Multiple Regression Model

Multiple Regression Model:

The statistical technique that use Several explanatory variables to Predict the Outcome of the response variable. The Goal of multiple linear regression (sums billion) is to model the Relationship Between the explanatory and response variables.
Multiple linear regression attempts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable y. The population regression line for p explanatory variables x₁, x₂, ... , x_p is defined to be _y = ₀ + ₁x₁ + ₂x₂ + ... + _px_p. This line describes how the mean response _y changes with the explanatory variables. The observed values for y vary about their means _y and are assumed to have the same standard deviation . The fitted values b₀, b₁, ..., b_p estimate the parameters ₀, ₁, ..., _p of the population regression line.
Since the observed values for y vary about their means _y, the multiple regression model includes a term for this variation. In words, the model is expressed as DATA = FIT + RESIDUAL, where the "FIT" term represents the expression ₀ + ₁x₁ + ₂x₂ + ... _px_p. The "RESIDUAL" term represents the deviations of the observed values y from their means _y, which are normally distributed with mean 0 and variance . The notation for the model deviations is .
Formally, the model for multiple linear regression, given n observations, is
y_i = ₀ + ₁x_i1 + ₂x_i2 + ... _px_ip + _i for i = 1,2, ... n.
In the least-squares model, the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0). Because the deviations are first squared, then summed, there are no cancellations between positive and negative values. The least-squares estimates b₀, b₁, ... b_p are usually computed by statistical software.
The values fit by the equation b₀ + b₁x_i1 + ... + b_px_ip are denoted _i, and the residuals e_i are equal to y_i - _i, the difference between the observed and fitted values. The sum of the residuals is equal to zero.
The variance ² may be estimated by s² = , also known as the mean-squared error (or MSE).
The estimate of the standard error s is the square root of the MSE.