Kendall's tau
ORDINAL DATA: Association and Inference
A. Kendall's Tau (data with ties): b = fa - fi
MV
Where:
fa = same as for gamma
fi = " " " "
M = ^{1}/_{2}[N(N - 1)] - ^{1}/_{2}[ (row total)(row total - 1)]
V = ^{1}/_{2}[N(N - 1)] - ^{1}/_{2}
[ (column total)(column total - 1)]
Example: Data with ties:
Overall Happiness by Health
VERY HAPPY | 31 | ||||
PRETTY HAPPY | 58 | ||||
NOT TOO HAPPY | 12 | ||||
101 |
SOURCE: GSS91 SURVEY SUBSAMPLE
Q: What is the association between overall happiness and health?
A: M = ^{1}/_{2} (101)(100) - ^{1}/_{2}
[31(30) + 58(57) + 12(11)] = 2,866
V = ^{1}/_{2} (101)(100) ^{1}/_{2}
[32(31) + 46(45) + 17(16) + 6(5)] = 3,368
b = 1,475 - 482 = .32
---(2,866)(3,368)
Interpretation: Use the scale below: "There is a ______
association between (variable 1) and (variable 2)."
From example above: b = .32 = "moderately small positive association between health and happiness"
B. Tau (without ties on data):
1. Arrange first variable in order from smallest to largest. Rearrange second variable to correspond.
2. For each score on the second variable, its contribution to p (p is a variable representing place on list) is the number of scores located below it which are larger than it is.
3. Interpret same as for b: a = {2(P) / [^{1}/_{2}
N(N -1)]} - 1
Example:
Person | Occupational. Status | Attractiveness Score | P |
D | 1 | 1 | 11 |
C | 2 | 5 | 7 |
A | 3 | 2 | 9 |
B | 4 | 6 | 6 |
K | 5 | 7 | 5 |
H | 6 | 3 | 6 |
I | 7 | 4 | 5 |
E | 8 | 10 | 2 |
L | 9 | 11 | 1 |
G | 10 | 8 | 2 |
F | 11 | 9 | 1 |
J | 12 | 12 | 0 |
55=P |
Q: What is the association between occupational status and attractiveness score?
A: a = = 1.67 - 1 = .67
There is a moderately large positive assosiation between occupational
status and attractiveness score.
C. Test of significance of Tau: Z = || / S
Where: S ("standard error of tau") = [2(2N + 5)] / [9N(N - 1)]
Assumptions: N 10
*P - value of Z on bottom in appendix.
Example: Using data above: occupational status/attractiveness
score = .67, N=12
Q: Can this be generalized to the whole population?
A: S = 2(2(12) + 5) = .22 :-----Z = |.67
| = 3.0
---------9(12)(11) ------------------.22
Looking this up on p - value table, find p < .01; so yes, one
can generalize.
D. Partial Tau: _{12.3}= _{12} - (_{13})( _{23})
[1 - (_{13})^{2}
][1 -(_{23} )^{2}
]
Where: _{12} = tau between variables 1 and 2.
_{13} = " " " 1 and 3.
_{23} = " " " 2 and
3.
Example:
Person | Occ. Status | Income | Age |
A | 1 | 5 | 5 |
B | 2 | 3 | 3 |
C | 3 | 4 | 4 |
D | 4 | 1 | 2 |
E | 5 | 2 | 1 |
Q: What is the association between occupational status
and income with age held constant?
A: Variable held constant (age) is #3; let #1 = occupational
status and #2 = income.
_{12} = [2(2)] / [^{1}/_{2}(5)(4)]
- 1 = -.60
_{ 13} = -.80
_{23}
= +.80
_{12.3} = (-.60) - [(-.80)(+.80)]
=
.04 = .11
---------[1 - (-.80)^{2}
][1 - (+.80)^{2} ]
.36
Interpretation: Scale from -1 to +1, same as for Tau. From example: "Small positive relationship between occupational status and income with age held constant."
Congratulations! You've completed all of the statistical work now for the SPSS project.