Linear Regression, GLMs and GAMs with R demonstrates how to use R to extend the basic assumptions and constraints of linear regression to specify, model, and interpret the results of generalized linear (GLMs) and generalized additive (GAMs) models. The course demonstrates the estimation of GLMs and GAMs by working through a series of practical examples from the book Generalized Additive Models: An Introduction with R by Simon N. Wood (Chapman & Hall/CRC Texts in Statistical Science, 2006). Linear statistical models have a univariate response modeled as a linear function of predictor variables and a zero mean random error term. The assumption of linearity is a critical (and limiting) characteristic. Generalized linear models (GLMs) relax this assumption of linearity. They permit the expected value of the response variable to be a smoothed (e.g. non-linear) monotonic function of the linear predictors. GLMs also relax the assumption that the response variable is normally distributed by allowing for many distributions (e.g. normal, poisson, binomial, log-linear, etc.). Generalized additive models (GAMs) are extensions of GLMs. GAMs allow for the estimation of regression coefficients that take the form of non-parametric smoothers. Nonparametric smoothers like lowess (locally weighted scatterplot smoothing) fit a smooth curve to data using localized subsets of the data. This course provides an overview of modeling GLMs and GAMs using R. GLMs, and especially GAMs, have evolved into standard statistical methodologies of considerable flexibility. The course addresses recent approaches to modeling, estimating and interpreting GAMs. The focus of the course is on modeling and interpreting GLMs and especially GAMs with R. Use of the freely available R software illustrates the practicalities of linear, generalized linear, and generalized additive models.
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Learning objectives
- Understand the assumptions of ordinary least squares (ols) linear regression.
- Specify, estimate and interpret linear (regression) models using r.
- Understand how the assumptions of ols regression are modified (relaxed) in order to specify, estimate and interpret generalized linear models (glms).
- Specify, estimate and interpret glms using r.
- Understand the mechanics and limitations of specifying, estimating and interpreting generalized additive models (gams).
- Understand the assumptions of ordinary least squares (ols) linear regression.
- Specify, estimate and interpret linear (regression) models using r.
- Understand how the assumptions of ols regression are modified (relaxed) in order to specify, estimate and interpret generalized linear models (glms).
- Specify, estimate and interpret glms using r.
- Understand the mechanics and limitations of specifying, estimating and interpreting generalized additive models (gams).
Syllabus
Students are introduced to the topics and focus of the course and will understand the basis of linear regression.
Introduction to Course
Preliminaries: Installing R, RStudio, R Commander, Course Materials and Exercise
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Syllabus
The term "linear" refers to the fact that we are fitting a line. The term model refers to the equation that summarizes the line that we fit. The term "linear model" is often taken as synonymous with linear regression model.
Assumptions of Linear Models (regression):
- The residuals are independent
- The residuals are normally distributed
- The residuals have a mean of 0 at all values of X
- The residuals have constant variance
In statistics, the generalized linear model (GLM) is a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.
Proportion data has values that fall between zero and one. Naturally, it would be nice to have the predicted values also fall between zero and one. One way to accomplish this is to use a generalized linear model (glm) with a logit link and the binomial family.
In statistics, Poisson regression is a form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables.
Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function as the assumed probability distribution of the response.
Log-linear analysis is a technique used in statistics to examine the relationship between more than two categorical variables.
In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear predictor depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. GAMs were originally developed by Trevor Hastie and Robert Tibshirani to blend properties of generalized linear models with additive models.
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Activities
Explore the fundamentals of generalized linear models
Show steps
Begin your learning journey by reviewing the foundational concepts of generalized linear models, setting the stage for a deeper understanding of the course material.
View
Generalized Additive Models: An Introduction...
on Amazon
Show steps
-
Read Chapters 1 and 2 of the book
-
Work through the exercises at the end of each chapter
Reinforce regression model assumptions
Show steps
Strengthen your understanding of the assumptions underlying linear regression models by engaging in targeted practice exercises.
Browse courses on
Linear Regression
Show steps
-
Solve practice problems on testing for normality, homoscedasticity, and independence
-
Analyze real-world datasets to identify potential violations of assumptions
Engage in peer discussions
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Foster a collaborative learning environment by actively participating in peer discussions, sharing insights, and seeking support from fellow learners.
