A linear model is a mathematical equation that describes the relationship between one or more independent variables and a dependent variable using a specific format. In statistics, linear models are widely used for prediction and to understand how predictors influence an outcome. Examples include simple and multiple linear regression.
A linear regression model follows a specific form: all terms in the equation are either a constant or a parameter multiplied by an independent variable. The equation is constructed by adding and subtracting these terms, producing the following form:
Dependent variable = constant + parameter × independent variable + … + parameter × independent variable
This setup means the model is linear in the parameters, even if the independent variables themselves are transformed. Importantly, linear models can still capture curvature in the data by including terms like squared independent variables (to model U-shaped patterns), logarithmic transformations, or inverse terms. While the independent variable may be transformed, as long as the parameters enter the equation linearly, the model remains a linear model.
For instance, if you square an independent variable, the model can follow a U-shaped curve.
For example, a researcher might use a linear model to predict a student’s exam score based on the number of hours they studied, the number of practice problems completed, and possibly the square of study hours to account for diminishing returns at high study levels.
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