In this presentation, we will talk about simple linear regression, from defining it, to visualizing it for sample seed data and for the dataset mtcars.

• Simple linear regression is a linear regression model with that estimates a relationship between one independent and one dependent variable. It is a statistical method to model the relationship between two continuous variables and as I mentioned earlier, it assumes a linear relationship between an independent variable \(x\) and a dependent variable \(y\).

• The model can be expressed using the formula:

\[ y = \beta_0 + \beta_1 x + \varepsilon \]

Where:

• \(y\) = Dependent variable
• \(x\) = Independent variable
• \(\beta_0\) = Intercept
• \(\beta_1\) = Slope
• \(\varepsilon\) = Error

• Parameters \(\beta_0\) and \(\beta_1\) are estimated using the formula:

\[ \min_{\beta_0, \beta_1} \sum_{i=1}^n (y_i - \beta_0 - \beta_1 x_i)^2 \]

library(ggplot2)
library(plotly)

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

library(tidyr)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

# Sample data
set.seed(123)
x <- 1:100
y <- 2 + 3 * x + rnorm(100, mean = 0, sd = 30)
data <- data.frame(x, y)

## 
## Call:
## lm(formula = y ~ x, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -73.607 -16.571  -1.039  19.455  62.846 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.90788    5.52862   0.164     0.87    
## x            3.07533    0.09505  32.356   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.44 on 98 degrees of freedom
## Multiple R-squared:  0.9144, Adjusted R-squared:  0.9135 
## F-statistic:  1047 on 1 and 98 DF,  p-value: < 2.2e-16

The intercept is the starting point where x=0. The slope shows, how much y changes with a one unit increase in x. The R squared shows how well our line fits to the data.

##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(color = "blue") +
  labs(
    title = "MPG vs. Car Weight",
    x = "Weight (1000 lbs)",
    y = "Miles Per Gallon (MPG)"
  ) +
  theme_minimal()

## Fitting Linear regression on them

mtcars_model <- lm(mpg ~ wt, data = mtcars)
summary(mtcars_model)

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

## `geom_smooth()` using formula = 'y ~ x'

## A marker object has been specified, but markers is not in the mode
## Adding markers to the mode...

library(ggplot2); small_plot <- ggplot(mtcars, aes(x = wt, y = mpg)) + geom_point(color = "blue", size = 2) + geom_smooth(method = "lm", se = FALSE, color = "darkgreen") + labs(title = "MPG vs. Weight: Summary Plot", x = "Weight (1000 lbs)", y = "Miles Per Gallon (MPG)") + theme_minimal(base_size = 5); small_plot

## `geom_smooth()` using formula = 'y ~ x'

• Simple Linear Regression allows us to model the relationship between two continuous variables.
• In the mtcars dataset, weight has a significant negative impact on MPG.
  • Heavier cars tend to have lower fuel efficiency.
• Regression tools help in making predictions into data patterns.