Multiple linear regression is a powerful statistical technique used to model relationships between variables. Unlike simple linear regression which uses only one predictor, multiple linear regression incorporates two or more independent variables to predict a single dependent variable. This allows us to capture more complex relationships and improve prediction accuracy.
The mathematical foundation of multiple linear regression is expressed through a linear equation. The general form shows Y as the dependent variable equal to a base value beta zero plus the sum of each independent variable multiplied by its coefficient, plus an error term. For example, in predicting house prices, we might use size and number of bedrooms as predictors, where each coefficient represents the expected change in price for a one-unit increase in that variable.
Multiple linear regression follows a systematic process. First, we collect data for all variables. Then we specify the model by defining the linear relationship. Next, we estimate parameters using methods like Ordinary Least Squares to find the best-fitting coefficients. We evaluate the model using statistical measures like R-squared and p-values. Finally, we use the fitted model to make predictions on new data.
Evaluating multiple linear regression models requires several key metrics. R-squared measures the proportion of variance explained by the model, ranging from zero to one. Adjusted R-squared accounts for the number of predictors and penalizes unnecessary variables. P-values test the statistical significance of each coefficient, with values less than zero point zero five typically considered significant. Residual analysis helps check model assumptions by examining patterns in the prediction errors.
To summarize what we have learned about multiple linear regression: It is a powerful statistical technique that models relationships between one dependent variable and multiple independent variables using a linear equation. The process involves systematic steps from data collection through model evaluation. We assess model quality using metrics like R-squared and p-values, and it has wide applications across many fields for prediction and analysis.