# Pearson correlation - sample independence assumption

I have a couple of specific questions regarding the underlying assumptions of the pearson's correlation. I have googled this question but I have not managed to find a consistent answer. In particular whether there is an assumption of sample by sample independence. I will give an example to clarify as I may not be using the correct nomenclature. Assume, I have taken 50 measurements of height and weight from 40 people. These measurements are:

-Measurement of weight and height of 10 people. -Measurement of weight and height of the same 10 people but repeated at a different point in time. -Measurement of weight and height of a different group of 30 people.

If I conduct a pearson's correlation for height and weight, using all 50 measurements, is there an underlying assumption of sample independence that is being violated? If there is no underlying assumption of independence being violated, how can generalisation of the result be interpreted?

Any responses are appreciated.

## 1 Answer

Well, as far as I'm aware there is no strict assumption of sample-by-sample independence, the way you mean it. The closest thing I can find is the assumption of no auto-correlation in a variable, meaning one sample should not depend on its previous values (e.g. as is the case in time series).

The main problem in the example you described is that your data, while measuring the same thing, originates from different distributions. Which will hurt your outcome in any case (not just linear regression.

• Thank you for your response. Can you clarify what are the different distributions you are referring to as I am not sure I understand this part of your answer. – David Whiting Apr 12 '18 at 5:16
• I mean that in the case of a simple height/weight measurment, your data should follow a normal distribution. When measuring the height of a single person over 10 years your distribution should be uniform (height stays the same on adults) while the weight should have ups and downs (this case will definitely have auto-correlation). If your data consisted samples generated by these two techniques your variables would be multimodal. – user204007 Apr 12 '18 at 15:38
• Thanks. Would doing a boot strapped pearson correlation be appropiate? – David Whiting Apr 12 '18 at 18:44
• I can't answer that for sure... – user204007 Apr 12 '18 at 21:27
• I've read about a little bit and although no auto-correlation is a requirement for regression it does not appear to be a requirement for conducting pearson correlation. – David Whiting Apr 12 '18 at 23:14