Introduction

Understanding the assumptions underlying regression analysis is crucial for accurate interpretation and meaningful results. In this chapter, we will explore the key assumptions that differentiate single and multiple regression. Single regression involves a single explanatory variable, while multiple regression incorporates multiple explanatory variables. Let’s delve into the specific assumptions that guide these two approaches and their implications for regression analysis.

Assumptions of Single Regression

Single regression focuses on the relationship between a dependent variable and a single explanatory variable. The primary assumptions for single regression include:

• Linearity: The relationship between the dependent and explanatory variables is assumed to be linear. This means that a change in the explanatory variable corresponds to a constant change in the dependent variable.

• Independence of Errors: The errors (residuals) are assumed to be independent of each other, meaning that the error term for one observation does not affect the error term for another observation.

• Homoscedasticity: The variance of the errors is constant across all levels of the explanatory variable. In simpler terms, the spread of residuals should be consistent.

• Normality: The errors follow a normal distribution. This assumption is important for making statistical inferences and constructing confidence intervals.

Assumptions of Multiple Regression

Multiple regression extends the concepts of single regression to accommodate multiple explanatory variables. In addition to the assumptions of single regression, multiple regression introduces new considerations:

• Multicollinearity: Explanatory variables should not be highly correlated with each other. High multicollinearity can make it difficult to isolate the individual effects of each variable.

• No Perfect Multicollinearity: There should be no perfect linear relationship between the explanatory variables. Perfect multicollinearity can lead to unstable coefficient estimates.

• Matrix Rank: The matrix of explanatory variables should have full rank. In other words, no variable can be expressed as a linear combination of other variables.

• Heteroscedasticity: The assumption of homoscedasticity is extended to multiple explanatory variables. Residuals should have constant variance across all levels of multiple variables.

Implications and Examples

Differentiating between single and multiple regression assumptions is essential for accurate model specification and interpretation. For example, in single regression, the linearity assumption implies that the relationship between temperature and ice cream sales is consistent for different temperature changes. In multiple regression, multicollinearity warns against including highly correlated variables, such as both age and years of experience, in the same model.