vectors and orthogonality

[#1]

I have a basic question.

If I have a 2^2 factorial design like this (4 runs)

X1   X2

-1    -1

-1     1

1     -1

1      1

the column vectors are orthogonal, but just what are they vectors of? What are the basis vectors? I'm trying to draw this on a graph (OK, really hard with four dimensions, but even trying two dimensions for a one-factor experiment). Or is there a much more general concept of "vector" going on here?

[#2]

Hi @del ,

I'd say 'general concept' might serve sufficiently here for most folks.  You're talking experiments, so the X1 & X2 are better represented by what you mean to control in your experiment, e.g. pH and temperature, your experimental factors (each a 'column vector' for the mathspeak).  

Ideally, if you detect an effect of one of the factors via stats techniques like regression, you'd like to know the effect is solely from that factor (e.g. pH), and not due to the correlation of the factor with another (e.g. temp) induced by the design.  Hence the desire for the design to provide orthogonality of the factors (or 'column vectors'); no correlation, effect source is clear.  Not sure where you saw 'vector', but likely framing the [n,1] representation of the data for their dot product to assess orthogonality.  

Hopefully that helps.  I expect you'll find more along these lines in either (both?) the free, self-paced 1) ANOVA and Regression and 2) Custom Design of Experiments courses offered by our Education folks here: https://www.jmp.com/en_us/training/overview.html

As for drawing it, JMP's got the 3D Scatterplot under Graph, in case that helps you out.

[#3]

Thanks for the response. I think maybe my question wasn't clear. I've attached a sketch with which I try to illustrate. The test points for the four test runs can be plotted on axes of X1 and X2. But what's said to be orthogonal in the orthogonal arrays is not the square shape of the test points but the angle between the two column vectors. If I had a four-dimensional piece of paper to draw some vectors on what would define the axes? It isn't the factors since all the -1 and 1 would cancel and leave a vector of zero length.