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Kindly advise if the value of beta is obtained in the results when a Regression analysis, Pearson Correlation or Partial Correlation is conducted.

I understand that beta reveals the strength of a latent variable, that may have a direct or indirect effect on the dependent variable. I am not familiar conducting PLS, but would like to know how to interpret the value of the beta obtained, along with the correlation coefficient.

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  • $\begingroup$ By beta do you mean the regression coefficients that is used for prediction? $\endgroup$ – theGD May 15 '18 at 19:54
  • $\begingroup$ I am aware of the correlation or regression coefficient R which is between -1 and +1. I believe beta is something different in PLS $\endgroup$ – Vyas May 15 '18 at 22:50
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In PLSR, the "beta" values are the regression coefficients that can be multiplied by your X data to give you your predicted Y data. You can plot the beta values associated with each x variable to give an indication of how strongly each x variable contributes to each latent variable.

For example, say you are modelling tree age (y) based on height(x1) and trunk width(x2), and density(x3) using two latent variables. If you have centered your data before PLSR, then you would get beta values along the lines of B=[b11,b12;b21,b22]. Your predicted y vector would be y=B*X. If you plot the two points (b12,b22) and (b21,b22), then the distance of each of these two points along each axis will show how strongly each variable contributes to the component associated with that axis (e.g. component 1 = x-axis; component 2 = y-axis).

The correlation coefficient (r) would be something you use afterwards, to compare how well your model explains the actual variation in the tree age (correlation between actual age and predicted age). Or you could look at the relationships between the individual X variables. The partial correlation coefficient between two variables measures what the correlation would be if all other influencing variables were held constant at their means. For example, you could find the partial correlation coefficient between x1 and x2 while controlling for the effect of x3.

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