{"id":1734,"date":"2013-01-25T19:00:37","date_gmt":"2013-01-25T10:00:37","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=1734"},"modified":"2014-08-05T17:01:10","modified_gmt":"2014-08-05T08:01:10","slug":"pearson-product-moment-correlation-coefficient-r-and-t-test-on-it","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=1734","title":{"rendered":"Pearson product-moment correlation coefficient (r) and t-test on it"},"content":{"rendered":"

The index of relation between x and y is correlation coefficient or Pearson product moment correlation coefficient as formula below. Range of correlation coefficient is between -1 and 1. <\/p>\n

\\displaystyle r = \\frac{\\sum_{i=1}^n(x_i-\\bar x)(y_i - \\bar y)}{\\sqrt{\\sum_{i=1}^n(x_i - \\bar x)^2}\\sqrt{\\sum_{i=1}^n(y_i - \\bar y)^2}}<\/p>\n<\/p>\n

\\bar x and \\bar y are average of x and y, respectively. i is number of sample (incremental variable). n is number of sample. <\/p>\n

Correlation coefficient (r) of 2 variables randomly extracted from population follows t-distribution. T-statistics of r is calculated formula as below and follows t-distribution with degree of freedom n-2, n is number of sample. When correlation coefficient of population is \\rho, null hypothesis is described that “\\rho = 0″. If t-statistics calculated from number of sample (n) and correlation coefficient (r) is greater than that of significance level (\\alpha), null hypothesis is rejected. <\/p>\n

\\displaystyle t = r\\sqrt{\\frac{n - 2}{1 - r^2}}<\/p>\n

\n

The test of significance for this important null hypothesis H (ρ = 0) is equivalent to that for the null hypothesis H (β1<\/sub> = 0) or H (β2<\/sub> = 0). It now follows that if x and y have a joint bivariate normal distribution, then the test for the null hypothesis H (ρ = 0) is obtained by using the fact that if the null hypothesis under test is true, then<\/p>\n

\\displaystyle F = \\frac{(n-2)Z^2}{XY-Z^2} = \\frac{(n-2)r^2}{1-r^2}\\vspace{0.1in}\\\\ X = \\sum(x - \\bar{x})^2\\vspace{0.1in}\\\\ Y = \\sum(y - \\bar{y})^2\\vspace{0.1in}\\\\ Z = \\sum(x - \\bar{x})(y - \\bar{y})\\vspace{0.1in}\\\\ r^2 = \\frac{Z^2}{XY}<\/p>\n

has the F distribution with 1, n – 2 d.f. An equivalent test of significance for the null hypothesis is obtained by using the fact that if the null hypothesis is true, then<\/p>\n

\\displaystyle t = \\frac{r\\sqrt{n - 2}}{\\sqrt{1 - r^2}}<\/p>\n

has “Student’s” distribution. with n – 2 d.f.<\/p>\n

For any non-zero null hypothesis about ρ there is no parallelism between the correlation coefficient ρ and the regression coefficients β1<\/sub> and β2<\/sub>. In fact, no exact test of significance is available for testing readily non-zero null hypothesis about ρ. Fisher has given an approximate method for such null hypothesis, but we do not consider this here.<\/p>\n<\/blockquote>\n