6 Questions
What is the purpose of simple linear regression?
To predict a continuous outcome variable based on a single predictor variable.
What is an assumption of simple linear regression?
The variance of the residuals should be constant across all levels of x.
What is the equation for simple linear regression?
y = β0 + β1x + ε
What does the coefficient of determination (R²) measure?
The proportion of the variance in y that is predictable from x
What is an additional assumption of multiple linear regression?
No or little multicollinearity between the predictor variables
What is the main difference between simple and multiple linear regression?
The number of predictor variables used
Study Notes
Linear Regression
Simple Linear Regression
- Definition: Simple linear regression is a statistical method that attempts to predict the value of a continuous outcome variable (y) based on a single predictor variable (x).
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Assumptions:
- Linearity: The relationship between x and y should be linear.
- Independence: Each data point should be independent of the others.
- Homoscedasticity: The variance of the residuals should be constant across all levels of x.
- Normality: The residuals should be normally distributed.
- No or little multicollinearity: The predictor variable should not be perfectly correlated with the intercept.
- Equation: y = β0 + β1x + ε, where β0 is the intercept, β1 is the slope, x is the predictor variable, and ε is the residual.
- Coefficient of determination (R²): Measures the proportion of the variance in y that is predictable from x.
- Hypothesis testing: Used to determine whether the slope (β1) is significantly different from zero.
Multiple Linear Regression
- Definition: Multiple linear regression is a statistical method that extends simple linear regression by using multiple predictor variables to predict the outcome variable (y).
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Assumptions: Same as simple linear regression, with the addition of:
- No multicollinearity: The predictor variables should not be highly correlated with each other.
- Equation: y = β0 + β1x1 + β2x2 + … + βnxn + ε, where β0 is the intercept, β1, β2, …, βn are the coefficients of the predictor variables, and ε is the residual.
- Coefficient of determination (R²): Measures the proportion of the variance in y that is predictable from the set of predictor variables.
- Hypothesis testing: Used to determine whether each predictor variable has a significant effect on the outcome variable.
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Model evaluation:
- Backward elimination: Starts with all predictor variables and removes the least significant ones.
- Forward selection: Starts with no predictor variables and adds the most significant ones.
- Stepwise regression: Combines backward elimination and forward selection.
This quiz covers the basics of simple and multiple linear regression, including assumptions, equations, coefficient of determination, hypothesis testing, and model evaluation techniques.
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