2014 Nonparametic Test

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NONPARAMETRIC TEST


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I. Introduction

II. Two related-samples rank sum test

III. Two independent-samples rank sum test

IV. independent-samples rank sum test


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Part one

Introduction


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1. One sample t test

Model assumptions

When σ is unknown andn


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2 two independent-samples t test

Model assumptions

1. The data from two

samples must follow theassumption of normal

distribution.

2. Two population variancesare e ual.

Test statistics

22

21 σ σ =

)11

(21

2

21

nnS

X X t

c +

−=

2

11

21

2

2

21

2

12

−+

−+−=

nn

nS nS S c

!e"ree of freedom


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3 Paired-samples t test

Model assumptions

The differences amon"

each paired#samples

must come from normal

distribution population.

Test statistics

1,/

0 −=−

= nn s

d t

d

υ

$umbers of pairs


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%. &ne wa' ($&)(Model assumptions

1 The k samples are independent

2 *ach of the populations isnormall' distributed.

+ *ach of the populations has thesame variances.

Test statistics

W

A

W

A

MS

MS

SS

SS F ==

W

A

W

A

MS

MS

SS

SS F ==


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The methods of inferential statisticspresented in previous chapters are called

parametric methods because the' are based

on samplin" from a population with specificparameters, such as the mean , standard

deviation σ, or proportion p.


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Those parametric methods usuall' mustconform to some fairl' strict conditions,

such as a re uirement that the sampledata come from a normall' distributionpopulation and the population variance

amon" "roups are e ual.


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-n this chapter, we would introducenonparametric methods, which do not have suchstrict re uirements.

arametric tests have re uirements about thenature or shape of the populations involved.

$onparametric tests do not re uire thatsamples come from populations with normaldistributions or have an' other particulardistributions. /onse uentl', nonparametric testsare called distribution#free tests.


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(dvanta"es of $onparametric ethods/an be applied to a wide variet' of situationsbecause the' do not have the more ri"idre uirements of the correspondin" parametric

methods. nlike parametric methods, nonparametricmethods can often be applied to nominal leveldata.

suall' involve simpler computations than thecorrespondin" parametric methods and aretherefore easier to understand and appl'.


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!isadvanta"es of $onparametric ethods

Tend to waste information because e actnumerical data are often reduced to a ualitative

form.$ot as efficient as parametric tests, so with anonparametric test we "enerall' need stron"er

evidence 3such as a lar"er sample or "reaterdifferences4 before we re ect a null h'pothesis.


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Comparison Rank sum Test Test statistics

! relatedsamples

Wilco on si"nedrank test T 3n65047 3n 504

! independentsamples

ann#whitne' testZ

k Independentsamples

8ruskal#Wallis Test H or 2 χ

T'pes of rank sum test.


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Part two

Two related-samples rank sumtest


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* ample 1 The uric from 12 workers were split to two parts

and the concentration of uric 9" were measured b'two methods. 9'dronium e chan"in" method anddistillation method. lease compare whether theconcentration of uric 9" is different in twomethods:


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1;

aired samples h'pothesis test

n is the number of pairs


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(1) Set up hypothesis and confirm α

Ho: Md= 0

H1: Md≠ 0

α= 0.05

(2) ompute !

! " =#5 ! $=%%& so choose !=%%

The median of difference


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/ompute the difference of each pairs

/ompute absolute value of the difference

(ffi => or ? to each of rank

/ompute sum of positive and ne"ative ranks

@elect the smaller ranks3T4

Aank the difference in ascendin" order

n650/ompute B

nC50


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T

T T z σ

µ 5.0−−=

24/)12)(1(

4/)1(

++=

+=

nnn

nn

T

T

σ

µ

48

)(

24)12)(1(

5.04/)1(3∑ −−++

−+−=

j j

ct t nnn

nnT z

/orrection formula

"e deleted w#en n

is lar$er t#an %&&'

T

T T

z σ

µ −=

t is the units of the same ranks

-f nC50,we should compute 7-f nC50,we should compute 7


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1!

+


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(3) Look for critical value

nD12, we should look up T#value table.Eecause ++ is in theran"e from1+ to ;5,

so P> 0.05. -n FD0.05level, we can notre ect 90. @o we thinkthe concentration ofuric 9" is same intwo methods.


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Gou can compute the difference ofeach pairs usin" @ @@ b' selectin"H

Analyze > transform > Computevariable…

!ata 1


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$ormalit' tests (nal'BeI !escriptive @tatisticsI * plore

!ependent listI difference

lotsIJ$ormalit' plots with tests

/ontinue

&8

P


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Two related samples rank sum test

(nal'BeI nonparametric testI two#related sampleLTest pairsI M1#M2

Test t'pe I J wilco on

&8


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9ow to write thereport:

The difference of two samples wasnKt chosen from normal population

3W testHP D0.0104 at the si"nificant level 0.05, so wilco on si"nedrank test was used which showed there was no difference in theconcentration of the uric 9" detected b' two methods 3Z D0.%N1P D0.;+O4.

Table 1 the concentration of the uric 9" with two methods in 12

workers


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Part three

Two independent-samples

rank sum test


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The followin" table is the concentration ofhemosiderin in patients with pneumonia and

normal people. lease test whether theconcentration of hemosiderin is different in

different people.

