In partial least squares regression, what is the difference between the regression coefficients and the loadings for each independent variable in each component? Specifically, I understand in evety component, each of the independent variables has a coresponding loading. Does each variable also have a regression coefficient? What is the relationship between the loading vector and the coefficients?

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#### Best Answer

Assuming your independent variable matrix is $mtimes n$, that you have $m$ observations and $n$ variables.

For each PLS component (AKA latent variable), you get a loading vector ($n times 1$), so for $h$ components the size of loading matrix ($P$) is $n times h$. These loadings are calculated for both interpretation and algorithmic purposes but **they have no use for prediction**.

On the other hand, SIMPLS algorithm (I believe the most popular PLS flavor) also involves calculation of weight matrix ($W$), which has the same size as loading matrix. This orthogonal matrix $W$ is used to calculate $X$ scores ($T$):

$T = Xcdot W$

which is then multiplied by $Y$ loadings ($Q$) for prediction:

$hat{Y} = T cdot Q'$

Therefore, the regression coefficients ($hat{B}$ that is $ntimes1$ for a single dependent variable) that can be used to predict $Y$ directly from $X$ can be calculated:

$hat{B} = W cdot Q'$

All in all, one obtains a loading vector for each component whereas for different number components a same sized yet different regression coefficients are produced.

As far as I know, a similar logic applies to other PLS algorithms too.

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