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TESTING THE SIGNIFICANCE OF r

Just as with G, it is possible to test whether or not any value of Pearson’s r is the result of sampling error, or reflects a real covariation in the larger population. In the case of r, the null hypothesis is that, within certain confidence limits, we should predict that the real coefficient of correlation in the population of interest is actually zero. In other words, there is no relation between the two variables.

We need to be particularly sensitive to the possible lack of significance of sample statistics when we deal with small samples—which is a lot of the time, it turns out. To simplify testing the confidence limits of r, I have constructed table 21.18, which you can use to get a ball-park reading on the significance of Pearson’s r. The top half of table 21.18 shows the 95% confidence limits for representative samples of 30, 50, 100, 400, and 1,000, where

Table 21.18 Confidence Limits for Pearson's rfor Various Sample Sizes

Pearson's r

Sample size

30

50

100

400

1,000

0.1

ns

ns

ns

ns

.04-.16

0.2

ns

ns

.004-.40

.10-.29

.14-.26

0.3

ns

.02-.54

.11-.47

.21-.39

.24-.35

0.4

.05-.67

.14-.61

.21-.55

.32-.48

.35-.45

0.5

.17-.73

.25-.68

.31-.63

.42-.57

.45-.54

0.6

.31-.79

.39-.75

.45-.71

.53-.66

.56-.64

0.7

.45-85

.52-.82

.59-.79

.65-.75

.67-.73

0.8

.62-.90

.67-.88

.72-.86

.76-.83

.78-.82

0.9

.80-.95

.83-.94

.85-.93

CO

CO

1

CD

ГО

.89-.91

Top half of table: 95% confidence limits

0.1

ns

ns

ns

ns

.02-.18

0.2

ns

ns

ns

.07-.32

.12-.27

0.3

ns

ns

.05-.51

.18-.41

.23-.45

0.4

ns

.05-.80

.16-.59

.28-.50

.33-.46

0.5

.05-.75

.17-.72

.28-.67

.40-.59

.44-.56

0.6

.20-.83

.31-.79

.41-.74

.51-.68

.55-.65

0.7

.35-.88

.46-.85

.55-.81

.63-.76

.66-.74

0.8

.54-.92

.62-.90

.69-.88

.75-.84

.77-.83

0.9

.75-.96

.80-.95

.84-.94

.87-.92

.88-.91

Bottom half of table: 99% confidence limits

the Pearson’s r values are .1, .2, .3, etc. The bottom half of table 21.18 shows the 99% confidence limits.

Reading the top half of table 21.18, we see that at the 95% level the confidence limits for a correlation of 0.20 in a random sample of 1,000 are 0.14 and 0.26. This means that in fewer than 5 tests in 100 would we expect to find the correlation smaller than 0.14 or larger than 0.26. In other words, we are 95% confident that the true r for the population (written p, which is the Greek letter rho) is somewhere between 0.14 and 0.26.

By contrast, the 95% confidence limits for an r of 0.30 in a random sample of 30 is not significant at all; the true correlation could be 0, and our sample statistic of 0.30 could be the result of sampling error.

The 95% confidence limits for an r of 0.40 in a random sample of 30 is statistically significant. We can be 95% certain that the true correlation in the population (p) is no less than 0.05 and no larger than 0.67. This is a statistically significant finding, but not much to go on insofar as external validity is concerned. You’ll notice that with large samples (like 1,000), even very small correlations are significant at the .01 level (box 21.4).

BOX 21.4

STATISTICAL SIGNIFICANCE

On the other hand, just because a statistical value is significant doesn't mean that it's important or useful in understanding how the world works. Looking at the lower half of table 21.18, we see that even an r value of 0.40 is statistically insignificant when the sample is as small as 30. If you look at the spread in the confidence limits for both halves of table 21.18, you will notice something very interesting: A sample of 1,000 offers some advantage over a sample of 400 for bivariate tests, but the difference is small and the costs of the larger sample could be very high, especially if you're collecting all your own data.

Recall from chapter 6, on sampling, that to halve the confidence interval you have to quadruple the sample size. Where the unit cost of data is high—as in research based on direct observation of behavior or on face-to-face interviews—the point of diminishing returns on sample size is reached quickly. Where the unit cost of data is low—as it is with mailed questionnaires or with telephone surveys—a larger sample is worth trying for.

 
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