COURSE 5: REGRESSION ANALYSIS: SIMPLIFY COMPLEX DATA RELATIONSHIPS

Module 2: Simple Linear Regression

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TABLE OF CONTENT

INTRODUCTION – Simple Linear Regression

In this section, participants will embark on a comprehensive exploration of modeling complex data relationships, with a specific focus on correlation relationships. The curriculum provides an in-depth understanding of how to employ models to describe intricate data interconnections and examines the nuances of correlation in the context of data analysis. Practical application takes center stage, as participants delve into building a simple linear regression model using the Python programming language.

Through hands-on exercises, learners will gain proficiency in constructing regression models, enabling them to interpret and derive meaningful insights from their results. By the conclusion of this section, participants will have honed their skills in using models to unravel complex data relationships and will be well-equipped to apply these techniques in real-world scenarios.

Learning Objectives

  • Identify different commonly used model evaluation metrics
  • Use EDA to evaluate whether linear regression is appropriate based on the model assumptions
  • Recall the main model assumptions of simple linear regression
  • Explain the relationship between variance, degrees of freedom, and residuals
  • Describe the Ordinary Least Square Regression Estimation method
  • Define simple linear regression

PRACTICE QUIZ: TEST YOUR KNOWLEDGE: FOUNDATIONS OF LINEAR REGRESSION

1. Fill in the blank: The best fit line is the line that fits the data best by minimizing some _____.

  • residual values
  • regression function
  • loss function (CORRECT)
  • predicted values

Correct: The best fit line is the line that fits the data best by minimizing some loss function. To find the best fit line, it’s necessary to measure error, which is the difference between the observed values and the predicted values generated by the model.

2. What is the sum of the squared differences between each observed value and the associated predicted value?

  • Sum of squared residuals (CORRECT)
  • Ordinary least squares
  • Sum of squared predicted values
  • Residual least squares

Correct: The sum of squared residuals is the sum of the squared differences between each observed value and the associated predicted value. Data professionals use this sum to capture a summary of total error in the model.

3. What does the circumflex symbol, or “hat” (^), indicate when used over a coefficient?

  • The coefficient is a residual
  • The coefficient is an “actual” value (not predicted)
  • The coefficient is an estimate or predicted value (CORRECT)
  • The coefficient is a population parameter value

PRACTICE QUIZ: TEST YOUR KNOWLEDGE: ASSUMPTIONS AND CONSTRUCTION IN PYTHON

1. How does a data professional determine if a linearity assumption is met?

  • They confirm whether data on the X-Y coordinate falls along a straight line. (CORRECT)
  • They confirm whether data on the X-Y coordinate falls along an upward curved line.
  • They confirm whether data on the X-Y coordinate falls along a downward curved line.
  • They confirm whether data on the X-Y coordinate resembles a random cloud.

Correct: A data professional determines if a linearity assumption is met by confirming whether data on the X-Y coordinate falls along a straight line. A linearity assumption is passed when each predictor variable X is linearly related to the outcome variable Y.

2. Which of the following statements accurately describes the normality assumption?

  • The normality assumption can only be confirmed while a model is being built.
  • The normality assumption can only be confirmed before a model is built.
  • The normality assumption can only be confirmed after a model is built. (CORRECT)
  • The normality assumption can be confirmed anytime during model building.

Correct: The normality assumption can only be confirmed after a model is built. It focuses on the model errors, which can be estimated by the residuals.

3. A data professional is using a scatterplot to plot residuals and predicted values from a regression model to check for homoscedasticity. What does this scenario represent?

  • Cone
  • Random cloud (CORRECT)
  • Curved line
  • Straight line

Correct: This scenario represents a random cloud. Random clouds are used to validate the homoscedasticity assumption. They confirm the variation of residuals is consistent or similar across the model.

4. What type of visualization uses a series of scatterplots that show the relationships between pairs of variables?

  • Residual matrix
  • Scatterplot residuals
  • Linear matrix
  • Scatterplot matrix (CORRECT)

Correct: A scatterplot matrix uses a series of scatterplots that show the relationships between pairs of variables. This helps data professionals assess whether there is a linear relationship between the independent and dependent variables

PRACTICE QUIZ: TEST YOUR KNOWLEDGE: EVALUATE A LINEAR REGRESSION MODEL

1. What is the area surrounding a regression line, which describes the uncertainty around the predicted outcome at every value of X?

