Correlation

Correlation is a statistical measure of the relationship, or association, between two or more variables. There are many different types of correlations, each of which measures particular statistical relationships among and between quantitative variables. Examples of different types of correlations include Pearson's correlation (sometimes called "product-moment correlation"), Spearman's correlation, Kendall's correlation, intra-class correlation, point-biserial correlation and others. The nature of the data (e.g. continuous versus dichoto-mous), the kind of information desired, and other factors can help determine the type of correlation measure that is most appropriate for a particular analysis.

The value of the correlation between any two variables is typically given by a correlation coefficient, which can take on any value between and including − 1.00 (indicating a perfect negative relationship) up to and including +1.00 (indicating a perfect positive relationship). A positive correlation between two variables means that as the value of one variable increases, the value of the second variable tends to increase. A negative correlation means that as the value of one variable increases, the value of the second variable tends to decrease. A correlation that is equal to zero means that as one variable increases or decreases, the other does not exhibit a tendency to change at all.

One frequently used measure of correlation is Pearson's correlation; it measures the linearity of the relationship between two variables. The Pearson's correlation coefficient is calculated by dividing the covariance of two variables by the product of the standard deviation of each variable. That is, for n pairs of variables x and y, the value of the Pearson's correlation is

For instance, as part of a study on smokers' health and demographics, a survey researcher might collect data on smokers' annual household income and the average number of cigarettes smoked daily. The data for 10 smokers—sorted in ascending order of income—might look like Table 1.

In this case, simple inspection reveals that the correlation is negative. That is, as income increases, the average number of cigarettes smoked daily tends to decrease. The value of the Pearson's correlation between these variables equals −0.484, confirming that the relationship between the two variables is, in fact, negative and moderately linear. A scatter plot of these variables visually illustrates the nature of this relationship, as shown in Figure 1 (next page).

While correlation analysis describes one aspect of the quantitative relationship between variables, it certainly has its limitations. First, it cannot be used to infer the extent of a causal relationship. For example, the preceding example shows only that income and average number of cigarettes smoked daily for these 10 individuals are related in a negative, somewhat linear fashion. It does not mean that increasing a smoker's income would cause a reduction in the number of cigarettes smoked or that smoking fewer cigarettes would cause an increase in an individual's income.

A second important limitation is that correlation analysis does not provide any information about the magnitude—or the size—of the relationship between variables. Two variables may be highly correlated, but the magnitude of the relationship might, in fact, be very small. For instance, the correlation of −0.484 between income and average number of cigarettes smoked daily in the example says only that the relationship is negative and that the relationship is somewhat linear. It does not provide any information regarding how many fewer cigarettes are related to an increase in income. That is, every extra dollar of income could be associated with a decrease in average number of cigarettes that is very large, very small, or anywhere in between.

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