## Regression

### Meaning

Regression is the study of the nature of relationship between the variables so that one may be able to predict the unknown value of one variable for a known value of another variable.### Comparison between Correlation and Regression

- Correlation study the degree of relationship between variables while the regression study the nature of relationship between variables.
- Correlation need not imply cause and effect relationship between variables whereas regression implies cause and effect relationship between variables.
- There may be non-sense correlation between variables whereas in case of regression there is nothing like non-sense regression.
- The correlation coefficient is independent of change of origin and scale but regression coefficients are independent of only change of origin but not of scale.
- Correlation coefficient cannot be used for prediction. But regression lines are used for prediction.

### Two lines of regression

Two lines of regression |

#### Regression equation of line Y on X

When we try to depict the change in 'y' for a given change in 'x', then the regression line of y on x is used.The regression equation of y on x is given by

y-ybar = byx (x-xbar)

Where byx = Cov(x,y)/Var(x)

byx is called the regression coefficient of y on x.

#### Regression equation of line X on Y

When we try to depict the change in 'x' for a given change in 'y'. then the regression line of x on y is usedThe regression equation of x on y is given by

x-xbar = bxy(y-ybar)

Where bxy= Cov(xy)/Var(y)

bxy is called the regression coefficient of x on y

#### Relation between correlation and regression coefficient

- byx.bxy = r^2

This implies that multiplication of both regression coefficient is equal to the square of correlation coefficient and the value of r^2 lies between 0 and 1

- Correlation coefficient is geometric mean of both regression coefficients i.e.

r = +/- square root( byx. bxy)

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