Split Plot Design

Introduction

In the previous discussion of several types of environmental designs for controlling experimental errors, we were only faced with one type of Experiment Unit for all treatments and one randomization process for placing the treatment into the experiment unit. However, in factorial experiments sometimes we are faced with other situations where there are several types of experimental units, and the extent of the experimental factors is placed sequentially and the randomization procedure is carried out separately.  For example, from the two factors we tried, we create an experimental unit size that is larger in size for one of the factors, then for each of these plots we divide it again into several plots of smaller size which are units of experiments for the level of the second factor.  This procedure is nothing but the principle of the Split-Plot experiment.  The plot of the experimental unit that is larger in size and in which there are sub plot is called the Main Plot, while the second plot of the experimental unit that is smaller in size and placed randomly on the Main Plot is called the sub plot.

Thus, split-plot experiments are superimposed of two types of experimental units where the environmental design for both can be the same or different.  The experimental units for the main plot can be designed with the basic design of CRD, RCBD, and LATIN.  Likewise, the sub plot experiment unit can be designed with all three basic designs.  A combination of designs that are often used in agriculture is RCBD for both the main plot and the sub plot.  In the next description, only the RCBD design for the basic design of the sub plot is discussed.

In the Split-plot design, not only the size and degree of accuracy for the two different factors, but here we are also faced with two different experimental units so that the comparison of the variance of experimental errors is different.  In the SPD design, the measurement of the effect of the main factor is sacrificed, on the contrary, the effect of the sub plot factor and the interaction of the sub plot with the main plot is more appropriate than the usual complete Block design. 

Some of the reasons for choosing the SPD design are as follows:

1. Degree of Accuracy
   • For example, a study is aimed at assessing 10 soybean varieties with three levels of fertilization in a 10 x 3 factorial experiment, if the researcher expects higher accuracy for soybean variety comparisons than for fertilization responses. Thus, the researcher will make the variety as a sub plot factor and fertilization as the main plot factor.
   • However, an agronomist studying the fertilization responses of the 10 soybean varieties developed by the researcher will probably want higher accuracy for fertilization responses than for varieties and will place the varieties on the main plot and fertilization on the sub plot.
2. Relative Measures of Main Effects.
   • From the previous information, it is known that there are greater differences in response between some levels of certain factors than some other levels. Combinations of treatments from factors that give rise to large response differences can be treated randomly on the main plot (Steel and Torrie, 1991).
   • One factor is more important than another.  If the main effect of one factor is expected to be greater and easier to see than the other, then one of those factors can be placed as the main plot, and the other factor as the sub plot (Gomez & Gomez, 1995).  This important factor may be a new discovery or new ways or another cause, so that one factor receives more attention than another. The factors that are not important can be caused because these factors already have quite a lot of information or have been repeated experiments.
   • For example, we want to research planting distances on several plant varieties. From previous experiments, information about the variety is known, including its production potential. While in this experiment you want to know more deeply about the effect of planting distance on some of these varieties, then in this kind of experiment, SPD is used. Varieties are treated as the main plot factor (main plot factor), while planting distance is treated as a sub-plot factor, because it expects the effect of spacing treatment to be greater than the variety treatment factor.
   • Another case in point is that at the beginning of 1984, a Hidrazil substance was discovered that could increase crop production. It is certain that the thing about Hydrazil is rather limited when compared to the familiar Rustica fertilizer. If the experiment is carried out using Hidrazil and Rustica materials, then by itself the Hydrazil factor is more important than the Rustica factor.
3. Management Practices
   • The placement of treatment as the main plot is carried out based on practical considerations in the field, for example, one factor requires a large plot and is very difficult to do on a small plot, for example:
   • Plowing of land (tillage with a plow or tractor), while other factors such as fertilization, spacing, spraying, height of inundation and others can be carried out on small plots. In the implementation of the experiment, land plowing was carried out first, and then smaller plots were created for other factors. In this case a large plot (plowing factor) seems to be of little importance while a small plot (fertilization etc.) is an important factor.
   • In an experiment to assess the appearance of several rice varieties with varying degrees of fertilization, the researchers may place the main plots for fertilization to minimize the need for separation of maps that require different levels of fertilization.
4. This design can be used when another factor is added in the experiment.
   • For example, the effect of comparing several fungicides as a protector against leaf rust disease attacks, as well as several varieties that are known to have different resistance to the disease, in this case the variety is used as the main plot and fungicide in the sub plot (subplot) (Steel and Torrie, 1991).
5. An experiment using time/place as the main factor or multiple experiments that are the same were performed in different times/places.
   • This experiment is often called a split in time experiment or a split in space.
   • Thus time/place can be considered as a factor/treatment that is less important, while other factors/treatments are considered as an important factor/treatment.
   • Factors that are less important are called main factors or main treatment while the factor that is important is called additional factor (sub factor) or additional treatment (sub treatment). For the next discussion, the less important factor is given the symbol A (Factor A) with its levels, while the factor that is important is given the symbol B (Factor B) with its levels. And so on when using more than two factors or given a symbol that corresponds to the treatment being tried.

