Coefficients of Correlation, Alienation, and Determination

Notations and Definition

We have S observations, and for each observation, we have two measurements, denoted W and Y, with respective means MW and MY. For each observation, we define the cross product as the product of the deviations of each variable to its mean. The sum of these cross products, denoted SCPWY, is computed as

The sum of the cross products reflects the association between the variables. When the deviations tend to have the same sign, they indicate a positive relationship, and when they tend to have different signs, they indicate a negative relationship. The average value of the SCPWY is called the covariance (cov), and just like the variance, the covariance can be computed by dividing by S or (S– 1):

The covariance reflects the association between the variables, but it is expressed in the original units of measurement. In order to eliminate them, the covariance is normalized by division by the standard deviation of each variable (σ). This defines the coefficient of correlation, denoted rW .Y, which is equal to

Rewriting the previous formula gives a more practical formula:

An Example: Correlation Computation

We illustrate the computation for the coefficient of correlation with the following data, describing the values of W and Y for S = 6 subjects:

Step 1: Computing the sum of the cross products

First compute the means of W and Y: and

and

The sum of the cross products is then equal to

The sum of squares of Ws is obtained as

The sum of squares of Ys is

Step 2: Computing rW.Y

The coefficient of correlation between W and Y is equal to

We can interpret this value of r =–.5 as an indication of a negative linear relationship between W and Y.

Some Properties of the Coefficient of Correlation

The coefficient of correlation is a number without a unit. This occurs because of dividing the units of the numerator by the same units in the denominator. Hence, the coefficient of correlation can be used to compare outcomes across different variables. The [Page 160]magnitude of the coefficient of correlation is always smaller than or equal to 1. This happens because the numerator of the coefficient of correlation (see Equation 4) is always smaller than or equal to its denominator (this property follows from the Cauchy-Schwartz inequity). A coefficient of correlation equal to +1 or −1 indicates that a plot of the observations will show that they are positioned on a line.

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