• What is Linear Regression?
  • Finds a linear relationship between variables and estimates missing values called interpolation.
• Why is it useful?
  • Helps predict outcomes based on past data.
    
  • Identifies trends and relationships between variables.
    
  • Provides insights for decision-making in various fields.
• Applications in finance, insurance, and healthcare.
  • Finance: Predicting stock prices, risk assessment.
    
  • Insurance: Estimating claim amounts, determining premiums.
    
  • Healthcare: Diagnosing diseases, predicting patient outcomes.

A simple linear regression model:

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

• \(Y\): Dependent variable
  
• \(X\): Independent variable
  
• \(\beta_0\): Intercept
  
• \(\beta_1\): Slope
  
• \(\epsilon\): Error term

##     x          y
## 1   1 -0.6047565
## 2   2  5.6982251
## 3   3 26.5870831
## 4   4 14.7050839
## 5   5 18.2928774
## 6   6 37.1506499
## 7   7 27.6091621
## 8   8 13.3493877
## 9   9 22.1314715
## 10 10 27.5433803
## 11 11 47.2408180
## 12 12 41.5981383
## 13 13 45.0077145
## 14 14 45.1068272
## 15 15 41.4415887

ggplot(data, aes(x = x, y = y)) +
  geom_point(color = "#8C1D40") + 
  geom_smooth(method = "lm", col = "steelblue") +
  labs(title = "Linear Regression Example", x = "X", y = "Y") +
  theme(plot.title = element_text(color = "#8C1D40"))

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

\[ y=4.8780+2.8307x \]

##             Estimate Std. Error  t value     Pr(>|t|)
## (Intercept) 4.878044  4.7474585 1.027506 0.3229217707
## x           2.830725  0.5221508 5.421278 0.0001168109

## The correlation coefficient is: 0.8326619

• Equation: \(y = 4.8780 + 2.8307x\)
  • Intercept: 4.8780 (not statistically significant)
    
  • Slope: 2.8307 (statistically significant, p < 0.05)
• Correlation: r = 0.83
  • Strong positive relationship between \(x\) and \(y\), indicating that as \(x\) increases, \(y\) also increases.

• What are Residuals?
  • Differences between observed and predicted values.
    
  • Formula: \(\text{Residual} = Y - \hat{Y}\), where \(Y\) is observed and \(\hat{Y}\) is predicted.
• Why are Residuals Important?
  • Model Assessment: Help evaluate how well the model fits the data.
    
  • Detecting Patterns: Patterns suggest the model might not capture all data aspects.
    
  • Outliers: Large residuals may indicate outliers affecting accuracy.
• What do Residuals Tell Us?
  • Random distribution suggests valid linear regression assumptions.

This plot lets you zoom in and hover over the residuals to view their values.

• What is the Residual Plot?
  • Shows the difference between observed and predicted values (residuals).
    
  • Residuals are on the vertical axis, and \(X\) is on the horizontal axis.
• What does it tell us?
  • Randomly scattered residuals suggest a good linear model.
    
  • Patterns in residuals indicate a possible need for a non-linear model.
    
  • In this case, no obvious pattern shows that the linear model fits well.

Since the residual plot shows no clear pattern, a linear model appears to be a good fit for this dataset.

• Linear Regression finds a straight-line relationship between variables.
• Benefits:
  • Simple and useful in finance, insurance, and healthcare.
• Limitations:
  • Assumes a linear relationship, which may not always be true.
  • Sensitive to outliers, which can affect results.
• Residuals help evaluate the model’s fit and identify problems, such as patterns that suggest a non-linear relationship or outliers.
  • Residual Plots helps evaluate the fit of a regression model by showing if the residuals are randomly scattered or if patterns suggest the need for a different model.