  • Ordinary least squares
  • Confidence band (CORRECT)
  • R squared
  • Confidence interval

Correct: The confidence band is the area surrounding a regression line, which describes the uncertainty around the predicted outcome at every value of X. The confidence band reveals the confidence interval for each point on a regression line.

2. Fill in the blank: R squared measures the _____ in the dependent variable, Y, that is explained by the independent variable, X.

  • proportion of variation (CORRECT)
  • coefficient of accuracy
  • proportion of accuracy
  • coefficient of variation

Correct: R squared measures the proportion of variation in the dependent variable, Y, that is explained by the independent variable, X. It is calculated by subtracting the sum of squared residuals (explained variance) divided by the total variance from 1.

3. Which linear regression evaluation metric is sensitive to large errors?

  • Mean squared error (MSE) (CORRECT)
  • Adjusted R squared
  • Mean absolute error (MAE)
  • The coefficient of determination

Correct: Mean squared error (MSE) is sensitive to large errors. The MSE is the average of the squared difference between the predicted and actual values.

PRACTICE QUIZ: TEST YOUR KNOWLEDGE: INTERPRET LINEAR REGRESSION RESULTS

1. Which of the following are best practices when communicating linear regression results? Select all that apply.

  • Always extrapolate to a larger or different group any data insights that apply only to a specific, smaller population.
  • Make the findings quickly understood without technical terms. (CORRECT)
  • Provide measures of uncertainty around estimated results. (CORRECT)
  • Use data visualizations to present the results. (CORRECT)

Correct: When communicating linear regression results, best practices include the following: Provide measures of uncertainty around estimated results, make the findings quickly understood without technical terms, and use data visualizations to present the results.

2. Which of the following statements accurately describe coefficients and p-values for regression model interpretation? Select all that apply.

  • P-values determine how changes in the independent variables are associated with changes in the dependent variable.
  • Coefficients demonstrate whether P-values are statistically significant.
  • Coefficients determine how changes in the independent variables are associated with changes in the dependent variable. (CORRECT)
  • P-values demonstrate whether coefficients are statistically significant. (CORRECT)

Correct: Coefficients determine how changes in the independent variables are associated with changes in the dependent variable. P-values demonstrate whether coefficients are statistically significant.

QUIZ: MODULE 2 CHALLENGE

1. A data professional determines the best fit line by calculating the difference between observed values and the predicted value of a regression line. What is this calculation?

  • Notion
  • Coefficient
  • Parameter
  • Residual (CORRECT)

2. In linear regression, what mathematical technique is used to calculate the best fit line?

  • Coefficient of determination
  • Sum of squared residuals
  • Hold out coefficient
  • Ordinary least squares (CORRECT)

3. A data professional testing for linear regression assumptions plots their dependent variable against their independent variable and notices that the graph appears as a repeating waveform. Which model assumption does this invalidate?

  • Independent observation
  • Normality
  • Linearity (CORRECT)
  • Homoscedasticity

4. Fill in the blank: A scatterplot matrix is a series of scatterplots that show the _____ between pairs of variables.

  • distances
  • discrepancies
  • relationships (CORRECT)
  • variability

5. A data professional at a toy manufacturer checks model assumptions while working on a project about potential new game concepts. They find no clear pattern in their scatterplot and can confirm constant variance along the values of the dependent variable. What does this scenario describe?

  • Independent observation
  • Normality
  • Linearity
  • Homoscedasticity (CORRECT)

6. Fill in the blank: A confidence band is the area surrounding a line that describes the uncertainty around the predicted outcome at every value of _____.

  • intercept
  • X (CORRECT)
  • Slope
  • Y

7. What is another term for R squared?

  • Residuals of determination
  • Error of residuals
  • Coefficient of determination
  • Coefficient of residuals (CORRECT)

8. Which of the following statements accurately describe running a randomized, controlled experiment? Select all that apply.

  • It is a study design that systematically and methodically assigns participants into groups.
  • The differences between the control and treatment groups must be observable and measurable. (CORRECT)
  • To be successful, data professionals must control for every factor in the experiment. (CORRECT)
  •  It is typically used when arguing for causation between variables. (CORRECT)

9. Fill in the blank: _____ is the difference between observed values and the predicted values of a regression line.

  • Coefficient
  • Residual (CORRECT)
  • Intercept
  • Error

10. A data professional minimizes the sum of squared residuals to estimate parameters in a linear regression model. What method are they using?

  • Residual coefficients
  • Mean absolute error
  • R squared
  • Ordinary least squares (CORRECT)

11. A data analytics professional working for a storage facility checks model assumptions while determining optimal storage space sizes. They notice that the model’s residuals appear in a cone-shaped pattern when plotted against the independent variable. Which model assumption does this invalidate?

