Statistical Dependence

Átírás

1 Statistical Dependence Petra Petrovics

2 Statistical Dependence Deinition: Statistical dependence exists when the value o some variable is dependent upon or aected by the value o some other variable. Independent variables Statistical Dependence/ Stochastic Dependence Functional relation

3 Types o Dependence association between two nominal data Yule (Y) Csuprov (T) Cramer (C) or (V) mixed between a nominal and a ratio data H; H or η; η correlation among ratio data

4 a) Yule-measure I. Association B (1) B (0) Total A (1) A (0) Total.1.0 n Where: 11, 10, 01, 00 the observed requencies 1., 0.,.1,.0 the marginal requencies Y Only when the number o categories o both variables is two! Y = 0 the variables are independent 0 Y 1 statistical dependence Y = 1 unctional relation

5 In case o statistical dependence: I the variables are independent:

6 Example: Suppose that a certain subect is oered to irst year and second year students on a pass-ail basis only. An advisor is interested in determining whether there is a relationship between the student s grade and year. Data or the test were obtained rom last semester s classes: Grade First year (1) Class standings Second year (0) Total Pass (1) Fail (0) Total Medium-strong dependence Y

7 b) Contingency table there are s categories o the row/column variable: A 1, A,, A s there are t categories o the row/column variable: B 1, B,, B t where s < t i B 1 B... B... B t A t 1. A t A i i1 i... i... it i A s s1 s... s... st s t n

8 The measure or statistical dependence in case o contingency table T measure, when s = t T n χ s - 1 t - 1 where C measure, when s < t χ s i1 t 1 i - i i T C where T 4 max T max 0 C 0.3 weak dependence 0.3 C 0.7 medium-strong dependence 0.7 C 1 strong dependence s t - 1-1

9 The variables are independent, when i i. i.. i. or i.e. i n n n. i. i expected requencies or case o independence i. n.

10 Example A manuacturer o printed circuit boards has determined that boards classiied as nonconorming nearly always have one o three deects: 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 o nonconormity and the machine, a sample o 500 nonconorming boards was obtained:

11 Machine Type o nonconormity missing damaged raised Total A B C Total Question: Is the type o nonconormity related to the machine used or production? s=3 t=3 T-measure

12 Type o nonconormity and machine Solution Missing, A Missing, B Missing, C Damaged, A Damaged, B Damaged, C Raised, A Raised, B Raised, C Total = T i i i - i i Medium-strong dependence

13 Exercise Is there any relationship between: - gender & employment category; - gender & current salary? Association Mixed

14 Association File / Open / Data / Employee data Analyze / Descriptive Statistics / Crosstabs: Gender Employment category i * i

15 Output View Symmetric Measures Nominal by Nominal N o Valid Cases Phi Cramer's V a. Not assuming the null hy pothesis. b. Value Approx. Sig.,409,000,409, 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 signiicance

16 Output View 4.6% o women are manager. 33.1% o people are male and clerical. The custodials are men. Number o people / cases

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

18 Dierences - variances d i total dierence: dierence between an employee s production and the grand mean d i x i W i within-column dierence: dierence between an employee s production and his group s mean W i x i - x B i between-column dierence: dierence between the group s mean and the grand mean - x B x - x

19 i d i = W i + B x - x x - x x - x i i i SS = SS W + SS B = W + B

20 Measures o mixed dependence SS SS σ σ B H B or Where: H = H = 0 the variables are independent H = H = 1 unctional relation 0 H 1 0 H 0.3 weak dependence SS SS 0.3 H 0.7 medium-strong dependence 0.7 H 1 strong dependence 0 H 1 σ σ B B H Statistical dependence

21 Example Marks I. II. III. Faculty Total Excellent (5) Good (4) Medium (3) Satisactory() Fail (1) Total Is there any dependence between the average marks and aculties?

22 σ σ σ W B Faculties Gazdaságtudományi Kar n x Faculty I Faculty II Faculty III Total σ n B n σ n σ W H n x or x 3. 1 n 455 σ B H 1. x x n % σ 0.09

23 Mixed dependence Analyze / Compare Means / Means

24 Output View Current Salary Gender Female Male Total Report Mean N Std. Dev iation $6, $7, $41, $19, $34, $17, This table shows you the central tendency & dispersion o the dependent variable (current salary) grouped by the independent variable (gender). ANOVA Table Current Salary * Gender Between Groups Within Groups Total (S) (S B ) (Combined) (S K ) Sum o Squares d Mean Square F Sig.,8E+010 1,79E ,798,000 1,1E ,5 1,4E Measures o Association Current Salary * Gender H Eta Eta Squared,450,0 H S S K %; proportion o variance in the dependent variable explained by dierences among groups

25 Thanks or your attention!