Another name for Split-Block Design is Strip-Plot Design . This design is suitable for a two-factor experiment where the accuracy of the interaction effect between factors is prioritized compared to the other two influences, the independent influence of factor A and factor B. This design is similar to Split Plot Design , only in split-blocks, treatment subunits placed in a line perpendicular to the main plot treatment. In split-blocks, the first factor is randomly assigned to the vertical line, while the second factor is randomly assigned to the horizontal line. Each path gets one treatment of factor A and one treatment of factor B.
Sub-discussion:
- Introduction
- Randomization and Layout of Split Block Experiment Design
- Split Block Linear Model
- Analysis of Variance:
- Standard Error
- Example of Split Block Design (Strip Plot)
- Calculation:
- Post Hoc
The full discussion can be read in the document below.
Introduction
Another name for a Split-Block Design is a Strip-Plot or Striped-Plot Design. This design is suitable for two-factor experiments where the accuracy of the effect of interaction between factors takes precedence over the other two effects, the independent effect of factor A and Factor B. This design is similar to the Split-plot design, except that in split-blocks, the treatment subunits are placed in a single direction perpendicular to the treatment of the main plot. In split-blocks, the first factor is randomly placed in a vertical direction, while the second factor is randomly placed on a horizontal direction. Each directions gets one factor A treatment and one factor B treatment.
Note the comparison of layout and randomization differences between split plot and split blocks for the same size, 5x4 (only shown for one block).
A3 | A2 | A1 | A5 | A4 |
| A3 | A2 | A1 | A5 | A4 |
B2 | B1 | B2 | B3 | B4 |
| B2 | B2 | B2 | B2 | B2 |
B1 | B3 | B1 | B2 | B3 |
| B4 | B4 | B4 | B4 | B4 |
B3 | B2 | B4 | B4 | B1 |
| B1 | B1 | B1 | B1 | B1 |
B4 | B4 | B3 | B1 | B2 |
| B3 | B3 | B3 | B3 | B3 |
Split-plot |
| Split-block or Strip-plot | ||||||||
In split-plots, the sub plot (B) is placed randomly (differently) on each main plot (A), while in the split-block, the placement of the sub plot (B) is in the same direction on the entire main plot (A). For example, in split-plots, the treatment of level B1 is located randomly for each level of Factor A, at the level of A3 it is in the 2nd row, while at the level of A2, it is located in line 1. In the split-block, the B1 treatment is on the 3rd row for all its main plots, so the subunit treatment will divide the blocks in a vertical, upper and lower direction. This is the reason why this design is called Split-Block! Another term for this design is strip-plot, because the treatment of factor A and factor B is placed in vertical and horizontal strips. Treatment A and B were placed randomly and freely in each block.
Here's why you chose the Split-block design:
- Ease in operation of its implementation (for example, tractor track, irrigation, harvesting)
- Heightens the degree of accuracy of the effect of the interaction between the two factors at the expense of their independent effect.
Randomization and Layout of Split-Block Experiments
The randomization procedure in the Split-Block design for both factors consist of two stages of randomization that are carried out freely for both, one for horizontal factors and one for vertical factors. The order is not very important.
For more details, consider the example of a factorial experiment to investigate the effect of Nitrogen Fertilization (Factor A) which consists of four levels, namely a1, a2, a3 and a4. The second factor (B) is in the form of varieties consisting of three varieties (3 levels), namely b1, b2, and b3. Factor A is placed in a vertical direction, while factor B is placed in a horizontal direction. The experiment was repeated three times.
Step 1: Divide the experimental area according to the number of tests. In this case it is divided into 3 blocks (blocks). The division of blocks is based on the consideration that variance in each of the same block is relatively homogeneous (see again discussion on RCBD)
Step 2: Each block is subdivided into a plot in a vertical direction, according to the level of Factor A. In this case example, each block is divided into 4 plots. Follow the randomization procedure for RCBD with treatment a = 4 and r = 3 replication and perform the 4th randomization of nitrogen levels on the vertical factor in each block separately and freely. Suppose that the result of randomization is as follows:
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a4 | a1 | a3 | a2 |
| a2 | a3 | a1 | a4 |
| a2 | a4 | a1 | a3 |
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3rd step: Each block is subdivided into b = 3 plots in a horizontal direction (horizontal direction). Follow the randomization procedure for RCBD with treatment b = 3 and r = 3 replications and perform the 3rd randomization of the Variety level on the horizontal direction in each block separately and freely. Suppose that the final structuring result is as follows:
I | II | III | ||||||||||||||
a4 | a1 | a3 | a2 | a2 | a3 | a1 | a4 | a2 | a4 | a1 | a3 | |||||
b2 | a4b2 | a1b2 | a3b2 | a2b2 | b1 | b3 | ||||||||||
b1 | a4b1 | a1b1 | a3b1 | a2b1 | b3 | b1 | ||||||||||
b3 | a4b3 | a1b3 | a3b3 | a2b3 | b2 | b2 |
Figure 5. Example of structuring a Split Block Design
Split-Block Linear Model
The linear model of additives for split-block design with RCBD environmental design is as follows:
Yijk = μ + ρk + αi + βj + γik + θjk + (αβ)ij + εijk
with i =1.2...,a; j = 1.2,...,b; k = 1.2,...,r
Yijk = observations on the kth experimental unit that obtained a combination of i-level treatment of factor A and j-th level of factor B
μ = actual average value (population average)
ρk = additive effect of the k-th block
αi = i-th level additive effect of factor A
βj = j-th level additive effect of factor B
(αβ) ij = effect of additives of the i-th level of factor A and the j-th level of factor B
γik = random effect that arises at the I-th level of factor A in the k-th block. Often called error (a). γik ~N(0.σγ2).
