Completely Randomized Design (CRD) Factorial

Introduction

In the previous discussion we have discussed the effect of single treatment on certain responses.  The single treatment is called a factor, and the level or level of the factor is called a level.  Factors are symbolized by capital letters while the degree of factors is symbolized by lowercase.  If we simultaneously observe the effect of several factors in the same study, then the experiment is called a factorial experiment. 

A factorial experiment is an experiment whose treatment consists of all possible combinations of several factors.  Experiments using f factors with level for each factor are symbolized by factorial experiments f^t.  For example, the factorial experiment 2² means that we use 2 factors, and the level of each factor consists of 2 levels.  Factorial experiments 2² are also often written in the form of 2x2 factorial experiments.   The latter is often used for factorial experiments where the levels of each factor are different, for example 2 levels for factor A and 3 levels for factor B, then the experiment is called a factorial experiment 2x3.  A 2x2x3 factorial experiment means a factorial experiment consisting of 3 factors with a level for each of its factors 2, 2, and 3 respectively.  Thus, in the factorial experiment, there are two stages that need to be carried out, firstly, namely the treatment design, as previously described, and then the stage of selecting the environmental design, which concerns experimental design forms such as CRD, RCBD, LATIN, Split-plot, Split-Block.

The purpose of the factorial experiment is to look at the interactions between the factors we are trying.  Sometimes the two factors synergize with each other  to the (positive) response, but sometimes also the existence of one factor hinders the  performance of other (negative) factors.  The existence of these two mechanisms tends to increase the effect of interactions between the two factors.  Interaction measures the failure of the effect of one factor to remain the same at every level of another factor or simply, the interaction between factors is whether the effect of a particular factor depends on the degree of another factor? For example, if the simple effect of N is the same at each level of fertilizer application P, the two factors are mutually independent and it is said that there is no interaction, while if the application of N has a different effect on each level of P, it is said that there is an interaction between Factor N and Factor P.

For example, if we want to observe the effect of nitrogen (N) administration consisting of 2 levels (n₀, and n₁) and phosphorus administration (P) consisting of 2 levels (p₀, p₁) on certain responses, with the following results:

Table 19. Simple effect, main effect, and Interaction effect

The difference between n1 – n0 and p1 – p0 is called simple effects symbolized by (se N) and (se P).  The average of simple effect is called the main effect, symbolized (me N) and (me P).

The approximate effect of the interaction and the main effect of the average treatment can be calculated by the following formula:

Simple effect (se):

 $\begin{matrix}se\ N\ \text{ on }\ p_0=n_1p_0-n_0p_0\\=48-40\\=8\\se\ N\ \text{ on } \ p_1=n_1p_1-n_0p_1\\=51-42\\=9\\\end{matrix}$          $\begin{matrix}se\ P\ \text{ on } \ n_0=p_1n_0-p_0n_0\\=42-40\\=2\\se\ P\ \text{ on }\ n_1=p_1n_1-p_0n_1\\=51-48\\=3\\\end{matrix}$

Main Effect (main effect, me):

$\begin{matrix}me\ N=\frac{1}{2}(se\ \ N\ \ on\ \ p0+se\ \ N\ \ on\ \ p1)\\=\frac{1}{2}\left[(n_1p_0-n_0p_0)+(n_1p_1-n_0p_1)\right]\\=\frac{1}{2}\left[(48-40)+(51-42)\right]\\=8.5\\\end{matrix}$        

$\begin{matrix}me\ P=\frac{1}{2}(se\ \ P\ \ on\ \ n0+se\ \ P\ \ on\ \ n1)\\=\frac{1}{2}\left[(p_1n_0-p_0n_0)+(p_1n_1-p_0n_1)\right]\\=\frac{1}{2}\left[(42-40)+(51-48)\right]\\=2.5\\\end{matrix}$

Interaction Effects:

 $\begin{matrix}interaction\ N\times P=\frac{1}{2}\left[(n_1p_0-n_0p_0)-(n_1p_1-n_0p_1)\right]\\=\frac{1}{2}\left[(48-40)-(51-42)\right]\\=-0.5\\\end{matrix}$

or

$\begin{matrix}=\frac{1}{2}\left[\left(p_1n_0-p_0n_0\right)-\left(p_1n_1-p_0n_1\right)\right]\\=\frac{1}{2}\left[\left(42-40\right)-\left(51-48\right)\right]\\=-0.5\\\end{matrix}$

This simple effect is required by the user (farmer, for example), if he only limits to the use of a certain degree of one of the factors.  For example, if farmers want to see the difference in the effect of N on each fertilization level of P, the simple effect of N at the level of p0 = 8 and at the level of p1 = 9. 

