Two Factor Experiments

The article is about the use and application of split plot design in Agriculture, here we will discuss the conditions in which split plot design should be used in agriculture, the related real-life examples of split plot design, and the model of the design. In factorial experiments, there are certain situations where it becomes difficult to handle all the combinations of different levels of the factors. This may be because of the following reasons:

• The nature of the factors may be such that levels of one factor require large experimental units as compared to the levels of other factors. For example, If the two factors are Rowing Methods and Nitrogen Levels”, then in the two-factor experiment the rowing methods require machinery, so they require large experimental units, and the nitrogen levels can be applied to the smaller units.
• Greater precision may be required for levels of one factor as compared to the levels of other factors. For example, If we want to compare two factors, varieties, and fertilizers, and more precision is required for fertilizers, then varieties would be in the larger units and the fertilizers would be in the smaller units.
• It may be that new treatments have to be introduced into an experiment that is already in progress.

Conditions in which Split Plot Design Used

The split plot design (and a variation, the split block) is frequently used for factorial experiments in which the nature of the experimental material or the operations involved makes it difficult to handle all factor combinations in the same manner.

• If irrigation is more difficult to vary on a small scale and fields are large enough to be split, a split-plot design becomes appropriate.
• Usually used with factorial sets when the assignment of treatments at random can cause difficulties, large-scale machinery can required for one factor but not another irrigation and tillage.
• Plots that receive the same treatment must be grouped.

• Degree of Precision: For greater precision for Factor $B$ than for factor $A$, the factor $B$ should be assigned to the subplot and factor $A$ to the main plot.
• Relative Size of the Main Effects: If the main effect of (say factor $B$) is much larger and easier to detect than that of the other factor (factor $A$), the factor $B$ can be assigned to the main plot, and factor $A$ to the subplot. This increases the chance of detecting the difference among levels of factor $A$ which has a smaller effect.
• Management Practices: The cultural practices required by a factor may dictate the use of large plots. For example, in an experiment to evaluate water management and variety, it may be desirable to assign water management to the main plot to minimize water movement between adjacent plots, facilitate the simulation of the water level required, and reduce border effects.

Split Plot Design in Agriculture: Irrigation and Fertilizer (Example 1)

In agricultural experiments involving two factors “irrigation” and “nitrogen” fertilizer. Sometimes, it is very convenient to apply different levels of irrigation to small neighbouring plots but there is no such difficulty for the application of different levels of nitrogen fertilizer. To meet such situations, it is desirable to have different sizes of the experimental units in the same experiment. For this purpose, we have two sizes of the experimental units. First, a design with bigger plots is taken to accommodate the factors that require bigger plots. Next, each of the bigger plots is split into as many plots as the number of treatments coming from the other factors.

The bigger plots are called main plots. The treatments allotted to them are called main plot treatments or simply main treatments. The consequent parts of the main plots are called sub-plots or split plots and the treatments allotted to them are called sub-plot treatments. The different types of treatments are allotted at random to their respective plot. Such a design is called split-plot design.

Split Plot Design in Agriculture: Irrigation and Fertilizer (Example 2)

Let there be 3 levels of irrigation prescribing 3 different amounts of water per plot and 4 doses of nitrogen fertilizer.

First, a randomized block design with a suitable plot is taken with 3 levels of irrigation as treatments say with 5 replications of the design. The irrigation treatments are then allotted at random to each five blocks, each consisting of 4 sub-plots.

Next, each of these main plots is split into 4 sub-plots to accommodate the 4 levels of nitrogen. The main 15 plots serve as 15 replications of the subplot treatments. Treatments are allotted at random to sub-plots of each of the main plots. The split-plot design is the combination of two or more randomized designs depending on several factors, such as the plots of one design from the block of another design. The main plot treatment or the levels of one factor or different factors each of which requires a similar plot size.

Model of Split Plot Design

\begin{align} y_{ijk} &= \mu + \tau_i + \beta_j + (\tau \beta){ij} + \gamma_k + (\tau \gamma){ik} + (\beta\gamma){jk}+(\tau \beta\gamma){ijk} + \varepsilon_{ijk}\\
i &= 1,2,\cdots, a \text{ levels of factor } A\\
j &= 1,2,\cdots, b \text{ levels of factor } B\\
k &= 1,2,\cdots, c \text{ levels of factor } C
\end{align}

Model Terms

• Linear Terms
  • $\mu$: Overall mean
  • $\tau_i$: Effect of $i$th level of $A$
  • $\beta_j$: Effect of $j$th level of $B$
  • $\gamma_k$: Effect of $k$th level of $C$
• Interactions Terms
  • $(\tau \beta){ij}$: Interaction effect of $A$ and $B$\ $(\tau \gamma){ik}$: Interaction effect of $A$ and $C$\
  • $(\beta\gamma){jk}$: Interaction effect of $B$ and $C$\ $(\tau\beta\gamma){ijk}$:Interaction effect of $A$, $B$ and $C$ \item \textbf{Error} $\varepsilon{ijk}$: Random error at $i$th level of $A$, $j$th level of $B$ and $k$th level of $C$\
  • $\varepsilon_{ijk} \sim NID(0,\sigma_{\varepsilon}^2)$
• Response
  • $y_{ijk}$: Response of $i$th level of $A$, $j$th level of $B$ and $k$th level of $C$

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The design in which the levels of one factor can be applied to large experimental units and the levels of other factors to the sub-units are known as “split plot design“.

