In the previous chapter we discussed how one could analyze for differences in treatments, each replicated on several plots, using an analysis of variance. Often there is more than one type of treatment we wish to use in an experiment. For example, we may want to know the effect of different plant populations on yield, but also how different levels of nitrogen application affect the yield. This chapter will build on the principle you learned in the last unit and add another factor to have two factors in the ANOVA.

Factorial Experiments

Factorial experiments involve more than one treatment factor. In the corn population experiment analyzed in chapter 7 on The Analysis of Variance (ANOVA), we were interested only in the response of one hybrid. However, corn hybrids respond differently to changes in plant population. What if we wanted to compare the response of three different hybrids to population changes? What are our options in designing this experiment?

Option 1

We could consider each hybrid in a separate experiment. For each hybrid, then, we would plant our plots using each of the three populations. We would need the following experimental units:

• hybrid A x 3 populations x 3 replications = 9 plots
• hybrid B x 3 populations x 3 replications = 9 plots
• hybrid C x 3 populations x 3 replications = 9 plots

The three experiments would require 27 (9+9+9) total experimental units, in this case plots. The results from this experiment would tell us the response of each hybrid to population changes. Each experiment would have an ANOVA table (Table 1) similar to that used previously.

Combining Factors

Yet, since each hybrid test is conducted as a separate experiment, we could not compare the yields of each hybrid. In other words, we would know which population is most appropriate for each hybrid. But, we could not predict which hybrid would produce the greatest yield with any confidence. The experiments were simply not designed in a manner to allow this.

Option 2

We could develop an experimental arrangement where these two factors are combined into a factorial experiment. In this case, all combinations of population and hybrid would occur within the same experiment. The total number of treatments would be 9 (3 hybrids x 3 populations). In this case, with three replications, we would have: 3 hybrids x 3 populations x 3 replications = 27 plots.

However, this factorial experiment will produce more useful information than the first design. Specifically, this experiment will tell us whether the effect produced by changing plant population is the same regardless of hybrid, or if each hybrid reacts differently to population. If each hybrid reacts differently to population, we will say that there is an interaction between the hybrids and plant populations. To analyze the results from this experiment, the ANOVA must be expanded to include the additional effects: another main factor and its interaction with the original main factor (Table 2).

Degrees of Freedom

The table is similar to the one used before, with the exception of these two new sources of variation. The degrees of freedom are calculated using the formulae in Table 3. Note that even though we have the same number of experimental units whether we evaluate the three hybrids separately (Table 3) or together (Table 3), our residual degrees of freedom and the sensitivity of our F-test increase dramatically with factorial design.

The ANOVA for factorial experiments can be completed using a “cookbook” method similar to that described in the previous chapter. However, since most experiments of this type are analyzed using computer software, we will not continue the example here. Rather, we will use the computer to analyze a similar experiment using R to calculate the statistics we need.

Interaction

Interaction is a differential response of one factor at different levels of another. In addition to maximizing the efficiency of an experiment by combining the treatments into one set of plots, a two-factor ANOVA introduces another effect; the interaction between the treatments. As introduced in the ANOVA Table 1, there is an additional source of variation in the analysis: the interaction between our two main treatment factors.

As is often realized, additional effects occur because of interaction between two treatments. Two factors cannot interact at all, interact positively, or interact negatively. These can be depicted well by graphs. Let’s consider two different varieties at three different levels of N.

Example 1: No Interaction

Here yield of the two varieties reacts similarly to N-rate. Thus, there is no interaction between N and variety (Fig. 1). Notice that the lines are parallel, so the effect of the variety is constant as N rate increases. Variety 1 is uniformly superior at each N level and higher levels of N result in improved yield of each variety.

Example 2: Positive Interaction

Yield of the two varieties differs based on level of N applied, but the variety 1 always yields better than variety 2 (Fig. 2). The response of each variety depends on the level of N applied. It is no longer constant as in the previous example. The type of interaction is sometimes referred to as a change in the magnitude of the response interaction. Variety 1 is still the highest yielding variety, but the magnitude of its response to N depends on the amount of N applied. With this type of interaction, you can’t evaluate the effect of N independent of variety because the response is different depending on which variety you are talking about. Whenever an interaction tests as significant in the ANOVA, it is important to ignore the main factor means and focus on the interaction means.

Example 3: Negative Interaction

The last type of interaction occurs when the yield response is completely opposite based on the level of N (Fig. 3). Note that the lines in the figure now cross each other. This changes the interpretation drastically because the best variety to grow now depends on the level of N that is applied. You would need to fertilize each variety differently based on the response. This is often called a crossover interaction and it is very important to pay attention when they occur because your recommendation about one factor depends heavily on the level of the other. Often when a crossover interaction occurs one or both main factors involved in the interaction tests as nonsignificant in the ANOVA. When this occurs, you should look at the results very carefully because failure to assess the interaction carefully can result in poor recommendations.

