Multiple Regression Equation

Introduction to multiple regression equation:

Multiple regression analysis is a statistical tool in which a mathematical model is developed to predict a dependent variable by two or more independent variables or in which atleast one predictor is non-linear. The principal advantage of multiple regression is that it allows us to utilize more of the information available to us to fit curves as well as lines.

Multiple Regression Model with Two Independent Variables:

The simplest multiple regression model is one constructed with two independent variables, where the highest power if either variable is one.

The model is given by  y = β0 + β1x1 + β2x2 + ε.
The constants and coefficients are estimated from sample information, resulting in the following model.

Y = b0 + b1x1 + b2x2

Multiple Regression Model Equation:

Multiple regression analysis is similar to simple regression analysis. However, it is more complex conceptually and computationally. The general equation for the probabilistic multiple regression model is given by


y = β0 + β1x1 + β2x2 …+ βkxk +  ε.

Where y = the value of the dependent variable

β0 = the regression constant

β1 = the partial regression coefficient for independent variable 1

β2 = the partial regression coefficient for independent variable 2

.....

.....

βk = the partial regression coefficient for independent variable k

k = the number of independent variables

In multiple regression analysis, the dependent variable, y, is some times reffered to as the responsive variable. The partial regression coefficient of an independent variable, βi represents the increase that will occur in the value of y from a one unit increase in that dependent variable if all other variables are held constant. The partial regression coefficient occur because more than one predictor is included in model.

In actuality, the partial regression coefficients and the regression constant of a multiple regression model are population values and are unknow. In virtually all research, these values are estimated y with sample information.Please express your views of this topic how to cross multiply by commenting on blog.

Y= b0 + b1x1 + b2x2 …+ bkxk

Where Y = the predicted value of y

b0  = the estimate of the regression constant

b1 = the estimate of the regression coefficient 1

b2 = the estimate of the regression coefficient 2

bk = the estimate of the regression coefficient k

k = the number of independent variables.

Determining the Multiple Regression Equation:

The procedure for determining formula to solve for multiple regression coefficients is similar to that of solving for simple regression coefficients. The formulas are established to meet an objective of minimizing the sum of squares of error for the model. Hence, the regression analysis shown here is reffered to as least square analysis. Methods of calculs are applied, resulting in K+1 unknowns for regression analysis with k dependent variables.

For multiple regression models with two independent variables, the result is three simultaneous equations with three unknowns( b0, b1 and b2).



The process of solving these equations is tedious and time consuming. Solving for the regression coefficients and regression constant in a multiple regression model with two independent variable requires `sum` x1, ∑ x2, ∑ y, ∑ x12, ∑ x22, ∑ x1x2, ∑ x1y  and ∑ x2y.