{"id":130,"date":"2018-10-31T18:08:14","date_gmt":"2018-10-31T22:08:14","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/simplestats\/?post_type=chapter&p=130"},"modified":"2019-11-15T18:37:45","modified_gmt":"2019-11-15T23:37:45","slug":"10-1-correlation","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/simplestats\/chapter\/10-1-correlation\/","title":{"raw":"10.1. Correlation","rendered":"10.1. Correlation"},"content":{"raw":"[latexpage]\r\n\r\nYou will recall from Section 7.2.3 that we use the coefficient of correlation (Pearson's) r<\/em> to examine associations between two continuous variables. The correlation coefficient r<\/em> varies between -1 and 1. The closer it is to either, the stronger the correlation, and the closer it is to 0, the weaker the correlation[footnote]The sign of r<\/em> is there only<\/em> to indicate the direction of the association: positive or negative, nothing else. Thus this is a reminder not to use r<\/em>'s sign as a measure of magnitude or strength of the association. Thus, for example, -0.9 is a stronger association than 0.2 because -0.9 is closer to -1 than 0.2 is to 1. (In fact, 0.2 is much closer to 0, or no association.) That is, a strong negative correlation is stronger<\/em> than a weak positive one, despite that -0.9<0.2.[\/footnote].\r\n\r\n \r\n\r\nWhere does r<\/em> come from though? What does it actually measure? I doubt you have lost sleep wondering about these questions which I left unanswered in Chapter 7, but here is your chance to learn this anyway (think of it as closure of sorts).\r\n\r\n \r\n\r\nThe correlation coefficient is, essentially,<\/span>\u00a0a ratio of the variabilities of the two variables[footnote]To be precise, the ratio is between the covariance of x<\/em> and y<\/em>\u00a0(i.e., their joint variability, sxy<\/sub><\/em>) and the product of their separate variances sx<\/sub><\/em> and sy<\/sub><\/em>:<\/span>\r\n\r\n \r\n\r\n$$r=\\frac{s_{xy}}{s_x s_y}$$ or\r\n\r\n \r\n\r\n$$\\rho=\\frac{\\sigma_{xy}}{\\sigma_x \\sigma_y}$$ if we apply it to a population instead of a sample. (Here\u00a0\u03c1<\/em> is the small-case Greek letter r<\/em>, pronounced [ROH].)\r\n\r\n[\/footnote]. <\/span>\r\n\r\nThe easiest way to calculate\u00a0<\/strong><\/span>r<\/em><\/strong>\u00a0between a variable x<\/em> and a variable y<\/em> is through the distances of the observations from the means of the two variables, or more accurately, the sums of squares<\/strong>[footnote]Recall that the sum of squares was the numerator in the formulas for the variance and the standard deviation. We take the distances of the observations from the mean, square them, and them add them altogether. (We square them <\/span>before<\/em> adding t<\/span>o turn them all positive, otherwise they'd cancel each other upon summation. See Section 4.3 (https:\/\/pressbooks.bccampus.ca\/simplestats\/chapter\/4-3-variance\/<\/a>) for details.)<\/span>\u00a0[\/footnote]:\u00a0<\/span>\r\n\r\n \r\n\r\n$$r=\\frac{\\Sigma{(x-\\overline{x})(y-\\overline{y})}}{\\sqrt{\\Sigma{(x-\\overline{x})^2}\\Sigma{(y-\\overline{y})^2}}}$$\r\n\r\n \r\n\r\nFrom Section 4.3, we know that $\\Sigma{(x-\\overline{x})}^2$ is the sum of squares of the variable x\u00a0<\/em>(so, SSx<\/sub><\/em>)<\/span>; by analogy,\u00a0<\/span> $\\Sigma{(y-\\overline{y})}^2$ will be the sum of squares of the variable y<\/em> (so, SSy<\/sub><\/em>). When the distances between an observation and the two means are \"cross-multiplied\" before summing (like in the numerator), they are called the sum of products (SPxy<\/sub><\/em>).