Polynomial Regression in R | R Tutorial 5.12 | MarinStatsLectures
TLDRIn this video, Mike Marin explores polynomial regression, a technique for modeling nonlinear relationships in data. He demonstrates how to fit and assess these models in R using lung capacity data. Starting with a simple linear regression, he shows the limitations and the correct way to include polynomial terms like height squared. The video compares models with and without these terms using R-squared, residual standard error, and visual plots. It also covers the partial F-test to statistically determine if the inclusion of polynomial terms improves the model fit, concluding with a brief on alternative approaches to handle nonlinearities.
Takeaways
- 📚 The video introduces the concept of polynomial regression, which is a form of linear regression that models the relationship between variables using a polynomial equation.
- 📈 Polynomial regression is used when the relationship between x and y is nonlinear, yet it is still considered a special case of multiple linear regression.
- 🔍 The video uses lung capacity data to illustrate the process of fitting and assessing polynomial regression models in R.
- 📊 A scatter plot and simple linear regression model are initially used to visualize the relationship between lung capacity and height, revealing a potential nonlinearity.
- 📉 The simple linear regression model has an R-squared of about 75% and a residual standard error of 1.292, indicating the model's fit to the data.
- ❌ Directly entering variables like 'height squared' into the model formula in R will not include them in the model; R ignores such terms without warning.
- 🔢 The correct method to include polynomial terms involves using the `I()` function in R or creating a new variable for the polynomial term before including it in the model.
- 🔄 The `poly()` function in R can be used to include polynomial terms, with the degree argument set to the desired polynomial degree.
- 📈 After including height squared, the model's R-squared increases to about 77%, and the residual standard error decreases to 1.238, suggesting an improvement in model fit.
- 📊 The polynomial model with height squared visually fits the data better than the linear model, as shown by adding the model line to the plot.
- 🧐 A partial F test is used to formally compare models, with a small p-value indicating that the model with height squared is significantly better than the linear model.
- 🚫 Beyond x squared or x cubed, including higher polynomial terms is generally not recommended without careful consideration.
- 📝 The video concludes by noting other approaches to handling nonlinearities, such as variable transformation, categorization, or using nonlinear regression methods, each with its own advantages and disadvantages.
Q & A
What is the main focus of the video by Mike Marin?
-The video focuses on the concept of polynomial regression, how to fit and assess these models in R, and demonstrates this using lung capacity data.
Why is polynomial regression considered a special case of linear regression?
-Polynomial regression is a special case of linear regression because it models the relationship between x and y using a polynomial equation, which can represent nonlinear relationships, yet still fits within the framework of multiple linear regression.
What is the 'lung capacity data' mentioned in the script, and where can it be found?
-The 'lung capacity data' is a dataset used in the video to demonstrate polynomial regression. It can be downloaded from a link provided in the video description.
What does the R-squared value of about 75% indicate about the initial linear regression model?
-An R-squared value of about 75% indicates that the initial linear regression model explains about 75% of the variability in the lung capacity data based on height.
What is the residual standard error of 1.292 in the context of the initial model?
-The residual standard error of 1.292 measures the average distance that the observed values fall from the regression line, indicating the precision of the model's predictions.
Why might the relationship between lung capacity and height appear curved or nonlinear in the scatter plot?
-The relationship may appear curved or nonlinear because the actual relationship between lung capacity and height may not be linear, suggesting that a polynomial model could provide a better fit.
What is the incorrect way to include height squared in the model as mentioned in the script?
-The incorrect way is to directly include 'height squared' in the model call without using the 'I()' function, which causes R to ignore the term.
How can you correctly include height squared in the model in R?
-To correctly include height squared, you should use the 'I(height^2)' function within the model formula in R, ensuring that the term is recognized and included in the model.
What is the purpose of the 'Poly' function in R when dealing with polynomial regression?
-The 'Poly' function in R is used to generate polynomial terms for a variable, allowing the model to include terms like height, height squared, and higher degree polynomial terms if needed.
What does the partial F test assess when comparing two regression models?
-The partial F test assesses whether there is a statistically significant difference between two models, specifically whether the inclusion of additional terms, like height squared, provides a significantly better fit to the data.
Why is it important to include all lower order terms when adding higher order polynomial terms to a model?