A split plot experiment is a blocked experiment in which blocks serve as experimental units. After blocking the levels of other factors are randomly applied to large units within blocks, often called whole plots or main plots.

The split plot design are specifically suited for two factors designs that have more treatment to be accommodated by a complete block designs. In split plot design all the factors are not of equal importance. For example, in an experiment of varieties and fertilizers, the variety is less important and the fertilizer is more important.

In these design, the experimental units are divided into two parts, (i) Main plot and (ii) sub-plot. The levels of one factor are assigned at random to large experimental units (main plot) and the levels of the other (second) factor are applied at random the the sub-units (sub-plot) within the large experimental units. The sub-units are obtained by dividing the large experimental units.

Note that the assignment of a particular factor to either the main plot or to the subplot is extremely important, it is because the plot size and precision of measurement of the effects are not the same for both factors.

The sub-plot treatments are the combination of the levels of different factors.

The split plot design involves assigning the levels of one factor to main plots which may be arranged in a “CRD”, “RCBD” or “LSD”. The levels of the other factor are assigned to subplots within each main plot.

Split Plot Design Layout Example

If there are 3 varieties and 3 fertilizers and we want more precision for fertilizers then with the RCBD with 3 replication, the varieties are assigned randomly to the main plots within 3 blocks using a separate randomization for each. Then the levels of the fertilizers are randomly assigned to the subplots within the main plots using a separate randomization in each main plot. The layout is

Another Split Plot Design Example

Suppose we want to study the effects of two irrigation methods (factor 1) and two different fertilizer types (factor 2) on four different fields (“whole plots”). While a field can easily be split into two for the two different fertilizers, the field cannot easily be split into two for irrigation: One irrigation system normally covers a whole field and the systems are expensive to replace.

Advantages and Disadvantages of Split Plot Design

Advantages of Split Plot Design

• More Practical
  Randomizing hard-to-change factors in groups, rather, than randomizing every run, is much less labor and time intensive.
• Pliable
  Factors that naturally have large experimental units can be easily combined with factors having smaller experimental units.
• More powerful
  Tests for the subplot effects from the easy-to-change factors generally have higher power due to partitioning the variance sources.
• Adaptable
  New treatments can be introduced to experiments that are already in progress.
• Cheaper to Run
  In case of a CRD, implementing a new irrigation method for each subplot would be extremely expensive.
• More Efficient
  Changing the hard-to-change factors causes more error (increased variance) than changing the easy-to-change factors a split-plot design is more precise (than a completely randomized run order) for the subplot factors, subplot by subplot interactions and subplot by whole-plot interactions.
• Efficient
  More efficient statistically, with increased precision. It permits efficient application of factors that would be difficult to apply to small plots.
• Reduced Cost
  They can reduce the cost and complexity of manipulating factors that are difficult or expensive to change.
• Precision
  The overall precision of split-plot design relative to the randomized complete block design may be increased by designing the main plot treatment in a Latin square design or in an incomplete Latin square design.

Disadvantages of Split Plot Design

• Less powerful
  Tests for the hard-to-change factors are less powerful, having a larger variance to test against and fewer changes to help overcome the larger error.
• Unfamiliar
  Analysis requires specialized methods to cope with partitioned variance sources.
• Different
  Hard-to-change (whole-plot) and easy-to-change (subplot) factor effects are tested against different estimated noise. This can result in large whole-plot effects not being statistically significant, whereas small subplot effects are significant even though they may not be practically important.
• Precision
  Differential in the estimation of interaction and the main effects.
• Statistical Analysis
  Complicated statistical analysis.
• Sources of Variation
  They involve different sources of variation ad error for each factor.
• Missing Data
  When missing data occurs, the analysis is more complex than for a randomized complete block design.
• Different treatment comparisons have different basic error variances which make the analysis more complex than with the randomized complete block design, especially if some unusual type of comparison is being made.

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The layout of a factorial design is typically organized in a table format. Each row of the table represents an experimental run, while each column represents a factor or the response variable. The levels of factors are indicated by symbols such as + and – for high and low levels, respectively. The response variable values corresponding to each experimental condition are recorded in the form of a sign table.

Consider a simple example layout for a two-factor factorial design with factors $A$ and $B$.

Run	Factor A	Factor B	Response
1	–	–	$Y_1$
2	+	–	$Y_2$
3	–	+	$Y_3$
4	+	+	$Y_4$

Layout of the Factorial Design: Two Factor in $n$ Replicates

Consider there are two factors and each factor has two levels in $n$ replicates. The layout of the factorial design will be as described below for $n$ replicates.

