Example of a Split Plot Design

Applied Example

Experiment: Effect of the combination of NPK fertilization and rice genotype on rice yield (kg/plot).  The effect of the NPK fertilization combination (A) consists of 6 levels placed as the main plot and the rice genotype (B) consists of 2 levels placed as sub plot (subplot).  The main plots are arranged using the basic design of the RCBD.  The experiment was repeated 3 times.  The data from the experiment are given in the following table.

Table 4.    Effect of the combination of NPK fertilization and rice genotype on rice yield (kg/plot)

Calculation by Hand:

Step 1: Calculate the Correction Factor

 $$ CF=\frac{Y...^2}{abr}=\frac{(1906.3)^2}{6\times2\times4}=75707.9102$$

Step 2: Calculate the Sum of The Total Squares

 $$\begin{matrix}SSTOT=\sum_{i,j,k}{Y_{ijk}}^2-CF\\=(20.7)^2+(32.1)^2+...+(45.6)^2-75707.9102\\=2273.93979\\\end{matrix}$$

Create a Fertilizer Table x Blocks:

Step 3: Calculate the Sum of Squares of Blocks

 $$\begin{matrix}SSR=\frac{\sum_{k}{(r_k)^2}}{ab}-CF\\=\frac{(449)^2+(482.6)^2+(461.5)^2+(513.2)^2}{6\times2}-75707.9102\\=197.110625\\\end{matrix}$$

Step 4: Calculate the Sum of Squares of Factor A

 $$\begin{matrix}SS(A)=\frac{\sum_{i}{(a_i)^2}}{rb}-CF\\=\frac{(244.7)^2+(259.5)^2+...+(365.9)^2}{4\times2}-75707.9102\\=1674.79604\\\end{matrix}$$

Step 5: Calculate the Sum of Squares of Main Plot Errors (Ea)

 $$\begin{matrix}SS(Ea)=\frac{\sum_{i,k}{(a_ir_k)^2}}{b}-CF-SSR-SS(A)\\=\frac{(48.4)^2+(65.1)^2+...+(86.5)^2+(92.2)^2}{2}-75707.9102-197.110625-1674.79604\\=267.728125\\\end{matrix}$$

Create A Table For Total Treatment:

Step 6: Calculate the Sum of The Squares of Factor B

 $$\begin{matrix}SS(B)=\frac{\sum_{j}{(b_j)^2}}{ra}-CF\\=\frac{(953.8)^2+(952.5)^2}{4\times6}-75707.9102\\=0.03520833\\\end{matrix}$$

Step 7: Calculate the Sum of Squares of AB Interactions

 $$\begin{matrix}SS(AB)=\frac{\sum_{i,j}{(a_ib_j)^2}}{r}-CF-SS(A)-SS(B)\\=\frac{(120.0)^2+(124.7)^2+...+(187.6)^2+(178.3)^2}{4}-75707.9102-1674.79604-0.03520833\\=78.5910417\\\end{matrix}$$

Step 8: Calculate the Sum of Squares of Sub plot Errors (Eb)

 $$\begin{matrix}SS(Eb)=SSTOT\ -\ SS(Other)\ \\=SSTOT\ -\ SSB\ -\ SS(A)\ -\ SSEa-SS(B)\ -SS(AB)\\=2273.93979-197.110625-1674.79604-267.728125-0.03520833-78.5910417\\=55.67875\\\end{matrix}$$

Step 9: Create an Analysis of variance Table with its F-Values

Table 5      Anova Table of Rice Yield

 $$\begin{matrix}CV(a)=\frac{\sqrt{MS(Ea)}}{\bar{Y}...}=\frac{\sqrt{17.8485}}{39.715}\\=10.64\%\\\end{matrix}$$

 $$\begin{matrix}CV(b)=\frac{\sqrt{MS(Eb)}}{\bar{Y}...}=\frac{\sqrt{3.09326}}{39.715}\\=4.43\%\\\end{matrix}$$

Step 10: Make a Conclusion

First, we check whether the Interaction Effect is significant or not? If it is significant, then examine the simple effect of the interaction, and ignore the main effect , even if the main effect is significant! Why? Take another look at the discussion about the effect of interactions and main effects!  Main effect testing (if significant) is only performed when the effect of the interaction is not significant.

Effect of AB Interactions

Since Fstat (5.08) > 2,773 then we reject H₀: μ₁ = μ₂ = ... at a confidence level of 95% (usually given one asterisk mark (*), which indicates a noticeable difference)

Main Effects

Since the effect of the interaction is significant, then its main effect does not need to be discussed further.

