Linear regression is a statistical method used to model the relationship between a dependent variable (target) and one or more independent variables (predictors) by fitting a linear equation to observed data. To apply linear regression effectively, several basic assumptions should be made about the data and the model. These assumptions are important to ensure the validity and reliability of the regression analysis:
Linearity: The relationship between the independent variables and the dependent variable should be linear. This means that the change in the dependent variable should be proportional to changes in the independent variables. You can assess linearity using scatterplots.
Independence: The observations or data points should be independent of each other. In other words, the value of the dependent variable for one data point should not be influenced by the value of the dependent variable for another data point. This assumption is often satisfied by collecting data through random sampling or experimental design.
Homoscedasticity (Constant Variance): The variance of the residuals (the differences between the observed and predicted values) should be constant across all levels of the independent variables. In other words, the spread of the residuals should be roughly the same for all values of the predictors. You can check for homoscedasticity using residual plots.
Normality of Residuals: The residuals should be approximately normally distributed. This means that the errors should follow a bell-shaped, symmetric distribution. Departures from normality can affect the validity of hypothesis tests and confidence intervals. You can assess normality through methods like Q-Q plots or histogram checks of residuals.
No or Little Multicollinearity: In multiple linear regression (with more than one predictor), the independent variables should not be highly correlated with each other. High multicollinearity can make it difficult to distinguish the individual effects of predictors on the dependent variable. You can calculate correlation coefficients between predictors to detect multicollinearity.
No Perfect Predictors: There should be no perfect linear relationships among the predictors. Perfect predictors lead to problems in estimating the coefficients of the model.
No Endogeneity: Endogeneity occurs when the independent variables are correlated with the error term in the regression model. This can lead to biased coefficient estimates. Care should be taken to address potential sources of endogeneity.
No Autocorrelation (for Time Series Data): In time series regression, there should be no autocorrelation in the residuals. Autocorrelation implies that the residuals at one time point are correlated with the residuals at previous time points. Special techniques like autoregressive models may be needed to address autocorrelation.
It's important to assess these assumptions when performing linear regression analysis and take appropriate steps to address violations if they occur. Failure to meet these assumptions can lead to biased or inefficient parameter estimates and can undermine the validity of statistical tests and confidence intervals associated with the regression model. Various diagnostic tools and statistical tests are available to assess these assumptions and make necessary adjustments when they are violated.