Linear Additive Model for Two-Factor ANOVA

The linear model for a factorial is a simple extension of the linear model. The addition of factors and their possible interaction produce further complexity in the ANOVA. But the effects can be added just as the previous effects were to the linear model.

The linear model for a two-factor ANOVA is:

[latex]Y_{ijk}=\mu+A_i+\beta_j+A\beta_{ij}+\epsilon_{(ij)k}[/latex]

[latex]\text{Equation 1}[/latex]

where:
[latex]Y_{ijk}[/latex] = response observed for the [latex]ijk^{th}[/latex] experimental unit,
[latex]\mu[/latex] = overall mean,
[latex]A_i[/latex] = effect of the [latex]i^{th}[/latex] level of factor A,
[latex]\beta_j[/latex] = effect of the [latex]j^{th}[/latex] level of factor B,
[latex]A\beta_{ij}[/latex] = effect of the interaction between the [latex]i^{th}[/latex] level of factor A and the [latex]j^{th}[/latex] level of factor B,
[latex]\epsilon_{(ij)k}[/latex] = effect associated with the [latex]ijk^{th}[/latex] experimental unit; commonly referred to as error.

True Sources of Variation

Notice that the only change in the linear model from equation 8 is that the treatment structure is modified. We now have sources for factor A, factor B and their interaction. The error structure is not changed; we still have only a single random error for the experiment. Factorial refers to the treatment structure, not the assignment of treatments to experimental units.

For the corn population x hybrid experiment we can rewrite the model to indicate the true sources of variation as:

[latex]\text{Yield}_{ijk}=\mu+POP_i+HYBRID_j+PH_{ij}+PLOT_{(ij)k}[/latex]

[latex]\text{Equation 2}[/latex]

where:
[latex]\text{Yield}_{ijk}[/latex] = corn yield observed for the [latex]ijk^{th}[/latex] plot,
[latex]\mu[/latex] = average yield for the experiment,
[latex]POP_i[/latex] = effect of the [latex]i^{th}[/latex] plant population,
[latex]HYBRID_j[/latex] = effect of the [latex]j^{th}[/latex] hybrid,
[latex]PH_{ij}[/latex] = effect of the interaction between the [latex]i^{th}[/latex] population and the [latex]j^{th}[/latex] hybrid,
[latex]PLOT_{(ij)k}[/latex] = random error effect of the [latex]ijk^{th}[/latex] plot.

Try: Running an ANOVA for a Two-factor CRD in the next screens

Ex. 1: Running an ANOVA for a Two-Factor CRD

R Code Functions

• setwd()
• aov()
• summary()
• <-
• attach()
• subset()
• read.csv()
• detach()
• pf()
• head()
• as.factor()
• interaction.plot()
• as.data.frame()
• aov()

The Scenario

You are an employee for a maize development company in charge of developing new high-yielding maize hybrids for use in central Iowa. For the past 2 years you have been developing A-lines and now you want to test their general combining ability with the company’s elite R-line. Unfortunately, (because it’s Iowa), your plots are savaged by a tornado and bowling ball-sized hail and only 2 of your candidate inbreds remain. You cross these ridiculously fortunate inbreds to the R-line to produce seed for 2 hybrids which are planted the following spring in 3 locations across central Iowa utilizing a randomized complete design with 3 reps at each location. You also plant a standard maize hybrid along with the two you developed to serve as a check within your field (the check is Hybrid C). With the help of your intern, you harvest each plot and calculate the projected bushels/acre.

Source data: The yield data from the hybrids can be found in the file ANOVA 2factorCRD [XLSX]

Ex. 1: Data Set

In this data set, you can see that we have columns for treatment, location, hybrid, replication, and the yield in t/ha (Fig. 4). It’s good to remember that while you have 3 hybrids and 3 locations, your total number of treatments is 9, not just the 3 hybrids or the 3 locations. In this activity you will learn how to run a 2-factor ANOVA that will help you choose which of the 3 hybrids you will select and move on to the next stage of testing in your program.

Activity Objectives

• Build and run an ANOVA for a 2 factor model that accounts for population, hybrid, and the interaction between population and hybrid
• Assess whether each individual hybrid is significantly affected by location

Ex. 1: Run the ANOVA

Ex. 1: Make Adjustments

Ex. 1: Two-Way ANOVA

Run the ANOVA using the aov() function, just like with one-way designs. The only difference is that now we want to include both factors and their interaction in our model. There are two equivalent ways to do this. The first way explicitly specifies each term; the second way is a shortcut.

str(Ex9.1)

‘data.frame’: 27 obs. of  5 variables:
$ Treatment: int  1 1 1 2 2 2 3 3 3 4 …
$ Location : int  1 1 1 1 1 1 1 1 1 2 …
$ Hybrid   : Factor w/ 3 levels “A”,”B”,”C”: 1 1 1 2 2 2 3 3 3 1 …
$ Rep      : int  1 2 3 1 2 3 1 2 3 1 …
$ Yield    : num  7.45 9.84 11.18 9.13 9.27 …

Or:

Ex9.1.outB = aov(Yield ~ Location*Hybrid, data = Ex9.1)

summary(Ex9.1.outB)

