Lesson 7: Confounding and Blocking in 2^k Factorial Designs

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Introduction

In Lesson 4 we discussed blocking as a method for removing extraneous sources of variation.  In this lesson we consider blocking in the context of 2k designs. We will then make a connection to confounding, and show a surprising application of confounding where it is beneficial rather than a liability.

Learning Objectives & Outcomes

By the end of this lesson, readers are supposed to understand:

• Concept of Confounding
• Blocking of replicated 2k factorial designs
• Confounding high order interaction effects of the 2k factorial design in 2p blocks
• How to choose the effects to be confounded with blocks
• That a 2k design with a confounded main effect is actually a Split Plot design
• The concept of Partial Confounding and its importance for retrieving information on every interaction effect

Blocking in Replicated Designs

In 2k replicated designs where we have n replications per cell and perform a completely randomized design we randomly assign all 2k times n experimental units to the 2k treatment combinations. Alternatively, when we have n replicates we can use these n replicates as blocks, and assign the 2k treatments to the experimental units within each of the n blocks. If we are going to replicate the experiment anyway, at almost no additional cost, you can block the experiment, doing one replicate first, then the second replicate, etc. rather than completely randomize the n times 2k treatment combinations to all the runs.

There is almost always an advantage to blocking when we replicate the treatments. This is true even if we only block using time due to the order of the replicates. However, there are often many other factors that we have available as potential sources of variation that we can include as a block factor, such as batches of material, technician, day of the week, or time of day, or other environmental factors. Thus if we can afford to replicate the design then it is almost always useful to block.

To give a simple example, if we have four factors, the 2k design has 16 treatment combinations, so say we plan to do just two replicates of the design. Without blocking, the ANOVA has 24 = 16 treatments, but with n = 2 replicates, the MSE would have 16 degrees of freedom. If we included a block factor, with two levels, the ANOVA would use one of these 16 degrees of freedom for the block, leaving 15 degrees of freedom for MSE.  Hence the statistical cost of blocking is really the loss of one degree of freedom for error, and the potential gain if the block explains significant variation would be to reduce the size of the MSE and thereby increase the power of the tests.

The more interesting case that we will consider next is when we have an unreplicated design.  If we are only planning to do one replicate, can we still benefit from the advantage ascribed to blocking our experiment?