# How can I combine different principal components into one index?

I'm a sport scientist working in football.

I have a large number of variables describing match performance. I ran a PCA in order to reduce the number of variables and I obtained 3 principal components.

I'd like to obtain only one index describing match performance in football. Is it possible to combine the 3 principal components I obtained? I read that there are 2 possible ways to do it:

1. Re-running a PCA on the new factors obtained

2. Multiplying the new factors with the variance explained. An example: PC1 explains 46% of the variance (0.46), PC2 30% (0.30) and PC3 12% (0.12). The total index will be calculated: Performance index= 0.46*PC1 + 0.30*PC2 + 0.12*PC3.

Are those two solutions appropriate? If not, is there a way to combine different components?

• You can do anything you like but there is a fallacy lurking here. PC1 is in this framework the best single summary of the data. You can't improve on it by mushing it together with other PCs. In fact you will get something that will perform less well and be harder to interpret. Jan 10, 2020 at 15:44
• Side notes: it is common to be sloppy about "components" and "factors", and that includes much widely used software. The results of PCA are components. The results of factor analysis are factors. No reason to regard the terms as interchangeable, just as association football, rugby football and other variants are distinct and you would be careful to use the precise word when it mattered. Jan 10, 2020 at 15:46
• I have to say both approaches strike me as nonsense, and I wonder whether who is proposing them and what their credentials are. A second PCA on PCs can't find more or deeper or better patterns. Jan 10, 2020 at 15:48
• There doesn't seem to be a unique solution because this question doesn't seem to be clearly formulated. Although the variables are ostensibly related to "match performance," exactly what does the latter mean? What is it intended to represent? A clear answer to those questions is needed to justify any proposed index.
– whuber
Jan 10, 2020 at 16:46
• You are trying to combine multiple attributes into some single number. Exactly how do you intend to use that number? This is the essence of the question. Although you may understand what you mean by "match performance," there is no information in your question or in your data that reveals what that is: you have to tell us.
– whuber
Jan 10, 2020 at 16:55

The first principal component is - by construction - the best single feature (or score, if you like) to explain your data. If you only want one element, you should simply use the First Principal Component.

The first approach you propose will return you exactly the same principal components you already have, since these are already built to maximise variance on their axes and to be orthogonal, running the decomposition again would issue the same results. The second approach would just give you some linear combination of your original features with no particular properties.

• (+1)The first approach might return negated or reversed versions of the originals, which is often confusing to those who have not seen it before. Jan 10, 2020 at 16:06

The beauty of (unrotated) principal components is that they are mutually orthogonal. There is no problem in just adding them up. There is also no surprise in how they will behave. The behavior of the sum will be exactly the same as if you included the three components as predictors but with equal regression weights, rather than allowing their regression weights to differ. Just add them together. You will them have a composite variable that accounts for a larger share of total variance and combines the functionality of the three components.

• If three PCs are better than one, why not just average them all? Jan 10, 2020 at 16:45
• Summing or averaging (dividing by a constant) should have the same effect in most (but not all) cases. If sample size were low, it could make sense to assign equal weights to the original predictors--that is, to just sum them all up (Dana and Dawes 2004). journals.sagepub.com/doi/pdf/10.3102/10769986029003317 Jan 10, 2020 at 18:19
• Thanks for the reply. I don't recognise the reference. I can't summon up any enthusiasm for this idea, if only because the scope for interpreting what you've done in substantive terms seems slight. Wanting that isn't universal but it's central to how I want to do statistics. Jan 10, 2020 at 18:21
• BTW, I too jumped to the conclusion that the aim was all preparatory to regression. Reading the question again, now I can't see any reason for that conclusion. Jan 10, 2020 at 18:29
• I did not make an assumption, though I rejected the idea that the aim is to build plausible proxies for unobserved variables. Whatever the aim, principal components are just slices of the information available. If you are doing the slicing, then you can put the slices on your sandwich any way you want. Jan 10, 2020 at 18:43