ASSOCIATION: one nominal, one interval variable

A. Eta n: ungrouped data:

n² = {[n_j (Ym_j - Ym)² ] } / {Y²-- [(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)	Y2u	Y=non-union (nu)	Y2nu
10	100	2	4
10	100	3	9
5	25	2	4
5	25	3	9
30 = Yu		10 = Ynu
nu= 4		nnu= 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
n² ={[4(7.50 - 5.00)2 ] + [4 (2.50 - 5.00)2]}/{276 - [(40)²/8]}

n² = .66

n = .66 = .31

Interpretation:

: Use scale "There is ______association between (variable 1) and (variable 2)."

Interpretation for n²: Convert to a percent and include in the statement, "_____% of the variance in (interval variable) can be explained by (nominal variable)." or 1 - n²: 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.

n² = .66: "66% of variance in income can be explained by union membership."

1 - n² = .34: "34% of variance in income cannot be explained by union membership."

B. Eta--grouped data

n² = {[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= n69	25	2= n71	13	5= n82	25

Ym₆₉ = 25/3 = 8.33; Ym₇₁ = 13/4 = 3.25; Ym₈₂ = 25/5 = 5.00

Total Sample:

Income	f	fm or Y	Y2	fY2
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

n² = [3(8.33 - 5.25)²] + [4(3.25 - 5.25)²] + [5(5.00 - 5.25)² ]

399 - [(63)² /12]

n² = .66

n= .81

Interpret these the same as for Eta for ungrouped data.

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