.}-{\overline{Y}}_{i..}-{\overline{Y}}_{.j.}+{\overline{Y}}_{...})+({\overline{Y}}_{..k}-{\overline{Y}}_{...})+(Y_{ijk}-{\overline{Y}}_{ij.})$$
| Definition | Calculation by Hand |
CF |
| $$\frac{Y...^2}{abr}$$ |
SSTOT | $$\sum_{i=1}\sum_{j=1}\sum_{k=1}{(Y_{ijk}-\bar{Y}...)^2}=\sum_{i=1}\sum_{j=1}\sum_{k=1}{Y_{ijk}}^2-\frac{Y...^2}{abr}$$ | $$\sum_{i,j,k}{Y_{ijk}}^2-CF$$ |
SS(R) | $$\sum_{i=1}\sum_{j=1}\sum_{k=1}{({\bar{Y}}_{..k}-\bar{Y}...)^2}=\sum_{i=1}\sum_{j=1}\sum_{k=1}\frac{{Y_{..k}}^2}{ab}-\frac{Y...^2}{abr}$$ | $$\frac{\sum_{k}{(r_k)^2}}{ab}-CF$$ |
SS(A) | $$\sum_{i=1}\sum_{j=1}\sum_{k=1}{({\bar{Y}}_{i..}-\bar{Y}...)^2}=\sum_{i=1}\sum_{j=1}\sum_{k=1}\frac{{Y_{i..}}^2}{br}-\frac{Y...^2}{abr}$$ | $$\sum_{i}\frac{{Y_{i..}}^2}{br}-CF=\frac{\sum_{i}{(a_i)^2}}{rb}-CF$$ |
SS(B) | $$\sum_{i=1}\sum_{j=1}\sum_{k=1}{({\bar{Y}}_{.j...}-\bar{Y}...)^2}=\sum_{i=1}\sum_{j=1}\sum_{k=1}\frac{{Y_{.j.}}^2}{ar}-\frac{Y...^2}{abr}$$ | $$\sum_{j}\frac{{Y_{.j.}}^2}{ar}-CF=\frac{\sum_{j}{(b_j)^2}}{ra}-CF$$ |
SS(AB) | $$\sum_{i=1}\sum_{j=1}\sum_{k=1}{({\bar{Y}}_{ij.}-{\bar{Y}}_{i...}-{\bar{Y}}_{..j.}+\bar{Y}...)^2}$$ | $$\sum_{i,j}\frac{{Y_{ij.}}^2}{r}-CF-SS(A)-SS(B)$$ $$=\frac{\sum_{i,j}{(a_ib_j)^2}}{r}-CF-SS(A)-SS(B)$$ |
SSE | $$\sum_{i=1}\sum_{j=1}\sum_{k=1}{({\bar{Y}}_{ijk}-{{\bar{Y}}_{ij.}}^2}$$ | SSTOT – SSB – SS(A) – SS(B) -SS(AB) |
The anova table of factorial experiments with two factors in the Randomized Complete Block Design is as follows :
Table 24. Anova Table of Two-Factorial Designs in A Randomized Complete Block Design
Sources of variance | Degree of freedom | Sum of Squares | Mean Square | F-stat | F-table |
Block | r-1 | SSB | MSB |
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Treatment | ab-1 | SST | MST | MST/MSE | F(α, df-P, df-G) |
A | a-1 | SS(A) | MS(A) | MS(A)/MSE | F(α, df-A, df-G) |
B | b-1 | SS(B) | MS(B) | MS(B)/MSE | F(α, df-B, df-G) |
AB | (a-1) (b-1) | SS(AB) | MS(AB) | MS(AB)/MSE | F(α, df-AB, df-G) |
Error | ab(r-1) | SS(G) | MSE |
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Total | abr-1 | SSTOT |
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If there is an effect of the interaction, then hypothesis testing of the main effect is not necessary. Testing of the main effect will be beneficial if the effect of the interaction is not significant. Reject Ho if the value of F > Fα(df1, df2), and vice versa accept Ho.
Table 25. Mean Squared Expectation Value Of Two-Factorial Design In Complete Block Random Design
Source Variance | Mean Square | E(MS) | |
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| Fixed factors A and B | Random factors A and B |
Block (R) | MS(K) | $$\sigma^2+ab\sigma_\rho^2$$ | $$\sigma^2+ab\sigma_\rho^2$$ |
A | MS(A) | $$\sigma^2+rb\sum_{i}{\alpha_i}^2/(a-1)$$ | $$\sigma^2+r{\sigma_{\alpha\beta}}^2+rb{\sigma_\alpha}^2$$ |
B | MS(B) | $$\sigma^2+ra\sum_{j}{\beta_j}^2/(b-1)$$ | $$\sigma^2+r{\sigma_{\alpha\beta}}^2+ra{\sigma_\beta}^2$$ |
AB | MS(AB) | $$\sigma^2+r\sum_{ij}{(\alpha\beta{)_{ij}}^2}/(a-1)(b-1)$$ | $$\sigma^2+r{\sigma_{\alpha\beta}}^2$$ |
Error | MSE | $$\sigma^2$$ | $$\sigma^2$$ |
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| Fixed A factor and random B | Fixed B factor and random A |
Block (R) | MS(K) | $$\sigma^2+ab\sigma_\rho^2$$ | $$\sigma^2+ab\sigma_\rho^2$$ |
A | MS(A) | $$\sigma^2+r{\sigma_{\alpha\beta}}^2+rb\sum_{i}{\alpha_i}^2/(a-1)$$ | $$\sigma^2+rb{\sigma_\alpha}^2$$ |
B | MS(B) | $$\sigma^2+ra{\sigma_\beta}^2$$ | $$\sigma^2+r{\sigma_{\alpha\beta}}^2+ra\sum_{j}{\beta_j}^2/(b-1)$$ |
AB | MS(AB) | $$\sigma^2+r{\sigma_{\alpha\beta}}^2$$ | $$\sigma^2+r{\sigma_{\alpha\beta}}^2$$ |
Error | MSE | $$\sigma^2$$ | $$\sigma^2$$ |
Standard Error Difference
The standard error for the difference among treatment averages is calculated by the following formula (Fixed Model/Model I):
Comparison of the two averages of Factor A:
$$ SED=S_{\bar{Y}}=\sqrt{\frac{2MSE}{rb}}$$
Comparison of the two averages of Factor B:
$$ SED=S_{\bar{Y}}=\sqrt{\frac{2MSE}{ra}}$$
Comparison of the interaction of the two average factors of AxB:
$$ SED=S_{\bar{Y}}=\sqrt{\frac{2MSE}{r}}$$
