Multiple Linear Regression

What is Multiple Linear Regression?

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 +…….. β_kX_k + ε

Where

Y is dependent variable
X is independent variable
β₀ is Y intercept
ε is residual also called error
β_kslope coefficient for each independent variable

β can also be compute in a such a way that minimizes the sum of squared errors

Multiple Linear Regression — 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.

Y̅ =354/8=44.25
p̅=61/8=7.625
q̅=38/8=4.75

b̂₀ = Y̅-b̂₁p̅- b̂₂q̅=31.37

The estimated regression line would be

y = 31.37+0.75p+1.5q