R - Linear Regression

Regression analysis is a very widely used statistical tool to establish a relationship model between two variables. One of these variable is called predictor variable whose value is gathered through experiments. The other variable is called response variable whose value is derived from the predictor variable.
In Linear Regression these two variables are related through an equation, where exponent (power) of both these variables is 1. Mathematically a linear relationship represents a straight line when plotted as a graph. A non-linear relationship where the exponent of any variable is not equal to 1 creates a curve.
The general mathematical equation for a linear regression is −
y = ax + b

Following is the description of the parameters used −
y is the response variable.
y is the response variable.
x is the predictor variable.
x is the predictor variable.
a and b are constants which are called the coefficients.
a and b are constants which are called the coefficients.
A simple example of regression is predicting weight of a person when his height is known. To do this we need to have the relationship between height and weight of a person.
The steps to create the relationship is −
Carry out the experiment of gathering a sample of observed values of height and corresponding weight.
Carry out the experiment of gathering a sample of observed values of height and corresponding weight.
Create a relationship model using the lm() functions in R.
Create a relationship model using the lm() functions in R.
Find the coefficients from the model created and create the mathematical equation using these
Find the coefficients from the model created and create the mathematical equation using these
Get a summary of the relationship model to know the average error in prediction. Also called residuals.
Get a summary of the relationship model to know the average error in prediction. Also called residuals.
To predict the weight of new persons, use the predict() function in R.
To predict the weight of new persons, use the predict() function in R.
Below is the sample data representing the observations −
# Values of height
151, 174, 138, 186, 128, 136, 179, 163, 152, 131

# Values of weight.
63, 81, 56, 91, 47, 57, 76, 72, 62, 48

This function creates the relationship model between the predictor and the response variable.
The basic syntax for lm() function in linear regression is −
lm(formula,data)

Following is the description of the parameters used −
formula is a symbol presenting the relation between x and y.
formula is a symbol presenting the relation between x and y.
data is the vector on which the formula will be applied.
data is the vector on which the formula will be applied.
x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)

# Apply the lm() function.
relation <- lm(y~x)

print(relation)
When we execute the above code, it produces the following result −
Call:
lm(formula = y ~ x)

Coefficients:
(Intercept)            x  
   -38.4551          0.6746 

x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)

# Apply the lm() function.
relation <- lm(y~x)

print(summary(relation))
When we execute the above code, it produces the following result −
Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q     Median      3Q     Max 
-6.3002    -1.6629  0.0412    1.8944  3.9775 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -38.45509    8.04901  -4.778  0.00139 ** 
x             0.67461    0.05191  12.997 1.16e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.253 on 8 degrees of freedom
Multiple R-squared:  0.9548,    Adjusted R-squared:  0.9491 
F-statistic: 168.9 on 1 and 8 DF,  p-value: 1.164e-06

The basic syntax for predict() in linear regression is −
predict(object, newdata)

Following is the description of the parameters used −
object is the formula which is already created using the lm() function.
object is the formula which is already created using the lm() function.
newdata is the vector containing the new value for predictor variable.
newdata is the vector containing the new value for predictor variable.
# The predictor vector.
x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)

# The resposne vector.
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)

# Apply the lm() function.
relation <- lm(y~x)

# Find weight of a person with height 170.
a <- data.frame(x = 170)
result <-  predict(relation,a)
print(result)
When we execute the above code, it produces the following result −
       1 
76.22869 

# Create the predictor and response variable.
x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
relation <- lm(y~x)

# Give the chart file a name.
png(file = "linearregression.png")

# Plot the chart.
plot(y,x,col = "blue",main = "Height & Weight Regression",
abline(lm(x~y)),cex = 1.3,pch = 16,xlab = "Weight in Kg",ylab = "Height in cm")

# Save the file.
dev.off()
When we execute the above code, it produces the following result −

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