-It is important to include all lower order terms to maintain the polynomial's mathematical integrity and to ensure that the model captures all relevant aspects of the nonlinear relationship between the variables.
What alternative approaches to dealing with nonlinearities are mentioned in the video?
-The video mentions transforming the x or y variable, converting x to a categorical variable or factor, and using nonlinear regression methods as alternative approaches to dealing with nonlinearities.
What was the conclusion of the partial F test when comparing the model with height cubed to the model without it?
-The conclusion was that the p-value was large, indicating no statistically significant difference between the models, suggesting that including height cubed was not necessary.
Outlines
📈 Polynomial Regression and Model Assessment in R
In this video, Mike Marin introduces the concept of polynomial regression, a form of linear regression that models the relationship between variables using a polynomial function. He demonstrates how to fit and evaluate these models in R using lung capacity data. The video begins with a simple linear regression model, showing how to plot the data and assess the model's fit with R-squared and residual standard error. Mike then explains the limitations of linear models for non-linear relationships and proceeds to include polynomial terms, specifically height squared, in the model. He illustrates the correct method to include polynomial terms in R and compares different approaches, such as manually creating a new variable or using the 'poly' function. The video also covers the use of partial F-tests to statistically compare models and concludes that including height squared improves the model fit.
🔍 Exploring Higher-Degree Polynomials and Model Comparison
Continuing the discussion on polynomial regression, the second paragraph delves into the inclusion of higher-order terms, such as height cubed, in the model. It emphasizes the necessity of including all lower-order terms when higher-degree polynomials are used. Mike fits a model with height squared and height cubed, visually comparing it to the simpler model without the cubic term. He uses the 'lines' command in R to plot the models and a legend to differentiate them. The video then employs the partial F-test to determine if the addition of height cubed significantly improves the model fit, which it does not, according to the test's large p-value. Mike concludes by mentioning alternative methods for handling non-linearities, such as variable transformation, categorization, or using non-linear regression techniques, and encourages viewers to consider the pros and cons of each approach. The video ends with a prompt to subscribe to the channel, like on Facebook, and visit the website for more statistical lectures.
Mindmap
Keywords
💡Polynomial Regression
💡Linear Regression
💡Lung Capacity Data
💡Scatter Plot
💡R-squared
💡Residual Standard Error
💡Abline Function
💡Polynomial Terms
💡Partial F Test
💡ANOVA Command
💡Orthogonal Polynomials
💡Model Assumptions
💡Nonlinear Regression
Highlights
Polynomial regression is a special case of linear regression where the relationship between x and y is modeled using a polynomial rather than a line.
Polynomial regression can be used when the relationship between x and y is nonlinear.
The video discusses how to fit and assess polynomial regression models in R.
The lung capacity data set is used as an example to model the relationship between lung capacity and height.
A scatter plot and a simple linear regression model are created to analyze the relationship between lung capacity and height.
The simple linear regression model has an R-squared of about 75% and a residual standard error of 1.292.
The relationship between lung capacity and height appears to be curved or nonlinear.
Residual plots can help assess linearity and check model assumptions.
Including polynomial terms like height squared in the model is one approach to dealing with nonlinearities.
R does not include height squared in the model if entered directly without using the I() function.
Creating a new variable called height squared and including it in the model produces the same result as using the I() function.
The poly() function in R can be used to include polynomial terms for a variable by setting the degree argument.
The model including height squared has an improved R-squared of about 77% and a lower residual standard error of 1.238 compared to the model with only height.
Visually, the model with height squared appears to provide a better fit to the data.
The partial F test can be used to formally compare two models and determine if including height squared significantly improves the model fit.
The p-value from the partial F test is small, indicating that the model with height squared provides a statistically significantly better fit than the model without it.
Including polynomial terms beyond x squared or x cubed is usually not recommended.
When including higher order terms like height cubed, all lower order terms must also be included in the model.
The partial F test shows no statistically significant difference between the models with and without height cubed, suggesting it is not necessary to include in the model.
Other approaches to dealing with nonlinearities include transforming variables, converting x to a categorical variable, or using nonlinear regression methods.
The video provides a comprehensive discussion on polynomial regression, its applications, and comparison with other methods for dealing with nonlinear relationships.
Transcripts
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