Multiple linear regression is an extension to methodology of simple linear regression. Simple linear regression is to study the two variables in which one variable is independent variable (X) and the other one is dependent variable (Y). In other words predict the change in dependent variable according to change in independent variable
When to Use Multiple Linear Regression
Multiple linear regression is to study more than two variables. In fact the basic difference between simple and multiple regression is in terms of explanatory variables. In multiple regression unlike simple linear regression there are more than one independent variable (X), these independent variables used to predict a single dependent variable(Y). Predict the change in dependent variable (Y) according to change in independent variables.
Example: The house price (Dependent variable Y) depends on the various Independent variables (X) like locality, number of bed rooms, number of bathrooms, age of the house and also square foot of the house.
Notes about Multiple Linear Regression
Y is the linear transformation of the X variables and subjected to the condition that the sum of squared deviations of the observed and predicted Y is minimized, in other words the sum of squared errors is minimized
Residual also called error is the difference between the actual observed values of dependent variable Y and the predicted values that we get as a linear transformation of the X variables.
The coefficient of determination is R2. It is the proportion of the explained variation divided by the total variation. When numbers of predictors are adding to the model then R2 will also increases, despite the fact that predictors have no relation with output variable.
Likewise r2 (the linear coefficient of determination) R2 (the multiple coefficient of determination) take values in the interval:
0≤ R2 ≤1
If the value of R2 is 0 then outcome cannot be predicated, where as if R2 is 1 outcome can be predicated and it is error free from the independent variables (X), but same it does not mean a great model
The computation in case of multiple regression is complex due to the number of explanatory variables in the model. However because of interrelationship among the variables the interpretation also changes accordingly
Assumptions of Multiple Linear Regression
Independent Residuals
No Multicollinearity – Not too high correlation between the independent variables
Residuals must be normally distributed
Furthermore relationship between each predictor variable and the outcome variable is linear
Formula to calculate Multiple Linear Regression
A first order
linear model
The formula for
two independent variables the prediction of Y is
Y= β0+β1X1+β2X2 +…….. βkXk + ε
Where
Y is dependent variable
X is independent variable
β0 is Y intercept
ε is residual also called error
βk slope coefficient for each independent variable
β can also be compute in a such a way that minimizes the sum of squared errors
ANOVA Table for Multiple Regression
Where k is the
number of predictor variables
And estimated regression line shall be y = b̂0+b̂1X1+b̂2X2
Formulas to
calculate estimates of parameters betas’
b̂0 =
Y̅-b̂1X̅1– b̂2X̅2
A Second –Order Linear Model (Two Predictor Variables)
Y= β0+β1X1+β2X2+ β3 X1X2+ β4 X12++ β5 X22+ε
Example of Multiple Linear Regression in DMAIC
Multiple Linear Regression will be used in Analyze phase of DMAIC to study more than two variables. In a laboratory chemist recorded the yield of the process which will be impacted by the two factors. Chemist wants to model the first order regression.
