Statistical Dependence

• Gazdaságtudományi Kar •

Gazdaságelméleti és Módszertani Intézet

Statistical Dependence

Petra Petrovics



Statistical Dependence Definition: Statistical dependence exists when the value of some variable is dependent upon or affected by the value of some other variable.

Statistical Dependence/ Stochastic Dependence

Independent variables

Functional relation



Types of Dependence • association – between two nominal data – Yule (Y) – Csuprov (T) – Cramer (C) or (V)

• mixed – between a nominal and a ratio data – H; H2 or η; η2

• correlation – among ratio data



I. Association a) Yule-measure B (1) f11 f01 f.1

A (1) A (0) Total

B (0) f10 f00 f.0

Total f1. f0. n

Where: f11, f10, f01, f00 the observed frequencies f1. , f0. , f.1 , f.0 the marginal frequencies f  Y  11 f 11 

f 00  f 10  f 01 f 00  f 10  f 01

• Y=0 • 0  Y  1 • Y = 1

Only when the number of categories of both variables is two!

the variables are independent statistical dependence functional relation



• In case of statistical dependence: f 11 f 01  f 10 f 00



f 11  f 00  f 10  f 01

• If the variables are independent: f 11 f01  f 10 f00

 f 11  f00  f 10  f01  f 11  f00  f 10  f01  0



Example: • Suppose that a certain subject is offered to first year and second year students on a pass-fail basis only. An advisor is interested in determining whether there is a relationship between the student’s grade and year. • Data for the test were obtained from last semester’s classes: Class standings Grade

First year (1)

Second year (0)

Total

Pass (1)

8

12

20

Fail (0)

10

70

80

Total

18

82

8  70  12  10 440 Y   0.65 8  70  12  10 680

100 Medium-strong dependence



b) Contingency table • there are s categories of the row/column variable: A1, A2, … , As • there are t categories of the row/column variable: B1, B2, … , Bt where s < t j i

A1 A2 ... Ai ... As 

B1

B2

...

Bj

...

Bt



f11 f21 ... fi1 ... fs1 f.1

f12 f22 ... fi2 ... fs2 f.2

... ... ... ... ... ... ...

f1j f2j ... fij ... fsj f.j

... ... ... ... ... ... ...

f1t f2t ... fit ... fst f.t

f1. f2. ... fi. ... fs. n



The measure for statistical dependence in case of contingency table • T – measure, when s = t 2

χ T n s - 1 t - 1

where



• C – measure, when s < t C

T Tmax



2  s t f ij - f ij 2 χ    f ij i 1 j 1

where Tmax 

0  C  0.3  weak dependence 0.3  C  0.7  medium-strong dependence 0.7  C  1  strong dependence

4

s-1 t -1



The variables are independent, when…

f ij

f i.  f .j n

 ij

or

f ij f i.



f .j n

i.e.

f ij 

f i.  f .j n

f  expected frequencie s for case of independence 

f i.  f.j n



Example A manufacturer of printed circuit boards has determined that boards classified as nonconforming nearly always have one of three defects: a component on the board is either missing, damaged or raised (installed improperly). The boards are produced on three machines (A, B and C). To determine whether there is a relationship between the type of nonconformity and the machine, a sample of 500 nonconforming boards was obtained:


Machine


Type of nonconformity missing

damaged

raised

Total

A

50

80

120

250

B

60

55

10

125

C

65

45

15

125

Total

175

180

145

500

Question: • Is the type of nonconformity related to the machine used for production? s=3 t=3

T-measure



Solution  f ij - f ij 2

Type of nonconformity and machine

f ij

f ij

Missing, A

50

87.50

16.071

Missing, B

60

43.75

6.036

Missing, C

65

43.75

10.321

Damaged, A

80

90.00

1.111

Damaged, B

55

45.00

2.222

Damaged, C

45

45.00

0.000

Raised, A

120

72.50

31.121

Raised, B

10

36.25

19.009

Raised, C

15

36.25

12.457

Total

500

500.00

2 = 98.35

T

98.35  0.3136 500  3 - 1  3 - 1

 f ij

Medium-strong dependence



Exercise Is there any relationship between: - gender & employment category; Association

