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, ΣX1X= 12005, ΣX2Y = 6485, n=10.

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

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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 x1x+ b2Σx22

or, 6485= 441b0 + 12005b1 + 19461b2    …………(iii)

To find b, 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, \(D = \begin{vmatrix}10 & 272 & 441\\ 272 & 7428 & 12005\\ 441 & 12005 & 19461\end{vmatrix}\)

= 10(144556308 – 144120025) – 272(5293392 – 5294205) + 441(3265360 – 3275748)

= 2858

\(D_1 = \begin{vmatrix}147 & 272 & 441\\ 4013 & 7428 & 12005\\ 6485 & 12005 & 19461\end{vmatrix}\)

= 147(144556308 – 144120025) – 272(78096993 – 77852425) + 441(48176065 – 48170580)

= 29990

\(D_2 = \begin{vmatrix}10 & 147 & 441\\ 272 & 4013 & 12005\\ 441 & 6485 & 19461\end{vmatrix}\)

= 10(78096993 – 77852425) – 147(5293392 – 5294205) – 441(1763920 – 1769733)

= 1658

\(D_1 = \begin{vmatrix}10 & 272 & 147\\ 272 & 7428 & 4013\\ 441 & 12005 & 6485\end{vmatrix}\)

= 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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