Try implicit regression - I developed this method for co-dependent relationships. It includes non-response analysis (1=ax+by+cz+interactions) and rotational analysis (x=f(x,y,z), y=g(x,y,z) and z=h(x,y,z)) - as the assumption of independence is not required, to measure the dependent/independent relationship, you would need to measure the degree of separation and to measure model quality, you would track the variable(s) of interest and measure the correlation between the variable estimate and the observed data.

https://arxiv.org/ftp/arxiv/papers/1602/1602.00158.pdf

https://arxiv.org/ftp/arxiv/papers/1512/1512.05307.pdf

Dear Daniel,

Rd Wooten's approach is appropriate. Your own approach is also appropriate. You can use the dominant PCA score in your three sets of PCA scores as the dependent variable, and then the other two as independen tvariables. The dominant PCA is the one with the largest eigenvalue or variance.

Unfortulately, your data does not seem to be designed orthogonally correct from the beginning - since they are correlated. A proper factorial design would have solved your question immediately.

Hi Mathias, thanks for your suggestion. But PCA will usually do varimax rotation and make the suppled data orthogonal. That is its advantage.

Harold,

That is true PCA will make your data orthogonal, but you will not get any regression coefficients and PCA is ill-suited for such a small dataset. But as the experiments have already been performed, it is probably the only way to get a picture of the relative importance of the factors.

PCA will do varimax rotation and actually confirm the actual number of orthogonal components. They may end up in just one or two orthogonal components if indeed the original three sets of variables were not orthogonal.

If indeed the components are orthogonal, their resulting scores are considered as new variables. Therefore, any regression analysis carried out with them will give the coefficients of the component scores.

Agree with suggestions given by Harold Chike.