Eta
ASSOCIATION: one nominal, one interval variable
A. Eta n: ungrouped data:
n^{2} = {[n_{j} (Ym_{j} - Ym)^{2} ] } / {Y^{2}-- [(Y)^{2} / N]}
Where: Y = score of interval variable
Ymj = mean for nominal category
Ym = mean for total sample
nj = number in a nominal category
N = total in sample
k = number of nominal categories
Example: Ungrouped data
Union income | Non-union income |
10 | 2 |
10 | 3 |
5 | 2 |
5 | 3 |
Q: What is the association between union membership and
income?
A:
Y=union (u) | Y^{2}u | Y=non-union (nu) | Y^{2}nu |
10 | 100 | 2 | 4 |
10 | 100 | 3 | 9 |
5 | 25 | 2 | 4 |
5 | 25 | 3 | 9 |
30 = Yu | 10 = Ynu | ||
n_{u}= 4 | n_{nu}= 4 |
n_{u} = 4 ----n_{nu} = 4
Y_{u} = 30/4 = 7.50 -----Y_{nu} = 10/4 = 2.50
Total Sample: Y = Y_{u} + Y_{nu} = 40
N = nu + nnu = 8
Y = Y/N = 40/8 = 5.0
Y2 = 276
---Union category -----non-union category
n^{2} ={[4(7.50 - 5.00)2 ] + [4 (2.50
- 5.00)2]}/{276 - [(40)^{2}/8]}
n^{2} = .66
n = .66 = .31
Interpretation:
: Use scale "There is ______association between (variable 1) and (variable 2)."
Interpretation for n^{2}: Convert
to a percent and include in the statement, "_____% of the
variance in (interval variable) can be explained by (nominal
variable)." or 1 - n^{2}:
Convert to a percent and include in the statement, "_____%
of the variance in (interval variable) cannot be explained
by (nominal variable)."
Example: .31 = a moderately small association between income and union membership.
n^{2} = .66: "66% of variance in income can be explained by union membership."
1 - n^{2} = .34: "34% of
variance in income cannot be explained by union membership."
B. Eta--grouped data
n^{2} = {[n_{j}(Ym_{j} - Ym)^{2} ]} / {f(Y)^{2} -- [(fY)^{2} / N]}
Where: f = frequency
Y = score of interval variable
Ymj = mean for nominal category
Ym = mean for total sample
nj = number in a nominal category
N = total in sample
k = number of nominal categories
Hint: Ym = fm/N;
Yj = fm/n_{j}
Example: grouped data
Income | Class '69 | Class '71 | Class '82 |
f | f | f | |
10 | 2 | 0 | 0 |
5 | 1 | 1 | 5 |
4 | 0 | 1 | 0 |
2 | 0 | 2 | 0 |
Q: What is the association between graduating class and
income?
A:
Income | Class '69 | Class '71 | Class '82 | |||
f | fm | f | fm | f | fm | |
10 | 2 | 20 | 0 | 0 | 0 | 0 |
5 | 1 | 5 | 1 | 5 | 5 | 25 |
4 | 0 | 0 | 1 | 4 | 0 | 0 |
2 | 0 | 0 | 2 | 4 | 0 | 0 |
3= n_{69} | 25 | 2= n_{71} | 13 | 5= n_{82} | 25 |
Ym_{69} = 25/3 = 8.33; Ym_{71} = 13/4 = 3.25;
Ym_{82 }= 25/5 = 5.00
Total Sample:
Income | f | fm or Y | Y^{2} | fY^{2} |
10 | 2 | 20 | 100 | 200 |
5 | 7 | 35 | 25 | 175 |
4 | 1 | 4 | 16 | 16 |
2 | 2 | 4 | 4 | 8 |
12 = N | 63 | 399 |
Ym_{total} = fm/N = 63/12 = 5.25
n^{2} = [3(8.33 - 5.25)^{2}] + [4(3.25 - 5.25)^{2}] + [5(5.00 - 5.25)^{2} ]
399 - [(63)^{2} /12]
n^{2} = .66
n= .81
Interpret these the same as for Eta for ungrouped data.