- gender & current salary? Mixed



Association File / Open / Data… / Employee data

Analyze / Descriptive Statistics / Crosstabs: Gender – Employment category

fij f*ij



Output View Symmetric Measures

Nominal by Nominal

Phi Cramer's V

N of Valid Cases

Value ,409 ,409 474

Approx. Sig. ,000 ,000

a. Not assuming the null hy pothesis. b. Using t he asy mptotic standard error assuming the null hy pothesis.

0  C  0,3  weak dependence 0,3  C  0,7  medium-strong dependence 0,7  C  1  strong dependence

There is a medium-strong dependence between gender & employment category. We can accept that statement at every significance



Output View

4.6% of women are manager.



33.1% of people are male and clerical.

The custodials are men.

Number of people / cases



Mixed dependence Analysis of Variance • One-way analysis of variance is a technique used to compare means of two or more samples. • In case of a qualitative and a quantitative variable.



Differences - variances • dji  total difference: difference between an employee’s production and the grand mean

d ji  x ji - x

• Wji  within-column difference: difference between an employee’s production and his group’s mean Wji  x ji - x j • Bji  between-column difference: difference between the group’s mean and the grand mean

Bj  x j - x



dji = Wji + Bj 

2 2 2       x x  x x  x x   ji   ji  j j j i

j i

 SS = SSW + SSB  2 = 2W + 2B

j



Measures of mixed dependence H

SSB  SS

σ 2B σ2

or

SSB σ 2B H   2 SS σ 2

Where: • H = H2 = 0 the variables are independent • H = H2 = 1 functional relation • 0H1 0  H  0.3  weak dependence 0.3  H  0.7  medium-strong dependence 0.7  H  1  strong dependence

• 0  H2  1

Statistical dependence



Example Marks

I.

II.

III.

Faculty

Total

Excellent (5)

20

20

20

60

Good (4)

30

50

40

120

Medium (3)

25

35

55

115

Satisfactory(2)

20

35

80

135

0

5

20

25

95

145

215

455

Fail (1) Total

Is there any dependence between the average marks and faculties?



Faculties

n

x

σ2

Faculty I.

95

3.53

1.09

Faculty II.

145

3.31

1.18

Faculty III.

215

2.81

1.27

Total

455

3.12

1.29

or

x

n  x n j

j

σ  2 W

2 n  σ  j j

n

j



 n j x j  x 

2

σ 

n

j



95  3.53  145  3.31  215  2.81  3.12 455

95 1.09  145  1.18  215  1.27   1.2 455

σ 2  σ 2B  σ 2W 2 B

j

σ 2B  1.29 - 1.2  0.09

953.53  3.12  1453.31  3.12  2152.81  3.12   0.09 455 2

0.09 H  0.2641 1.29

2

H 2  6.81 %

2



Mixed dependence Analyze / Compare Means / Means…



Output View Report Current Salary Gender Female Male Total

Mean $26,031.92 $41,441.78 $34,419.57

N 216 258 474

Std. Dev iat ion $7,558.021 $19,499.214 $17,075.661

This table shows you the central tendency & dispersion of the dependent variable (current salary) grouped by the independent variable (gender).

ANOVA Table

Current Salary * Gender Between Groups (Combined) (SK) Within Groups (SB) Total (S)

Sum of Squares 2,8E+010 1,1E+011 1,4E+011

df 1 472 473

Mean Square 2,792E+010 233046530,5

F 119,798

Sig. ,000

Measures of Association

H Et a Current Salary * Gender

,450

Et a Squared ,202

H2 

SK S

%; proportion of variance in the dependent variable explained by differences among groups



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Statistical Dependence

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