What is an orthogonal design?

Question

I am a little bit confused about what an orthogonal design is and how it relates to the model matrix. It appears that there are many perspectives into the definition of orthogonal inside this post but none exactly helps me understand my problem. I am seeking an explanation with an experimental design context.

In Bailey 2008 pg 179, It introduces two factors G and F as orthogonal iff the subspaces $V_G \cap(V_F \cap V_G)^\bot$ and $V_F \cap(V_F \cap V_G)^\bot$ are both orthogonal to each other (or $V_G \cap V_{G \wedge F}^\bot$ and $V_F \cap V_{G \wedge F}^\bot$ is orthogonal)

A more intuitive theorem in the same book says (in terms of factors and levels) that

F and G on the same set are orthogonal to each other iff

• every F-class meets every G-class
• all these interesections have size proportional to the product of the sizes of the relevant F-class and G-class

However, the problem with these definitions is that it does help with 3 factors being orthogonal to each other or or when continuous covariates are included in your model. I was guessing that perhaps the model matrix when encoded can reveal something about orthogonality. However, my guess is that this is not true because It is obvious that when dummy coding the columns of the model matrix is not orthogonal to each other.

So my question is,

1. What is a general definition of an orthogonal design? Is there a more general definition of orthogonal design? Including continuous covariates
2. What are the advantages of orthogonal designs?
3. Does the model matrix reveal anything about orthogonality?

Answer 1

1. Perhaps you haven't fully grasped the definition yet. The requirements for orthogonal designs are that the blocking is orthogonal and the treatment is orthogonal. This simply means that crossproducts total to zero, whether the blocking or treatment is continuous, pseudo-continuous, polytomous, or binary. As @whuber correctly points out, statisticians often call dot products cross products, and furthermore often assume blocking and treatment factors have mean 0. So any blocking factor or treatment factor "crossed" with any other will come out to 0.

2. Efficiency.

3. Absolutely. We would expect that cross products between any two columns of the design matrix will total out to zero.