# R - Multiple Regression

Multiple regression is an extension of linear regression into relationship between more than two variables. In simple linear relation we have one predictor and one response variable, but in multiple regression we have more than one predictor variable and one response variable.

The general mathematical equation for multiple regression is −

```y = a + b1x1 + b2x2 +...bnxn
```

Following 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.

We create the regression model using the 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")]
```

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
```
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