                Df Sum Sq Mean Sq F value Pr(>F)
Location         2   9.45   4.726   2.109 0.1504
Hybrid           2  13.09   6.544   2.920 0.0797 .
Location:Hybrid  4  30.25   7.562   3.374 0.0316 *
Residuals       18  40.34   2.241
—
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1

Ex. 1: Run Individual ANOVAs

Ex. 1: Simple Main Effects

Ex. 1: Plot the Interaction

Ex. 1: Interaction Plot

Looking at this interaction plot, you can see that the rank of each line changes depending on the location (Fig. 5). This indicates that there is an interaction between location and environment, but is this interaction significant? Go back to the first ANOVA that you ran and check to see if the interaction is significant, and in this case it is. Based on this information you have gathered from the ANOVAs and your interaction plot, which hybrid would you advance to further testing? Would you choose any hybrid at all? What else do you need to know in order to make this decision?

Ex. 1: ANOVA for 10 Hybrids

Ex. 1: Interaction Plot of 10 Hybrids

Looking at the plot, you can see that many of the hybrids change rank in different locations (Fig. 6), but if you look at the ANOVA you can see that despite this the interaction is not significant. In real life, you will likely have even larger data sets with many individuals in which case the interaction plot will look very messy and even more difficult to read. This is why it is important to always check the ANOVA and not just rely on the plot.

Ex. 1: Review Questions

1. What information can you get from an ANOVA?
2. How can you use the ANOVA to make selection decisions?
3. What is an interaction?

ANOVA and Experimental Design

Experimental Design and Analysis

You will recall from Chapter 1 on Basic Principles that experimental design refers to the manner in which treatments are assigned to experimental units (plots). In this chapter, we have assumed that all treatments are applied at random to the entire set of experimental units used in the experiment. This is what is known as a Completely Random Design (CRD). In a CRD every experimental unit has the same chance of receiving any given treatment.

We use a CRD when we expect the magnitude of the plot effect to be similar among all the plots used in the experiment. Statisticians refer to this condition as homogeneity, and it is assumed when we use a CRD. The CRD is a common design (Fig. 7); there are many cases where its use is appropriate. For example, in a growth chamber, we might have 25 flats filled with sand, each planted with 30 wheat seeds, 5 reps of each of 5 varieties.

However, whenever there are not enough plots with similar characteristics to accommodate all the treatments and replications in an experiment, alternative designs should be considered to improve the precision of the experiment. We will learn more about this as we study other designs in subsequent chapters.

Randomly Assign Treatments

There are many ways to randomly assign treatments to experimental units. The use of a table of random numbers is described in the text. This method involves listing all the treatments and their replications and assigning a random number from a table of random numbers to each one. This random order is then matched to a predetermined order of experimental units. A far easier approach is to use a computer to randomize treatments. You can use Excel or R to randomize an experiment.

Ex. 2: Randomized Complete Design using R

R Code Functions

• setwd()
• paste()
• sample()
• <-
• data.frame()
• write.csv()
• read.csv()
• write.table()
• as.data.frame()
• head()
• matrix()

The Scenario

You are an employee for WinField Solutions interested in studying the impact of Japanese beetles (Popillia japonica) on soybeans and recently two new insecticides have come on the market. To test their effectiveness against Japanese beetles, you choose to test the insecticides on the three most commonly grown cultivars in Iowa. You want to design an experiment where each insecticide is paired with each cultivar and replicated four times. To account for differences within your field you want to test these pairs in a completely randomized design. Ultimately you wish to be able to show clients which insecticide/cultivar pairs are the most effective at preventing damage from Japanese beetles.

Ex. 2: Activity Objectives

Ex. 2: Randomize as Pairs

Ex. 2: Randomize the Order

Ex. 2: Matrix Form

Ex. 2: Visualizing Results in Excel

This way you can save your results and you can do further visualization in programs like Excel.

For example (Fig. 8):

You can start with this and add ranges and row labels to get a clearer idea of what the field layout will be like Fig. 9.

Ex. 2: R Code Glossary

Error Structure

Once the randomization is completed and the experiment properly conducted according to plan, the error structure for the experiment is, at least partially, determined. For the CRD, for example, we do not have a blocking factor because treatments are assigned to experimental units completely at random. In chapter 11 on Randomized Complete Block Design, we will introduce another type of design, the Randomized (Complete) Block Design, or RCBD, in which some restrictions are put on the randomization. For the RCBD, there is yet another recognizable source of variation in the ANOVA, that for blocks, as you will see in chapter 11.

Summary

Factorial Experiments

• Involve more than one treatment factor.
• Allow exploration of combinations of factors, interaction
• Refers to the treatment structure, not how treatments are assigned to plots.

Interaction

• Differential response of one factor at different levels of another.

Linear Model

• Factorial Linear Model is simple extension for more detail on treatment factors and interaction.

Experimental Design

• Determines model and ANOVA.
• CRD has treatments assigned to experimental units completely at random.