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Let´s say I want to run a correlation between "eye spherical defect" and height and I want to use only individuals with myopia, whose "spherical defect" goes from 0 to -20 or so. Whereas the sphericity of eyes may be a normally distributed variable with a mean of 0, if I use only myopic individuals I would be using a sample containing only half the values (i.e. from 0 to -20) of the normal distribution of that variable. My question: How can I run a correlation using such a variable? Would I be able to use parametric tests for this? (e.g. Pearson's $r$)

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You're focusing on a subset of the data (myopic subjects with scores -20 to zero ). Nothing wrong with using Pearson but run a histogram on those cases to ensure the bulk of the data form a distribution with central location. It could be skewed toward the left, and if so, run Spearman rank correlation.

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    $\begingroup$ The last line is contentious. I don't see that any kind of skewness means ipso facto that you should rule out using Pearson correlation and jump to Spearman correlation. You should naturally always plot your data and think about whether Pearson correlation is a good idea, but symmetry of marginal distribution isn't an essential. If $x$ is 1,2,4,7,11 and $y$ is twice that, then Pearson correlation is fine, and many less trivial examples, including those with real data, have the same flavour. $\endgroup$
    – Nick Cox
    Commented Mar 2, 2015 at 16:54
  • $\begingroup$ Hi @NickCox, I am aware that parametric assumptions should always be checked. In this case however, I expect my variable to be skewed not because of my sample characteristics, but because of the characteristics of the population of myopics. If the variable was skewed, 1) it would not be appropriate to use pearson's correlation, right? 2) Would it make sense to transform this variable, given that it is skewed because it represents half the population of a normally distributed variable? 4) Any methods for handling truncated variables that may be useful or any writings you may recommend reading? $\endgroup$
    – Irving
    Commented Mar 2, 2015 at 23:34
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    $\begingroup$ 1) is a question I've already answered and 2) is similar. There could be grounds for transforming your data, not visible to us. But being skew doesn't oblige a transformation: it's nonlinearity, not skewness, that you need to focus on. 4) is too big to answer here, but there are many resources in this forum alone. The term "parametric assumptions" is odd; Spearman's correlation too is a parameter being estimated. $\endgroup$
    – Nick Cox
    Commented Mar 3, 2015 at 9:19

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