1.2 Simple Linear Regression: Concept and Assumptions

Simple linear regression is a powerful tool for modeling relationships between two variables. It helps us understand how one variable affects another, allowing us to make predictions and spot trends in data.

This method assumes a straight-line relationship between variables and requires certain conditions to be met. By understanding these concepts, we can use simple linear regression to solve real-world problems and make informed decisions.

Simple Linear Regression

Concept and Purpose

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• Simple linear regression is a statistical method used to model the relationship between two continuous variables, where one variable (the independent or predictor variable) is used to predict the value of the other variable (the dependent or response variable)
• The purpose of simple linear regression is to find the best-fitting straight line (regression line) that describes the relationship between the independent and dependent variables, allowing for the prediction of the dependent variable based on the value of the independent variable
• The regression line is represented by the equation $y = \beta_0 + \beta_1x$, where $y$ is the dependent variable, $x$ is the independent variable, $\beta_0$ is the y-intercept (the value of $y$ when $x = 0$), and $\beta_1$ is the slope of the line (the change in $y$ for a one-unit change in $x$)
• The coefficients $\beta_0$ and $\beta_1$ are estimated using the least squares method, which minimizes the sum of the squared differences between the observed values of the dependent variable and the values predicted by the regression line
• Simple linear regression is used for various purposes, such as:
  • Describing the strength and direction of the linear relationship between two variables (height and weight)
  • Predicting the value of the dependent variable for a given value of the independent variable (sales based on advertising expenditure)
  • Identifying outliers or unusual observations that deviate significantly from the regression line
  • Assessing the goodness of fit of the model and determining how well the independent variable explains the variation in the dependent variable

Assumptions of Linear Regression

Data Requirements

• Linearity: The relationship between the independent and dependent variables is linear, meaning that the change in the dependent variable is proportional to the change in the independent variable
• Independence: The observations are independent of each other, meaning that the value of one observation does not influence the value of another observation
• Homoscedasticity: The variance of the residuals (the differences between the observed and predicted values of the dependent variable) is constant across all levels of the independent variable
• Normality: The residuals follow a normal distribution with a mean of zero

Model Considerations

• No multicollinearity: In simple linear regression, there is only one independent variable, so multicollinearity (high correlation between independent variables) is not a concern. However, this assumption becomes relevant when extending to multiple linear regression
• No autocorrelation: The residuals are not correlated with each other, meaning that there is no pattern or dependence among the residuals
• Measurement error: The independent variable is measured without error, and any measurement error in the dependent variable is uncorrelated with the independent variable

Linear Regression Appropriateness

Data Suitability

• Determine if the research question or problem involves the relationship between two continuous variables, where one variable is used to predict the other
• Check if the assumptions of simple linear regression are reasonably met by the data:
  • Visually inspect a scatterplot of the dependent variable against the independent variable to assess linearity
  • Examine the residual plot (residuals vs. fitted values) to check for homoscedasticity and linearity
  • Assess the normality of residuals using a histogram, Q-Q plot, or statistical tests like the Shapiro-Wilk test
  • Verify independence of observations based on the study design and data collection process

Practical Considerations

• Consider the sample size and power: Simple linear regression typically requires a larger sample size than other statistical methods to achieve adequate power and precise estimates of the regression coefficients
• Evaluate the presence of outliers or influential observations that may significantly impact the regression results, and consider appropriate methods for handling them, such as removal or robust regression techniques (median regression)
• Assess the practical significance of the relationship between the variables, considering the context and domain knowledge, to determine if simple linear regression is a meaningful approach for the given problem (predicting housing prices based on square footage)

Modeling with Linear Regression

Model Formulation

• Identify the dependent variable ($y$) and the independent variable ($x$) based on the research question or problem statement
• Obtain a sample of data containing observations for both the dependent and independent variables
• Create a scatterplot of the dependent variable against the independent variable to visually assess the linearity of the relationship and identify any outliers or unusual patterns
• Estimate the regression coefficients ($\beta_0$ and $\beta_1$) using the least squares method:
  • Calculate the means of the dependent variable ($\bar{y}$) and the independent variable ($\bar{x}$)
  • Calculate the variance of the independent variable ($S_{xx}$) and the covariance between the dependent and independent variables ($S_{xy}$)
  • Estimate the slope coefficient: $\beta_1 = S_{xy} / S_{xx}$
  • Estimate the y-intercept: $\beta_0 = \bar{y} - \beta_1 * \bar{x}$

Model Interpretation

• Write the estimated regression equation in the form: $\hat{y} = \beta_0 + \beta_1x$, where $\hat{y}$ represents the predicted value of the dependent variable for a given value of the independent variable
• Assess the goodness of fit of the model using metrics such as the coefficient of determination ($R^2$), which represents the proportion of variance in the dependent variable explained by the independent variable
• Interpret the estimated regression coefficients in the context of the problem, considering the units of measurement and the practical significance of the relationship
  • The slope coefficient ($\beta_1$) indicates the change in the dependent variable for a one-unit increase in the independent variable (a $\beta_1$ of 0.5 means that for every additional year of experience, salary increases by $500)
  • The y-intercept ($\beta_0$) represents the predicted value of the dependent variable when the independent variable is zero (the base salary for an employee with no experience)