• A method to model the relationship between two continuous variables.
• Predicts a dependent variable (Y) using one independent variable (X).

\[ Y = \beta_0 + \beta_1 X + \varepsilon \]

Where: - \(\beta_0\) = intercept
- \(\beta_1\) = slope
- \(\varepsilon\) = error term

set.seed(123)
x <- rnorm(100, mean = 50, sd = 10)
y <- 5 + 0.8 * x + rnorm(100, sd = 5)
data <- data.frame(x, y)
head(data)

##          x        y
## 1 44.39524 36.96416
## 2 47.69823 44.44300
## 3 65.58708 56.23621
## 4 50.70508 43.82635
## 5 51.29288 41.27621
## 6 67.15065 58.49538

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

summary(model)

## 
## Call:
## lm(formula = y ~ x, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5367 -3.4175 -0.4375  2.9032 16.4520 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.79778    2.76324   2.098   0.0385 *  
## x            0.77376    0.05344  14.479   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.854 on 98 degrees of freedom
## Multiple R-squared:  0.6815, Adjusted R-squared:  0.6782 
## F-statistic: 209.7 on 1 and 98 DF,  p-value: < 2.2e-16

The least squares method minimizes the sum of squared residuals:

\[ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]

\[ \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \]

• Linear regression finds the best linear relationship between two variables.
• Useful for prediction and identifying trends.
• Can be extended to multiple regression and other models.