{"id":850,"date":"2025-02-04T16:44:58","date_gmt":"2025-02-04T10:59:58","guid":{"rendered":"https:\/\/pokharelsugam.com.np\/bioinformatics\/?p=850"},"modified":"2025-02-06T22:49:52","modified_gmt":"2025-02-06T17:04:52","slug":"correlation","status":"publish","type":"post","link":"https:\/\/pokharelsugam.com.np\/bioinformatics\/correlation\/","title":{"rendered":"Correlation"},"content":{"rendered":"\n<script type=\"text\/javascript\" async \n  src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\">\n<\/script>\n\n\n\n<p class=\"wp-block-paragraph\">Correlation is a statistical measure that expresses the extent to which two variables change together. It shows the relationship between two variables, indicating whether an increase in one variable results in an increase or decrease in the other.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Types of Correlation<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Based on Direction:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Positive Correlation:<\/strong> Both variables move in the same direction. Example: Height and weight.<\/li>\n\n\n\n<li><strong>Negative Correlation:<\/strong> One variable increases while the other decreases. Example: Price and demand.<\/li>\n\n\n\n<li><strong>No Correlation:<\/strong> No apparent relationship between the two variables.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Based on Strength:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Perfect Correlation (+1 or -1):<\/strong> One variable changes proportionally with the other.<\/li>\n\n\n\n<li><strong>High Correlation (Close to +1 or -1):<\/strong> A strong relationship but not perfect.<\/li>\n\n\n\n<li><strong>Low Correlation (Close to 0):<\/strong> A weak relationship.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Based on Nature:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Linear Correlation:<\/strong> The relationship between variables forms a straight-line pattern.<\/li>\n\n\n\n<li><strong>Non-Linear (Curvilinear) Correlation:<\/strong> The relationship does not follow a straight-line pattern.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Methods for Measuring Correlation<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Measuring Simple Correlation <\/li>\n\n\n\n<li>Measuring Partial Correlation<\/li>\n\n\n\n<li>Measuring Multiple Correlation <\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Measuring Simple Correlation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">It measure relationship between two variables only.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">1. Scatter Diagram<\/h4>\n\n\n\n<h4 class=\"wp-block-heading\">2. Karl Pearson\u2019s Coefficient of Correlation (r)<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Let X and Y be two correlated variable then Karl Pearson&#8217;s Correlation Coefficient denoted by \\(r_{xy}\\) or r(X,Y) or simply r. It is also called product moment correlation coefficient or simple correlation coefficient or simply a correlation. It is defined as follows.<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(r = \\frac{COV(X,Y)}{\\sqrt{Var(X)}. \\sqrt{Var(Y)}}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r = \\frac{\\sum(x-\\bar{x})(y-\\bar{y})}{\\sqrt{\\sum(x-\\bar{x})^2} \\sqrt{\\sum(y-\\bar{y})^2}}\\) &#8230;&#8230;&#8230;&#8230;.. i<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r = \\frac{n \\sum xy -\\sum x . \\sum y}{\\sqrt{n \\sum x^2 &#8211; (\\sum x)^2} . \\sqrt{n \\sum y^2 &#8211; (\\sum y)^2}} \\)  &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. ii<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where n = number of paired observation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">You can also calculate correlation coefficient by taking assumed mean.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let U = X-A, V = X-B, Where A and B are assumed mean of X and Y respectively. Then formula become,<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(r = \\frac{n \\sum uv -\\sum u . \\sum v}{\\sqrt{n \\sum u^2 &#8211; (\\sum u)^2} . \\sqrt{n \\sum v^2 &#8211; (\\sum v)^2}} \\) &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. ii<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Interpretation of r:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If r = 0 to 0.5, there is low degree of correlation.<\/li>\n\n\n\n<li>If r = 0.51 to 0.699, there is moderate degree of correlation<\/li>\n\n\n\n<li>If 0.7 to 0.999, there is high degree of correlation.<\/li>\n\n\n\n<li>If r = 1, there is perfectly positive correlation.<\/li>\n\n\n\n<li>If r = -1, there is perfectly negative correlation.<\/li>\n\n\n\n<li>If r = 0, there is no correlation, the variables are independent.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">3. Spearman\u2019s Rank Correlation<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Ranks are the assignments of orders or priorities according to their status or importance. Karl Pearson&#8217;s correlation coefficient is specially useful when the data are quantitatively measured. There are some variables like beauty, knowledge, intelligence, honesty etc. which cannot be measured quantitatively directly. These types of variables can be measured by assigning ranks or some sorts of ratings. Then the degree of association that exists between the two sets of rank is known as rank correlation. <\/p>\n\n\n\n<h3 class=\"wp-block-heading\">a. When rank is not repeated<\/h3>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(r_s = 1- \\frac{6 \\sum d^2}{n(n^2-1)}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where, d = difference between rank and n = number of paired data.