The disadvantages of the Split-plot design are as follows:

1. The main effect of the main plot is thought to be with a lower degree of precision than the effect of the interaction and the main effect of the sub plot. Therefore, this analysis is not recommended for experiments that require the same level of estimation accuracy between two factors
2. The analysis is more complex than the factorial design, especially if applied in RCBD. Although computer engineering is the solution, the interpretation of the output is not easy.

Randomization and Layout of SPD Experiments

SPD trials  can be used both in laboratories, greenhouses, and in the field.  Experimental units for the main plot and its sub plots can be designed with a combination of basic CRD, RCBD, and LATIN designs.  The randomization procedure is carried out in 2 stages, namely randomization on the main plot, then continued with randomization on the sub plot.  Here, it will only be discussed the randomization process and the layout of the SPD with the basic plot design mainly CRD, RCBD, and LATIN, while the basic design for the sub plot is the same, namely RCBD.

Completely Randomized Design (CRD)

In this experiment, CRD was aimed at the layout of its main factors, meaning that the main factor plots were randomly designed to be complete, then these main plots were divided (split) into additional factor plots that were randomized in the main factor plots. For more details, consider the example of a factorial experiment to investigate the effect of A as a less important factor (Main Plot) which consists of three levels, namely a1, a2 and a3.  The second factor is B which is a more important factor (sub plot) in the form of a variety consisting of two varieties (2 levels), namely b1, and b2. The experiment was repeated three times.

Thus, the design of the treatment:

Factor A          : 3 levels

Factor B          : 2 levels

Replication      : 3 times.

The randomization procedure and layout of the Split-plot experiment with the basic design of CRD on its main plot are as follows: 

Step 1: Divide the experimental area into rxb of the experimental unit, according to the level of Factor A and the number of tests.  In this case it is divided into 3x3=9 plots. 

Step 2.  Randomization of the main plot.

In this case, randomization for the placement of factor A was carried out synchronously on 9 plots. The randomization procedure can be seen again in the discussion of randomization on the CRD.  Suppose from the randomization process we get the following results:

Step 3.  Divide each of the main plot above into b plots, according to the level of Factor B.  In this case, each main plot is divided into 2 plots.  Next, perform Sub plot Randomization on each main plot separately and freely.  Thus, there are 9 times the randomization process separately and freely.  For example, the randomization results are as follows: 

Figure 1.        Example of structuring a Split Plot Design using the basic design of CRD

Randomized Complete Block Design (RCBD)

The main plot randomization procedure in the SPD design with the basic RCBD design is the same as the RCBD randomization procedure.  It's just that, the SPD is continued with randomization for the placement of sub plots on each main plot.  To facilitate the understanding of the randomization process and the layout of the SPD with the basic RCBD design on the main plot, here is taken back the same case example as in the CRD case above.  Suppose factor A consists of 3 levels and Factor B 2 levels is repeated 3 times.

The design of the treatment:

Factor A       : 3 levels

Factor B       : 2 levels

Blocks          : 3 Blocks

The randomization procedure and layout of the Split-plot experiment with the basic RCBD design on the main plot are as follows:

Randomization on the main plot

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, according to the level of Factor A.  In this case example, each Block is divided into 3 plots, so there are 9 plots in total.

Step 3.  Randomize the Main Plot on each Block separately.

