Lesson 4: Multi-Factor ANOVA

Overview Section

Introduction to Multi-factor ANOVA: The Treatment Design

Researchers often identify more than one experimental factor of interest. So what to do? One alternative is to set up separate, independent experiments in which a single treatment is used in each experiment, and each experiment might be analyzed as we have done so far using a one-way ANOVA.

This approach might have the advantage of concentrated focus on the single treatment of interest and simplicity of computations for the ANOVA. However, there are several disadvantages that outweigh the advantages to this approach.

  • First, environmental changes or changes in the experimental material may be changing during the process. This could distort the assessment of the relative importance of different treatments on the response variable.
  • Secondly, it is inefficient. Setting up and running multiple separate experiments usually will involve more work and resources.
  • Last, and probably most importantly, is that in the one-at-time approach we don’t have any way to evaluate how responses to one treatment behave with respect to the levels of other treatments.

As a result, multi-factor experiments are extremely common in practice. To accommodate this situation we will be modifying our simple ANOVA in some interesting ways, and it is essential that we structure the analysis so it is technically correct. Here is Dr. Rosenberger and Dr. Shumway talking about some of the things to look out for as you work your way through this lesson.

To put it into perspective, let’s take a look at the phrase ‘Experimental Design’, a term that you often hear. We are going to take this colloquial phrase and divide it into two formal components:

  1. The Treatment Design
  2. The Randomization Design

We will use the treatment design to address the nature of the experimental factors under study, and Randomization to address how treatments are assigned to experimental units. An experimental unit is defined to be that which receives a treatment level. Simply, this would be a single plant in the greenhouse receiving fertilizer. So the experimental unit here is the plant. (we will discuss later in the course how this is not always quite so clear!)

An ANOVA model often contains elements from both the treatment design and the randomization process, and working with these separately will help us devise the correct ANOVA model for a given experimental situation.

The following figure illustrates this conceptual division:

Experimental Design

Treatment Design


How many treatments are there?


How many levels of each treatment are there?


If there is more than 1 treatment, how are they related?

Crossed: each level of treatments occur with all levels of other treatments. (Factorial)

Nested: levels of a treatment are unique to different levels of another treatment.


Are treatments fixed or random effects?


Are there continuous covariates? (ANOCOVA)

Randomization Design

In Multi-factor experiments, combinations of treatments are applied to experimental units.


Upon completion of this lesson, you should be able to:

  • Familiarity with Factorial, Nested, and Cross-nested Treatment Designs
  • Proficiency in the use of SAS and Minitab to:
    • Run the different treatment designs
    • Conduct complete (with lettering) mean comparisons.