Elementary Statistics and Computer Application

Assumptions and Properties of Regression coefficients

Assumptions

1. The x’s are non-random or fixed constants
2. At each fixed value of X the corresponding values of Y have a normal distribution about a mean.
3. For any given x, the variance of Y is same.
4. The values of y observed at different levels of x are completely independent.
Properties of Regression coefficients

1. The correlation coefficient is the geometric mean of the two regression coefficients
2. Regression coefficients are independent of change of origin but not of scale.
3. If one regression coefficient is greater than unit, then the other must be less than unit but not vice versa. ie. both the regression coefficients can be less than unity but both cannot be greater than unity, ie. if b1>1 then b2<1 and if b2>1, then b1<1.
4. Also if one regression coefficient is positive the other must be positive (in this case the correlation coefficient is the positive square root of the product of the two regression coefficients) and if one regression coefficient is negative the other must be negative (in this case the correlation coefficient is the negative square root of the product of the two regression coefficients). ie.if b1>0, then b2>0 and if b1<0, then b2<0.
5. If θ is the angle between the two regression lines then it is given by