Eta-squared, written ^^{2} or eta^{2}, is a PRE measure that tells you how much better you could do if you predicted the separate means for chunks of your data than if you predicted the mean for all your data. Figure 21.8 graphs the hypothetical data in table 21.19. These data show, for a sample of 20 people, ages 12-89, their ‘‘number of close friends and acquaintances.”

FIGURE 21.8.

Number of friends by age.

The dots in figure 21.8 are the data points from table 21.19. Respondent #10, for example, is 45 years of age and was found to have approximately 55 friends and acquaintances. The horizontal dashed line in figure 21.8 marked y is the global average for these data, 41.15.

Clearly: (1) The global average is not of much use in predicting the dependent variable; (2) Knowing a person’s age is helpful in predicting the size of his or her social network; but (3) The linear regression equation is hardly any better than the global mean at reducing error in predicting the dependent variable. You can see this by comparing the mean line and the regression line (the slightly diagonal line running from upper left to lower right in figure 21.8). They are not that different.

What that regression line depicts is the correlation between age and size of network, which is a puny — 0.099. But if we inspect the data visually, we find that there are a couple of natural ‘‘breaks.’’ It looks like there’s a break in the late 20s, and another somewhere in the 60s. We’ll break these data into three age chunks from 12 to 26, 27 to 61, and 64

Table 21.19 Hypothetical Data on Number of Friends by Age

Person

Age

Number of friends

1

12

8

2

18

14

y = 19.25

3

21

30

4

26

25

5

27

56

6

30

43

7

36

61

8

39

41

9

42

82

y_{2} = 57.00

10

45

55

11

47

70

12

49

75

13

51

41

14

55

38

15

61

65

16

64

52

17

70

22

18

76

28

y_{3} = 23.80

19

80

11

20

89

6

y = 41.15

to 89, take separate means for each chunk, and see what happens. I have marked the three chunks and their separate means on table 21.19.

Like r, which must be squared to find the variance accounted for, is a measure of this and is calculated from the following formula:

where y_{c} is the average for each chunk and y is the overall average for your dependent variable. For table 21.19, rf^{2}is:

which is the proportionate reduction of error in predicting the number of friends people have from the three separate averages of their age, rather than from the global average of their age. This shows a pretty strong relation between the two variables, despite the very weak Pearson’s r.