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In health sciences many variables may exhibit half-normal distribution. For example an inflammation process regarding one cell type or population in a tissue sample. The lack of the inflammatory cells describes the lack of disease and with increasing cell count in a sample the disease process becomes more severe and finally terminal. Moreover, a mild disease (small cell count) is more common than severe disease (high cell count) and thus the distribution of cell count is half-normal.

Assuming I have a variable with half-normal distribution and another variable which is normally distributed. Is it valid to use Pearson correlation coefficient with these two variables?

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    $\begingroup$ What is intended by "valid" there? It's possible for there to be a linear relationship (in that if $X$ is half normal and $Y$ be conditionally normal, its possible to have $E(Y|X)$ be linear - easily so, indeed I just generated some data like that now, though it looks as though $E(X|Y)$ will be curved). However, I wouldn't necessarily expect that you'd be able to get the marginal distribution of $Y$ to be normal with a linear relationship. If the relationship is not linear, the Pearson will only capture part of the relationship. $\endgroup$ – Glen_b Jul 27 '15 at 8:11
  • $\begingroup$ @Glen_b thanks! I guess with 'valid' I intend to mean I am not violating any critical assumptions needed to meet. My data has its llimitations (half-normal) and I guess I must deal with some amount of uncertainty. I was unsure whether to choose pearson or non-parametric methods. What would be the consequence of non-normal marginal distribution for Y. $\endgroup$ – arkiaamu Jul 29 '15 at 8:42
  • $\begingroup$ It's not that the marginal for Y being non-normal is a problem; it's that if you're saying that for you the marginal distribution for $Y$ is normal, and that for $X$ is half-normal, the conditional distribution for $Y$ won't be normal (and that's not necessarily a problem either; the bigger issue is whether the relationship you're trying to measure is really linear, or (perhaps) whether you're trying to measure the linear part of a non-linear relationship, or indeed whether you want to measure something else, such as monotonic relationship. $\endgroup$ – Glen_b Jul 29 '15 at 9:29
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The Pearson correlation coefficient is valid for comparing two samples drawn from distributions with finite means and variances, in that it estimates a linear relationship between them.

If the two samples are drawn from a bivariate normal distribution then the Pearson correlation coefficient has nice properties, otherwise there may be some bias in the estimator (unless the sample sizes are very large).

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