REGRESSION LINES

If we take the case of two variables X and Y, we shall have two regression lines as the regression of X an Y and the regression of Y on X. The regression line of Y on X gives the most probable values of Y for given values of X and the regression line of X on  Y gives the most probable value of X for given values of Y. However, when there is either perfect positive or perfect negative correlation between the two variables (r = + 1) the two regression lines will coincide, i.e., we will have only one line. The farther the two regression lines from each other, the lesser is the degree of correlation and the nearer the two regression lines to each other, the higher is the degree of correlation. If the variables are independent, r is zero and the lines of regression are at right angles, i.e., parallel to OX and OY.

It should be noted that the regression lines cut each other at the point of average of X and Y, i.e., if from the point where both the regression lines cut each other a perpendicular is drawn on the X- axis, we will get the mean value of X and if from that point a horizontal line is drawn on the Y-axis, we will get the mean value of Y.

It is important to note that the regression lines are drawn on least squares assumption which stipulates that the sum of squares of the deviations of the observed ‘Y’ values from the fitted lines shall be minimum. The total of the squares of the deviations of the various points is minimum only from the line of best fit. The deviations from the points to the line of best fit can be measured in two ways –vertical, i.e., parallel to Y-axis, and horizontal, i.e., parallel to X-axis. For minimizing the total of the squares separately it is essential to have two regression lines. The regression line of Y on X is drawn in such a way that it minimizes total of squares of the vertical deviation and the regression line of X on Y minimizes the total squares of the horizontal deviations.

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