{"id":4015,"date":"2018-04-24T10:38:36","date_gmt":"2018-04-24T09:38:36","guid":{"rendered":"http:\/\/www.blopig.com\/blog\/?p=4015"},"modified":"2018-04-24T10:46:57","modified_gmt":"2018-04-24T09:46:57","slug":"measuring-correlation","status":"publish","type":"post","link":"https:\/\/www.blopig.com\/blog\/2018\/04\/measuring-correlation\/","title":{"rendered":"Measuring correlation"},"content":{"rendered":"

Correlation<\/strong> is defined as how close two variables are to having a dependence relationship with each other. At first sight, it looks kind of simple, but there are two main problems:<\/p>\n

    \n
  1. Despite the obvious situations (i.e. correlation = 1), it is difficult to say whether 2 variables are correlated or not (i.e correlation = 0.7). For instance, would you be able to say if the variables X<\/em> and Y<\/em> from the following to plots are correlated?<\/li>\n
  2. There are different ways of measure of correlation that may not agree when comparing different distributions. As an example, which plot shows a higher correlation? The answer will depend on how <\/em>you do measure the correlation since if you use Pearson correlation, you would pick A whereas if you choose Spearman correlation you will take B<\/li>\n<\/ol>\n

    \"\"<\/p>\n

    Here, I will explain some of the different correlation measures you can use:<\/p>\n

    Pearson product-moment correlation coefficient<\/strong><\/p>\n

      \n
    • What does it measure?<\/em> Only linear dependencies between the variables.<\/li>\n
    • How it is obtained?<\/em> By dividing the covariance of the two variables by the product of their standard deviations. (It is defined only if both of the standard deviations are finite and nonzero). \"\\rho<\/li>\n<\/ul>\n
        \n
      • Properties:<\/em><\/li>\n<\/ul>\n
          \n
        1. \u03c1 (X,Y) = +1 : perfect direct (increasing) linear relationship (correlation).<\/li>\n
        2. \u03c1 (X,Y) = -1 : perfect decreasing (inverse) linear relationship (anticorrelation).<\/li>\n
        3. In all other cases,\u00a0\u03c1 (X,Y) indicates the degree of linear dependence between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated).<\/li>\n
        4. Only gives a perfect value when X and Y are related by a linear function.<\/li>\n<\/ol>\n
            \n
          • When is it useful<\/em>? For the case of a linear model with a single independent variable, the coefficient of determination (R squared) is the square of r, Pearson’s product-moment coefficient.<\/li>\n<\/ul>\n

             <\/p>\n

            Spearman’s rank correlation coefficient<\/strong>:<\/p>\n

              \n
            • What does it measure?<\/em> How well the relationship between two variables can be described using a monotonic function (a function that only goes up or only goes down).<\/li>\n
            • How it is obtained<\/em>? Pearson correlation between the rank values of the two variables.<\/li>\n<\/ul>\n

              \"{\\displaystyle<\/p>\n

              Only if all n<\/i> ranks are distinct integers<\/i>, it can be computed using the popular formula.<\/p>\n

              \"{\\displaystyle<\/p>\n

              W<\/span>here di<\/em> is the difference between the two ranks of each observation.<\/p>\n

                \n
              • Properties<\/em>:<\/li>\n<\/ul>\n
                  \n
                1. rs (X,Y) = +1:\u00a0 X and Y are related by any increasing monotonic function.<\/li>\n
                2. rs (X,Y) = -1:\u00a0 X and Y are related by any decreasing monotonic function.<\/li>\n
                3. The Spearman correlation increases in magnitude as X and Y become closer to being perfect monotone functions of each other.<\/li>\n<\/ol>\n
                    \n
                  • When is it useful?<\/em> It is appropriate for both continuous and discrete ordinal variables. It can be use for looking for non-linear dependence relationships.<\/li>\n<\/ul>\n

