Editing Correlation (section)

==Sensitivity to the data distribution==
{{Further|Pearson product-moment correlation coefficient#Sensitivity to the data distribution}}

The degree of dependence between variables {{mvar|X}} and {{mvar|Y}} does not depend on the scale on which the variables are expressed.   That is, if we are analyzing the relationship between {{mvar|X}} and {{mvar|Y}}, most correlation measures are unaffected by transforming {{mvar|X}} to {{math|''a'' + ''bX''}} and {{mvar|Y}} to {{mvar|''c'' + ''dY''}}, where ''a'', ''b'', ''c'', and ''d'' are constants (''b'' and ''d'' being positive).  This is true of some correlation [[statistic]]s as well as their [[Population (statistics)|population]] analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant to [[monotone function|monotone transformations]] of the marginal distributions of {{mvar|X}} and/or {{mvar|Y}}.

[[File:correlation range dependence.svg|300px|right|thumb|[[Pearson product moment correlation coefficient|Pearson]]/[[Spearman's rank correlation coefficient|Spearman]] correlation coefficients between {{mvar|X}} and {{mvar|Y}} are shown when the two variables' ranges are unrestricted, and when the range of {{mvar|X}} is restricted to the interval (0,1).]]Most correlation measures are sensitive to the manner in which {{mvar|X}} and {{mvar|Y}} are sampled.  Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations.<ref>{{cite book|last=Thorndike|first=Robert Ladd|title=Research problems and techniques (Report No. 3)|year=1947|publisher=US Govt. print. off.|location=Washington DC}}</ref>

Various correlation measures in use may be undefined for certain joint distributions of {{mvar|X}} and {{mvar|Y}}.  For example, the Pearson correlation coefficient is defined in terms of [[moment (mathematics)|moments]], and hence will be undefined if the moments are undefined.  Measures of dependence based on [[quantile]]s are always defined.  Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being [[bias of an estimator|unbiased]], or [[consistent estimator|asymptotically consistent]], based on the spatial structure of the population from which the data were sampled.

Sensitivity to the data distribution can be used to an advantage. For example, [[scaled correlation]] is designed to use the sensitivity to the range in order to pick out correlations between fast components of [[time series]].<ref name = "Nikolicetal">{{cite journal | last1 = Nikolić | first1 = D | last2 = Muresan | first2 = RC | last3 = Feng | first3 = W | last4 = Singer | first4 = W | year = 2012 | title = Scaled correlation analysis: a better way to compute a cross-correlogram | journal = European Journal of Neuroscience | volume = 35| issue = 5| pages = 1–21 | doi = 10.1111/j.1460-9568.2011.07987.x | pmid = 22324876 | s2cid = 4694570 }}</ref> By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.