Randomize Block 1 to place a Factor A level, then do a re-randomization for the 2nd Block and the 3rd Block.  Thus, there are 3 times the randomization process separately and freely.  For example, the randomization results are as follows: 

Randomization of sub plot

Step 4.  Divide each of the main plot above into b plots, according to the level of Factor B.  In this case, each main plot is divided into 2 plots.  Next, perform Sub plot Randomization on each main plot separately. Thus, there are 9 times the randomization process separately and freely.  For example, the randomization results are as follows: 

Figure 2.        Example of structuring a Split Plot Design using a RCBD base design

Latin Square Design (LSD)

The main plot randomization procedure in the SPD design with the basic LATIN design is the same as the RSBL randomization procedure.  It's just that, the SPD is continued with randomization for the placement of sub plots on each main plot.  In this case example, an example of the treatment design on the CRD and RCBD above was reused, namely Factor A consisting of 3 levels and Factor B 2 levels repeated 3 times. Note, if the Main Plot is designed using the basic design of the LATIN, then the level of factor A (the main plot) must be equal to the number of repeats, while the level of factor B can be different.  In the example case above, the level of factor A = the degree of repeat.

The design of the treatment:

Factor A   : 3 levels

Factor B    : 2 levels

Blocks       : 3 Blocks

Randomization on the Main Plot:

Step 1: Choose a basic LATIN design for a 3x3 size. 

Step 2: Randomize the row direction and then the column direction. Suppose the result is as follows: 

Step 3: Replace the above code with the treatment code of factor A. In this case example: A = a1; B = a2; C = a3. The result is as follows, which is nothing but a layout for the main plot arranged with the LATIN pattern:

Randomization On Sub Plot:

Step 4: Divide each unit of experiment on the main plot according to the level of Factor B (in this case each main plot is divided into 2, because the level of factor B = 2), bringing the total to 9x2 = 18 units of experiment. Perform randomization separately on each of the main plots (in the case above, there are 9 randomizations).  Remember, each level B must be contained on each main plot. For example, the results are as follows (note, the 2 levels B, b1 and b2, are found at each level of Factor A):  

Figure 3.        Example of structuring a Split Plot Design using the basic design of LATIN 

Split-Plot Linear Model

The linear model of additives for  split-plot design with its environmental design complete random design is as follows :

Y_ijk = μ + α_i + β_j + γ_ik + (αβ)_ij + ε_ijk

with i =1,2, ..., a; j = 1, 2, ..., b;  k = 1, 2, ..., r

Y_ijk      =   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)           

α_i            =   i-th level additive effect of factor A

β_j        =   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

γ_ik       =   random effect of the main plot, which appears at the I-th level of the factor A in the kth replication.  γ_ik ~N(0.σ_γ²).

ε_ijk         =   random effect of the kth experimental unit that obtained a combination of ij treatments.  ε_ijk ~ N(0.σ_ε²).

The linear model of additives for split-plot design in RCBD is as follows :

Y_ijk = μ + ρ_k + α_i + β_j + γ_ik + (αβ)_ij + ε_ijk

with i =1, 2, ..., a; j = 1, 2, ..., b;  k = 1, 2, ..., r

Y_ijk      =   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)           

ρ_k           =   additive effect of the k-th Block

α_i            =   i-th level additive effect of factor A

β_j        =   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

γ_ik       =   random effect of the main plot, which appears at the I-th level of factor A in the k-th Block.  Often called the main plot error.  γ_ik ~N(0.σ_γ²).

ε_ijk         =   random effect of the kth experimental unit that obtained a combination of ij treatments.  It is often called a sub plot error.  ε_ijk ~ N(0.σ_ε²).

Assumption:

Hypothesis:

The hypotheses tested in the Split-plot design are:

Analysis of variance:

In Split-Plot there are two types of errors, namely Main Plot error and Subplot error.  Main Plot Error is often referred to as Error A, the calculation procedure is the same as the Interaction of Main Plot x Replication and in the RCBD model it is the same as the Interaction of Main Plot x Block.  Sub plot Error, often referred to as Error B, is measured from the interaction [Sub plot x Replication + Main Plot x Sub Plot x Replication].  This 2nd error is used to measure the significance level of the sub plot effect and the plot x Sub plot Interaction effect.

