# Principal Component Analysis, should I interpret the component matrix or the component score?

I am doing a Principal Component Analysis on some demographic data. I have extracted four principal components and I hope to find out how each component is characterized by the original variables. I have got component matrix and the component score from SPSS output. Which one should I look at? From what I have seen, the component score is also interpreted as factor loading, and shows the correlation betweeen the original variable and the principal component, is the one that I am looking for. But the component score shows the "weight" of the variables in the principal component, or the new "variable".

Can anyone explain to me the difference and help me choose between the two? Thanks!  • fwiw, I disagree with the vote to close this as off-topic. But you can probably find your answers in the SPSS documentation. If nowhere else, then in the Help...Tutorial. – rolando2 Oct 20 '18 at 20:09

## 1 Answer

Essentially you have two things: PC scores and PC loadings. Each PC loadings live in the same space as you data. For each observation in your sample, you can get the PC score corresponding to each PC loading (let's call it $$k$$-th PC score). If your dataset is of size $$n=100$$, then the $$k$$-th PC scores is a vector of size $$100$$, each entries of which corresponds to one observation. If your data is a sample from $${\bf X} = (X_1, \ldots, X_p)' \in \mathbb{R}^p$$, then your $$k$$-th PC score, $$Y_k$$, is going to be a linear combination of the variables $$X_1,\ldots, X_p$$, with the weights of the linear combination given by the entries of your PC loading. Remember, the $$k$$-th PC loading $${\bf e}_k = (e_{k1}, \ldots, e_{kp})' \in \mathbb{R}^p$$ is given by the $$k$$-th eigenvector of the covariance matrix of $$\bf X$$ (associated with the $$k$$-th largest eigenvalue), and it "lives" in the same space as the sample. Using the notation introduced, $$Y_k = \sum_{i=1}^p e_{ki} X_i.$$ Therefore looking at the values of $$e_{ki}$$ will tell you how much variable $$X_i$$ contributes to PC score $$Y_k$$.

I'm a bit confused about the tables you have included above. Usually PC loadings are rescaled to be vectors of length 1. I'm also confused with the second table. Have you tried looking at the help function of the software you are using?

I hope this helps.