Which assumptions are important for making inferences from a bivariate regression model?
The correct answer and explanation is:
In making inferences from a bivariate regression model, several key assumptions must be met to ensure the validity of the results. These assumptions include:
- Linearity: The relationship between the independent variable (X) and the dependent variable (Y) should be linear. This means that the effect of X on Y is constant across the range of X values. If the relationship is non-linear, the regression model may not accurately capture the true nature of the relationship.
- Independence of Errors: The residuals, or errors (the differences between the observed and predicted values), should be independent of each other. This means that the error for one observation should not depend on the error for another observation. If errors are correlated, it can lead to biased estimates and incorrect inferences.
- Homoscedasticity: The variance of the errors should be constant across all levels of the independent variable. This means that the spread of residuals around the regression line should be the same for all values of X. If the variance changes (heteroscedasticity), it can affect the reliability of hypothesis tests and confidence intervals.
- Normality of Errors: The residuals should be approximately normally distributed. This assumption is especially important when performing hypothesis tests on the regression coefficients. If the residuals deviate significantly from normality, the significance tests may not be valid.
- No Multicollinearity: Although this is more relevant in multiple regression, if you are extending the model to include more than one predictor, it is crucial that the independent variables are not highly correlated with each other. Multicollinearity can cause issues with the estimation of coefficients and their interpretation.
- No Measurement Error: The independent variable (X) should be measured without error. Measurement error in X can lead to biased estimates of the regression coefficients.
Meeting these assumptions ensures that the inferences made from the bivariate regression model, such as confidence intervals and hypothesis tests, are valid and reliable.