Correlation. Barrow, Statistics for Economics, Accounting and Business Studies, 4 th edition Pearson Education Limited 2006
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1 Correlation Lecture 5
2 Correlation Correlation examines the relationships between pairs of variables, for example between the price of doughnuts and the demand for them between economicgrowth and life expectancy between hair colour and hourly wage between rankings Such analyses can be useful for formulating policies
3 Positive Correlation income expenditure E.g. income and food expenditure
4 Negative Correlation price quantity E.g. demand and price
5 Zero (absence of) Correlation sales of cameras Price of fish E.g. sales of cameras and the price of fish
6 The Correlation Coefficient, r Measures the strength of association between two variables, and Positivecorrelation: r > 0 Negativecorrelation: r < 0 Zero correlation: r 0 1 r +1 The closer r is to +1 (or 1), the closer the points lie to a straight line with positive (negative) slope
7 Positive Correlation r = E.g. income and food expenditure
8 Negative Correlation r = E.g. demand and price
9 Zero (absence of) Correlation r = 0 E.g. sales of cameras and the price of fish
10 Formula for the Correlation Coefficient Use either r r or, equivalently ) ) ( )( ) ( ( n n n r
11 Why the Formula Works More +ve than -ve points, hence r > 0
12 Calculation of r Between Growth and Birth Rates Country Birth rate GNP growth Brazil Colombia Costa Rica India , Mexico , Peru , Philippines , Senegal , South Korea Sri Lanka Taiwan Thailand Total , ,139.7
13 Calculation of r Between Growth and Birth Rates (cont.) Using the second formula, r ( n n ( ) )( n ( ) ) we obtain r 11, ( )(1 1, ) 0.84
14 Chart of Birth Rate Against Growth Rate rate Birth Growth rate
15 Notes About r The correlation between and is the same as between and it does not matter which variable is labelled and which r is independent of units of measurement If the birth rate were measured as births per 100 population (3.0,.9,...) r would still be 0.84 Correlation does not imply pycausality
16 Is the Result Statistically Significant? The correlation coefficient r is just like any other summary statistic H 0 : = 0 versus H 1 : 0where is the population correlation coefficient The null asserts no genuine association between and ; the sample correlation observed ed is just stdue eto (bad) luck The test statistic is t r n 1 r ~ tn
17 Hypothesis Testing Choose = 5%. This implies t* 10 =.8 Cl Calculate l the test statistic i t r n r Hence we reject H 0. There does seem to be genuine association between growth and birth rates
18 Spearman Rank Correlation Coefficient, r s Used for examining relationships between ranks of variables e.g. rankings of school performance and spending per pupil Can be useful if data contains extreme values that may distort themeans and other summarystatisticsstatistics Based on looking at differences in ranks by each variable
19 Country Birth rate GNP growth Rank Rank d d Brazil Colombia Costa Rica India Mexico Peru Philippines Senegal South Korea Sri Lanka Taiwan Thailand Total r s 6 d n( n 1) (144 1) 0.676
20 Hypothesis Testing With r s r s does not follow a standard distribution so we have to use special tables H 0 : s =0 vs H 1 : s 0 so we can reject the null n 10% 5% % 1% See Table A6 in Barrow for full version
21 Summary Correlation measures the association between two variables correlation coefficient, r, if we have data values Spearman rank correlation coefficient, if just have ranks, or have extreme values Note that association does not infer causation
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