Non Parametric Statistics PPT @ BEC DOMS
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Transcript of Non Parametric Statistics PPT @ BEC DOMS
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Nonparametric Statistics
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Chapter Goals
After completing this chapter, you should be
able to:
Recognize when and how to use the Wilcoxonsigned rank test for a population median
Recognize the situations for which theWilcoxon signed rank test applies and be able
to use it for decision-making
Know when and how to perform a Mann-Whitney U-test
Perform nonparametric analysis of variance
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Nonparametric Statistics
Nonparametric Statistics
Fewer restrictive assumptions about data
levels and underlying probabilitydistributions
Population distributions may be skewed
The level of data measurement may only be
ordinal or nominal
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Wilcoxon Signed Rank Test
Used to test a hypothesis about one population
median
th
e median is th
e midpoint of th
e distribution: 50% below,50% above
A hypothesized median is rejected if sample
results vary too much from expectations
no highly restrictive assumptions about the shape of the
population distribution are needed
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The W Test Statistic
Performing the Wilcoxon Signed Rank Test
Calculate the test statistic W using these steps:
Step 1: collect sample data
Step 2: compute di = difference between each
value and the hypothesized median
Step 3: convert di values to absolute
differences
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The W Test Statistic
Performing the Wilcoxon Signed Rank Test
Step 4: determine the ranks for each di
value
eliminate zero di values
Lowest di value = 1
For ties, assign each
th
e average rank of th
e tiedobservations
(continued)
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The W Test Statistic
Performing the Wilcoxon Signed Rank Test
Step 5: Create R+ and R- columns
for data values greater than the hypothesized
median, put the rank in an R+ column
for data values less th
an th
eh
ypoth
esizedmedian, put the rank in an R- column
(continued)
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The W Test Statistic
Performing the Wilcoxon Signed Rank Test
Step 6: the test statistic W is the sum of the
ranks in the R+ column
Test the hypothesis by comparing the
calculated W to the critical value from thetable in appendix P
Note that n = the number of non-zero di values
(continued)
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Example
Rank the absolute differences:
| di | Rank
5
6
6
17
21
26
38
55
1
2.5
2.5
4
5
6
7
8
tied
(continued)
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Example
Put ranks in R+ and R- columnsand find sums:
Class
size = xi
Difference
di = xi4
0
| di | Rank R+ R-
23
45
34
78
34
66
61
95
-17
5
-6
38
-6
26
21
55
17
5
6
38
6
26
21
55
4
1
2.5
7
2.5
6
5
8
1
7
6
5
8
4
2.5
2.5
7=27 7
=9
(continued)
These three
are below
the claimed
median, the
others are
above
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Completing the Test
H0: Median = 40
HA: Median 40Test at the E = .05 level:
This is a two-tailed test and n = 8, so find WL and WU in
appendix P: WL = 3 and WU = 33
The calculated test statistic is W = 7R+ = 27
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Completing the Test
H0: Median = 40
HA: Median 40W
L
= 3 and WU
= 33
WL < W < WU so do not reject H0
(there is not sufficient evidence to conclude that the median
class size is different than 40)
(continued)
WL = 3do not reject H0reject H0
W=7R+ = 27
WU = 33reject H0
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If the Sample Size is Large
The W test statistic approaches a normal
distribution as n increases
For n > 20, W can be approximated by
241)1)(2nn(n
4
1)n(nW
z
!
