Matakuliah
Tahun
: A0392 – Statistik Ekonomi
: 2006
Pertemuan 11
Analisis Varians Data Nonparametrik
1
Outline Materi :
 Uji Kruskal Wallis
 Pembuatan peringkat data
 Statistik uji Kruskal Wallis
2
Analysis of Variance:Data
Nonparametric
(continued)
• Kruskal-Wallis Rank Test for Differences in
c Medians
• Friedman Rank Test for Differences in c
Medians
3
Kruskal-Wallis Rank Test
• Assumptions
– Independent random samples are drawn
– Continuous dependent variable
– Data may be ranked both within and among
samples
– Populations have same variability
– Populations have same shape
• Robust with Regard to Last 2 Conditions
– Use F test in completely randomized designs
and when the more stringent assumptions
hold
4
Kruskal-Wallis Rank Test
Procedure
• Obtain Ranks
– In event of tie, each of the tied values gets
their average rank
• Add the Ranks for Data from Each of the c
Groups
– Square to obtain Tj2
c T2 
 12
j
H 
  3(n  1)

 n(n  1) j 1 n j 
n  n1  n2 
 nc
5
Kruskal-Wallis Rank Test
Procedure
• Compute Test Statistic
–
c T2 
 12
j
H 
  3(n  1)

 n(n  1) j 1 n j 
–
n  n1  n2 
(continued)
 nc
– n j  # of observation in j –th sample
– H may be approximated by chi-square
distribution with df = c –1 when each nj >5
6
Kruskal-Wallis Rank Test
Procedure
• Critical Value for a Given a
(continued)
– Upper tail
• Decision Rule 
2
U
– Reject H0: M1 = M2 = ••• = Mc if test statistic
2
H > U
– Otherwise, do not reject H0
7
Kruskal-Wallis Rank Test:
Example
As production manager, you
want to see if 3 filling
machines have different
median filling times. You
assign 15 similarly trained &
experienced workers,
5 per machine, to the
machines. At the .05
significance level, is there a
difference in median filling
times?
Machine1 Machine2
Machine3
25.40
26.31
24.10
23.74
25.10
23.40
21.80
23.50
22.75
21.60
20.00
22.20
19.75
20.60
20.40
8
Example Solution: Step 1
Obtaining a Ranking
Raw Data
Machine1 Machine2
Machine3
25.40
26.31
24.10
23.74
25.10
23.40
21.80
23.50
22.75
21.60
Ranks
Machine1 Machine2
Machine3
20.00
22.20
19.75
20.60
20.40
14
15
12
11
13
65
9
6
10
8
5
38
2
7
1
4
3
17
9
Example Solution: Step 2
Test
Statistic Computation
2


T
c
j 
 12
H 
 3(n  1)


n(n  1) j  1 n

j


 12
 652 382 17 2  

   3(15  1)



5
5 
15(15  1)  5



 11.58
10
Kruskal-Wallis Test Example
Solution
H0: M1 = M2 = M3
H1: Not all equal
a = .05
df = c - 1 = 3 - 1 = 2
Critical Value(s):
a = .05
0
5.991
Test Statistic:
H = 11.58
Decision:
Reject at a = .05.
Conclusion:
There is evidence that
population medians are
not all equal.
11
Kruskal-Wallis Test in PHStat
• PHStat | c-Sample Tests | Kruskal-Wallis
Rank Sum Test …
• Example Solution in Excel Spreadsheet
12
Friedman Rank Test for
Differences in c Medians
• Tests the equality of more than 2 (c)
population medians
• Distribution-Free Test Procedure
• Used to Analyze Randomized Block
Experimental Designs
• Use 2 Distribution to Approximate if the
Number of Blocks r > 5
– df = c – 1
13
Friedman Rank Test
• Assumptions
– The r blocks are independent
– The random variable is continuous
– The data constitute at least an ordinal scale of
measurement
– No interaction between the r blocks and the c
treatment levels
– The c populations have the same variability
– The c populations have the same shape
14
Friedman Rank Test:
Procedure
 Replace the c observations by their
ranks in each of the r blocks; assign
average rank for ties
c
12
2
 Test statistic: FR 
R
  j  3r  c  1
rc  c  1
j 1
 R.j2 is the square of the rank total for group j
 FR can be approximated by a chi-square
distribution with (c –1) degrees of freedom
 The rejection region is in the right tail
15
Friedman Rank Test: Example
As production manager, you
want to see if 3 filling
machines have different
median filling times. You
assign 15 workers with varied
experience into 5 groups of 3
based on similarity of their
experience, and assigned
each group of 3 workers with
similar experience to the
machines. At the .05
significance level, is there a
difference in median filling
times?
Machine1 Machine2
Machine3
25.40
26.31
24.10
23.74
25.10
23.40
21.80
23.50
22.75
21.60
20.00
22.20
19.75
20.60
20.40
16
Friedman Rank Test:
Computation Table
Machine 1
25.4
26.31
24.1
23.74
25.1
Timing
Machine 2 Machine 3
23.4
20
21.8
22.2
23.5
19.75
22.75
20.6
21.6
20.4
Machine 1
3
3
3
3
3
R. j
15
R.2j
225
Rank
Machine 2 Machine 3
2
1
1
2
2
1
2
1
2
1
9
6
81
36
c
12
2
FR 
R

 j  3r  c  1
rc  c  1 j 1

12
 342   3  5 4   8.4
 5 3 4 
17
Friedman Rank Test Example
Solution
H0: M1 = M2 = M3
H1: Not all equal
a = .05
df = c - 1 = 3 - 1 = 2
Critical Value:
a = .05
0
5.991
Test Statistic:
FR = 8.4
Decision:
Reject at a = .05
Conclusion:
There is evidence that
population medians are
not all equal.
18
Chapter Summary
(continued)
• Discussed Kruskal-Wallis Rank Test for
Differences in c Medians
• Illustrated Friedman Rank Test for
Differences in c Medians
19