In our March 2, 2017 Tip of the Week, we discussed 2-Factor (aka 2-Way) ANOVA. We said that:
For example, in the left diagram, above, both detergents behave the same way to a change in water temperature -- they show no change in their Effect -- cleanliness. Likewise, the middle and right diagrams show both detergents behaving the same -- a parallel increase in Effect.
But what if we got something like the two graphs below? In the example at left, Detergent #1 shows a substantial increase in effectiveness as the water temperature is increased. But for Detergent #2, heating the water has the opposite effect: its effectiveness is decreased.
In the example on the right, both detergents show an increase in effectiveness as water temperature increases. But Detergent #2's increase is fairly minor. In fact, its increase may not be Statistically Significant and the Interaction may not be Statistically Significant..
In either case, we do have reason to suspect an Interaction. so 2-Way ANOVA Without Replication cannot be used. We must use 2-Way ANOVA With Replication.
The With Replication method repeats (Replicates) the experiment several times for each combination of Factor A and B Values. This can provide sufficient data to quantify an Interaction.
The number of Replications required to achieve a specified level of accuracy is determined by the methods of Design of Experiments, DOE. The Design also specifies the levels of each Factor to be used in each replication, the order of replication and other specifics of the experiment. The book has a 3-part series on DOE, and eventually there may be a video series on it as well.
There is currently a video on the book's YouTube channel with more information about the subject of this post: ANOVA -- Part 4 (of 4): 2-Way (aka 2-Factor).
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.