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Basic Question - Central Composite Design

PunksNotDead

Hello, I wondered if I could ask a question about orthogonality of the squared term columns in a Central Composite Design (CCD) with two factors A and B. The dot product of any pair of columns in the model matrix is always zero - I always understood this to mean that the column (vectors) of say A with B or B with AB or B with B^2 are orthogonal (at right angles to) each other so their beta's can be calculated independently. The one exception is column A^2 with B^2 - their dot product can never be zero as you have -1 and/or +1 in 4 rows and they all become +1 when you square them which makes me think that their effects (the betas) would be aliased (so wouldn't be able to determine the coefficients for the A^2 and B^2 term independently from each other). I suspect I'm misunderstanding the meaning of orthogonality and that you can estimate these parameters independently of each other and of the other terms in the model otherwise what would be the point of the CCD design. Can anyone help with my lack of understanding please? Thankyou

Accepted answers

DonMcCormack

The first column/row is the intercept and is ignored when considering orthogonality. I am fairly certain that for common designs it will always be correlated with one or more effects in the model.

All comments

statman

Read this:

https://www.jmp.com/support/help/en/18.1/?os=mac&source=application#page/jmp/choose-a-design.shtml%23

PunksNotDead

Hi Statman, Thanks for the link. It was an interesting read but I couldn't find an answer to my question. I thought I'd draw a 2 factor CCD to explain. As you can see, any pair of columns are orthogonal to each other (dot product is zero, i always thought that was the criterion for orthogonality but maybe it's not - see below). The only exception is that for the A^2 and B^2 columns the dot product can never be zero and so I'm interpreting this (probably incorrectly) that those factors are not orthogonal. However, if I plot A^2 and B^2 as a pair of points, they all sit at the 4 corners of a square and if you try to fit a trend line you get an R value of zero. Maybe I've answered my question - Is it the case that it's the R=0 which means A^2 and B^2 are orthogonal to each other and so their betas can be independently estimated. So a non zero dot product of two column vectors doesn't mean non-orthogonality

DonMcCormack

For this to happen, the axial points have to be at ±Sqrt(2) in a 2 factor RSM. Ignore my earlier post, sorry for the confusion.

DonMcCormack

Reading my response I realize I didn't answer your question. Rather than muddling things even more by editing it, let me add another comment. Orthogonality is a function of the entire design and not just any subset of columns. If X is the design matrix then the design is orthogonal if all off-diagonal elements of Inverse(Transpose(X)*X) are zero except those associated with the intercepts. In this case * corresponds to matrix multiplication.