Exer;Measures of Association_11213

STAT 166: STATISTICS FOR SOCIAL SCIENCES Exercise: Measures of Association

CorrelationAnalysis Aneconomist Aneconomist isinterested isinterested todeterminewheth todeterminewhetherthereis erthereisa a linear ear rela elations onship betwe tween the total tal food food expen xpend ditur tures ( foof ) and total family expenditures (totex ). A random sampleoftenhouseholdswereobservedandthefollowing datawererecorded: foof 38 , 7 7 4 44 , 5 4 8 53 , 1 5 9 95 , 6 2 9 33 , 1 4 1

1.

2.

3.

totex 52 , 6 37 71 , 6 68 1 06 , 4 75 3 01 , 2 05 54 , 4 72

foof 152,606 62,783 16,408 41,131 96,671

totex 21 9 , 63 0 1 4 5 , 07 2 2 9 , 50 0 9 1 , 56 7 2 4 3 , 81 3

Assuming that the data follows bivariate normal distribution, compute and interpret the Pearson’s correlationcoefficient. SupposethedatadonotfollowtheNormaldistribution, compute for the Spearman’s rank-order correlation coefficient. Also, compute for the Kendall Tau rankordercorrelationcoefficient. Test Testwh wheth ether erthe there reis isan an assoc associa iati tion onbet betwe ween enthe the total total food food expen expendit diture ures s ( foof ) ) and and total total family family expend expenditu itures res (totex ) )

STAT 166 Statistics for Social Sciences First Semester 2012-2013 JCYnion

1


1.

EstimatedPearson’scorrelationcoefficient foof

n m e an stdd tddev

totex

10 10 6 3 , 4 8 5 .0 0 1 3 1 , 6 0 3 .9 0 405 053 38.2756 564 4 92929 29..14506

n

∑ X Y i

i

=

1.11948E + 11

i =1

n

∑ X Y − n X Y s xy

=

s

=

xy

ρ ρ ˆ

(

1.11948E + 11 − 10 * 6348 63485. 5.00 00 * 1316 131603 03.9 .9

i i

i =1

=

n − 1

10 − 1

3155445241

=

r

=

s XY s x s y

3155445241 =

40538.27564*92929.14506

=

0.837

using STATA: . pwcorr foof totex, sig | foof totex -------------+-----------------foof | 1.0000 | | totex | 1.0000 | 0.0025

STAT 166 Statistics for Social Sciences First Semester 2012-2013 JCYnion

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2.

Spearman’srank-ordercorrelationcoefficient foof 38 , 7 74 ( 3 ) 44 , 5 48 ( 5 ) 53 , 1 59 ( 6 ) 95 , 6 29 ( 8 ) 33 , 1 41 ( 2 ) 1 5 2, 6 0 6 ( 1 0 ) 62 , 7 83 ( 7 ) 16 , 4 08 ( 1 ) 41 , 1 31 ( 4 ) 96 , 6 71 ( 9 )

totex 52,637 (2) 71,668 (4) 106,475 (6) 3 01 , 2 05 ( 1 0 ) 54,472 (3) 219,630 (8) 145,072 (7) 29,500 (1) 91,567 (5) 243,813 (9)

d

i

2

d

i

1 1 0 -2 -1 2 0 0 -1 0

1 1 0 4 1 4 0 0 1 0

n

∑ d

2

6 r

s

=

1−

i

i 1 =

3

n

=

− n

1−

6*12 3

10

=

− 10

1 − 0.07273

=

0.9273

usingSTATA: . spearman foof totex Number of obs =

10

Test of Ho: foof and totex are independent Prob > |t| = 0.0001 STAT 166 Statistics for Social Sciences First Semester 2012-2013 JCYnion

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3.

KendallTaurank-ordercorrelationcoefficient a. ObservetheranksofXandY. Household (X) foof (X) (Y) totex (Y)

1 3 2

2 5 4

3 4 6 8 6 10

5 6 2 10 3 8

7 7 7

8 1 1

9 10 4 9 5 6

b. Arrange ArrangetheranksofX theranksofXinascen inascending dingorderwhile orderwhileranksofY ranksofY followX’sarrangement. Household (X) foof (X) totex (Y) (Y)

8 1 1

5 2 3

1 3 2

9 4 5

2 5 4

3 6 6

7 4 10 6 7 8 9 10 7 10 6 8

c. ObservedthenaturalorderoftheranksofY. Household 8 5 1 9 2 3 1 2 3 4 5 6 (X) foof (X) 1 3 2 5 4 6 (Y) totex (Y) 1 + + + + + 3 + + + 2 + + + 5 + 4 + 6 7 Number of positive signs = 40 10 Number of negative signs = 5 6 S=40–5=35 STAT 166 Statistics for Social Sciences First Semester 2012-2013 JCYnion

7 4 10 6 7 8 9 10 7 10 6 8 + + + + + + + + + + + + + + + + + + + + + + + + + + + - 4


ˆxy

=

T xy

S =

! $ # n & " 2 %

=

2S n (n − 1)

=

2*35 10(10 − 1)

=

0.7778

. ktau foof totex Number of obs Kendall's tau-a Kendall's tau-b Kendall's score SE of score

= = = = =

10 0.7778 35 11.180

Test of Ho: foof and totex are independent Prob > |z| = 0.0024 (continuity corrected)

Pearson Spearman Kendall-Tau Ho : ρ=0 ρs = 0 τ xy=0 Ha: ρ≠0 ρs ≠ 0 τ xy≠0 Test Proc: t-test Spearman Kendall-Tau Test Statistic: r rs s Decision RejectHoifp-value<alpha, Rule: Otherwise,failtorejectHo Computation: 0 .0 0 2 5 0 .0 0 0 1 0.0024 Decision: since p-value < alpha, we reject Ho At alpha = 5%, we have sufficient Conclusion: evidence to say that there is an associationbetween foof and andtotex . . STAT 166 Statistics for Social Sciences First Semester 2012-2013 JCYnion

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Exer;Measures of Association_11213

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