Statistical Dependence

Miskolci Egyetem Gazdaságtudományi KarÜzleti Információgazdálkodási és Módszertani Intézet

Petra PetrovicsPhD Student

Statistical Dependence


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 DependenceIndependent Functional

variables relation


Types of dependence

• association – between two nominal data– Yule (Y)– Csuprov (T)

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

• correlation – among ratio data


I. Association

Where: f11, f10, f01, f00 the observed frequencies

f1. , f0. , f.1 , f.0 the marginal frequencies

• Y = 0 the variables are independent• 0 Y 1 statistical dependence Y = 1 functional relation

A (1) B (0) Total

X (1) f11 f10 f1.

Y (0) f01 f00 f0.

Total f.1 f.0n

01100011

01100011

f f f f

f f f f Y

a) Yule-measure

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


• In case of statistical dependence:

• If the variables are independent:

f f f f f

f

f

f01100011

00

01

10

11

0 f f f f f f f f f

f

f

f0110001101100011

00

01

10

11


Example:• Suppose that a certain elective is offered to freshmen and

sophomores on a pass-fail basis only. An advisor is interested in determining whether there is a relationship between the student’s grade and class standings.

• Data for the test were obtained from last semester’s classes:

GradeClass standings

Freshman (1) Sophomore (0) Total

Pass (1) 8 12 20

Fail (0) 10 70 80

Total 18 82 100

0.65 680

440

10 12 70 8

10 12 70 8 Y

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 B1 B2... Bj

... Bt

A1 f11 f12... f1j

... f1t f1.

A2 f21 f22... f2j

... f2t f2.

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

Ai fi1 fi2... fij

... fit fi.

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

As fs1 fs2... fsj

... fst fs.

f.1 f.2... f.j

... f.tn


The measure for statistical dependence in case of contingency table

• T – measure, when s = t

• C – measure, when s < t

s

1i

t

1j ij

2ijij2

2

f

f -f χe wher

1-t 1-s n

χ T

1-t1-s

T where T

T C 4max

max

0 C 0,3 weak dependence0,3 C 0,7 medium-strong dependence0,7 C 1 strong dependence


The variables are independent, when…

n

f f f i.e.

n

f

f

f or

nf

f

f .ji.ij

.j

i.

iji.

.j

ij

n

f f sfrequencie expected f

.ji.ij


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:


Question:• Is the type of nonconformity related to the

machine used for production?

MachineType of nonconformity

Totalmissing damaged raised

A 50 80 120 250

B 60 55 10 125

C 65 45 15 125

Total 175 180 145 500

s=3t=3 T-measure


Type of nonconformity and machine

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

Solution ijf

ij

2ijij

f

f -fijf

0.3136 1-3 1-3 500

98.35 T

Medium-strong

dependence


Mixed dependenceAnalysis 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

• Bji within-column difference: difference between an employee’s production and his group’s mean

• Kji between-column difference: difference between the group’s mean and the grand mean

x -x d jiji

x -x B jjiji

x -x K jj


dji = Bji + Kj

SS = SSB + SSK

2 = 2

B + 2K

x -x x -x x -x 2

jj

2

j ijji

2

j iji


Measures of mixed dependence or

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 dependence0,7 H 1 strong dependence

• 0 H2 1

σ

σ

SS

SS H

2

2KK

σ

σ

SS

SS H

2

2KK2

Statistical dependence


Example

MarksI. II. III.

TotalFaculty

Excellent (5) 20 20 20 60

Good (4) 30 50 40 120

Medium (3) 25 35 55 115

Satisfactory(2) 20 35 80 135

Fail (1) 0 5 20 25

Total 95 145 215 455

Is there any dependence between the average marks and faculties?


Faculties n

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

x 2σ

1.2 455

1.27 215 1.18 145 1.09 95

n

σ n σ

j

2jj2

B

0.09 1.2 - 1.29 σ σ σ σ 2K

2B

2K

2

% 6.81 H 0.2641 1.29

0.09 H 2

455

12.381.221512.331.314512.353.395n σ

2222j

K

j

j

n

xxor

or 12.3455

81.221531.314553.395nx j

j

j

n

x


Exercise 1

• The workers of a company are grouped according to their position and sex:

• Is there a relationship between position and sex?

Man Woman Total

White-collar worker 20 30 50

Blue-collar worker 230 170 400

Total 250 200 450


Exercise 2

• In a town doctors are grouped in the following way:

• Find the type of statistical dependence and determine the strength of the relationship.

Man Woman Total

Internist 122 101 223

Gynecologist 92 10 102

Optometrist 7 35 42

Others 806 776 1,582

Total 1,027 922 1,949


Exercise 3

Calculate H and H2-measures.

Number of passengers (million)

Distance of the trip (km)

Mean Standard deviation

Bus 458 15.4 4.5

Train 60 34.6 15

Private bus 6 182 70

Total 524 … …


Exercise 4• In a supermarket there was a survey among those who buy

chips. 33 of the 100 persons who were asked bought Chio chips, one quarter of them bought Pom Bär, one tenth of them bought Lay’s. They spent 104 HUF on the average on chips. They spent 98 HUF on Chio chips and 74 HUF on Cerbona on the average. The 20 persons who bought Chee-tos all chose the same chips on sale which cost 120 HUF. The standard deviation of the money spent on Chio is 23 HUF, spent on Pom-Bär is 30 HUF, spent on Lay’s is 8 HUF, i.e. 13.56% and in case of Cerbona it’s 14.86%.

• Create a table using these data and fill in the gaps.• Determine the strength of relationship between the type of

the chips and the money spent on them.


Exercise 5• In a shoe-factory the relationship between the sex and the education of the

2,500 employees was examined.• 60% of the workers is man, 16% of the men has university degree and 24% of

them has primary qualification. Half of those who has primary education is man, for those who has secondary education the principle of indipendence is realized.

• Fill in the following table and determine the relationship between sex and qualification!

EducationSex Total

Man Woman

Primary

Secondary

University

Total


Exercise 6

• The following table shows the distribution of workers in a company:

• What can you say about the strength of the relationship between sex and position?

Men Women Total

Managers 4,110 7,545 11,655

Workers 1,576 463 2,039

Total 5,686 8,008 13,694


Exercise 7

• The tourists of hotels in 1996 were groupped in the following way:

• Is the type of tourists related to the type of hotels?

Type of hotelTourists

Five and four

Three Two and one

Total

stars hotels

Hungarian 57 503 601 1,161

Foreign 789 894 494 2,177

Total 846 1,397 1,095 3,338


Thanks for your attention!

Documents

Statistical Dependence