Show steps
-
Participate in online forums and discussion groups
-
Engage in peer review and feedback sessions
Five other activities
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Show all eight activities
Explore generalized additive modeling techniques
Show steps
Expand your knowledge of generalized additive models by following guided tutorials, deepening your comprehension of their applications and capabilities.
Browse courses on
Generalized Additive Models
Show steps
-
Follow online tutorials on fitting GAMs using R
-
Experiment with different smoothing methods and basis functions
-
Interpret the results of GAM analyses
Practice fitting GLMs
Show steps
Gain proficiency in fitting generalized linear models through dedicated practice, enhancing your ability to apply them to various data scenarios.
Browse courses on
Generalized Linear Models
Show steps
-
Work through practice problems on fitting GLMs for different distributions
-
Analyze real-world datasets using GLMs and interpret the results
Apply GAMs to a real-world dataset
Show steps
Solidify your understanding of GAMs by applying them to a real-world dataset, gaining practical experience in model building and interpretation.
Browse courses on
Generalized Additive Models
Show steps
-
Identify a suitable dataset for GAM analysis
-
Fit a GAM to the data and evaluate its performance
-
Interpret the results and draw meaningful conclusions
Delve into advanced GAM concepts
Show steps
Extend your understanding of generalized additive models by exploring advanced concepts and techniques, further empowering your modeling capabilities.
View
Generalized Additive Models: An Introduction...
on Amazon
Show steps
-
Read Chapters 7-9 of the book
-
Work through the exercises and examples provided in the chapters
Contribute to open-source GAM projects
Show steps
Engage in the open-source community by contributing to GAM-related projects, deepening your understanding and fostering collaboration.
Browse courses on
Generalized Additive Models
Show steps
-
Identify open-source GAM projects
-
Explore the project source code and documentation
-
Make code contributions or report issues
Explore the fundamentals of generalized linear models
Show steps
Begin your learning journey by reviewing the foundational concepts of generalized linear models, setting the stage for a deeper understanding of the course material.
View
Generalized Additive Models: An Introduction...
on Amazon
Show steps
- Read Chapters 1 and 2 of the book
- Work through the exercises at the end of each chapter
Reinforce regression model assumptions
Show steps
Strengthen your understanding of the assumptions underlying linear regression models by engaging in targeted practice exercises.
Browse courses on
Linear Regression
Show steps
- Solve practice problems on testing for normality, homoscedasticity, and independence
- Analyze real-world datasets to identify potential violations of assumptions
Engage in peer discussions
Show steps
Foster a collaborative learning environment by actively participating in peer discussions, sharing insights, and seeking support from fellow learners.
Show steps
- Participate in online forums and discussion groups
- Engage in peer review and feedback sessions
Explore generalized additive modeling techniques
Show steps
Expand your knowledge of generalized additive models by following guided tutorials, deepening your comprehension of their applications and capabilities.
Browse courses on
Generalized Additive Models
Show steps
- Follow online tutorials on fitting GAMs using R
- Experiment with different smoothing methods and basis functions
- Interpret the results of GAM analyses
Practice fitting GLMs
Show steps
Gain proficiency in fitting generalized linear models through dedicated practice, enhancing your ability to apply them to various data scenarios.
Browse courses on
Generalized Linear Models
Show steps
- Work through practice problems on fitting GLMs for different distributions
- Analyze real-world datasets using GLMs and interpret the results
Apply GAMs to a real-world dataset
Show steps
Solidify your understanding of GAMs by applying them to a real-world dataset, gaining practical experience in model building and interpretation.
Browse courses on
Generalized Additive Models
Show steps
- Identify a suitable dataset for GAM analysis
- Fit a GAM to the data and evaluate its performance
- Interpret the results and draw meaningful conclusions
Delve into advanced GAM concepts
Show steps
Extend your understanding of generalized additive models by exploring advanced concepts and techniques, further empowering your modeling capabilities.
View
Generalized Additive Models: An Introduction...
on Amazon
Show steps
- Read Chapters 7-9 of the book
- Work through the exercises and examples provided in the chapters
Contribute to open-source GAM projects
Show steps
Engage in the open-source community by contributing to GAM-related projects, deepening your understanding and fostering collaboration.
Browse courses on
Generalized Additive Models
Show steps
- Identify open-source GAM projects
- Explore the project source code and documentation
- Make code contributions or report issues
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