\u00a0<\/span>\r\n\r\n \r\n\r\nThus we can restate the formula above in the following simplified (and easier to remember) way[footnote]Note that other \"versions\" of the formula for<\/span>\u00a0<\/span>r<\/em>\u00a0<\/span>exist. All of them calculate the same r<\/em>, but are just restated in different term. The two \"versions\" presented in the text above are the simplest. For example, one of the most common ways to express r<\/em> you may find elsewhere (but which is rather hard on the eyes and for purposes of calculation by hand) is this:<\/span>\r\n\r\n \r\n\r\n$$r=\\frac{N\\Sigma{xy} -\\Sigma{x}\\Sigma{y}}{\\sqrt{N\\Sigma{x^2}-(\\Sigma{x})^2)(N\\Sigma{y^2}-(\\Sigma{y})^2}}$$<\/span><\/span>\r\n\r\n[\/footnote]<\/span>:<\/span>\r\n\r\n \r\n\r\n$$r=\\frac{SP_{xy}}{\\sqrt{SS_x SS_y}}$$\r\n\r\n \r\n\r\nExample 10.1(A) provides an empirical application of r<\/em>'s calculation.\r\n\r\n \r\n
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Example 10.1(A) Education and Parental Education, GSS 2018<\/em><\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n \r\n\r\nTable 10.1 lists the years of schooling (our variable y<\/em>) of seven respondents in the GSS 2018<\/em> (NORC, 2019) and the years of schooling of their respective fathers (our variable x<\/em>)[footnote]Here parental education<\/em> is the independent variable and respondent's education<\/em> is the dependent variable, so they are denoted as x<\/em> and y<\/em>, respectively, according to convention. [\/footnote]. While inference with N<\/em>=7 is not a serious proposition, the small observation count allows for a quick calculation for demonstration purposes only. (After all, we already know the correlation coefficient of these exact same two variables from Section 7.2.3; there the SPSS-calculated r <\/em>was equal to\u00a00.413.)\r\n\r\n \r\n\r\nThe rest of the columns in Table 10.1 list the necessary computations (obtaining distances from the mean, squaring distances, summing distances, etc.) to produce SSx<\/sub>, SSy<\/sub><\/em>, and SPxy<\/sub>.<\/em>\r\n\r\n \r\n\r\nTable 10.1 Calculating Pearson's r<\/em>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
$x$<\/td>\r\n$y$<\/td>\r\n\u00a0$(x-\\overline{x})$<\/td>\r\n$(x-\\overline{x})^2$<\/td>\r\n$(y-\\overline{y})$<\/td>\r\n$(y-\\overline{y})^2$<\/td>\r\n\u00a0$(x-\\overline{x})(y-\\overline{y})$<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n8<\/td>\r\n(12-12.4) = -0.4<\/td>\r\n-0.42\u00a0<\/sup>= 0.2<\/td>\r\n(8-13.6) = 5.6<\/td>\r\n5.62\u00a0<\/sup>= 31.4<\/td>\r\n(-0.4)(5.6)=2.2<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n12<\/td>\r\n(6-12.4) = -6.4<\/td>\r\n-6.42\u00a0<\/sup>= 41<\/td>\r\n(12-13.6) = -1.6<\/td>\r\n-1.62\u00a0<\/sup>= 2.6<\/td>\r\n(-6.4)(1.6)=10.2<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n19<\/td>\r\n(12-12.4) = -0.4<\/td>\r\n-0.42\u00a0<\/sup>= 0.2<\/td>\r\n(19-13.6) = 5.4<\/td>\r\n5.42\u00a0<\/sup>= 29.2<\/td>\r\n(-0.4)(5.4)=-2.2<\/td>\r\n<\/tr>\r\n
16<\/td>\r\n16<\/td>\r\n(16-12.4) = 3.6<\/td>\r\n3.62\u00a0<\/sup>= 13<\/td>\r\n(16-13.6) = 2.4<\/td>\r\n2.42\u00a0<\/sup>= 5.8<\/td>\r\n(3.6)(2.4)=8.6<\/td>\r\n<\/tr>\r\n
