Definition of Correlation

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Correlation

Residuals

Suppose that the average lifespan for people who smoke is:

Packs Per Week	Life Span
1	72
2	70
3	69
5	68

We can calculate the least squares regression line:

y = 73 - 1.3x

We define the first residual to be the difference between the first lifespan and the first estimated lifespan:

72 - (73 - 1.3(1)) = 0.3

the second residual as:

70 - (73 - 1.3(2)) = -0.4

the third as:

69 - (73 - 1.3(3)) = -0.1

and the fourth as

68 - (73 - 1.3(5)) = 1.5

in general we have the residual is

y_i - y = y_i - (a + bx_i)

Coefficient of determination: r²

We define the coefficient of determination as an indication of how linear the data is. r² has the following properties:

Properties of the Coefficient of Determination

r² is between 0 and 1.
If r² = 1 then all points lie on a line. (perfectly linear)
If r² = 0 then the regression line is a useless indicator for predicting y values.

Construction

To compute r², do the following:

Compute the sum of the squares of the residuals: SSResid
Compute Sy² and (Sy)². We say that

SSTo = Sy² - (S y)²/n
Compute

1 - SSResid/SSto

This is r² .

If we multiply r² by 100%, we arrive at the percent of the observed variation attributable to the linear relationship.

Correlation: r

If we want to determine not just if they are linearly related, but also want to know whether there is a positive relationship or a negative relationship (b> 0 or b<0) and want the calculation unitless, we compute Pearson's correlation coefficient r

\[ r = \frac{ \sum xy - \sum x \sum y /n}{ \sqrt{ (\sum x^{2} - (\sum x)^2 /n)(\sum y^{2} - (\sum y)^2 /n)}} \]

We have

r² = r²

that is the square of the correlation coefficient is equal to the coefficient of determination.

If r < 0 then they are negatively correlated.
If r > 0 then they are positively correlated.

We say that the correlation is

strong if |r| >.8
middle if .5 < |r| < .8 and
weak otherwise.

Correlation does not imply causation. For example there may be a strong correlation between grayness in hair and wrinkles, but having gray hair does not cause one to have wrinkles.

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