Principal Component Analysis, should I interpret the component matrix or the component score? - Cross Validated most recent 30 from stats.stackexchange.com 2019-09-19T17:26:17Z https://stats.stackexchange.com/feeds/question/372904 https://creativecommons.org/licenses/by-sa/4.0/rdf https://stats.stackexchange.com/q/372904 0 Principal Component Analysis, should I interpret the component matrix or the component score? jian pang https://stats.stackexchange.com/users/218637 2018-10-20T19:21:47Z 2018-10-26T16:21:34Z <p>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". </p> <p>Can anyone explain to me the difference and help me choose between the two? Thanks!</p> <p><a href="https://i.stack.imgur.com/jx809.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jx809.png" alt="enter image description here"></a></p> <p><a href="https://i.stack.imgur.com/iUa3P.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iUa3P.png" alt="enter image description here"></a></p> https://stats.stackexchange.com/questions/372904/-/373912#373912 0 Answer by stavakol for Principal Component Analysis, should I interpret the component matrix or the component score? stavakol https://stats.stackexchange.com/users/224877 2018-10-26T16:21:34Z 2018-10-26T16:21:34Z <p>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 <span class="math-container">$k$</span>-th PC score). If your dataset is of size <span class="math-container">$n=100$</span>, then the <span class="math-container">$k$</span>-th PC scores is a vector of size <span class="math-container">$100$</span>, each entries of which corresponds to one observation. If your data is a sample from <span class="math-container">${\bf X} = (X_1, \ldots, X_p)' \in \mathbb{R}^p$</span>, then your <span class="math-container">$k$</span>-th PC score, <span class="math-container">$Y_k$</span>, is going to be a linear combination of the variables <span class="math-container">$X_1,\ldots, X_p$</span>, with the weights of the linear combination given by the entries of your PC loading. Remember, the <span class="math-container">$k$</span>-th PC loading <span class="math-container">${\bf e}_k = (e_{k1}, \ldots, e_{kp})' \in \mathbb{R}^p$</span> is given by the <span class="math-container">$k$</span>-th eigenvector of the covariance matrix of <span class="math-container">$\bf X$</span> (associated with the <span class="math-container">$k$</span>-th largest eigenvalue), and it "lives" in the same space as the sample. Using the notation introduced, <span class="math-container">$$Y_k = \sum_{i=1}^p e_{ki} X_i.$$</span> Therefore looking at the values of <span class="math-container">$e_{ki}$</span> will tell you how much variable <span class="math-container">$X_i$</span> contributes to PC score <span class="math-container">$Y_k$</span>.</p> <p>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?</p> <p>I hope this helps.</p>