(2^k)r Factorial Designs with Replication

Shortcoming of 2^k design:

Cannot estimate experimental error because no experiment is repeated.

Solution: Replication

Let's repeat each experiment (i.e., same factor levels) in a 2^k design r times, yielding (2^k)r observations.

This is called the (2^k)r design.

We will discuss k=2.

We will not discuss Sections 18.7 to 18.9: visual tests, multiplicative models, and k>2. However, you should read these sections on your own.

(2^k)r Factorial Design for k=2

Model

Thus we have added an error term, denoted e, representing experimental error.

Why Use Statistical Analysis

Holding factor levels constant and repeating experiment r times yields samples of response y_i.

Statistical analysis of y_i's yields: