Assumptions for PCR and PLS

Question

I am writing up a report on fitting Principal Component Regression (PCR) and Partial Least Squares (PLS) to my data-set.

A similar question: Model assumptions of partial least squares (PLS) regression only covers PLS.

According to Springer texts in Statistics "An Introduction to Statistical Learning with Applications in R" Section 6.3.1:

The principal components regression (PCR) approach involves constructing the firstM principal components, Z1, . . ., ZM, and then using these components as the predictors in a linear regression model that is fit using least squares.

Section 6.3.2:

Like PCR, PLS is a dimension reduction method, which first identifies squares a new set of features Z1, . . ., ZM that are linear combinations of the original features, and then fits a linear model via least squares using these M new features.

For both of the methods it says that the new features are used in the linear regression model.

Question

Does this mean that the new regression model has to satisfy the assumptions of a linear model, mainly:

1. Independent observations
2. Normality of errors
3. Errors having a mean of zero
4. Errors having constant variance

Are there any other assumptions I would have to take into account when fitting these models?

Answer 1

First, using PCR or PLS does not remove any of the usual assumptions of regression that you list. There are lots of posts on when and why these assumptions are needed (not all are needed for all uses of regression). I'm not going to repeat those here.

PCR and PLS also require that the underlying analysis be met. Both of those are based on covariance matrices, so those matrices have to make sense. Laird Statistics has a detailed list, but, briefly (and this is my paraphrase of Laird):

1. The variables should be measured at the interval level, although ordinal is possible with modifications.

2. The relationships between the pairs of variables should be linear.

3. Sample size large enough to assure a reliable result.

4. The relations among the variables should be strong enough that variable reduction makes sense.

5. No outliers.

For PLS, the matrix include the dependent variable. For PCA it only includes the independent variables.