  • Normality
  • Homoscedasticity (CORRECT)
  • Independent observation
  • Linearity

12. A data professional determines how much of the variation in the X variable explains the variation in the Y variable. Which model evaluation metric enables this determination?

  • Mean absolute error (MAE)
  • Mean squared error (MSE)
  • P-value
  • R squared (CORRECT)

13. Fill in the blank: A scatterplot _____ is a series of scatterplots that show the relationships between pairs of variables.

  • succession
  • matrix (CORRECT)
  • array
  • progression

14. Which of the following statements accurately describe a randomized, controlled experiment? Select all that apply.

  • As the study is conducted, the only expected similarity between the control and experimental groups is the outcome variable being studied.
  • The differences between the control and treatment groups must be observable and measurable. (CORRECT)
  • It is a study design that randomly assigns participants into an experimental group or a control group. (CORRECT)
  • To be successful, data professionals must control for every factor in the experiment. (CORRECT)

15. In linear regression, what mathematical technique is used to calculate beta zero hat and beta one hat?

  • Coefficient R squared
  • Mean squared error
  • Ordinary least squares (CORRECT)
  • Coefficient of determination

16. Fill in the blank: A scatterplot matrix is a series of scatterplots that show the relationships between pairs of _____.

  • models
  • coordinates
  • variables (CORRECT)
  • lines

17. What is the difference between observed or actual values and the predicted values of a regression line?

  • Beta
  • Slope
  • Residual (CORRECT)
  • Parameter

18. Fill in the blank: A _____ is the area surrounding a line that describes the uncertainty around the predicted outcome at every value of X.

  • confidence band (CORRECT)
  • confidence slope
  • interval band
  • interval slope

19. What measures the proportion of variation in the dependent variable Y explained by the independent variable X?

  • R squared (CORRECT)
  • P-value
  • Mean absolute error (MAE)
  • Mean squared error (MSE)

20. Fill in the blank: A scatterplot _____ is a series of scatterplots that show the relationships between pairs of variables.

  • succession
  • array
  • progression
  • matrix (CORRECT)

21. Fill in the blank: A _____ is the area surrounding a line that describes the uncertainty around the predicted outcome at every value of X.

  • interval slope
  • confidence band (CORRECT)
  • confidence slope
  • interval band

22. Fill in the blank: A confidence band is the area surrounding a line that describes the _____ around the predicted outcome at every value of X.

  • Uncertainty (CORRECT)
  • certainty
  • accuracy
  • inaccuracy

23. What term describes the difference between observed or actual values and the predicted values of the regression line?

  • Residuals (CORRECT)
  • Best fit lines
  • Ordinary least squares
  • Predicted values

Correct: Residuals describe the difference between observed or actual values and the predicted values of the regression line. Residual equals observed value minus predicted value.

24. There are four assumptions of simple linear regression, including linearity, normality, and independent observations. What is the fourth assumption?

  • Homoscedasticity (CORRECT)
  • Independant observations
  • Heteroscedasticity
  • Dependant observations

Correct: The four assumptions of simple linear regression are linearity, normality, independent observations, and homoscedasticity. Linearity assumes that each predictor variable Xi is linearly related to the outcome variable Y. Normality assumes that the residual values are normally distributed. Independent observation assumes that each observation in the dataset is independent. And homoscedasticity assumes the values have the same variance.

25. In a linear regression model, what is the area surrounding the regression line that describes the uncertainty around the predicted outcome at every value of X?

  • sum of squared residuals
  • p-value
  • confidence interval
  • confidence band (CORRECT)

Correct: In a linear regression model, a confidence band is the area surrounding the regression line that describes the uncertainty around the predicted outcome at every value of X. A confidence band reveals the confidence interval for each point on a regression line. It is another way to report findings responsibly.

CONCLUSION – Simple Linear Regression

In conclusion, this section has equipped participants with essential knowledge and practical skills to navigate the intricacies of modeling complex data relationships. By delving into correlation relationships and hands-on exercises involving the construction of a simple linear regression model in Python, learners have gained valuable insights into the application of these techniques.

The emphasis on real-world scenarios ensures that participants not only comprehend theoretical concepts but also develop a proficiency for interpreting and deriving meaningful insights from their analyses. This comprehensive overview stands as a valuable resource for those seeking to master the art of modeling data relationships and serves as a crucial foundation for advanced data analytics.

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