Regression

Understanding Regression

  • Regression refers to the statistical process that allows us to study the relationship between two or more variables.
  • Simple linear regression is a form of regression analysis where there is just one independent variable and one dependent variable.
  • The dependent variable is the one that we want to predict or estimate, while the independent variable is the one we use to make predictions.
  • Regression estimation results in a line of best fit that minimises the distance between the projected points and the actual points in the dataset.

Simple Linear Regression Equation

  • The simple linear regression equation is commonly written in the format: y = a + bx. Here, ‘y’ is the dependent variable, ‘x’ is the independent variable, ‘a’ is the y-intercept, and ‘b’ is the slope of the line.
  • The slope (b) indicates the rate at which ‘y’ changes for each unit change in ‘x’.
  • The y-intercept (a) is the value of ‘y’ when ‘x’ is zero.

Calculating the Line of Best Fit

  • To calculate the line of best fit, we use the formulas: b = Σ [(xi - X̄)(yi - Ȳ)] / Σ(xi - X̄)² and a = Ȳ - bX̄. Here, X̄ is the mean of the x values, Ȳ is the mean of the y values, xi and yi are the individual x and y values.
  • The Residual sum of squares (RSS) measures the total difference between the observed dependent variable values and the values predicted by the regression model. The best model minimises the RSS.

Interpreting Regression Output

  • The slope (b) provides an estimate of how much the dependent variable will change with each one-unit increase in the independent variable, assuming all other variables are kept constant.
  • The y-intercept (a) is the estimated value of ‘y’ when ‘x’=0. It represents the starting value.
  • The R Squared statistic, or coefficient of determination, shows the proportion of variation in the dependent variable that can be explained by the independent variable. It ranges from 0 to 1, with 1 indicating that the independent variable explains all the variability in the dependent variable.

Applications of Regression

  • Regression analysis can be used to forecast or predict future values by looking at the trend in the historical data.
  • Regression is important in data analysis and used in various fields such as economics, biology, and engineering to better understand relationships between variables and make informed decisions.
  • Care should be taken to avoid over-fitting the model to the data, as this can lead to inaccurate predictions when applied to new data.