where W= sum of the R+ ranks
d = numberof non-zero di values
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Nonparametric Tests for Two PopulationCenters
Nonparametric
Tests for Two
Population Centers
Wilcoxon
Matched-Pairs
SignedR
ank Test
Mann-Whitney
U-test
Large
Samples
Small
Samples
Large
Samples
Small
Samples
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Mann-Whitney U-Test
Used to compare two samples from two
populations
Assumptions:
The two samples are independent and random
The value measured is a continuous variable
The measurement scale used is at least ordinal
If they differ, the distributions of the two populations will differ
only with respect to the central location
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Consider two samples
combine into a singe list, but keep track of which
sample each value came from
rank the values in the combined list from low to
high For ties, assign each the average rank of the tied values
separate back into two samples, each valuekeeping its assigned ranking
sum the rankings for each sample
Mann-Whitney U-Test(continued)
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If the sum of rankings from one sample differs
enough from the sum of rankings from the
other sample, we conclude there is a
difference in the population medians
Mann-Whitney U-Test(continued)
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(continued)
Mann-Whitney U-Test
Mann-Whitney U-
Statistics
! 111
211 2
1
R
)n(n
nnU
!2
22
212
2
1R
)n(nnnU
where:
n1 and n2 are the two sample sizes
R1 and R2 = sum ofranks forsamples 1 and 2
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(continued)
Mann-Whitney U-Test
Claim: Median class size forMath is larger
than the median class size forEnglish
Arandom sample of 9 Math and 9 English
classes is selected (samples do not have to
be of equal size)
Rank the combined values and then splitthem back into the separate samples
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Suppose the results are:
Class size (Math, M) Class size (English, E)
23
45
34
78
34
66
62
95
81
30
47
18
34
44
61
54
28
40
(continued)
Mann-Whitney U-Test
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Size Rank
18 1
23 2
28 3
30 4
34 6
34 6
34 6
40 8
44 9
Size Rank
45 10
47 11
54 12
61 13
62 14
66 15
78 16
81 17
95 18
Ranking forcombined samples
tied
(continued)
Mann-Whitney U-Test
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Split back into the original samples:Class size (Math,
M)Rank
Class size
(English, E)Rank
23
45
34
78
34
66
62
95
81
2
10
6
16
6
15
14
18
17
30
47
18
34
44
61
54
28
40
4
11
1
6
9
13
12
3
8
7 = 104 7 = 67
(continued)
Mann-Whitney U-Test
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H0: MedianM MedianE
HA: MedianM >
MedianE
Claim: Median class size for
Math is largerthan the
median class size forEnglish
221042
(9)(10)(9)(9)R
2
1)(nnnnU 1
11211 !!
!
5967
2
(9)(10)(9)(9)R
2
1)(nnnnU 2
22212 !!
!
Note: U1 + U2 = n1n2
(continued)
Mann-Whitney U-Test
Math:
English:
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The Mann-Whitney U tables in Appendices L
and M give the lower tail of the U-distribution
For one-tailed tests like this one, check thealternative hypothesis to see if U1 or U2should be used as the test statistic
Since the alternative hypothesis indicates thatpopulation 1 (Math) has a higher median, use
U1 as the test statistic
(continued)
Mann-Whitney U-Test
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Use U1 as the test statistic: U = 22
Compare U = 22 to the critical value UE from
the appropriate table
For sample sizes less than 9, use Appendix L
For samples sizes from 9 to 20, use Appendix M
If U < UE, reject H0
(continued)
Mann-Whitney U-Test
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Since U UE, do not reject H0
Use U1 as the test statistic: U = 19
UE from Appendix M forE = .05, n1 = 9 and
n2
= 9 is UE = 7
(continued)
Mann-Whitney U-Test
UE = 7
U= 19
do not reject H0reject H0
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Mann-Whitney U-Test forLarge Samples
The table in Appendix M includes UE values
only for sample sizes between 9 and 20
The U statistic approaches a normaldistribution as sample sizes increase
If samples are larger than 20, a normal
approximation can be used
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Mann-Whitney U-Test forLarge Samples
The mean and standard deviation for Mann-
Whitney U Test Statistic:
(continued)
2
nn 21!Q
12)1nn)(n)(n( 2121 !W
Where n1 and n2 are sample sizes from populations 1 and 2
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Mann-Whitney U-Test forLarge Samples
Normal approximation for Mann-Whitney U
Test Statistic:
(continued)
12)1nn)(n)(n(
2
nnU
z
2121
21
!
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Large Sample Example
We wish to test
Suppose two samples are obtained:
n1 = 40 , n2 = 50
When rankings are completed, the sum of
ranks for sample 1 is 7R1 = 1475 When rankings are completed, the sum of
ranks for sample 2 is 7R2
= 2620
H0: Median1 u Median2HA: Median1 < Median2
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U statistic is found to be U = 655
134514752
(40)(41)(40)(50)R
2
1)(nnnnU 1
11211 !!
!
65526202
(50)(51)(40)(50)R
2
1)(nnnnU 2
22212 !!
!
Since the alternative hypothesis indicates that
population 2 has a highermedian, use U2 as the test
statistic
Compute the U statistics:
Large Sample Example(continued)
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Since z= -2.80 < -1.645, we reject H0
645.1z !E
Reject H0
0MedianMedian:H
0MedianMedian:H
21A
210
"
e
80.2
12
)15040)(50)(40(
1000655
12
)1nn)(n)(n(
2
nnU
z
2121
21
!
!
!
