Pearson's correlation coefficient R is a useful descriptor ...



Pearson's correlation coefficient R is a useful descriptor of the degree of linear association between two variables. A positive R implies correlation in the same direction and negative R implies an opposite correlation. When R is close to 0, there is no correlation, but as it approaches -1 or +1 there is a strong negative or positive relationship respectively between the variables. But how do we know when a correlation is sufficiently different from zero to assert that a real relationship exists?

To answer this question, we need some estimate of how much variation in R can be attributed to random chance. That is, we need to construct a sampling distribution for R and determine its standard error. All variables are correlated to some extent; rarely will a correlation be exactly zero. What we need is to be able to draw a line that tells us that above that line a correlation will be considered as a real correlation and below that line the correlation will be considered as probably due to chance alone. This is how Pearson’s R is used in statistical hypothesis testing.

Spearman's ( is based on ranks. It is calculated in almost the same way as Pearson's R, except that whereas Pearson's R uses the actual values of both variables, Spearman's ( correlates the rankings of those variables.

Therefore Pearson's is appropriate for parametric data at interval or ratio level, where the intervals are meaningful and can be uniformly measured (height, score on IQ test, etc). If the data is non-parametric of and of ordinal level only (responses to a 5 point scale of satisfaction) the Spearman's ( is used.

Here is an example of hypothesis test for Pearson’s R:

A correlation coefficient of 0.933 is obtained for annual income and education obtained from a national random sample of 20 employed adults. At ( = 0.05, is the correlation significant in the population of employed adults?

 

H0: There is no correlation between annual income and education for employed adults

Ha: There is a significant correlation between annual income and education for employed adults

Degrees of freedom = n - 2 = 20 - 2 = 18

Critical 2-tailed t- score = 2.101

Decision Rule: Reject H0 if the test t- score < -2.101 or > 2.101

t = R * ([(n - 2)/(1 - R^2) = 0.933 * ([(20 - 2)/(1 - 0.933^2)] = 11

Since 11 > 2.101, we reject H0 and accept Ha

Conclusion: There is a significant correlation between annual income and education for employed adults.

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