1.  The relationship of two quantitative variables

2.  What is relationship?  Going/moving together: cooccurrance  Causal effect, dependence  Independence

3.  Example I 35 40 45 50 55 12345 Birth weight (kg) Birth height (cm)

4.  Example II 115 120 125 130 135 140 145 202530354045 Body weight at 10 (kg) Height at 10

5.  The problem of prediction If Mom is 50 kg at 30, what will be the weight of his 10 years old son?

6.  Prediction by means of a line 20 25 30 35 40 45 4050607080 Mom’s body weight (kg) Son’s weight at 10

7.  20 25 30 35 40 45 4050607080 Which is the best predicting line? Mom’s body weight (kg) Son’s weight at 10

8.  The best line is the one that lies closest to the points of the diagram The general formula of a line : f(x) = a + bx

9.  0 80 160 240 320 400 012345 Variable X Variable Y a y = a + bx  parameter ‘a’ = intercept parameter ‘b’ = slope The parameters of a line

10.  Basic terms of prediction  Predicted (dependent) variable: Y  Predicting (independent) variable: X  Linear prediction: Ŷ = a + bX  True Y-value belonging to value x: y  Prediction belonging to x: ŷ = a + bx  Error of prediction for one subject: (y - ŷ) 2  For the best line E((Y - Ŷ) 2 ) is minimal

11.  Basic terms of regression  Thge best predicting line: Regression line  The y =  +  x formula of the regression line: Linear regression function  Determining the regression line: Regression problem  Error of regression = Error variance: Res = E((Y - Ŷ) 2 )  ,  parameters: Regression coefficients

12.  How strong is the relationship between X and Y?  The more X is informative for Y, the smaller Res will be relative to Var(Y), that is the smaller will be Res/Var(Y).  But the greater will be the coefficient of determination:

13.  The coefficient of determination  0  Det(X,Y)  1  A measure of explained variance  Important: Det(X,Y) = Det(Y,X).  Shows the strenght of the linear relationship between X and Y.

14.  The independence of two random variables QUESTION: Does the height of a person depend on gender?

15.  Does birth height depend on birth weight? 35 40 45 50 55 12345 Birth weight (kg) Birth height (cm)

16.  Does variable Y depend on variable X? 20 50 80 205080 0 0,5 1 0 1 YY X X

17.  Does variable Y depend on X? 2 -303 X Y

18.  The independence is mutual IMPORTANT: If Y is independent from X, then X is independent from Y as well.

19.  The covariance  DEFINITION: Cov(X,Y) = E(X·Y) - E(X)·E(Y)  If X and Y are independent, then Cov(X,Y) = 0  The reverse is not always true.

20.  The correlation coefficient  Standardized covariance = correlation coefficient:

21.  Relationship between correlation coefficient and coefficient of determination (  (X,Y)) 2 = Det(X,Y)

22.  Some characteristics of  (X, Y)  -1   (X,Y)  1  If X and Y are independent then  (X,Y) = 0.  If  (X,Y) = 0, that is X and Y are uncorrelated, then X and Y can still be related to each other (U shaped relationship).

23.  Prediction and correlation IQ of father = 130. IQ of son = ??? z(IQ/father) = 2. z(IQ/son) = ??? z(predicted) =  z(predictor) z ŷ =  z x

24.  

25.  

26.  

27.  

28.  

29.  Sample correlation coefficient  Notation: r XY or r  Formula:

30.  (X,Y)-sample H 1 :  XY < 0 H0H0 H 2 :  XY > 0 Condition: X and Y are bivariate normals r  - r 0.05 r  r 0.05 |r| < r 0.05 Significance test of correl. coefficient H 0 :  XY = 0 Computation of r xy (df = n  2)