E = .05
Do not reject H0
0
Large Sample Example(continued)
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The Wilcoxon T Test Statistic
Performing the Small-Sample WilcoxonMatched Pairs Test (for n < 25)
Calculate t
he test statistic T using t
hese steps:
Step 1: collect sample data
Step2
: compute di = difference between th
esample 1 value and its paired sample 2 value
Step 3: rank the differences, and give eachrank the same sign as the sign of thedifference value
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The Wilcoxon T Test Statistic
Performing the Small-Sample WilcoxonMatched Pairs Test (for n < 25)
Step 4: The test statistic is the sum of theabsolute values of the ranks for the group withthe smaller expected sum
Look at the alternative hypothesis to determine
the group with the smaller expected sum
For two tailed tests, just choose the smaller sum
(continued)
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Small Sample Example
Paired samples, n = 9:
Value (before) Value (after)
38
45
34
58
30
46
42
55
41
30
47
18
34
34
31
24
38
40
baA
ba0
MedianMedian:H
MedianMedian:H
u
Claim: Median
value is smallerafter than before
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Small Sample Example
Paired samples, n = 9:Value
(before)
Value
(after)
Difference
d
Rank
of d
Ranks with smaller
expected sum
36
45
34
58
30
4642
55
41
30
47
18
54
38
3124
62
40
6
-2
16
4
-8
1518
-7
1
4
-2
8
3
-6
79
-5
1
2
6
5
7 = T =13
(continued)
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The calculated T value is T = 13
Complete the test by comparing the calculated
T value to the critical T-value from AppendixN
For n = 9 and E = .025 for a one-tailed test,
TE = 6
Since T TE, do not reject H0
TE = 6
T= 13
do not reject H0reject H0
Small Sample Example(continued)
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Wilcoxon Matched Pairs Testfor Large Samples
The table in Appendix N includes TE values
only for sample sizes from 6 to 25
The T statistic approaches a normaldistribution as sample size increases
If the number of paired values is larger than
25, a normal approximation can be used
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The mean and standard deviation for
Wilcoxon T :
(continued)
4
)1n(n !Q
24)1n2)(1n)(n( !W
where n is the numberof paired values
Wilcoxon Matched Pairs Testfor Large Samples
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Mann-Whitney U-Test forLarge Samples
Normal approximation for the Wilcoxon T
Test Statistic:
(continued)
24)1n2)(1n(n
4
)1n(nT
z
!
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Tests the equality ofmore than 2populationmedians
Assumptions:
variables have a continuous distribution.
the data are at least ordinal.
samples are independent.
samples come from populations whose onlypossible difference is that at least one may have adifferent central location than the others.
Kruskal-Wallis One-Way ANOVA
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Kruskal-Wallis Test Procedure
Obtain relative rankings for each value
In event of tie, each of the tied values gets the
average rank
Sum the rankings for data from each of the k
groups
Compute the H test statistic
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Kruskal-Wallis Test Procedure
The Kruskal-Wallis H test statistic:(with k 1 degrees of freedom)
)1N(3nR
)1N(N12H
k
1i i
2
i
! !
where:
N = Sum of sample sizes in all samplesk = Numberof samples
Ri = Sum ofranks in the ith sample
ni = Size of the ith sample
(continued)
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Complete the test by comparing the calculated
H value to a critical G2 value from the chi-
square distribution with k 1 degrees of
freedom
(The chi-square distribution is Appendix G)
Decision rule Reject H
0if test statistic H > G2E
Otherwise do not reject H0
(continued)
Kruskal-Wallis Test Procedure
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Do different departments have different class
sizes?
Kruskal-Wallis Example
Class size
(Math, M)
Class size
(English, E)
Class size
(History, H)
23
45
54
7866
55
60
72
4570
30
40
18
3444
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Do different departments have different class
sizes?
Kruskal-Wallis Example
Class size
(Math, M)R
anking
Class size
(English, E)R
anking
Class size
(History, H)R
anking
23
41
54
78
66
2
6
9
15
12
55
60
72
45
70
10
11
14
8
13
30
40
18
34
44
3
5
1
4
7
7 =44 7 =56 7 =20
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The H statistic is(continued)
Kruskal-Wallis Example
72.6)115(35
20
5
56
5
44
)115(15
12
)1N(3n
R
)1N(N
12H
222
k
1ii
2
i
!
!
! !
equalareMedianspopulationallotN:H
MedianMedianMedian:H
A
HEM0 !!
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Since H = 6.72