What is Multiple Linear Regression (MLR)? From following information of variables X1 , X2 , and Y.

ΣX₁ = 272, ΣX₂= 441, ΣY= 147, ΣX₁²= 7428, ΣX₂²=19461, ΣY² = 2173, ΣX1Y = 4013, ΣX₁X₂= 12005, ΣX₂Y = 6485, n=10.

Fit a regression equation Y on X1 and X2. Interpret the regression coefficients.

Question

What is Multiple Linear Regression (MLR)? From following information of variables X1 , X2 , and Y.

ΣX₁ = 272, ΣX₂= 441, ΣY= 147, ΣX₁²= 7428, ΣX₂²=19461, ΣY² = 2173, ΣX1Y = 4013, ΣX₁X₂= 12005, ΣX₂Y = 6485, n=10.

Fit a regression equation Y on X1 and X2. Interpret the regression coefficients.

Suresh Chand · Accepted Answer

It is a linear function of one dependent variable with two or more independent variables. With the help of two or more independent variables the value of dependent variable is predicted. For example, if we wish to test the hypothesis the whether that whether or not 'pass grade' of students depends on many causes such as previous task mark, study hours, IQ, .... then we can test a regression of cause (pass grade) with effect variables. This test will give us which causes are really significant is generating effect variable and among the significant cause variables their relative value responsible to generate the effect variables. If we assume more than one cause (called Y or dependent variable), it is known as multiple regression. If we assume that the relation between Y and X's is linear it is called multiple linear regression. Let us consider three variable Y, X1 and Y2 in which Y is dependent variable, X1 and Y2 are independent variables, then the mathematical form of the linear relationship of Y with X1 and X2 is expressed as Y = b0 + b1X1 + b2X2 + ε Where Y = Dependent variable X1 and X2 = Independent variable or explanatory variable or regressors b0 = Intercept and is called average value of Y and X1 and X2 are zero b1 = Regression coefficient of Y on X1 keeping X2 constant. It measures the amount of change in Y per unit change in X1 holding X2 constant. b2 = Regression coefficient of Y on X2 keeping X1 constant. It measures the amount of change in Y per unit change in X2 holding X1 constant. ε = Random error Random error(ε) is not created from mistake. It is a technical term that denotes the excess of value from real by model estimation. error is also called Residual. To fit regression equation y = b0 + b1x1 + b2x2 Σy = nb0 + b1Σ x1 + b2Σ x2 or, 147 = 10b0 + 272b1 + 441b2    ............(i) Σyx1 = b0Σx1 + b1Σx12 + b2Σ x1x2 or, 4013 = 272b0 + 7428b1 + 12005b2    ............(ii) Σyx2 = b0Σx2 +  b1Σ x1x2 + b2Σx22 or, 6485= 441b0 + 12005b1 + 19461b2    ............(iii) To find b0 , b1 and b2 using Cramer's rule Coefficient of b0 Coefficient of b1 Coefficient of b2 Constant 10 272 441 147 272 7428 12005 4013 441 12005 19461 6485 Now, (D = begin{vmatrix}10 & 272 & 441 272 & 7428 & 12005 441 & 12005 & 19461end{vmatrix}) = 10(144556308 - 144120025) - 272(5293392 - 5294205) + 441(3265360 - 3275748) = 2858 (D_1 = begin{vmatrix}147 & 272 & 441 4013 & 7428 & 12005 6485 & 12005 & 19461end{vmatrix}) = 147(144556308 - 144120025) - 272(78096993 - 77852425) + 441(48176065 - 48170580) = 29990 (D_2 = begin{vmatrix}10 & 147 & 441 272 & 4013 & 12005 441 & 6485 & 19461end{vmatrix}) = 10(78096993 - 77852425) - 147(5293392 - 5294205) - 441(1763920 - 1769733) = 1658 (D_1 = begin{vmatrix}10 & 272 & 147 272 & 7428 & 4013 441 & 12005 & 6485end{vmatrix}) = 10(48170580 - 48176065) - 272(1763920 - 1769733) + 147(3265360 - 3275748) = -750 Now, b0 = (frac{D_1}{D} = frac{29990}{2825}) = 10.493 b1 = (frac{D_2}{D} = frac{1685}{2825}) = 0.58 b2 = (frac{D_3}{D} = frac{-750}{2825}) = -0.262 Hence regression equation is y = b0 + b1x1 + b2x2 = 10.493 + 0.58x1 - 0.262x2 Here, b1 = 0.58 it means y changes by 0.58 per unit change x1 keeping x2 constant. Here, b2 = -0.262 it means y changes(decreases) by 0.262 per unit change x2 keeping x1 constant.

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