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INTRODUCTION DEFINITION TEST STATISTICS KRC TABLE EXAMPLES PROPERTIES

Topic :- Kendall Rank correlation

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Sir Maurice George Kendall

Sir Maurice George Kendall, FBA (A british Academy) (6 September 1907 – 29 March 1983) was a British statistician, widely known for his contribution to statistics. The Kendall tau rank correlationis named after him

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Prof. Maurice George Kendall

Sir Maurice Kendall (1907-1983) President of the IASC (International Accounting

standard committee) (1979-1981)London, UK

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IN STATISTICS, THE KENDALL RANK CORRELATION COEFFICIENT, COMMONLY REFERRED TO AS KENDALL'S TAU COEFFICIENT (AFTER THE GREEK LETTER Τ), IS A STATISTIC USED TO MEASURE THE ORDINAL ASSOCIATION BETWEEN TWO MEASURED QUANTITIES

Kendall Rank correlation

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Hypothesis

H0: There is no association between two variables H1: There is an association between two variables

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Test Statistics

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Continued Test StatisticsWhere nc is the number of concordant pairs and nd

is the number of discordant pairsAnd  no. of possible pairs i.e : a,b,c,d is an arrange data so the no. of

possible pairs are  =6as shown below(a,b),(a,c),(a,d),(b,c), (b,d), (c,d)For these pairs we subtract 2nd value from 1st, if the value become positive so it will be concordant and if negative then the pair is said to be discordant

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Critical values of τ for α= 0.05 and α =0 .01.

N 4 5 6 7 8 9 10

.05 1 .8000 .7333 .6190 .5714 .5000 .4667

.01 - 1 .8667 .8095 .7143 .6667 .6000

Decision rule: if cal is greater than tab then we reject Ho,

OR)

If cal > tab (Reject Ho)

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WHERE =

For n>10 we use normal

approximation

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SPEARMAN'S RANK CORRELATION IS SATISFACTORY FOR TESTING A NULL

HYPOTHESIS OF INDEPENDENCE BETWEEN TWO VARIABLES BUT IT IS DIFFICULT TO

INTERPRET WHEN THE NULL HYPOTHESIS IS REJECTED. KENDALL'S RANK

CORRELATION IMPROVES UPON THIS BY REFLECTING THE STRENGTH OF THE

DEPENDENCE BETWEEN THE VARIABLES BEING COMPARED.

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Example

Assume that we are ranking the grades and IQ levels for a group of students

Students

Grades IQ

a 1 1b 2 4e 5 2c 3 3d 4 5

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1. Testing of hypothesis: H0: The IQ is independent of Grades

H1: The IQ is not independent of Grades

2. Level of significance α=0.05

3. Test statistics:

Where nc is the number of concordant pairs and nd is the number of discordant pairs

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Calculations4.Calculations: Sort these in an order

such that one of the variables is in the proper order, let do this for grades we will then examine the variable IQ for the value of Tau (),

Students

Grades IQ

a 1 1b 2 4c 3 3d 4 5e 5 2

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Calculations Continued

 Pairs (1,4), (1,3), (1,5) ,(1,2), (4,3), (4,5), (4,2), (3,5), (3,2), (5,2)Where Nc=6 and Nd=4 and =10

So =6-4/10 =0.2

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Critical Region And Conclusion :

Critical Region: Tab(5,0.05)=0.8000Where cal= 0.2Conclusion : Since cal is not greater than tab value so we donot reject our Ho and conclude that the IQ is independent of grades

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1. T H E  D E N O MI N AT O R  I S T H E T O TA L N U MB ER O F PA I R C O M B I N AT I O N S, S O T H E C O EF F I C I E N T MU S T B E I N T H E R A N G E − 1  ≤   Τ   ≤   1 .

2. I F T H E A G R E EM EN T B E T W EE N T H E T W O R A N K I N G S I S P ER F EC T ( I . E . , T H E T W O R A N K IN GS A R E T H E S A ME) T H E C O EF F IC IE N T H A S VA LU E 1 .

3. I F T H E D I S A G R EE ME N T B E T W E EN T H E T W O R A N K I N G S I S P E R F E C T ( I . E . , O N E R A N K IN G I S T H E R EV E R S E O F T H E O T H E R ) T H E C O E F F I C IE N T H A S VA LU E − 1 .

4. I F  X   AN D  Y   A R E   I N D EP EN D EN T, T H E N W E W O U L D

EX P E C T T H E C O E F F IC IE N T T O B E A P P R OX I MAT E LY Z ER O.

Kendall Rank correlationProperties