Example of RCBD Factorial

Example 1: Factorial RCBD (Not significant interaction)

Table 26. Experiment on the Effect of Tillage and Organic Fertilizers on the Aggregate Stability Index.

Calculation Analysis of Variance:

Step 1: Calculate the Correction Factor

 $$ CF=\frac{Y...^2}{abr}=\frac{(5864)^2}{3\times4\times3}=955180.44$$

Step 2: Calculate the Sum of The Total Squares

 $$\begin{matrix}SSTOT=\sum_{i,j,k}{Y_{ijk}}^2-CF\\=(154)^2+(151)^2+...+(182)^2-955180.44\\=9821.56\\\end{matrix}$$

Step 3: Calculate the Sum of Squares of Groups

 $$\begin{matrix}SSR=\frac{\sum_{k}{(r_k)^2}}{ab}-CF\\=\frac{(1975)^2+(1931)^2+(1958)^2}{3\times4}-955180.44\\=82.06\\\end{matrix}$$

The Sum of Squares of Treatment needs to be decomposed into the Sum of the Squares of its components.

Create a Table For Total Treatment

Step 4: Calculate the Sum of Squares of Factor A

 $$\begin{matrix}SS(A)=\frac{\sum_{i}{(a_i)^2}}{rb}-CF\\=\frac{(2075)^2+(1890)^2+(1899)^2}{3\times4}-955180.4444\\=1813.39\\\end{matrix}$$

Step 5: Calculate the Sum of Squares of Factor B

 $$\begin{matrix}SS(B)=\frac{\sum_{j}{(b_j)^2}}{ra}-CF\\=\frac{(1317)^2+(1443)^2+(1482)^2+(1622)^2}{3\times3}-955180.44\\=5258.00\\\end{matrix}$$

Step 6: Calculate the Sum of Squares of AB Interactions

 $$\begin{matrix}SS(AB)=\frac{\sum_{i,j}{(a_ib_j)^2}}{r}-CF-SS(A)-SS(B)\\=\frac{(470)^2+(492)^2+...+(491)^2+(515)^2}{3}-955180.44-1813.39-5258.00\\=463.50\\\end{matrix}$$

Note: SST = SS(A) + SS(B) + SS(AB)

Step 7: Calculate the Sum of Squares of Errors

 $$\begin{matrix}SSE=\ SSTOT\ -\ SSB\ -\ SS(A)\ -\ SS(B)\ -SS(AB)\\=9821.56-82.06-1813.39-5258.00-463.50\\=2204.61\\\end{matrix}$$

Step 8: Create a Variance Analysis Table along with its F-tables.

Table 27.  ANOVA Table of Two Factor Factorial Design in Completely Randomized Block Design

F_(0.05,2,22) =3.443

F_{(0.0 1,2,22)} = 5,719

F_(0.05,3,22) = 3,049

F_{(0.0 1,3,22)} = 4,817

F_(0.05,6,22) = 2,549

F_{(0.0 1,6,22)} = 3,758

Step 9: Make a Conclusion

Interaction Effect: not significant

Since F-stat (0.77) ≤ 2.549 then we fail to reject H₀:μ₁ = μ₂ = .... at a confidence level of 95%.  This means that at a confidence level of 95%, there is no difference in the effect of the interaction on the observed response.

Effect of Factor A: significant

Since F-stat (9.05) > 3,443 then we reject H₀: μ₁ = μ₂ = .... at a confidence level of 95%.  This means that at a confidence level of 95%, there is one or more of the average different treatments from others.  Or in other words, a decision to reject Ho can be made, meaning that there is a difference in the effect of Factor A on the observed response.

Effect of Factor B: significant

Since F-stat (9.05) > 3,443 then we reject H₀: μ₁ = μ₂ = .... at a confidence level of 95%.  This means that at a confidence level of 95%, there is one or more of the average different treatments from others.  Or in other words, a decision can be made to reject Ho, meaning that there is a difference in the effect of Factor B on the observed response.

Since Interaction is not significant (significant), then we proceed to the examination of its main effect. Both of its main effect are significant, so we need to further investigate which treatments are the same and which are different. Perform further testing to compare treatment averages, both treatment mean differences for Factor A and Factor B.

Post-Hoc

Based on the analysis of variance, the effect of the interaction between Factor A and Factor B is not significant, while both main effect are significant so that further testing is only carried out on the main effect of the two factors we are trying.

In this further test, the differences between the average pairs of treatments were carried out using the LSD test.

