Does the sign of scores or of loadings in PCA or FA have a meaning? May I reverse the sign?

Question

I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the PCA scores differed in sign. How can it be?

Consider this:

set.seed(999)
prcomp(data.frame(1:10,rnorm(10)))$x

            PC1        PC2
 [1,] -4.508620 -0.2567655
 [2,] -3.373772 -1.1369417
 [3,] -2.679669  1.0903445
 [4,] -1.615837  0.7108631
 [5,] -0.548879  0.3093389
 [6,]  0.481756  0.1639112
 [7,]  1.656178 -0.9952875
 [8,]  2.560345 -0.2490548
 [9,]  3.508442  0.1874520
[10,]  4.520055  0.1761397

set.seed(999)
princomp(data.frame(1:10,rnorm(10)))$scores
         Comp.1     Comp.2
 [1,]  4.508620  0.2567655
 [2,]  3.373772  1.1369417
 [3,]  2.679669 -1.0903445
 [4,]  1.615837 -0.7108631
 [5,]  0.548879 -0.3093389
 [6,] -0.481756 -0.1639112
 [7,] -1.656178  0.9952875
 [8,] -2.560345  0.2490548
 [9,] -3.508442 -0.1874520
[10,] -4.520055 -0.1761397

Why do the signs (+/-) differ for the two analyses? If I was then using principal components PC1 and PC2 as predictors in a regression, i.e. lm(y ~ PC1 + PC2), this would completely change my understanding of the effect of the two variables on y depending on which method I used! How could I then say that PC1 has e.g. a positive effect on y and PC2 has e.g. a negative effect on y?

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In addition: If the sign of PCA components is meaningless, is this true for factor analysis (FA) as well? Is it acceptable to flip (reverse) the sign of individual PCA/FA component scores (or of loadings, as a column of loading matrix)?

Answer 1

PCA is a simple mathematical transformation. If you change the signs of the component(s), you do not change the variance that is contained in the first component. Moreover, when you change the signs, the weights (prcomp( ... )$rotation) also change the sign, so the interpretation stays exactly the same:

set.seed( 999 )
a <- data.frame(1:10,rnorm(10))
pca1 <- prcomp( a )
pca2 <- princomp( a )
pca1$rotation

shows

                 PC1       PC2
X1.10      0.9900908 0.1404287
rnorm.10. -0.1404287 0.9900908

and pca2$loadings show

Loadings:
          Comp.1 Comp.2
X1.10     -0.99  -0.14 
rnorm.10.  0.14  -0.99 

               Comp.1 Comp.2
SS loadings       1.0    1.0
Proportion Var    0.5    0.5
Cumulative Var    0.5    1.0

So, why does the interpretation stays the same?

You do the PCA regression of y on component 1. In the first version (prcomp), say the coefficient is positive: the larger the component 1, the larger the y. What does it mean when it comes to the original variables? Since the weight of the variable 1 (1:10 in a) is positive, that shows that the larger the variable 1, the larger the y.

Now use the second version (princomp). Since the component has the sign changed, the larger the y, the smaller the component 1 -- the coefficient of y< over PC1 is now negative. But so is the loading of the variable 1; that means, the larger variable 1, the smaller the component 1, the larger y -- the interpretation is the same.

Possibly, the easiest way to see that is to use a biplot.

library( pca3d )
pca2d( pca1, biplot= TRUE, shape= 19, col= "black"  )

shows

The same biplot for the second variant shows

pca2d( pca2$scores, biplot= pca2$loadings[,], shape= 19, col= "black" )

As you see, the images are rotated by 180°. However, the relation between the weights / loadings (the red arrows) and the data points (the black dots) is exactly the same; thus, the interpretation of the components is unchanged.

Answer 2

This question gets asked a lot on this forum, so I would like to supplement @January's excellent answer with a bit more general considerations.

In both principal component analysis (PCA) and factor analysis (FA), we use the original variables $x_1, x_2, ... x_d$ to estimate several latent components (or latent variables) $z_1, z_2, ... z_k$. These latent components are given by PCA or FA component scores. Each original variable is a linear combination of these components with some weights: for example the first original variable $x_1$ might be well approximated by twice $z_1$ plus three times $z_2$, so that $x_1 \approx 2z_1 + 3z_2$. If the scores are standardized, then these weights ($2$ and $3$) are known as loadings. So, informally, one can say that $$\mathrm{Original\: variables} \approx \mathrm{Scores} \cdot \mathrm{Loadings}.$$

From here we can see that if we take one latent component, e.g. $z_1$, and flip the sign of its scores and of its loadings, then this will have no influence on the outcome (or interpretation), because $$-1\cdot -1 = 1.$$

The conclusion is that for each PCA or FA component, the sign of its scores and of its loadings is arbitrary and meaningless. It can be flipped, but only if the sign of both scores and loadings is reversed at the same time.

Answer 3

This was well answered above. Just to provide some further mathematical relevance, the directions that the principal components act correspond to the eigenvectors of the system. If you are getting a positive or negative PC it just means that you are projecting on an eigenvector that is pointing in one direction or $180^\circ$ away in the other direction. Regardless, the interpretation remains the same! It should also be added that the lengths of your principal components are simply the eigenvalues.

Answer 4

It is easy to see that the sign of scores does not matter when using PCA for classification or clustering. But it seems to matter for regression. Consider a case where you have just one principal component or one common factor underlying several variables. Then lm(y ~ PC1) will give you different predictions of y compared to lm(y ~ -PC1). If y and PC1 have a positive linear relationship, y and -PC1 have a negative linear relationship. Maybe for regression, you should consider other alternatives discussed here, for example lasso regression.