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The relationship of two quantitative variables

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What is relationship?

Going/moving together: cooccurrance Causal effect, dependence Independence

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Example I

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Birth weight (kg)

Bir

th h

eig

ht

(cm

)

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Example II

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130

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145

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Body weight at 10 (kg)

Hei

gh

t at

10

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The problem of prediction

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

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Prediction by means of a line

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Mom’s body weight (kg)So

n’s

wei

gh

t at

10

7

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45

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Which is the best predicting line?

Mom’s body weight (kg)So

n’s

wei

gh

t at

10

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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

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0

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Variable X

Var

iab

le Y

a

y = a + bx

parameter ‘a’ = interceptparameter ‘b’ = slope

The parameters of a line

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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

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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

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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:

Det X Ys

Var YVar Y s

Var Y( , )

Re( )

( ) Re( )

1

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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.

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The independence of two random variables

QUESTION:Does the height of a person

depend on gender?

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Does birth height depend on birth weight?

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45

50

55

1 2 3 4 5

Birth weight (kg)

Bir

th h

eig

ht

(cm

)

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Does variable Y depend on variable X?

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80

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0

0,5

1

0 0,5 1

Y Y

X X

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Does variable Y depend on X?

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-3 0 3

X

Y

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The independence is mutual

IMPORTANT:

If Y is independent from X,

then X is independent from Y as well.

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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.

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The correlation coefficient

Standardized covariance = correlation coefficient:

( , )( , )

( ) ( )X Y

Cov X YD X D Y

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Relationship between correlation coefficient and coefficient of

determination

((X,Y))2 = Det(X,Y)

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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).

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Prediction and correlation

IQ of father = 130. IQ of son = ???

z(IQ/father) = 2. z(IQ/son) = ???

z(predicted) = z(predictor)

zŷ = zx

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Sample correlation coefficient

Notation: rXY or r Formula:

XYXY

X Yr

ss s

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(X,Y)-sample

H1: XY < 0 H0H2: XY > 0

Condition: X and Y arebivariate normals

r -r0.05 r r0.05|r| < r0.05

Significance test of correl. coefficient

H0: XY = 0

Computation of rxy (df = n 2)