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I have 4 metrics, three of them measured on the scale 0 to 1, and one measured on the scale 0 to 6. When I stored my data, I converted the fourth one by dividing it by 10, so that I can get values between 0 and 0.6.

Now I am using Principal Component Analysis (PCA) to analyze my data with OriginLab software. Should I use correlation matrix or covariance matrix with PCA? I used correlation Matrix, but I am not sure if what I am doing is correct. The fourth metric makes me confused ..

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  • $\begingroup$ I can't comment yet (not enough rep), but I believe your question is answered here: Should I use correlation matrix or covariance matrix with PCA? $\endgroup$ – ralph Oct 21 '19 at 9:26
  • $\begingroup$ Tags edited. Tags that are very general such as "mathematical-statistics" will routinely fail to attract the experienced users you want to read your question. $\endgroup$ – Nick Cox Oct 21 '19 at 10:54
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You are confusing two issues that are easily confused if new to this field. But first off, why did you divide by 10? Why not 6?

The two issues are

  1. The range of each variable is what you are looking at. 3 variables have range 0 to 1 and one doesn't. Dividing the 4th variable by 6 would fix that if identical range is required, but it's not required for PCA.

  2. The SD, or equivalently the variance, of each variable, is what bites with PCA. Using a correlation matrix is equivalent to standardizing variables to mean 0 and SD or variance 1. But then the range is irrelevant. In practice it is likely that variables ranging between 0 and 1 have similar SDs (although there is no guarantee) and that a variable ranging between 0 and 6 has a larger SD (ditto).

There is no right or wrong answer here without knowing

  • Why the variable with range up to 6 is different

  • What you are imagining the PCA will do for you

It's common advice to use the correlation matrix when variables are on different scales, and that's usually better than mixing mice and giraffes together, but that still leaves PCA likely to overweight some variables and underweight others compared with their substantive importance.

People often seem to think that PCA is a kind of washing machine that takes dirty data and emits clean components, but unless you have a bundle of variables that belong together, and simple latent structure, the results often disappoint.

With 4 variables a scatter plot matrix will often be informative.

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  • $\begingroup$ Thank a lot. Yes , I am completely new to this field! $\endgroup$ – Paulo Oct 21 '19 at 10:35
  • $\begingroup$ I am using PCA to see if these metrics are interchangeable, since they are used to measure the same population . I use different population sizes and see how much these metrics relate to each other.The variable up to 6 is Shannon index . I actually divide it by 10 in case it got more than 6 . The metrics are all diversity metrics , Shannon Index, Simpson Index ... etc $\endgroup$ – Paulo Oct 21 '19 at 10:38
  • $\begingroup$ Thanks for the extra comment, but it doesn't answer my questions. Why is one variable 0 to 6? Doesn't a scatter plot matrix show how variables relate to each other? PCA is not a good technique for beginners, I have to say. I wish someone had told me that when I was a beginner. $\endgroup$ – Nick Cox Oct 21 '19 at 10:42
  • $\begingroup$ The problem is that, I am using different population sizes (30,60,80,100), If I use scatter plots , I will need to report many graphs in my research. I needed something that is more concise. Then I read about PCA . I agree , It is pretty confusing. $\endgroup$ – Paulo Oct 21 '19 at 10:44
  • $\begingroup$ Sorry, I don't understand the different population sizes. Nothing about that in your question. But I know something about diversity measures and there are many questions here about them. Some like Shannon have upper limits depending on the number of species (or whatever it is) and some like Simpson don't. I think you need to ask a new question here that is explicit about your data, the different population sizes, the different diversity measures and use tags to attract several very experienced ecologists here. Asking about PCA here hides the underlying issue of what you want. $\endgroup$ – Nick Cox Oct 21 '19 at 10:51

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