Ultimately, simple linear regression models a relationship between: - a predictor X (what we know) - and a response Y (what we want to predict)
Example: Hours spent jump-roping (X) -> Vertical jump (Y)
My Statistics Topic: Simple linear regression
SLR mathematical model / equation:
\[Y_i = \beta_0 + \beta_1 X_i + \epsilon_i\]
Example data (Randomly Generated)
| x | y |
|---|---|
| 2.655087 | 59.63320 |
| 3.721239 | 63.36479 |
| 5.728534 | 77.09591 |
| 9.082078 | 89.66829 |
| 2.016819 | 53.93474 |
| 8.983897 | 81.69062 |
| 9.446753 | 89.97450 |
| 6.607978 | 81.04311 |
Example scatterplot with regression line
Interpreting the slope from previous two slides
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\] - Intercept: 50.65 - Slope: 4.01 - Each extra hour jumping rope increases the predicted vertical jump by about 4.01 centimeters.
Residual plot
\[e_i = y_i - \hat{y}_i\]
The classic least squares idea
\[SSE = \sum_{i=1}^{n}(y_i - \hat{y}_i)^2\]
Plotly 3D example
R code example
fit <- lm(y ~ x, data = df) summary(fit) predict(fit, newdata = data.frame(x = c(2, 5, 9)))
Conclusion!!
- Linear regression models a straight-line relationship.
- The slope shows how the response variable changes when you alter the predictor variable.
- Residuals help check if the model is reasonable, and help to visualize accuracy of the model.