θjk = random effect that arises at the j-th level of factor B in the k-th block. Often called error (b). θjk ~ N(0,σθ 2).
εijk = random effect of the kth experimental unit that obtained a combination of ij treatments. Often called error (c). εijk ~ N(0.σε2).
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 the Split-Block 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 effects 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 effects 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 Anova in Split-block is divided into three parts, namely horizontal factor analysis, vertical factor analysis, and interaction analysis, so that in Split-Block there are three types of errors, error (a), error (b), and error (c), respectively. Main Plot Error is often referred to as Error A, the calculation procedure is the same as the Main Plot Interaction x Replication and in the RCBD model it is the same as the interaction of Main Plot x Replication. Sub Plot Error, often referred to as Error B, is measured from the interaction [Subplot x Replication + Main Plot x Subplot x Replication]. This 2nd error was used to measure the significance level of the subplot effect and the interaction of the Subplot x Main plot.
An error (a) that is nothing but an interaction between the Main Plot (Factor A) x Replication. Error (a) is a divisor on test F for the main effect of Factor A. Error (b) is an interaction between Subplot (Factor B) x Replication. Error (b) is a divisor on test F for the main effect of Factor B. Ea and Eb are symmetrical. This is easy to understand, considering that in the split design the two factors are similar in randomization and are symmetrical.
Error (b) is a parsing of the subplot error, error (c). Thus, error C will have a smaller value compared to a subplot error in a Split-Plot design. Error (c) is used to test the AxB interaction. Thus, it appears that parsing such errors will increase the accuracy of the effect of AxB interactions.
The data representation of the linear model Yijk = μ + ρk + αi + βj + γik + θjk + (αβ)ij + εijk is as follows:
$$\begin{matrix}Y_{ijk}={\overline{Y}}_{...}+({\bar{Y}}_{..k}-{\bar{Y}}_{...})+({\overline{Y}}_{i..}-{\overline{Y}}_{...})+({\bar{Y}}_{i.k}-{\bar{Y}}_{i..}-{\bar{Y}}_{..k}+{\bar{Y}}_{...})+({\bar{Y}}_{.jk}-{\bar{Y}}_{j..}-{\bar{Y}}_{..k}+{\bar{Y}}_{...})\\+({\overline{Y}}_{.j.}-{\overline{Y}}_{...})+({\overline{Y}}_{ij.}-{\overline{Y}}_{i..}-{\overline{Y}}_{.j.}+{\overline{Y}}_{...})+(Y_{ijk}-{\overline{Y}}_{ij.}-{\overline{Y}}_{i.k}-{\overline{Y}}_{.jk}+{\overline{Y}}_{i..}+{\overline{Y}}_{.j.}+{\overline{Y}}_{..k}-{\overline{Y}}_{...})\\\end{matrix}$$
Based on such a linear model:
| Definition | Workmanship |
CF |
| $$\frac{Y...^2}{abr}$$ |
SSTOT | $$\sum_{i,j,k}{(Y_{ijk}-\bar{Y}...)^2}$$ | $$\sum_{i,j,k}{Y_{ijk}}^2-CF$$ |
SS(R) | $$ ab\sum_{k}{({\bar{Y}}_{..k}-\bar{Y}...)^2}$$ | $$\sum_{k}\frac{{Y_{..k}}^2}{ab}-CF=\frac{\sum_{k}{(r_k)^2}}{ab}-CF$$ |
SS(A) | $$ rb\sum_{i}{({\bar{Y}}_{i..}-\bar{Y}...)^2}$$ | $$\sum_{i}\frac{{Y_{i..}}^2}{br}-CF=\frac{\sum_{i}{(a_i)^2}}{rb}-CF$$ |
SS(Ea) | $$ b\sum_{i,k}{({\bar{Y}}_{i.k}-{\bar{Y}}_{i..}-{\bar{Y}}_{..k}+{\bar{Y}}_{...})^2}$$ | $$\sum_{i,k}\frac{{Y_{i.k}}^2}{b}-CF-SSR-SS(A)$$ $$=\frac{\sum_{i,k}{(a_ir_k)^2}}{b}-CF-SSR-SS(A)$$ |
SS(B) | $$ ra\sum_{j}{({\bar{Y}}_{.j.}-\bar{Y}...)^2}$$ | $$\sum_{j}\frac{{Y_{.j.}}^2}{ar}-CF=\frac{\sum_{j}{(b_j)^2}}{ra}-CF$$ |
