• Definition
  • A statistical method to understand the connection between two continuous variables.
• Variables
  • Independent variable (x): Predictor
    
  • Dependent variable (y): Response
• Purpose
  • Fits a straight line to data
    
  • Estimates the relationship between x and y
    
  • Predicts y values based on x values
• Key Features
  • Assumes a linear relationship.
  • Can predict future outcomes (y) from given inputs (x).

The equation: \[ y = \beta_0 + \beta_1x + \epsilon \]

Where:

• \(\beta_0\): intercept

• \(\beta_1\): slope

• \(\epsilon\): error term.

Dataset contains height (in cm) and weight (in kg) for 100 individuals:

Scatter plot shows the relationship between height and weight:

Model equation: \[ \hat{y} = \beta_0 + \beta_1x \]

Where:

• \(\hat{y}\) is the predicted weight,

• \(x\) is the height.

Summary that provides intercept, slope, and R-squared value:

## 
## Call:
## lm(formula = weight ~ height, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12.3483  -2.7408  -0.1779   3.2188  10.4538 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -3.88638    5.50755  -0.706    0.482    
## height       0.77480    0.03136  24.704   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.573 on 98 degrees of freedom
## Multiple R-squared:  0.8616, Adjusted R-squared:  0.8602 
## F-statistic: 610.3 on 1 and 98 DF,  p-value: < 2.2e-16

Confidence intervals for the slope and intercept:

##                   2.5 %    97.5 %
## (Intercept) -14.8159242 7.0431722
## height        0.7125575 0.8370354

A 3D plot of height, weight and the interaction between them:

The residuals that represent the difference between actual and predicted values:

plot(model$residuals, main="Residuals Plot", 
  ylab="Residuals", xlab="Fitted values")
abline(h=0, col="blue")