                    Kendall’s tau coefficient<\/strong><\/p>\n

                      \n
                    • What does it measure? <\/em>The ordinal association between two measured quantities.<\/li>\n
                    • How it is obtained<\/em>?<\/li>\n<\/ul>\n

                      \"{\\displaystyle<\/p>\n

                      Any pair of observations (xi , yi)\u00a0 and (xj, yj) are said to be concordant if the ranks for both elements agree. That happens if xi-xj and yi-xj have the same sign. If their sign are different, they are considered as discordant pairs<\/p>\n

                        \n
                      • Properties:<\/em><\/li>\n<\/ul>\n
                          \n
                        1. \u03c4 (X,Y) = +1: The agreement between the two rankings is perfect (i.e., the two rankings are the same)<\/li>\n
                        2. \u03c4 (X,Y) = -1: The disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other)<\/li>\n
                        3. If X and Y are independent, then we would expect the coefficient to be approximately zero.<\/li>\n<\/ol>\n
                            \n
                          • When is it useful?<\/em> It is appropriate for both continuous and discrete ordinal variables. It can be use for looking for non-linear dependence relationships.<\/em><\/li>\n<\/ul>\n

                            Distance correlation:<\/strong><\/p>\n

                              \n
                            • What does it measure?<\/em> Both linear and nonlinear association between two random variables or random vectors.<\/li>\n
                            • How is it obtained?<\/em> By dividing the variable’s distance covariance by the product of their distance standard deviations:<\/li>\n<\/ul>\n

                              \"\\operatorname<\/p>\n

                              The distance covariance is defined as:<\/p>\n

                              \"{\\displaystyle<\/p>\n

                              Where:<\/p>\n

                              \"{\\displaystyle<\/p>\n

                              \"{\\begin{aligned}a_{{j,k}}&=\\|X_{j}-X_{k}\\|,\\qquad<\/p>\n

                              where || \u22c5 || denotes Euclidean norm.<\/p>\n

                                \n
                              • Properties:<\/em><\/li>\n<\/ul>\n
                                  \n
                                1. dCor (X,Y) = 0 if and only if the random vectors are independent.<\/li>\n
                                2. dCor (X,Y) = 1: Perfect dependence between the two distributions.<\/li>\n
                                3. dCor <\/span><\/span><\/span><\/span>(<\/span><\/span>X<\/span><\/span>,<\/span><\/span>Y<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> is defined for X<\/span><\/span><\/span><\/span><\/span> and Y<\/span><\/span><\/span><\/span><\/span> in arbitrary dimension.<\/li>\n<\/ol>\n
                                    \n
                                  • When is it useful?<\/em> It is appropriate to find any kind\u00a0 dependence relationships between the 2 variables. Also if X and Y have different dimensions.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

                                    Correlation is defined as how close two variables are to having a dependence relationship with each other. At first sight, it looks kind of simple, but there are two main problems: Despite the obvious situations (i.e. correlation = 1), it is difficult to say whether 2 variables are correlated or not (i.e correlation = 0.7). […]<\/p>\n","protected":false},"author":52,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","wikipediapreview_detectlinks":true,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"ngg_post_thumbnail":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[10,15],"tags":[],"ppma_author":[534],"class_list":["post-4015","post","type-post","status-publish","format-standard","hentry","category-groupmeetings","category-technical"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"authors":[{"term_id":534,"user_id":52,"is_guest":0,"slug":"javier","display_name":"Javier Pardo D\u00edaz","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/fc152623b8f763cda0c87f20eab1dfc3a69a2067beac85560657a717ed373605?s=96&d=mm&r=g","0":null,"1":"","2":"","3":"","4":"","5":"","6":"","7":"","8":""}],"_links":{"self":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/4015","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/users\/52"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/comments?post=4015"}],"version-history":[{"count":11,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/4015\/revisions"}],"predecessor-version":[{"id":4034,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/4015\/revisions\/4034"}],"wp:attachment":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/media?parent=4015"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/categories?post=4015"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/tags?post=4015"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/ppma_author?post=4015"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}