SS(Eb) | $$ a\sum_{i,k}{({\bar{Y}}_{.jk}-{\bar{Y}}_{.j.}-{\bar{Y}}_{..k}+{\bar{Y}}_{...})^2}$$ | $$\sum_{j,k}\frac{{Y_{.jk}}^2}{a}-CF-SSR-SS(B)$$ $$=\frac{\sum_{j,k}{(b_lr_k)^2}}{a}-CF-SSR-SS(B)$$ |
SS(AB) | $$ r\sum_{i,j}{({\bar{Y}}_{ij.}-{\bar{Y}}_{i..}-{\bar{Y}}_{.j.}+\bar{Y}...)^2}$$ | $$\sum_{i,j}\frac{{Y_{ij.}}^2}{r}-CF-SS(A)-SS(B)$$ $$=\frac{\sum_{i,j}{(a_ib_j)^2}}{r}-CF-SS(A)-SS(B)$$ |
SS(Ec) | $\sum_{i,j,k}\begin{matrix}(Y_{ijk}-{\overline{Y}}_{ij.}-{\overline{Y}}_{i.k}-{\overline{Y}}_{.jk}+\\{\overline{Y}}_{i..}+{\overline{Y}}_{.j.}+{\overline{Y}}_{..k}-{\overline{Y}}_{...})^2\\\end{matrix}$ | Difference = SSTOT – Other SS |
The table Analysis of Variance of split-block in the RCBD design is as follows:
Table 11. Anova Table of Split-Block
Source of Variance | Degree Of Freedom | Sum Squares | Squares Mean | F-stat | F-table |
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Block | r-1 |
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Factor A (Vertical) |
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A | a-1 | SS(A) | MS (A) | MS(A)/MSEa | F(α, db-A, db-Ga) |
Error a (Ea) | (a-1)(r-1) | SS (Ea) | MSEa |
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Factor B (Horizontal) |
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B | b-1 | SS(B) | MS(B) | MS(B)/MSEb | F(α, db-B, db-Gb) |
Error b (Eb) | (b-1)(r-1) | SS (Eb) | MSEb |
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Interaction |
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AB | (a-1) (b-1) | SS(AB) | MS(AB) | MS(AB)/MSEc | F(α, db-AB, db-Gc) |
Error c (Ec) | (a-1)(r-1)(b-1) | SS (Ec) | MSEc |
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Total | rab-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(db1, db2), and vice versa accept Ho.
Standard error
To compare the mean value of the treatment, it is necessary to first determine the standard error of the split-block. In Split-block there are 4 different types of pairwise comparisons, there are 4 types of standard errors. The following table is a formula for calculating the exact standard error for the mean difference for each pairwise comparison type.
Table 12. Standard Error Split-Block
Types of Pairwise Benchmarking | Example | Standard Error (SED) |
Two vertical averages | a1 – a2 | $$\sqrt{\frac{2MS(Ea)}{rb}}$$ |
Two horizontal averages | b1 – b2 | $$\sqrt{\frac{2MS(Eb)}{ra}}$$ |
Two vertical treatment averages (ai) at the same level of horizontal factors (bi) | a1b1 – a2b1
| $$\sqrt {\frac{2[(b-1)MS(Ec)+MS(Ea)]}{rb}}$$ |
Two horizontal treatment averages (bi) at the same level of vertical factors (ai) | a1b1 – a1b2 | $$\sqrt {\frac{2[(a-1)MS(Ec)+MS(Eb)]}{ra}}$$ |
From the standard error table above, to compare its simple effects, two types of MS(Error) are used. The implication is that the ratio of the difference in treatment to standard error does not follow the distribution of t-students so it needs to be calculated t combined/weighted. If ta, tb and tc are successively t values obtained from the student table to a certain significant extent at error-degree of freedoms a, b and c, then the weighted t values are:
For two vertical treatment averages (ai) at the same level of horizontal factors (bi)
$$ t'=\frac{(b-1)(MS\ \ Ec)(t_c)+(MS\ \ Ea)(t_a)}{(b-1)(MS\ \ Ec)+(MS\ \ Ea)}$$
For two horizontal treatment averages (bi) at the same level of vertical factors (ai):
$$ t'=\frac{(a-1)(MS\ \ Ec)(t_c)+(MS\ \ Eb)(t_b)}{(a-1)(MS\ \ Ec)+(MS\ \ Eb)}$$
