 The relationship of two quantitative variables

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

 Example I Birth weight (kg) Birth height (cm)

 Example II Body weight at 10 (kg) Height at 10

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

 Prediction by means of a line Mom’s body weight (kg) Son’s weight at 10

 Which is the best predicting line? Mom’s body weight (kg) Son’s weight at 10

 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

 Variable X Variable Y a y = a + bx  parameter ‘a’ = intercept parameter ‘b’ = slope The parameters of a line

 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

 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

 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:

 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.

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

 Does birth height depend on birth weight? Birth weight (kg) Birth height (cm)

 Does variable Y depend on variable X? , YY X X

 Does variable Y depend on X? X Y

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

 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.

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

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

 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).

 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

 

 

 

 

 

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

 (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)