Main Effects of Tillage Factors (A)

Calculate LSD with the following Formula:

1.  $ LSD=t_{\alpha/2;db}\sqrt{\frac{2MSE}{rb}}$

• MSE = 100.21
• error-degree of freedom = 22
• Block (r) = 3; Level of Factor B (b) = 4
• t_(α/2.22) = t_(α/2.22) = 2,074 (See t-student table at a significant level, α = 0.05, and df = 22, or if using functions in MS Excel, write the formula "=tinv(0.05,22)"
• The above parameters  are entered into the formula:
•  $\begin{matrix}LSD=t_{\alpha/2;db}\sqrt{\frac{2MSE}{rb}}\\=t_{0.05/2;22}\sqrt{\frac{2(100.21)}{3\times4}}\\=2.074\times4.087\\=8.475\\\end{matrix}$

2. Create a Table of treatment averages for Factor A (the main effect of Factor A), and then sort from small to large values (ascending order).

• Sorted by tillage:

3. Compare the difference in the average treatment with the LSD value = 8,475.  To make the work easier, create a matrix table of average differences as in the following example:

4. After being given a letter notation, return the order based on the order of treatment (not the average order).  The end result is as follows:

Main Effect of Organic Fertilizer Factors (B)

Calculate LSD with the following Formula:

1.  $ LSD=t_{\alpha/2;db}\sqrt{\frac{2MSE}{ra}}$

• MSE = 100.21
• error-degree of freedom = 22
• Block (r) = 3; Factor Level A (a) = 3
• t_(α/2.22) = t_(α/2.22) = 2,074 (See t-student table at a significant level, α = 0.05, and df = 22, or if using functions in MS Excel, write the formula "=tinv(0.05,22)"
• The above parameters  are entered into the formula:
•  $\begin{matrix}LSD=t_{\alpha/2;db}\sqrt{\frac{2MSE}{ra}}\\=t_{0.05/2;22}\sqrt{\frac{2(100.21)}{3\times3}}\\=2.074\times4.719\\=9.787\\\end{matrix}$

2. Create a Table of treatment averages for Factor B (the main effect of Factor B), then sort from small to large values (ascending order).  It just so happens that in this example, the average value of the treatment has been sorted from small to large.

3. Compare the difference in the average treatment with the LSD value = 9.787.  To make the work easier, create a matrix table of average differences as in the following example:

4. After being given a letter notation, return the order based on the order of treatment (not the average order).  It just so happens that the order is as simple as according to the order of treatment.  The final result is as follows:

Calculation Using SmartstatXL Excel Add-In

Example 2: Factorial RCBD

The data is provided as follows:

Analysis of Variance Calculation:

 $$ CF=\frac{Y...^2}{abr}=\frac{(400)^2}{2\times2\times4}=10000$$

 $$\begin{matrix}SSTOT=\sum_{i,j,k}{Y_{ijk}}^2-CF\\=(12)^2+(15)^2+...+(37)^2-10000\\=1170\\\end{matrix}$$

 $$\begin{matrix}SSR=\frac{\sum_{k}{(r_k)^2}}{ab}-CF\\=\frac{(92)^2+(99)^2+(108)^2+(101)^2}{2\times2}-10000\\=32.5\\\end{matrix}$$

Create a Table For Total Treatment

 $$\begin{matrix}SS(A)=\frac{\sum_{i}{(a_i)^2}}{rb}-CF\\=\frac{(139)^2+(261)^2}{4\times2}-10000\\=930.25\\\end{matrix}$$

 $$\begin{matrix}SS(B)=\frac{\sum_{j}{(b_j)^2}}{ra}-CF\\=\frac{(173)^2+(227)^2}{4\times2}-10000\\=182.25\\\end{matrix}$$

 $$\begin{matrix}SS(AB)=\frac{\sum_{i,j}{(a_ib_j)^2}}{r}-CF-SS(A)-SS(B)\\=\frac{(54)^2+(85)^2+(119)^2+(142)^2}{4}-10000-930.25-182.25\\=4\\\end{matrix}$$

Note: SST = SS(A) + SS(B) + SS(AB)

 $$\begin{matrix}SSE=\ SSTOT\ -\ SSB\ -\ SS(A)\ -\ SS(B)\ -SS(AB)\\=1170-32.5-930.25-182.25-4\\=21\\\end{matrix}$$

Table 28.  Anova Table of Two-Factorial Designs In RCBD

Post-Hoc

Based on the analysis of variance, the effect of the interaction between Factor A and Factor B is not significant, while both main effects are significant so that further testing is only carried out on the main effect of the two factors we are trying.

In this follow-up test, the differences between the average pairs of treatments were carried out using the Duncan test.