Statistics Tip of the Week: Use the "With Replication" method of 2-Factor ANOVA if there is an Interaction between the 2 Factors.

In 2-Factor (aka "2-Way") ANOVA, we study the effect of 2 Nominal (named) Variables A and B, on the Dependent Numerical Variable y. (For more on Nominal (also known as "Categorical" Variables) see my blog post from November 23, 2016 .)

y = f(A,B) or y = f(A,B, AB)

A and B are the Factors influencing the value of y. "AB" -- which may or may not exist -- is the Interaction between A and B. It can be a 3rd Factor if it is Statistically Significant. 

In the example below,

• we start with y = f(A,B). y is a numerical measure of cleanliness.
• Factor A is the type of detergent. There are two values of A tested: "Detergent 1" and "Detergent 2".  Note that these values are names, since Factor A is a Nominal Variable.
• Factor B is the water temperature. There are three values of B tested: "Cold", "Warm" and "Hot". As you can see, Factor B is also a Nominal Variable.

We must use the WITH REPLICATION method of 2-Way ANOVA if there is a Statistically Significant Interaction.
 
But how do we know if there is? As my blog post of June 8, 2017  says, you can use a line chart to identify Interactions.  If the lines cross, that indicates an Interaction, and the With Replication method must be used. That method can determine if the Interaction is Statistically Significant.
 
In the left diagram below, neither type of detergent is affected by water temperature, so there is no Interaction between Factor A, detergent type, and Factor B, water temperature.  In the middle diagram, both detergents of Factor A are affected the same. Cleanliness is improved similarly with hotter water, no matter what detergent is used. So, again there is no Interaction. 

In the right diagram, they behave differently. Hotter water makes Detergent #1 more effective, but makes Detergent #2 less effective. The lines definitely cross, indicating a Statistically Significant Interaction. We'll need to perform a 2-Way ANOVA With Replication to make sure.
 
With the With Replication method, we replicate (repeat) the taking of measurements several times with different levels of each Factor (detergent type and water temperature). We need to use the discipline of Design of Experiments (DOE) to specify the number of Replications required and what values of each Factor comprise each Replication. The book has a 3-part series of articles on DOE, and I plan to eventually record videos on the subject.
 
For more on 2-Way ANOVA, both WIth and Without Replication, see my video:
ANOVA Part 4 (of 4) -- 2-Way (aka 2-Factor).