Outline l Correlation and Covariance l Bivariate Correlation
Outline l Correlation and Covariance l Bivariate Correlation Coefficient l Types of Correlation l Correlation Coefficient Formula l Correlation Coefficient Computation l Short-cut Formula l Linear Function (Intercept and Slope) (c) 2007 IUPUI SPEA K 300 (4392)
Correlation and Covariance l It asks how two variables are related l When x changes, how does y change? l Underlying information is covariance l Cov(x, y)=E[(x-xbar)(y-ybar)] l Cov(x, y)=Cov(y, x) l Cov(x, x)=Var(x), variance is a special type of covariance (covariance of a variable and itself) (c) 2007 IUPUI SPEA K 300 (4392)
Bivariate Correlation Coefficient l l l (Karl Pearson product moment) correlation coefficient Bivariate correlation coefficient (BCC) for two interval/ratio variables Differentiated from Spearman’s rank correlation coefficient (nonparametric) Differentiated from partial correlation coefficient that controls the impact of other variables No causal relationship imposed. X Y or Y X BCC is used for prediction (c) 2007 IUPUI SPEA K 300 (4392)
Bivariate Correlation Coefficient l l l BCC ranges from -1 to 1 (So does Gamma γ) Covariance component can be negative + means positive relationship; when x increases 1 unit, y increases r unit 0 means no relationship. - means negative relationship; when x increases 1 unit, y decreases r unit. http: //noppa 5. pc. helsinki. fi/koe/corr/cor 7. html (c) 2007 IUPUI SPEA K 300 (4392)
Positive relationship (c) 2007 IUPUI SPEA K 300 (4392)
Negative relationship (c) 2007 IUPUI SPEA K 300 (4392)
No relationship (c) 2007 IUPUI SPEA K 300 (4392)
Correlation Coefficient l Ratio of the covariance component of x and y to the square root of variance components of x and y (c) 2007 IUPUI SPEA K 300 (4392)
Correlation Coefficient (short-cut) Textbook suggests a short-cut formula below but it is not recommended. (c) 2007 IUPUI SPEA K 300 (4392)
Illustration: example 10 -2, p. 526 No x y (x-xbar) (y-ybar) (x-xbar)^2 (y-ybar)^2 (x-xbar)(y-ybar) 1 43 128 -14. 5 -8. 5 210. 25 72. 25 123. 25 2 48 120 -9. 5 -16. 5 90. 25 272. 25 156. 75 3 56 135 -1. 5 2. 25 4 61 143 3. 5 6. 5 12. 25 42. 25 22. 75 5 67 141 9. 5 4. 5 90. 25 20. 25 42. 75 6 70 152 12. 5 156. 25 240. 25 193. 75 Sum 345 819 561. 5 649. 5 541. 5 Mean 57. 5 137 SSxx Correlation coefficient 0. 8967 (c) 2007 IUPUI SPEA K 300 (4392) SSyy SPxy
Hypothesis Test l How reliable is a correlation coefficient? l r is a random variable drawn from the sample; ρ is its corresponding parameter l H 0: ρ =0, Ha: ρ ≠ 0 l TS follows the t distribution with df=n-2 l If H 0 is not rejected, r is not reliable regardless of its magnitude (ρ =0) (c) 2007 IUPUI SPEA K 300 (4392)
Illustration: Example 10 -3, p. 529 l Step 1. H 0: ρ =0, Ha: ρ ≠ 0 l Step 2. α=. 05, df=4 (=6 -2), CV=2. 776 l Step 3. TS=4. 059, r=. 897 l Step 4. TS>CV, reject H 0 at the. 05 level l Step 5. ρ ≠ 0 (c) 2007 IUPUI SPEA K 300 (4392)
Linear function l A function transforms input into output in its own way l Ex: y=square_root(x). Whey you put x (input) into the funciton square_root(), you will get y (output). l Linear function consists of a intercept and linear combinations of variables and their slops. Y= a + b. X + c. X 2… l Slopes are constant (c) 2007 IUPUI SPEA K 300 (4392)
Intercept and Slope of a function l A linear model: Y = a + b X l Dependent variable Y to be explained l Independent variable X that explains Y l Y-Intercept a: the coordinate of the point at which the line intersects Y axis. l Slope b: the change of dependent variable Y per unit change in independent variable X (c) 2007 IUPUI SPEA K 300 (4392)
Illustration (c) 2007 IUPUI SPEA K 300 (4392)
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