# Retaining second principal component as a single index

I am using PCA to create an index for my research. In my original dataset, I have 4 main items, (Suppose A,B,C,and D) and one of the items have 4 subitems (for instance, B has 4 subitems, let's say, 1b,2b,3b, and 4b. So, I have 7 items in my original dataset.

Then, I use PCA, and I retained 3 principal components based on Eigenvalues. My first component explained 38 percent of the variation, the second one explained 19 percent of the variation, and the third one explained 16 percent of the variation. My first component is correlated with 6 items in the original dataset (the one that is not correlated is 2b, and the second component is correlated with 5 items (4 subitems of B, and C) in the dataset. Correlation between the second component and 4 subitems of B is higher than that of component 1 and the subitems of B.

So I am wondering if it is possible to use the predicted scores of the second component as my index.

It is a curious question: you can use any PC's scores as your index. It depends on why you are deriving the index and what you are using the scores for. Based on the way you asked the question, it seems that having a separate index for B would make more sense than amalgamating everything. For more details, what is the cross-correlation structure among 'items' and particularly 'subitems'?

• Thanks, @Katya! Please see above for my another curious question. May 8, 2015 at 20:38
• I predicted the scores for 3 components & got the correlation among the components and original items. Correlation between the predicted scores of component 1 and the original items are higher than that of component 2, except the correlation between 2c and predicted scores of component 1. The cross-corr. is generally linear. Now I plan to use the predicted scores of components 2 and 3 as instruments for my variable of interest (IV); I assume there must be a bi-directional causality between my IV & DV. So do you think if it is possible to use components 2 and 3 as instrument variables? May 8, 2015 at 20:39
• @user179313: why do you 'assume there must be a bi-directional causality' and how does it relate to the decision to use secondary axes with such small proportion of variance explained? ps I apologize for my last question, which I will edit, I was meaning to ask whether they are cross-correlated (not sure what happened there) May 9, 2015 at 1:45
• Sorry for the confusion. I meant that the bi-directional causality may exist is just conceptually, nothing related to statistical inference. May 10, 2015 at 3:06

I wouldn't use PCA in your case. It's not working. When it's working you'll get the first component which has approximately equal parts from all original series with a big explained variance, much bigger than the next component.

I'd try using the average of all 7 components as an index.

• Why would you only consider that PCA is "working" if PC1 weights are approximately equal for all original variables? This seems strange to me. May 9, 2015 at 10:38
• Yes @amoeba, what Aksankal pointed out was strange to me as well. My first component explained about 39 percent of the variation, which is largest among all the components. In fact, it has the largest eigenvalue. Is it possible to average all 7 components as an index as I have never heard of doing that? In addition, all the components are uncorrelated so why would we average of all the components? May 10, 2015 at 3:02
• @user179313 Google "index construction". The simplest index is a sum of variables, e.g. sociology.about.com/od/Research-Tools/a/indexes-scales.htm I don't want to get into this whole another world of external validity and such. Always start with the simplest thing when you're not clear what to do. May 10, 2015 at 3:29
• @amoeba OP's talking about a very specific application of PCA: indexing. In this case you have to look at only PC1, otherwise it doesn't make any sense for the purpose. You're trying to squeeze 7 variables into one, if you want to be successful PC1 better get a big chunk of the variance. May 10, 2015 at 3:31