In many cases, the independent variable and the dependent variable are both continuous (taking on real values). This enables another important measure called the *Pearson correlation coefficient* (or *Pearson's r*). This estimates the amount of linear dependency between the two variables. For each subject , the treatment (or level) is applied and the response is . Note that in this case, there are no groups (or every subject is a unique group). Also, any treatment could potentially be applied to any subject; the index only denotes the particular subject.

The *r-value* is calculated as the estimated covariance between and when treated as random variables:

(12.7) |

in which and are the averages of and , respectively, for the set of all subjects. The denominator is just the product of the estimated standard deviations: .

The possible r-values range between and . Three qualitatively different outcomes can occur:

- : This means that and are
*positively correlated*. As increases, tends to increase. A larger value of implies a stronger effect. - : This means that and are
*uncorrelated*, which is theoretically equivalent to a null hypothesis. - : This means that and are
*negatively correlated*. As increases, tends to decrease. A smaller value of implies a stronger effect.

Steven M LaValle 2020-11-11