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I am analyzing weather data from 2 sites, and trying to determine if they are correlated. For the most part, I am able to run linear regressions and Pearson's correlations since the data are relatively normal. However, when looking at relative humidity, it obviously has an upper limit in place (100%; it would have a lower limit of 0 as well, but this is not applicable, as I'm only looking at maximum RH). One site reaches 100% RH frequently; the other occasionally does.

The Question: Is Spearman's \rho appropriate in the case of data with upper limits?

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Likert-type rating scales also have limits and moreover are discrete, but we frequently use Pearson r with them. Also, non-normality isn't an obstacle for r, albeit asymptotic method computation of its significance does assume bivariate normality in the population.

There are two reasons to use Spearman instead of Pearson: (a) your variables are metric but you admit nonlinear monotonic underlying relationship (and it is wise to admit especially when the distributions are of different shape, e.g. symmetric vs skewed); (b) you consider your variables ordinal, not metric.

P.S. Still another reason is when you have outliers but are unwilling to get rid of them.

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  • $\begingroup$ "non-normality isn't an obstacle for r" - Wouldn't you agree that 1) if a distribution is nonnormal because it contains outliers, they can seriously distort r? 2) lack of normality tends to attenuate correlations, and that part of the reason we often transform variables is to see how much stronger an r we can obtain that way? $\endgroup$
    – rolando2
    Apr 6, 2012 at 11:42
  • $\begingroup$ @rolando: (1) outliers is a different theme than normality (and not touched by the OP), they can distort any type of distribution. Still, I will add a phrase about them. (2) attenuation means that linear r underestimates the strength of association because the latter is nonlinear; this happens when the two marginal distributions are of different shape (rather than being non-normal) $\endgroup$
    – ttnphns
    Apr 6, 2012 at 13:30
  • $\begingroup$ Thanks @ttnphns. So, to verify: as long as I'm not concerned about the significance (p-value) of the correlation, but just the correlation coefficient (Pearson's r) itself, I can safely use Pearson's? What about Spearman's rho, as I do have some outliers I can't justify discarding? I assume that because it's similar but rank-based, it should be ok as well? $\endgroup$
    – Mog
    Apr 6, 2012 at 15:47
  • $\begingroup$ Of course, rho is insensitive to outliers, but it is still sensitive to shape of the cloud (see here). Apart from outliers, you can always use Pearson if you treat the variables as metric and are comfortable with the fact that r taps only linear portion of the association (which might be curvilinear) $\endgroup$
    – ttnphns
    Apr 6, 2012 at 15:59
  • $\begingroup$ Thanks again @ttnphns. Because of the outliers, I'll stick with Spearman's for this analysis, but that's excellent to keep in mind for future analyses. $\endgroup$
    – Mog
    Apr 6, 2012 at 17:50

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