Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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What is a Perfect Positive Linear Correlation?

– It occurs when everyone has the same exact score on two different variables.

– It also occurs when everyone’s score on one variable differs by a constant from their score on the other variable (e.g., everyone’s Final exam score is exactly 10 points higher than their midterm score, or exactly twice as much).

– It will occur if everyone occupies the same position in a normal distribution for one variable that they occupy for the other variable (i.e., if everyone has the same z score on both variables).

– Perfect negative correlation occurs when everyone has the same z score on both variables, but with opposite signs (e.g., if z = +1.5 for one variable, that person’s z will be –1.5 for the other variable.

Chapter 9: Linear Correlation

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen.

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The Pearson Correlation Coefficient (r)

– Pearson’s r ranges from -1.0 to +1.0, where• +1.00 = perfect positive correlation• –1.00 = perfect negative correlation• 0 = a total lack of correlation

– A formula that illustrates the relationship between z-scores and Pearson’s r is the following:

– The magnitude of the number represents the amount of correlation, while its sign represents the direction of the correlation.

N

zzr yx

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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Correlation coeffiecent r = +1.0

0

5

10

15

20

25

0 5 10 15 20 25

Correlation coeffiecent r = -1.0

0

5

10

15

20

25

0 5 10 15 20 25

Graphing a Correlation:

The Scatterplot

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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4

6

8

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10 20 30 40 50

Correlation r = .83

Correlation r = -.83

4

6

8

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22

10 20 30 40 50

The Graph of a Correlation That Is Less

Than Perfect

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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Calculating Pearson’s r• Computing formula in terms of population

standard deviations:

– The numerator is the biased estimate of the covariance. Dividing the covariance by the product of the biased standard deviations ensures that r will never be greater than +/– 1.0.

• Computing formula in terms of the unbiased covariance estimate divided by the unbiased standard deviations:

This formula always yields the same value for r as the preceding formula based on the biased covariance.

YX

YXNXY

r

YX ss

YXNXYNr

11

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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• Try this example…

Is education about other ethnicities correlated with tolerant attitudes towards others?

Education Score

Tolerance Score

XY

25 3 75 Xbar = 31.5 25 9 225 Ybar = 11.1 33 14 462 N = 10 35 11 385 sX = 6.222 38 13 494 sY = 3.414 36 14 504 σX = 5.903 31 12 372 σY = 3.239 29 12 348 22 9 198 41 14 574 315 111 3637

735.

242.21

611.15

242.21

5.496,3637,391

414.3*222.6

1.11*5.31*10637,3110

1

r

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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Testing Pearson’s r for Significance

– H0: ρ = 0; df = N – 2 – Using the t distribution:

– t.05 (8) = 2.306 < 3.07, so we can reject the null hypothesis for the correlation example from the previous slide.

– Using the table of critical values:• df = N – 2 (where N is the number of

pairs of scores).• Critical r for df = 8 is .632, for a .05, two-

tailed test. .632 < .735, so H0 can be rejected (consistent with t test).

– Effects of N on Critical r• For small N, you need a fairly large r to find

significance.• For very large N, even tiny sample rs can attain

statistical significance.• As N increases, r is a better estimate of ρ.

07.36782.

079.2

540.1

210735.

1

22

r

Nrt

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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Assumptions of the Test of Significance for Pearson’s r

– The sample has been obtained by independent random sampling.

– Both variables have been measured on interval or ratio scales.

– Both variables exhibit normal distributions in the population.

– The two variables jointly follow a bivariate normal distribution (see next slide).

– If one or both variables has been measured on an ordinal scale, or the distribution assumptions have been severely violated, consider calculating the Spearman rank-order correlation coefficient (rS) as an alternative (a special table must be used for its critical values when N is small).

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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Limitations and Cautions

• Pearson’s r measures only the degree of linear correlation (r can be small even though there is a very close curvilinear relationship between the two variables).

• Pearson’s r can underestimate the population correlation (ρ) if there are:– Restricted (truncated) ranges on one

or both variables.– Bivariate outliers.

• Correlation does not imply causation!

Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen

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Uses of Pearson’s r– Reliability

• Test-retest reliability• Split-half reliability• Inter-rater reliability

– Criterion validity (e.g., of a self-report measure).

– To measure the degree of linear association between two variables that are not obviously related, but are predicted by some theory or past research to have an important connection.

– To evaluate the results of experimental studies in which both the DV, and the levels of the manipulated variable (IV), have been measured on interval or ratio scales.