Post Hoc

Based on the analysis of variance, the effect of the interaction is significant so that the testing of the main effect of the treatment of the combination of fertilizers and the two rice genotypes does not need to be carried out.  The next step is to examine its simple effect because the interaction between the two factors is significant. 

Here are the steps to test Advanced Test by using LSD:

Test criteria:

Compare the absolute value of the difference between the two averages that we will see the difference with the LSD value with the following test criteria:

 $$ if\ \ \left|\mu_i-\mu_j\right|\ \ \left\langle\ \ \begin{matrix}>LSD_{0.05}\ reject\ H_0,\ two\ means\ are\ significantly\ different\\\le LSD_{0.05}\ accept\ H_0,\ two\ means\ are\ not\ significantly\ different.\\\end{matrix}\right.$$

Comparison of The Average of Sub Plot (between two rice genotypes on a specific fertilization combination):

Calculate the appropriate Comparative Value (LSD)

1. To compare two sub plot's averages (between rice genotypes) on the same main plot treatment (a combination of specific fertilizations), it is necessary to first determine the standard error (s_y) of the SPD using the formula: 

 $$s_{\bar{y}}=\sqrt{\frac{2MS(Eb)}{r}}$$

• Determine the t-student value:
• 
  
  

 $$\begin{matrix}LSD=t_{0.05/2;18}\cdot s_{\bar{Y}}\\=t_{0.05/2;18}\cdot\sqrt{\frac{2MS(Eb)}{r}}\\=2.101\times\sqrt{\frac{2(3.0933)}{4}}\\=2.6129\ \ kg\\\end{matrix}$$

Compare the difference in the average treatment with the LSD value = 2.6129. State it differently if the average difference is greater than the LSD value.  Since the level of rice genotypes is only two, a simple examination of the effect on the ratio of two averages of sub plot can be simplified.  Give the same letter (a) to the two genotypes when the average difference is ≤ LSD and a different letter when the average difference is > LSD.  The result is as follows:

Comparison of Main Plot Mean (between two fertilization combinations on the same or different genotypes):

Calculate the appropriate Comparative Value (LSD)

To compare two plot averages (average pairs of fertilization combinations) on the same or different main plot treatments, the standard error is calculated using the formula:

 $$s_{\bar{Y}}=2[(b-1)MS(Eb)+MS(Ea)]rb$$

From the formula, it can be seen that to compare the two average values of the main plots 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:

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

ta = t_(0.05/2.15) = 2.131

tb = t_(0.05/2.18) = 2.101

b = 2 (subplot/genotype level)

MS(Ea) = 17.8485

MS(Eb) = 3.0933

so that:

 $$\begin{matrix}t'=\frac{(b-1)(MS\ \ Eb)(t_b)+(MS\ \ Ea)(t_a)}{(b-1)(MS\ \ Eb)+(MS\ \ Ea)}\\=\frac{(2-1)(3.0933)(2.101)+(17.8485)(2.131)}{(2-1)(3.0933)+(17.8485)}\\=2.1266\\\end{matrix}$$

and

 $$sY=2[(b-1)MS(Eb)+MS(Ea)]rb=2[(18-1)(3.0933)+17.8485]4×2=2.288111$$

So:

 $$\begin{matrix}LSD=t' \times s_Y\\=2.1266\times2.2881\\=4.8667\ \ kg\\\end{matrix}$$

Comparison between main plots in sub plot of Genotype IR-64 plots

Comparison between main plots in subplot of Genotype S-969 plots

From the results of further tests of the simple effects above, the results can be summarized in the form of a Table of Fertilizer Interactions x Genotypes as below.

Information:

Letters in brackets are read in a horizontal direction, comparing between 2 G on the same P

Lowercase letters without brackets are read vertically, comparing between 2 P's on the same G

Interaction (Combined Factor)

Comparison of treatment combinations (???)

If the interaction is significant, a simple effect should be examined. However, if you want to compare the combinations (AxB), it is necessary to first determine the appropriate standard error (sy). In the standard error table for RPT, there is no suitable standard error formula. In the variance table, the error for AxB Interaction is error b, so the standard error that might be used is the standard error of the subplots, with the formula:

 $$\begin{matrix}LSD=t_{0.05/2;18}\cdot s_{\bar{Y}}\\=t_{0.05/2;18}\cdot\sqrt{\frac{2MS(Eb)}{r}}\\=2.101\times\sqrt{\frac{2(3.0933)}{4}}\\=2.6129\ \ kg\\\end{matrix}$$
From the previous calculation, using the standard error, the LSD value = 2.6129 is obtained. 

Calculation by SmartstatXL Excel Add-In

Anova

Post Hoc

Main Effect:

Interaction (Simple Effect)

Interaction (Combined Factor)

Anova Assumption: