Weighted Correlation

Excerpt

Weighted correlation is concerned with the use of weights assigned to the subjects in the calculation of a correlation coefficient (see Correlation Coefficient) between two variables X and Y . The weights can either be naturally available beforehand or chosen by the user to serve a specific purpose. For instance, if there is a different number of measurements on each subject, it is natural to use these numbers as weights and calculate the correlation between the subject means. On the other hand, if the variables X and Y represent, for instance, the ranks of preferences of two human beings over a set of n items, one might want to give larger weights to the first preferences, as these are more accurate. In another situation, if we want to calculate the correlation between two stocks in a stock exchange market during last year, we might want to favor (larger weight) the more recent observations, as these are more important for the present situation. Suppose that X _i and Y _i are the pair of values corresponding to observation i in each sample and w _i the weight attributed to this observation, such that \(\sum\nolimits_{i = 1}^n {wi} = 1\). Then, the sample weighted correlation coefficient is given by the formula where the sums are from i = 1 to n and \({\overline{X}}_{w} = \sum \nolimits {w}_{i}{X}_{i}\) and \({\overline{Y }}_{w} = \sum \nolimits {w}_{i}{Y }_{i}\) are the weighted means. When all the w _i are equal they cancel out, giving the usual formula for the Pearson product–moment correlation coefficient.

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