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>\\(x\\)<\/strong><\/td><td><strong>\\(y\\)<\/strong><\/td><td><strong>\\(R_x\\)<\/strong><\/td><td><strong>\\(R_y\\)<\/strong><\/td><td><strong>\\(d=R_x-R_y\\)<\/strong><\/td><td><strong>\\(d^2\\)<\/strong><\/td><\/tr><tr><td>25<\/td><td>40<\/td><td>4<\/td><td>5<\/td><td>-1<\/td><td>1<\/td><\/tr><tr><td>21<\/td><td>50<\/td><td>5<\/td><td>2<\/td><td>3<\/td><td>9<\/td><\/tr><tr><td>27<\/td><td>54<\/td><td>2<\/td><td>1<\/td><td>1<\/td><td>1<\/td><\/tr><tr><td>26<\/td><td>43<\/td><td>3<\/td><td>3<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td>31<\/td><td>42<\/td><td>1<\/td><td>4<\/td><td>-3<\/td><td>9<\/td><\/tr><tr><td><\/td><td><\/td><td><\/td><td><\/td><td><strong>\\(\\sum d = 0 \\)<\/strong><\/td><td><strong>\\(\\sum d^2 = 20 \\)<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Where n= 5 (Number of Paired Observation) , Then use following formula <\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(r_s = 1- \\frac{6 \\sum d^2}{n(n^2-1)}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 1- \\frac{6 \\times 20}{5(5^2-1)}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 1- \\frac{120}{5(25-1)}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 1- \\frac{120}{5 \\times 24}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 1- \\frac{120}{120}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 1- 1\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 0\\), There is no correlations <\/p>\n\n\n\n<h3 class=\"wp-block-heading\">b. When rank is repeated<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Formula<\/strong><\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(r_s = 1- \\frac{6 \\left[\\sum d^2+\\sum{\\frac{m(m^2-1)}{12}}\\right]}{n(n^2-1)}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Where \\(\\sum{\\frac{m(m^2-1)}{12}}\\) is correction factor for reprated rank.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If any rank is repeated calculated average rank for their rank repeatation as follows<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>\\(x\\)<\/strong><\/td><td><strong>\\(y\\)<\/strong><\/td><td><strong>\\(R_x\\)<\/strong><\/td><td><strong>\\(R_y\\)<\/strong><\/td><td><strong>\\(d=R_x-R_y\\)<\/strong><\/td><td><strong>\\(d^2\\)<\/strong><\/td><\/tr><tr><td>18<\/td><td>46<\/td><td>15<\/td><td>11<\/td><td>4<\/td><td>16<\/td><\/tr><tr><td>78<\/td><td>81<\/td><td>3<\/td><td>2<\/td><td>1<\/td><td>1<\/td><\/tr><tr><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">46<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">38<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">11<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">14.5<\/mark><\/td><td>-3.5<\/td><td>12.25<\/td><\/tr><tr><td>37<\/td><td>44<\/td><td>13<\/td><td>13<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">47<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">38<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">8<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">14.5<\/mark><\/td><td>-6.5<\/td><td>42.25<\/td><\/tr><tr><td>56<\/td><td>47<\/td><td>5<\/td><td>10<\/td><td>-5<\/td><td>25<\/td><\/tr><tr><td>82<\/td><td>75<\/td><td>1<\/td><td>4<\/td><td>-3<\/td><td>9<\/td><\/tr><tr><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">47<\/mark><\/td><td>56<\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">8<\/mark><\/td><td>9<\/td><td>-1<\/td><td>1<\/td><\/tr><tr><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">46<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">63<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">11<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">7.5<\/mark><\/td><td>3.5<\/td><td>12.25<\/td><\/tr><tr><td>28<\/td><td>71<\/td><td>14<\/td><td>5<\/td><td>9<\/td><td>81<\/td><\/tr><tr><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">46<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">63<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">11<\/mark><\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">7.5<\/mark><\/td><td>3.5<\/td><td>12.25<\/td><\/tr><tr><td>75<\/td><td>78<\/td><td>4<\/td><td>3<\/td><td>1<\/td><td>1<\/td><\/tr><tr><td>81<\/td><td>95<\/td><td>2<\/td><td>1<\/td><td>1<\/td><td>1<\/td><\/tr><tr><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">47<\/mark><\/td><td>45<\/td><td><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">8<\/mark><\/td><td>12<\/td><td>-4<\/td><td>16<\/td><\/tr><tr><td>50<\/td><td>67<\/td><td>6<\/td><td>6<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td><\/td><td><\/td><td><\/td><td><\/td><td><strong>\\(\\sum d = 0 \\)<\/strong><\/td><td><strong>\\(\\sum d^2 = 230 \\)<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Repeated Ranks<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>m1 for repeated rank 8 =3<\/li>\n\n\n\n<li>m2 for  repeated rank 11 =3<\/li>\n\n\n\n<li>m3 for repeated rank 7.5 = 2<\/li>\n\n\n\n<li>m4 for repeated rank 14.5 =2<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate correction factor<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(\\sum{\\frac{m(m^2-1)}{12}}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(\\frac{3(3^2-1)}{12}+\\frac{3(3^2-1)}{12}+\\frac{2(2^2-1)}{12}+\\frac{2(2^2-1)}{12} = 5 \\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Now<\/strong><\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(r_s = 1- \\frac{6 \\left[\\sum d^2+\\sum{\\frac{m(m^2-1)}{12}}\\right]}{n(n^2-1)}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 1- \\frac{6 \\left[230-5\\right]}{15(15^2-1)}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Or, \\(r_s = 0.5804\\). There is moderate degree of correlation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Correlation is a statistical measure that expresses the extent to which two variables change together. 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