Regression Equations

The   regression equations express the regression line. As there are two regression lines so there  are two  regression  equations. The regression equation X and Y describes the variation in the values of  X for  the  given  changes  in Y. And used for estimating the values  of X for the  given  value of Y. Similarly  the regression  equation  Y and X describes the variation  in the  values  of Y for the  given  changes in X  and is  used  for estimating  the  value of Y for the  given value of X.

Regression Equation of Y on X

The regression equation of Y on  X is  expressed  as follows:

Y= a + b X

It may be noted that in  this equation y is a dependent  variables i , its  values depends  on X....X is  independent  variables  i, e, we  can take a given values  of X  and compute the values of  Y.

A is  y intercept   because its  values is  the point  at which the regression  line cross the  Y axis  that is  the vertical  axis b is the slope  of line. It  represents  changes  in Y  variable  for a unit  change in  X variable.

A and  b in the equation  are called  numerical  constants  because  for any  given  straight line  their  value  does  not  change.

If the  values  of the  constants  a and b  are obtained the line is  completely  determined. But  the  question is how  to obtain  these  values. The answer is  provided by the  method of least squares which  states  that the  line should  be drawn  through the plotted points  in such  a manner that  the sum  of the square of the deviations of  the actual y values from the  computed Y values  is the least or in  other words in order  to obtain a line which  fits  the points  best ∑ ( Y - Y _c)2 should  be minimum. Such  a line is  known  as the  line  of best fit.

A straight  line  fitted by  least  squares  has the followings  characteristics;

a.It gives the best fit to  data in  the since that it  make the sum  of the  squared deviations  from the  line  ( Y- Y c)2 smaller than  they would  be from  any other  straight  line. This  property  accounts for the name least squares.

b.The deviation  above  the line  equal  those  below the line  on the average. This mean  that the  total  of the  positive  deviations is  zero or ∑( Y-Y_c)= 0

c. The straight line goes  through  the overall  mean of the  data( S Y).

d. When the  data  represent  a sample  from a large  population  the least  squares  lien is a best estimate of the population  regression line.

With a little  algebra and differential  calculus it  can be shown that the followings two equations if  solved simultaneously will yield  values of  the parameters a and b  such that  the least squares requirement  is fulfilled:

∑Y = Na + b ∑X

∑XY = a ∑X + b ∑X2

These equations  are usually  called  the normal  equations. In the equations ∑X ∑XY, ∑X22, indicate  totals  which  are computed from  the observed pairs of values  of two  variables  X and y  to which  the least  squares estimating.