The difference between simple effect and interactions graphically can be visualized as follows:

Figure 8.        The difference between simple effect and interaction

The possibilities that can occur between the main effect and the interaction are presented in the following Figure:

Figure 9.        Various possible forms of interaction between factors

Advantage:

1. More efficient in using existing sources
2. The information obtained is more comprehensive because we can study the main effect and interactions
3. The results of the experiment can be applied in a broader range of conditions because we study a combination of various factors

Disadvantage:

1. Statistical Analysis becomes more complex
2. There are difficulties in providing relatively homogeneous experimental units
3. the effect of a certain combination of treatments may not mean anything so that there is a waste of existing resources

Factorial Experiments in a Completely Randomized Design

Factorial experiments in a Completely Randomized Design are factorial experiments using a Completely Randomized Design as their environmental design. In principle it is the same as a Completely Randomized Design, but in this case the factors tried are more than one.

Randomization and Factorial Trial Plan in a Completely Randomized Design

The randomization method is the same as a complete random design. The placement of treatments that are a combination of the level of factors to be tried is carried out in the same way as the Completely Randomized Design.  Consider the following case example.  An experiment wanted to study the effect of Nitrogen and Variety fertilization on the yield of production carried out in the Greenhouse. Environmental conditions are assumed to be homogeneous.  The fertilization factor consists of 2 levels, namely 0 kg N / ha (n0) and 60 kg N / ha (n1).  The Variety Factor consists of two levels, namely the IR-64 Variety (v1) and the S-969 Variety (v2).  The experiment was designed using the basic CRD design which was repeated 3 times.  The experiment was a 2x2 factorial experiment so there were 4 treatment combinations: n0v1; n0v2; n1v1; and n1v2.  Because it is repeated 3 times, the unit of experiment consists of 4x3 = 12 units of experiments. 

Create 12 plots (experimental units) and the units of experiments are numbered from 1 to 12.  The randomization step is the same as randomization on a single CRD.  For example, the result of randomization is as follows:

Based on the results of the randomization, the experimental layout is as follows:

Figure 10.     Trial Layout of 2 x 2 Factorial Experiment with CRD Environmental Design

Linear Model of Factorial Design In CRD

The linear model of additives for a two-factor factorial design with its environmental design a complete random design is as follows :

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

with i =1.2...,a; j = 1, 2, ..., b;  c = 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         =   population mean

α_i            =   i-th degree effect of factor A

β_j        =   j-th degree effect of factor B

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

ε_ijk         =   random effect of the kth experimental unit that obtained a combination of ij treatments.  ε_ij ~ N(0.σ²).

Assumption:

Hypothesis:

The hypotheses tested in a factorial design consisting of two factors with a Completely Randomized Design environment design are:

Analysis of variance:

The linear model of factorial experiments with the basic design of CRD is as follows:

 $$\begin{matrix}Y_{ijk}=Model+Error\\Y_{ijk}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\varepsilon_{ijk}\\Y_{ijk}={\overline{Y}}_{...}+\left({\overline{Y}}_{i..}-{\overline{Y}}_{...}\right)+\left({\overline{Y}}_{.j.}-{\overline{Y}}_{...}\right)+\left({\overline{Y}}_{ij.}-{\overline{Y}}_{i..}-{\overline{Y}}_{.j.}+{\overline{Y}}_{...}\right)+\left(Y_{ijk}-{\overline{Y}}_{ij.}\right)\\\left(Y_{ijk}-{\overline{Y}}_{...}\right)=\left({\overline{Y}}_{i..}-{\overline{Y}}_{...}\right)+\left({\overline{Y}}_{.j.}-{\overline{Y}}_{...}\right)+\left({\overline{Y}}_{ij.}-{\overline{Y}}_{i..}-{\overline{Y}}_{.j.}+{\overline{Y}}_{...}\right)+\left(Y_{ijk}-{\overline{Y}}_{ij.}\right)\\\end{matrix}$$

If the two fields are squared, then we will get:

Table 20   . Mean Squared Expectation Value of Two-Factorial Design In Complete Random Design

Using the mean-squared expectation value above, we can compile its Analysis of variance Table.  The anova table of factorial experiments with two factors in the Completely Randomized Design is as follows:

Table 21.  Anova Table of Two-Factorial Designs in a Completely Randomized Design

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. 

Standard Error

The standard error for the difference among treatment averages is calculated by the following formula:

Comparison of the two averages of Factor A:

 $$ SED=S_{\bar{Y}}=\sqrt{\frac{2MSE}{rb}}$$

Comparison of the two averages of Factor B:

 $$ SED=S_{\bar{Y}}=\sqrt{\frac{2MSE}{ra}}$$

Comparison of the interaction of the two average factors of AxB:

 $$ SED=S_{\bar{Y}}=\sqrt{\frac{2MSE}{r}}$$