A4.3.1 Linear Regression in Predicting Continuous Outcomes (HL only) Notes

Predicting Continuous Outcomes

1. Dependent Variable ((Y)): The outcome we want to predict
2. Independent Variable ((X)): The input used to make predictions

The goal is to find a linear equation that best predicts the dependent variable based on the values of the independent variables.

The Linear Regression Equation

In its simplest form, the linear regression equation is:

$$ Y = \beta_0 + \beta_1X + \epsilon $$

1. $Y$: The dependent variable
2. $X$: The independent variable
3. $\beta_0$: The intercept, representing the value of $Y$ when $X$ is zero
4. $\beta_1$: The slope, indicating how much $Y$ changes for a one-unit change in $X$
5. $\epsilon$: The error term, accounting for the variation in $Y$ not explained by $X$

Relationship Between Independent and Dependent Variables

1. Independent Variables: These are the predictors. Their values are assumed to influence the dependent variable but are not influenced by it.
2. Dependent Variable: This is the response. Its values are assumed to depend on the independent variables.

Significance of the Slope and Intercept

The Intercept: $\beta_0$

1. Represents the expected value of the dependent variable when the independent variable is zero.
2. It is where the regression line crosses the y-axis.

The Slope: $\beta_1$

1. Showcases how much the dependent variable changes for each unit increase in the independent variable (indicates steepness).
2. A positive slope means the dependent variable increases with the independent variable, while a negative slope means it decreases (indicates direction).

How Well the Model Fits the Data

1. In regression, the model fit refers to how closely the model’s predicted values match the actual observed values
2. This fit indicates how well the model captures patterns or relationships in the data set
3. A good fit means the predicted values are close to the actual data points, while a poor fit means the predictions deviate significantly
4. To evaluate this fit, we can use a common metric known as the coefficient of determination, denoted as $r^2$ (R-squared)

Understanding R-squared

1. The formula for R-squared is: $r^2$ 1 - \frac{S_{\text{res}}}{S_{\text{tot}}}
   1. $SS_{\text{res}}$ is the sum of squared residuals, which measures the differences between observed and predicted values.
   2. $SS_{\text{tot}}$ is the total sum of squares, representing the differences between observed values and the mean of those values.
2. The $r^2$ values mean the following:
   1. A higher $r^2$ value indicates a better model fit, meaning the model explains more of the variability in the data.
   2. An $r^2$ value of 0 means the model explains none of the variability in the response variable.
   3. An $r^2$ value of 1 means the model explains all of the variability in the response variable.