# R - Multiple Regression

The general mathematical equation for multiple regression is −

y = a + b1x1 + b2x2 +...bnxnFollowing is the description of the parameters used −

**y**is the response variable.**a, b1, b2...bn**are the coefficients.**x1, x2, ...xn**are the predictor variables.

**lm()**function in R. The model determines the value of the coefficients using the input data. Next we can predict the value of the response variable for a given set of predictor variables using these coefficients.

## lm() Function

This function creates the relationship model between the predictor and the response variable.### Syntax

The basic syntax for**lm()**function in multiple regression is −

lm(y ~ x1+x2+x3...,data)Following is the description of the parameters used −

**formula**is a symbol presenting the relation between the response variable and predictor variables.**data**is the vector on which the formula will be applied.

## Example

### Input Data

Consider the data set "mtcars" available in the R environment. It gives a comparison between different car models in terms of mileage per gallon (mpg), cylinder displacement("disp"), horse power("hp"), weight of the car("wt") and some more parameters.The goal of the model is to establish the relationship between "mpg" as a response variable with "disp","hp" and "wt" as predictor variables. We create a subset of these variables from the mtcars data set for this purpose.

input <- mtcars[,c("mpg","disp","hp","wt")] print(head(input))When we execute the above code, it produces the following result −

mpg disp hp wt Mazda RX4 21.0 160 110 2.620 Mazda RX4 Wag 21.0 160 110 2.875 Datsun 710 22.8 108 93 2.320 Hornet 4 Drive 21.4 258 110 3.215 Hornet Sportabout 18.7 360 175 3.440 Valiant 18.1 225 105 3.460

### Create Relationship Model & get the Coefficients

input <- mtcars[,c("mpg","disp","hp","wt")] # Create the relationship model. model <- lm(mpg~disp+hp+wt, data = input) # Show the model. print(model) # Get the Intercept and coefficients as vector elements. cat("# # # # The Coefficient Values # # # ","\n") a <- coef(model)[1] print(a) Xdisp <- coef(model)[2] Xhp <- coef(model)[3] Xwt <- coef(model)[4] print(Xdisp) print(Xhp) print(Xwt)When we execute the above code, it produces the following result −

Call: lm(formula = mpg ~ disp + hp + wt, data = input) Coefficients: (Intercept) disp hp wt 37.105505 -0.000937 -0.031157 -3.800891 # # # # The Coefficient Values # # # (Intercept) 37.10551 disp -0.0009370091 hp -0.03115655 wt -3.800891

### Create Equation for Regression Model

Based on the above intercept and coefficient values, we create the mathematical equation.Y = a+Xdisp.x1+Xhp.x2+Xwt.x3 or Y = 37.15+(-0.000937)*x1+(-0.0311)*x2+(-3.8008)*x3

### Apply Equation for predicting New Values

We can use the regression equation created above to predict the mileage when a new set of values for displacement, horse power and weight is provided.For a car with disp = 221, hp = 102 and wt = 2.91 the predicted mileage is −

Y = 37.15+(-0.000937)*221+(-0.0311)*102+(-3.8008)*2.91 = 22.7104

*Table of contents:*1. R - Overview

2. R - Environment Setup

3. R - Basic Syntax

4. R - Data Types

5. R - Variables

6. R - Operators

7. R - Decision Making

8. R - Loops

9. R - Functions

10. R - Strings

11. R - Vectors

12. R - Matrices

13. R - Arrays

14. R - Factors

15. R - Data Frames

16. R - Packages

17. R - Data Reshaping

18. R - CSV Files

19. R - Excel Files

20. R - Binary Files

21. R - XML Files

22. R - JSON Files

23. R - Web Data

24. R - Database

25. R - Pie Charts

26. R - Bar Charts

27. R - Boxplots

28. R - Histograms

29. R - Line Graphs

30. R - Scatterplots

31. R - Mean, Median and Mode

32. R - Linear Regression

33. R - Multiple Regression

34. R - Logistic Regression

35. R - Normal Distribution

36. R - Binomial Distribution

37. R - Poisson Regression

38. R - Analysis of Covariance

39. R - Time Series Analysis

40. R - Nonlinear Least Square

41. R - Decision Tree

42. R - Random Forest

43. R - Survival Analysis

44. R - Chi Square Tests

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