Calculate: Regression

November 5, 2008 by  
Filed under How To Calculate, Regression

Think of regression as a 5-step process.
First, calculate the mean of X and the mean of Y.
Second, calculate the slope of the line (called b). To find the slope, divide the SSxy by the SS of the predictor. That is:
Third, calculate the Y intercept (called a). The formula is:
Fourth, make a prediction. Plug in your X. Since you’ve already calculated a and b, all you need is an X value and you can predict what Y will be. Use the formula for a straight line:
Fifth, estimate the accuracy of the prediction. Don’t worry, there’s a formula for that too. Here it is:
 
EXAMPLE
X            Y
2             5
4             7
6             6
9             8
12         14
15          11
15          12
 
First, calculate the mean of X and the mean of Y. Each mean = 9.
 
Second, calculate the slope of the line (called b). To find the slope, divide the SSxy by the SS of the predictor. The SSx is 164; the SSy is 68; and the SSxy 92. So the slope (b) = .56.
 
Third, calculate the Y intercept (called a). So a = 9 – (.56 * 9) = 3.95.
 
Fourth, make a prediction. Use the formula for a straight line: Let’s assume that the X value is 8, we would predict that Y (which will call Y-prime so we know it’s a prediction) equals 8.44.
 
The standard deviation of Y is 3.12, and r = .87. So the standard error of estimate is 1.81. This means that we’re 68% sure that the real score will be 8.44, plus or minus 1.81. In other words, we’re fairly sure the score will be between 6.63 and 10.25.