Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet Introduction to Multiple Correlation

Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet

Introduction to Multiple Correlation Roland Szilágyi Ph.D. Associate professor


Correlation describes the strength of a relationship, the degree to which one variable is linearly related to another

Regression shows us how to determine the nature of a relationship between two or more variables

• X (or X1, X2, … , Xp): known variable(s) / independent variable(s) / predictor(s) • Y: unknown variable / dependent variable

• causal relationship: X „causes” Y to change


Covariance • A measure of the joint variation of the two variables; • An average value of the product of the deviations of observations on 2 random variables from their sample means.

x  x    y  y   Cx, y   n 1

– – – –

ranges from -  to +; C = 0, when X and Y are uncorrelated; its sign shows the direction of correlation it doesn’t measure the degree of relationship!!!


Covariance-variance matrix • A covariance matrix describes correlation among variables. • The diagonal elements (covariances of variables with themselves) are always equal to the variances. • Shows the direction of the relationship.   c yy c1 y  c py   s yy 2     2   c1 y c11   c1 y s11 C              2  c  c  c c  s pp  1p pp   py  py


Coefficient of correlation Σd x d y C r = s xs y d 2x d 2y

• • • • •

Pearson correlation A measure of how closely related two data series are. Its sign shows the direction of correlation It measures the strength of correlation 0 < r  < 1  statistical dependence r = 0  X and Y are uncorrelated  r = -1  negative ☻ r = 1  positive ☺ • You can use only in case of linear relationship!


Matrix of correlation • A correlation matrix describes correlation among variables. • The diagonal elements (correlations of variables with themselves) are always equal to 1.00. • Direction & strength!  ryy   r1 y R   r  py

r1 y  r1 p   1   r11   r1 y         rpp   rpy

1 r1 p

       1 


Partial correlation coefficient ry1.2 

ry2.1 

ry1  ry2  r12 2 y2

2 12

(1 - r )(1 - r ) ry2  ry1  r12 2 y1

2 12

(1 - r )(1 - r )


Inverse of correlation matrix  q yy q  1y   1 R   q jy    q py

q y1 q11  q j1  q p1

     

ryj .1, 2..., p 

q yj q1 j  q jj  q pj  q yj q yy q jj

     

q yp   q1 p     q jp     q pp 


Coefficient of partial determination r

2

yj .1,2...,p

  q yj     q yy q jj   

2

The squared partial correlation coefficient is called coefficient of partial determination, which measures the marginal contribution of one X variable when all others are already included in the model. Hence, the proportion of variance in the dependent variable explained by Xj, which cannot be explained by X1, X2,…Xj-1, Xj+1,…,Xp variables.


Coefficient of Multiple Correlation • It expresses the combined effect of all the variables acting on the dependent variable.

Ry.1, 2 

r  r  2ry1ry 2 r12 2 y1

Ry.1, 2,...,p

2 y2

1 r

2 12

1  1 q yy


Coefficient of Multiple Determination • How many percent of the variation of the dependent variable can be explained by all independent variables.

R

2 y .1, 2 ,...,p

1  1 q yy

SSR S yˆ R   SST S y 2


5. Adjusted coefficient of multiple determination •

It enables to compare the multiple determination coefficient among populations/samples with different size and different number of independent variables as it controls for the number of samples/population size (n) and the number of independent variables (p)

n 1 2 R  1  (1  R ) n  p 1 2


Example y Turnover

x1-Capital

x2-Number of employees

1

35

54

98

2

27

52

120

3

42

50

95

4

47

58

145

5

53

82

184

6

45

72

106

7

61

120

240

8

58

108

175

9

65

92

165

10

77

122

202


C=

y

x1

x2

y

221.11

372.7

559.56

x1

372.67

799.3

1222.9

x2

559.56

1223

2367.8


y

R=

x1

x2

y

x1

x2

1

0.886

0.7733

1

0.8889

1


Partial correlation betwen Turnover and Capital controled for Number of employees

ry1.2  0.6853 vs. ry1  0.886


Partial correlation betwen Turnover and Number of employees controled forCapital

ry2.1  -0.069 vs. ry 2  0.773


R=

0.887

R2=

0.7868

R2 adj=

0.7259


Thanks for your attention!

Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet Introduction to Multiple Correlation

Recommend Documents