15<\/td>\r\n12<\/td>\r\n(15-12.4) = 2.6<\/td>\r\n2.62\u00a0<\/sup>= 6.8<\/td>\r\n(12-13.6) = -1.6<\/td>\r\n-1.62\u00a0<\/sup>= 2.6<\/td>\r\n(2.6)(-1.6)=-4.2<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n12<\/td>\r\n(12-12.4) = -0.4<\/td>\r\n-0.42\u00a0<\/sup>= 0.2<\/td>\r\n(12-13.6) = -1.6<\/td>\r\n-1.62\u00a0<\/sup>= 2.6<\/td>\r\n(-0.4)(-1.6)=0.6<\/td>\r\n<\/tr>\r\n
14<\/td>\r\n16<\/td>\r\n(14-12.4) = 1.6<\/td>\r\n1.62\u00a0<\/sup>= 2.6<\/td>\r\n(16-13.6) = 2.4<\/td>\r\n2.42\u00a0<\/sup>= 5.8<\/td>\r\n(1.6)(2.4)=3.8<\/td>\r\n<\/tr>\r\n
\"\\overline{x}\"12.4<\/td>\r\n\"\\overline{y}\"13.6<\/td>\r\n<\/td>\r\nSSx<\/sub><\/em>=63.7<\/strong><\/td>\r\n<\/td>\r\nSSy<\/sub><\/em>=79.7<\/strong><\/td>\r\nSPxy<\/sub><\/em>=19.3<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThen, according to the formula for r<\/em> we have:\r\n\r\n \r\n\r\n$$r=\\frac{SP_{xy}}{\\sqrt{SS_x SS_y}}=\\frac{19.3}{\\sqrt{63.7\\times79.7}}=\\frac{19.3}{71.3}=0.271$$\r\n\r\n \r\n\r\nObviously, this r<\/em>=0.271 is not the same as the SPSS-produced r<\/em>=0.413 we had from Section 7.2.3; in fact, it would be very surprising if they were the same, considering the former is based on N<\/em>=7 while the latter is based on N<\/em>=1,687. The exact value of r<\/em> in the above calculation (r<\/em>=0.271) doesn't matter, and doesn't serve any purpose and shouldn't be interpreted as it exists only as the end result of our demonstration.\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n\r\nFancy trying it out on your own?\r\n\r\n \r\n
\r\n

Do It! 10.1 Calculating Pearson's r<\/em><\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n \r\n\r\nHere are 7 more cases from the same GSS 2018<\/em> dataset. Fill out the table fully and produce r<\/em>.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
$x$<\/td>\r\n$y$<\/td>\r\n\u00a0$(x-\\overline{x})$<\/td>\r\n$(x-\\overline{x})^2$<\/td>\r\n$(y-\\overline{y})$<\/td>\r\n$(y-\\overline{y})^2$<\/td>\r\n\u00a0$(x-\\overline{x})(y-\\overline{y})$<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n12<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n14<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
13<\/td>\r\n13<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
13<\/td>\r\n16<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
14<\/td>\r\n20<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
20<\/td>\r\n16<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
21<\/td>\r\n18<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
\"\\overline{x}\"=<\/td>\r\n\"\\overline{y}\"=<\/td>\r\n<\/td>\r\nSSx<\/sub><\/em>=<\/strong><\/td>\r\n<\/td>\r\nSSy<\/sub><\/em>=<\/strong><\/td>\r\nSPxy<\/sub><\/em>=<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n \r\n\r\nEven if we dismiss the value of the N<\/em>=7 coefficients and go back to r<\/em>=0.413 based on N<\/em>=1,687, we still want to know if this correlation as observed in the sample<\/em> is statistically significant (i.e., generalizable to the population). Thus, we need to test r<\/em>, and we do that through a t<\/em>-test.\r\n\r\n \r\n\r\nThe t<\/em>-test for Pearson's r<\/em> is<\/strong> given by the following formula:\r\n\r\n \r\n\r\n$$t=\\frac{r\\sqrt{N-2}}{\\sqrt{1-r^2}}$$\r\n\r\n \r\n\r\nwith df<\/em>=N<\/em>-2.\r\n\r\n \r\n
\r\n

Example 10.1(B)\u00a0Testing the<\/em>\u00a0Education and Parental Education Correlation, GSS 2018<\/em><\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n \r\n\r\nAs usual, it helps to know what we are testing exactly:\r\n