CRD

The data representation of the linear model Y_ijk = μ + α_i + γ_ik + β_j + (αβ)_ij + ε_ijk  is as follows:

 $$\begin{matrix}Y_{ijk}={\overline{Y}}_{...}+({\overline{Y}}_{i..}-{\overline{Y}}_{...})+({\overline{Y}}_{i.k}-{\overline{Y}}_{i..})\\+({\overline{Y}}_{.j.}-{\overline{Y}}_{...})+({\overline{Y}}_{ij.}-{\overline{Y}}_{i..}-{\overline{Y}}_{.j.}+{\overline{Y}}_{...})+(Y_{ijk}-{\overline{Y}}_{ij.}-{\overline{Y}}_{i.k}+{\overline{Y}}_{i..})\\\end{matrix}$$

Based on such a linear model:

RCBD

The data representation of the linear model Y_ijk = μ + ρ_k + α_i + γ_ik + β_j + (αβ)_ij + ε_ijk  is as follows:

 $$\begin{matrix}Y_{ijk}={\overline{Y}}_{...}+({\bar{Y}}_{..k}-{\bar{Y}}_{...})+({\overline{Y}}_{i..}-{\overline{Y}}_{...})+({\bar{Y}}_{i.k}-{\bar{Y}}_{i..}-{\bar{Y}}_{..k}+{\bar{Y}}_{...})\\+({\overline{Y}}_{.j.}-{\overline{Y}}_{...})+({\overline{Y}}_{ij.}-{\overline{Y}}_{i..}-{\overline{Y}}_{.j.}+{\overline{Y}}_{...})+(Y_{ijk}-{\overline{Y}}_{ij.}-{\overline{Y}}_{i.k}+{\overline{Y}}_{i..})\\\end{matrix}$$

Based on such a linear model:

LATIN

The data representation of the linear model Y_ijk = μ + ρ_k + κ_l + α_i + γ_ik + β_j + (αβ)_ij + ε_ijk  is as follows:

 $$\begin{matrix}Y_{ijk}={\overline{Y}}_{...}+({\bar{Y}}_{..k}-{\bar{Y}}_{...})+({\bar{Y}}_{...l}-{\bar{Y}}_{...})+({\overline{Y}}_{i..}-{\overline{Y}}_{...})+(Y_{i.kl}-{\overline{Y}}_{.k.}-{\overline{Y}}_{...l}-{\overline{Y}}_{i...}+2{\overline{Y}}_{...})\\+({\overline{Y}}_{.j.}-{\overline{Y}}_{...})+({\overline{Y}}_{ij.}-{\overline{Y}}_{i..}-{\overline{Y}}_{.j.}+{\overline{Y}}_{...})+(Y_{ijk}-{\overline{Y}}_{ij.}-{\overline{Y}}_{i.k}+{\overline{Y}}_{i..})\\\end{matrix}$$

Based on such a linear model:

The Split-Plot analysis of variance table in a complete random design is as follows:

Table 1.    Anova Table of Split-Plot Design

The ANOVA formula for Split-Plot designed with RCBD and LATIN is similar to CRD, especially on Sub Plot, the formula is exactly the same.  The difference lies in the Main Plot formula, as can be seen in the following Table:

Table 2.    Anova Table of Split-Plot Design with CRD, LATIN and RCBD.

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 > F_{α(df1, df2)}, and vice versa accept Ho. To determine the magnitude of the variance in the main plot as well as the sub plot can use the following formula:

 $$ CV(a)=\frac{\sqrt{MS(Ea)}}{\bar{Y}...}\times100%$$

 $$ CV(b)=\frac{\sqrt{MS(Eb)}}{\bar{Y}...}\times100%$$

Standard Error

To compare the mean values of the treatment, it is necessary to first determine the standard error of the SPD.  In SPD there are 4 different types of paired 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 3      Standard error of SPD

From the standard error table above, it can be seen that to compare the two average values of the main plot on the same or different sub plot treatment, two types of MS(Error) are used, namely MS(Ea) and MS(Eb).  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 t_a and t_b are successively the values of t obtained from the table with a certain significant degree at error-degree of freedom and error-degree of freedom b, then the weighted value of t is:

 $$ t\prime=\frac{(b-1)(MS\ \ Eb)(t_b)+(MS\ \ Ea)(t_a)}{(b-1)(MS\ \ Eb)+(MS\ \ Ea)}$$