# Nonparametric Test

```Nonparametric Test
Distribution-Free Tests
1.
2.
3.
4.
No assumptions of normality
Focus on medians rather than means
Not affected by outliers
Des NOT really reduce power
Mann-Whitney U Test
1.
2.
3.
4.
5.
Test for two independent groups
Assumptions for t-test cannot be made
Data is neither ratio nor interval
Data must be at least ordinal
Tests if two groups have the same distribution
n1 = # of cases in the smaller of the groups
n2 = # of cases in the larger of the groups
E = experimental group membership
C = control group membership
e.g., E scores = 11,15,9 and C scores = 6,8,13,10
Rank order all scores with group ID
6C,8C,9E,10C,11E,13C,15E
Count # of Es that precede each C: U = 0 + 0 +1 + 2 = 3
Go top table for P under Null Hypothesis associated with the data; n1 = 3, n2 = 4, U = 3
U equal or greater than 3 p = .20 Thus, fail to reject at alpha .05.
More than two groups: Kruskal-Wallis (H) (for ordinal data)
One-Way ANOVA by ranks
Null Hypothesis: k samples come from same population or identical populations
with respect to central tendencies.
Rank order all N scores from the smallest to the largest.
e.g., Authoritarian Scores
G1
G2
G3
96
82
115
128
124
149
83
132
166
61
135
147
101
109
Transformed
4
9
3
1
5
2
8
10
11
6
7
13
14
12
K-W (H)
Ri2
12
H
 3( N  1)

N ( N  1) ni
K = # of groups
n = number of cases in a group
N = total number of cases
R = sum of ranks in a group
12
22 2 37 2 462
H
[


 3(14  1)
14(14  1) 5
5
4
= 6.4
This is tested against Chi-Squared (k-1) df.
Go to Chi-Squared Table , with 2 df and alpha .05
Friedman’s Test for k related samples
One-way repeated measures
Example
Patients
1
2
3
4
5
6
7
8
9
10
Totals
Psy(rank)
6(2)
4(1)
9(3)
5(2)
2(1)
6(1)
7(2)
4(1)
6(3)
72)
18
drug(rank) psy/drug(rank)
8(3)
5(1)
8(3)
6(2)
12
7(2)
4(1)
 2F 
R 2j  3N ( k  1)

Nk ( N  1)
4(1)
6(3)
7(3)
3(2)
N = # of patients
8(3)
7(2)
9(3)
5(1)
8(3)
5(2)
5(2)
4(1)
12
2
2
2

(
18

26

16
)  3(10)(4)
8(3)
6(1)
10(3)(4)
26
16
= 5.6
This is test against Chi-Squared (k-1) df.
Fail to reject at alpha .05.
```