Linear regression is a statistical method used to analyze and model the relationship between a dependent variable and one or more independent variables. The goal of linear regression is to find the best-fit line or hyperplane that represents the linear relationship between the variables.
The basic idea behind linear regression is to use a straight line to represent the relationship between two variables. The line is determined by finding the slope and intercept that minimize the distance between the observed data points and the predicted values on the line. The slope represents the rate of change in the dependent variable for a unit change in the independent variable, while the intercept represents the value of the dependent variable when the independent variable is zero.
The equation for a simple linear regression model with one independent variable is:
y = β0 + β1x + ε
where y is the dependent variable, x is the independent variable, β0 is the intercept, β1 is the slope, and ε is the error term. The error term represents the difference between the observed value of y and the predicted value based on the line.
The coefficients β0 and β1 are estimated using a method called least squares regression. The goal of least squares regression is to minimize the sum of the squared errors (SSE) between the observed data points and the predicted values on the line. The SSE is calculated as:
SSE = Σ(y – y’)^2
where y is the observed value, y’ is the predicted value based on the line, and Σ represents the sum over all data points.
Once the coefficients β0 and β1 are estimated, the line can be used to make predictions about the dependent variable based on the value of the independent variable. For example, if we have a new value of x, we can use the line to predict the corresponding value of y.
Simple linear regression can be extended to multiple linear regression, where there are multiple independent variables. The equation for a multiple linear regression model is:
y = β0 + β1×1 + β2×2 + … + βpxp + ε
where x1, x2, …, xp are the independent variables, β1, β2, …, βp are the coefficients, and ε is the error term. The goal of multiple linear regression is to estimate the coefficients that minimize the SSE between the observed data points and the predicted values based on the line.
Linear regression is a powerful tool for analyzing and modeling the relationship between variables. It can be used to make predictions, identify trends, and test hypotheses. However, it is important to note that linear regression assumes that there is a linear relationship between the variables and that the error term is normally distributed with constant variance. If these assumptions are not met, the results of the analysis may not be reliable.
In addition to simple and multiple linear regression, there are other types of regression models that can be used for more complex relationships between variables, such as polynomial regression, logistic regression, and time series regression. Each of these models has its own assumptions and limitations, and it is important to choose the appropriate model based on the characteristics of the data and the research question.