Split-Split Plot Design

Introduction

Split-Split Plot Design (SSPD) is an extension of Split Plot Design (SPD).  In SPD we only do experiments with 2 factors, while in SSPD we are dealing with 3 experimental factors.  The first, second, and third factors are placed as the main plot, sub-plot, and sub-sub-plot, respectively.  The principle is almost the same as that of SPD, that is, the factor placed on plots that are smaller in size is more important than plots that are larger in size.  Thus, the sub-sub plots are allocated as the most important factor, followed by the sub-plots and finally, the main plots that are not very important.

Likewise with the use of the basic design, the SSPD is still combined with the basic design of the CRD, RCBD, or LATIN.  In the next description, only SPD is discussed using the basic RCBD design for both the main plot, sub plot, and sub-sub plot.

Randomization and Trial Layout of SSPD

The randomization procedure in the SSPD is conducted in 3 stages, namely randomization on the main plot, then continued with randomization on the sub plot, and finally randomization on the sub-sub plot.  The main plot randomization procedure in the SPD design with the RCBD basic design is the same as the RCBD randomization procedure.  To facilitate the understanding of the randomization process and the layout of SSPD with the basic design of RCBD, imagine that there is a factorial experiment of 5 x 3 x 3 that is repeated 3 times.  The first factor is Nitrogen which consists of 5 levels as the main plot, the 2nd factor is management practice consisting of 3 levels and allocated as sub plots, the 3rd factor is rice varieties consisting of 3 levels as sub-sub plots.  The stages of randomization and layout of the SSPD experiment with the basic design of RCBD are as follows:

The design of the treatment:

Factor A       : 5 levels

Factor B        : 3 levels

Factor C        : 3 levels

Blocks           : 3 blocks

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 5 plots.   Then Perform a Main Plot Randomization on each block separately. Randomize block 1 to place a Factor A level, then do a re-randomization for the second block and the 3rd block.  Thus, there are 3 times the randomization process separately and freely.  For example, the randomization results are as follows:

Step 3.  Divide each of the main plots above into b sub plots, according to the level of Factor B. In this case, each main plot is divided into 3 sub plots.  Next, perform sub plot randomization on each main plot separately. Thus, there is a 3x5 = 15 times the randomization process separately and freely.  For example, the randomization results are as follows:

Step 4.  Divide each of the above sub plots into c = 3 sub-sub plots, according to the level of Factor C.  Next, perform Sub-sub plot Randomization on each sub plot separately. Thus, there is 15x3 =45 times the randomization process separately and freely.  For example, the randomization results are as follows:

Figure 4.        Example of structuring a Split-Split Plot Design

Linear Model SSPD

The linear model of additives for the SSPD design with its environmental design RCBD is as follows:

Y_ijk = μ + K_l + A_i + γ_il + B_j + (AB)_ij + δ_ijl + C_k + (AC)_ik + (BC)_jk + (ABC)_ijk + ε_ijkl

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

Y_ijkl     =   observations on the lth experimental unit obtaining a combination of i-level treatment from factor A, j-th level from factor B and k-th level from factor C

μ         =   actual average value (population average)           

K_l            =   effect of additives of the l-th block

A_i            =   i-th level additive effect of factor A

γ_il        =   random effect of the main plot, which arises at the i-th level of the factor A in the l-th block.  It is often called a main plot error or an A error.  γ_il ~ N(0.σ_γ²).

B_j        =   j-th level additive effect of factor B

(AB) _ij  =   effect of additives of the i-th level of factor A and the j-th level of factor B

δ_ijl          =   random effect of the lth-th experimental unit obtaining a combination of ij treatments.  It is often called a sub plot error or a b error.  δ_ijl ~ N(0.σ_δ²).

C_k       =   k-th level additive effect of factor C

(AC) _ik  =   effect of additives of the i-th level of factor A and the k-th level of factor C

(BC) _jk  =   j-th level additive effect of factor B and k-th level of factor C

ε_ijkl        =   random effect of the kth experimental unit that obtained a combination of ijk treatments.  It is often called a sub-sub plot error or a C error.  ε_ijkl ~ N(0.σ_ε²).

Analysis of Variance:

Analysis of variance for linear models:

Y_ijk = μ + K_l + A_i + γ_il + B_j + (AB)_ij + δ_ijl + C_k + (AC)_ik + (BC)_jk + (ABC)_ijk + ε_ijkl

is as follows:

*) SS = Sum of Square (SS)

The table of SPD  Analysis of Variance in RCBD is as follows:

Table 6.    Anova Table of Split-split Plot

*) SS = Sum of Square (SS); MS = Mean Square (MS) 

If the effect of the interaction of the three factors (ABC) is significant, then hypothesis testing is carried out on the interaction of the three factors, while the other effects do not need to be carried out.  However, if the interaction of the three factors is not significant, the next step is to check whether the interaction of 2 factors (AB, AC, BC) is significant or not.  If there is anything significant, a hypothesis assessment is conducted on the interaction of the two significant factors and ignores the testing of its main/independent effect. Finally, if there is no significant interaction, hypothesis testing is performed on a significant main effect (A, B, or C).  For example, the interaction of AB and AC is significant, hypothesis testing is only conducted on such interactions, while its main effect (A, B, C) is not required even though based on the Analysis of Variance is significant because it is already represented by its interaction.  What if AB, A, B, C are significant and everything else is insignificant? Testing was conducted only on AB interactions and main effect of C.  Main effect of A and B are not necessary since the effect of A will differ depending on the degree of factor B and vice versa.  Thus, if the source component of the variety is already represented by its interaction, then no testing is required on the component of the source of the variety. Reject H0 if the value of F_stats > F_{of the table}, and vice versa accept Ho.

Coefficient of Variance (CV)

To determine the magnitude of the Variance in the main plot, sub-plots and sub-sub plots can use the following formula:

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

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

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

The value of CV(a) indicates the degree of accuracy associated with the main effect of the main plot factor, the value of CV(b) indicates the degree of accuracy related to the main effect of the sub plot factor and its interaction with the main plot, and the value of CV(c) indicates the degree of precision related to the main effect of the sub-sub plot factor  and the combination  with other factors.  By and large, the coefficient of Variance of the main plot > the sub-plot > the sub-sub plot.

Standard Error

To compare the Mean values of the treatment, it is necessary to first determine the Standard Error.  In Split-split Plot there are 12 different types of paired comparisons, there are 12 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 7.    Standard Error Split-split Plot

Information:

E_a = Mean Squared Ea

E_b = Mean Squared Eb

E_c = Mean Square of Ec

As in Split-plot, it can be seen that to compare the difference in treatment mean there is a comparison of the average that has a standard error from the mean involving more than one Mean Square of the Error, so it is necessary to calculate the combined/weighted t so that the ratio of the difference in treatment to the standard error follows the distribution of t-students.  The following are the combined t values for comparison that correspond to the Standard Error.

Table 8.    Table of the weighted t value that is associated with the standard error in Table 7.

Information:

E_a = Mean Squared Ea

E_b = Mean Squared Eb

E_c = Mean Square of Ec