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I ran a PCA using 9 variables and want to run all possible additive linear models using the first three principal components. However when I run the 8 possible linear models the intercept and coefficient estimates are the same in each model. I expected the coefficient estimates for each dependent variable to change as dependent variables were added or removed from the model and so I'm confused as to why they stayed the same. If you've dealt with this issue or have any suggestions, I would really appreciate it!

The R code I used is below for some reference.

MOOPCA<-prcomp ( MOOSE [, -1], cor=T, scale=T )
MOOPCA
summary(MOOPCA)
PCApredict<-predict(MOOPCA)
PC1<-PCApredict[,1]
PC2<-PCApredict[,2]
PC3<-PCApredict[,3]

Full<-lm(Density~PC1+PC2+PC3)
summary(Full)

MOO1<-lm(Density~PC1)
summary(MOO1)

MOO2<-lm(Density~PC1+PC2)
summary(MOO2)
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How to "fix" it? Don't do PCA.

PCA makes your observations orthogonal (i.e., not correlated) to each other. If each coefficient is completely independent of the others, it's absence will not affect the value of the other coefficients.

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    $\begingroup$ You could try using an oblique rotation within PCA. This relaxes the orthogonality constraint and allows for correlation between the PCs. R has decent packages that allow you to run oblimin rotation. $\endgroup$ – ERT Jul 6 '18 at 22:05

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