It is used to describe the variation in values of Y for given changes in X. The regression equation of Y on X is expressed as:
Y = a + bX ------------ (2)
It may be noted that in equation (2), Y is a dependent variable [i.e., its value depends on X], X is an independent variable [i.e., we can take a given value of X and compute value of Y], 'a' is Y-intercept because it is the point at which the regression line crosses the Y-axis, i.e., vertical axis and 'b' is the slope of a line. It represents change in Y variable for a unit change in X variable.
In the equation (2), 'a' and 'b' are called numerical constants because for any given straight line, their values does not change. To determine the values of 'a and b', the following two normal equations are to be solved simultaneously:
ΣY = Na + bΣX and ΣXY = aΣX + bΣX2
Regression equation of Y on X when deviations taken from means of X and Y: The above method of finding out regression equation is too slow. In order to find out regression equation quickly, we take deviations of X and Y series from their respective means instead of dealing with actual values of X and Y. In such a case regression equation of Y on X is expressed as:
In the above equation, X is a mean of X series, Y is a mean of Y series and is called regression coefficient of Y on X and it is denoted by the symbol bYX.
- If the deviations are taken from actual means, then we need not calculate coefficient of correlation (r), sX and sY. Therefore, the regression coefficient of Y on X is given as:
- If the deviations are taken from assumed means (A), then the regression coefficient of Y on X is expressed as:
where dX = X – A, dY = Y – A and N = number of observations.
|Points to remember
|• With the help of two regression coefficients it is possible to calculate correlation coefficient (r) symbolically as:
|• Both the regression coefficients cannot exceed one.
|• One of the regression coefficients must be less than one.
|• Both the regression coefficients will have the same sign, i.e., either they will be positive or negative.
|• The coefficient of correlation (r) will have the same sign as that of regression coefficients.