* ample 2


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Patients wit# pneumonia normal people 1 177

68 172

2 7 4

174 47

457 1 2

492 54

199 47

515 52

599 47

2 8 294

68

4

277

44

4


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0%P05P1;+1

Two independent samples h'pothesis test


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(1) set up hypotheses and confirm α H

0: M1 = M2 ( t'o popu ation distri ution are same) H

1: M1 ≠ M2 ( t'o popu ation distri ution are different) α =0.05

(2) Compute test statistics *an+ a the o ser,ationsin ascendin- order and -i,e the order num er ( 0 is a so

ran+ed). f the o ser,ations are same& then -et the

a,era-e of order num er.hoose the ran+ sum 'ith sma er samp e si/e as !.S


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To compute t#e )ilco*on

test statistic+

,% Two samples are

com(ined

,! T#e com(ined alues are

ordered /rom low to #i$#+and

,0 T#e ordered alues are

replaced (1 ranks+ startin$

wit# % /or t#e smallest alue


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0 2ind P alue and draw conclusion.

T.@, 1O+.5, be'ond the ran"e from %2

to Q;, so we should re ect 90. we canthink the concentration of hemosiderinis hi"her in patients with pneumonia.


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formula of approximately normal distribution

1 If n is smaller(n1


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$ormalit' tests (nal'BeI !escriptive @tatisticsI * plore

!ependent listI GRactor list I "rouplotsIJ$ormalit' plots with tests

/ontinue

&8

(ack

P


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Two independent rank sum test (nal'BeI nonparametric testI two#independent samplesLTest variable listI '

Sroupin" variable I "roup !efine "roups "roup 1H1

"roup 2 H2Test t'pe I ann#Whitne' &8


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9ow to write the report:

The distribution of the concentration of hemosiderin wasskewed in normal "roup 3W testHP D0.0104, so Wilco on ranksum test of two independent samples was used. The resultssupported that there was a si"nificant difference in the twopopulation distributions Z D2.55Q P D0.011 .


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+, P/+ 3

One doctor treat c#ronic "ronc#itis patients of

different t*pes usin$ t#e same dru$. #e curati!e

effect are listed in ta"le - . uestion Is t#e curati!e effect different w#en

t#is dru$ was used to treat c#ronic "ronc#itis patients

of different t*pes.


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table 7-4 T#e curati e e//ect o/ some dru$ topatients wit# ! t1pes o/ (ronc#itis


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Set up the hypothesis and confirm α Ho: M1= M2

H1:

α= 0.05

ompute !.S(1) ompute the tota num er of each -roup confirm the ran+

ran-e of each -roup& then compute the a,era-e ran+ of each-roup

(2) ompute ran+ sum of each -roup(%) onfirm !.S !. !1= 112.5& !2= 10 %.5. So != 112.5

(#) ompute /$,a ue

21 M M ≠


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Ta(le 3-4 T#e curati e e//ect o/ some dru$ topatients wit# ! t1pes o/ (ronc#itis

5,%6%&% 7!

5,%&!6%3! 7! 5,%306%89 7!


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In this e a!"le, n 1#$8,n 2#118, so %.& %#8112.5. 'eshoul use *test.

224.1/

814.01+,1+,)2424()$1$1()101101(

1

)/()(1

104.1$.38842+

12/)11+,(118$8

5.02/)11+,($85.8112

3

333

33

==

=−−+−+−

−=

−−−=

==+××

−+×−=

∑

c z z

N N t t c

z

c

j j

3 4ind t#e re5ection re$ion and draw conclusion.

1.22#41. &P 60.05. So 'e can7t re8ect H0. !he curati,e effect is same

to patients 'ith 2 types of ronchitis.


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!ata IWei"ht cases Wei"ht cases b' fre uenc

(nal'Be I$onparametric tests I2 independent samplesLL

Test variables list I &utcome

Sroupin" variable I t'pe

!efine "roups "roup 1H1 "roup 2 H2

Test t'pe I ann#Whitne'

ok

!ata +

http://data4.sav/http://data4.sav/


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table 3 T#e curati e e//ect o/ some dru$ topatients wit# ! t1pes o/ (ronc#itis


Z D1.225,P D0.221, we can think the curative effect of this dru"is same to patients with two t'pes of bronchitis.


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Part four

k independent-samples

rank sum test


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E*ample 4

The followin" sample data were obtained fromthree populations ,please compare the

population distribution is same.


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Sample 1 Sample 2 Sample

29.7 11.0 72.9

65.5 4.2 12.6

28.6 29.9 42.8

12.9 24.4 6.5

8.8 8. 8.5

8. 9.7 52.8

42.4 14.9 7 .4

2.7 52.6 9.2

1 .6 4.7 11.

10.5 5.2 4.4

10. 4.0

15.2 14.

2.5 12.7


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To compute the 8ruskal#Wallis test statistic,314 (ll the samples are combined

324 The combined values are ordered from low tohi"h, and3+4 The ordered values are replaced b' ranks,startin" with 1 for the smallest value


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6et up t#e #*pot#esis and confirm 7

Ho: three popu ation distri ution are same

H1: three popu ation distri ution are different

α= 0.05

Compute .6(1) ompute the tota num er of each -roup confirm the ran+ ran-e

of each -roup& then compute the a,era-e ran+ of each -roup

(2) ompute ran+ sum of each -roup(%) onfirmH

4ind P !alue and draw conclusion


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∑ +−+= )1(3)1(12 2

N n

R

N N H

i

i

Total sample siBe+;

each sample siBe

1+,1+,10

Aank sum in each "roup225,2++,20O

82.0

)13(3)102081323313225()13(3 12

)1(3)1(

12

222

2

=

+−+++=

+−+

= ∑ N n

R N N

H i

i


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df D the number of "roup #1 D+#1D2

Conclusion: Not reject H 0 at 0.05 signifcant level and think

three population distribution are same.


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Sroup 1 was skewed 3W testHP D0.02;4, so8ruskal#wallis rank sum test was used. The

results supported that there was no si"nificantdifference in the three population distributions H D0.;O2 P D0.N11 .


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2014 Nonparametic Test

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