$y_{111}$ is the response from the first factor at the low level, the second factor at the low level, and the first replicate of the trial. Similarly, $y_{112}$ represents the second replicate of the same trial, and up to $n$th observation is $n$th trial at the same level of $A$ and $B$.

Geometrical Structure of Two-Factor Factorial Design

The geometrical structure of two factors (Factor $A$ and $B$), each factor has two levels, low ($-$) and high (+). Response 1 is at the low level of $A$ and a low level of $B$, similarly, response 2 is produced at a high level of $A$ and a low level of $B$. The third response is at a low level of $A$ and a high level of $B$, similarly, the 4th response is at a high level of $A$ and a high level of $B$.

Real Life Example

The concentration of reactant vs the amount of the catalyst produces some response, the experiment has three replicates.

Geometrical Structure of the Example

Factor Effects

\begin{align} A &=\frac{(a+ab)-((I) +b)}{2} = \frac{100+90-80-60}{2} = 25\\
B &= \frac{(b+ab) – ((I) +a) }{2} = \frac{60+90-80-100}{2} = -15\\
AB&=\frac{((I)+ab)-(a+b)}{2} = \frac{80+90-100-60}{2}=5
\end{align}

Minus 15 ($-15$) is the effect of $B$, which shows the change in factor level from low to high bringing on the average $-15$ decrease in the response.

Reference

Montgomery, D. C. (2017). Design and Analysis of Experiments. 9th ed, John Wiley & Sons.

R and Data Analysis

Test Preparation MCQs

The smallest case of a $2^K$ factorial experiment is one in which 2 factors are of interest and each factor has two levels. This design is known as a $2^2$ factorial design. We are interested in Factor effects or Effects of Factors.

The level of the factors (say $a$' and$b$’) may be called the low and high or presences and absences.

In a factorial design, in each complete trial (or replicate of the experiment), all possible combinations of the levels of the factors are investigated. For example, if Factor-A has $a$'' levels and Factor-B has$b$” levels, then each replicate contains all “$ab$” treatment combinations. Two factors each at 2 levels are:

Factor Effects (or Effect of Factors)

A change in the quantity of response due to the change in the level of a factor is called the effect of that factor. Here we mean average effect.

Main Effects

A main effect of a factor is defined as a measure of the average change in effect produced by changing the level of the factor. It is measured independently from the effect of other factors. The main effect is the effect of the factor only. Main effects are sometimes regarded as an interaction of zero order. Frequently, the main effect refers to the primary factors of interest in the experiment.

Interaction Effects

Factors are said to interact when they are not independent. Interaction in a factorial experiment is a measure of the extent to which the effect of changing the levels of one or more factors depends on the levels of the other factors. Interactions between two factors are referred to as first-order interactions, those concerning three factors, as second-order interactions, and so on.

Example: Consider a two-factor factorial experiment. Consider an investigation into the effect of the concentration of reactant (Factor $A$) and the presence of catalysts on the reaction time of the chemical process (Factor $B$).

Solution of Example

Main Effects

\begin{align}
\text{Main effect of A} & = \text{Average response at high level of $A$} – \text{Average response at low level of $A$}\\
&=\frac{45+60}{2}-\frac{20+35}{2}=25
\end{align}

The results indicate that Increasing Factor-A from the low level to the high level causes an average response increase of 25 units.

\begin{align}
\text{Main effect of B}&=\text{Average response at high level of $B$} -\text{ Average response at low level of $B$}\\
&=\frac{35+60}{2}-\frac{20+45}{2}=15
\end{align}

Increasing Factor B from the low level to the high level causes an average response increase of 15 units.

Effect of AB Interaction

It is possible that the difference in response between the levels of a factor is not the same at all levels of the other factor(s), then there is an interaction between the factors. Consider

\begin{align}
\text{The effect of Factor $A$ (at low level of Factor $B$)} &= 50 – 20 = 30\\
\text{The effect of Factor $A$ (at high level of Factor $B$)}&= 15 – 40 = -25\\
\text{The effect of Factor $B$ (at low level of Factor $A$)} &= 40 – 20 = 20\\
\text{The effect of Factor $B$ (at high level of Factor $B$)} &= 15 – 50 = -35
\end{align}

Because the effect of Factor-$A$ depends on the level chosen for Factor-$B$, we see that there is interaction between $A$ and $B$. One can computer Effect of $AB$ interaction as described below:

Effect of AB Interaction
= Average difference between effect of $A$ at high level of $B$ and the effect of $A$ at low level of $B$.

The magnitude of the interaction effect is the average difference in these two A effects, or $AB=\frac{-25-30}{2}=\frac{-55}{2}$.

OR

= Average difference between effect of $B$ at high level of $A$ and the effect of $B$ at low level of $A$.

The magnitude of the interaction effect is the average difference in these two B effects, or $AB = \frac{-35-20}{2} = \frac{-55}{2}$.

The interaction is large in this experiment.

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