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Linear Regression, GLMs and GAMs with R

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How to extend linear regression to specify and estimate generalized linear models and additive models.

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What Will I Learn?

- 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).

Requirements

- Students will need to install R and R Commander software but ample instruction for doing so is provided.

Description

* 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.

Who is the target audience?

- This course would be useful for anyone involved with linear modeling estimation, including graduate students and/or working professionals in quantitative modeling and data analysis.
- The focus, and majority of content, of this course is on generalized additive modeling. Anyone who wishes to learn how to specify, estimate and interpret GAMs would especially benefit from this course.

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Curriculum For This Course

Expand All 69 Lectures
Collapse All 69 Lectures
07:54:31

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Introduction to Course and to Linear Modeling
12 Lectures
01:14:57

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.

Preview
05:11

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

Assumptions of Linear Modeling (slides, part 2)

06:08

Desirable Properties of Beta-hat (slides, part 3)

07:19

Example: Estimate Age of Universe (slides)

04:39

Example: Estimate Age of Universe Live in R (part 1)

07:44

Example: Estimating Age of the Universe (part 3)

08:50

Finish Example and More Notes on Linear Modeling

08:31

Linear Modeling Exercises

01:48

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Generalized Linear Models (GLMs) Part 1
9 Lectures
01:00:00

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.

Introduction to GLMs (slides, part 1)

06:58

Introduction to GLMs (slides, part 2)

07:29

Introduction to GLMs (slides, part 3)

07:50

Introduction to GLMs (slides, part 4)

06:44

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.

Preview
07:50

Example: Binomial (Proportion) Model with Heart Disease (part 2)

07:26

Example: Binomial (Proportion) Model with Heart Disease (part 3)

08:16

GLM Exercises

01:05

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Generalized Linear Models Part 2
11 Lectures
01:05:41

Current Agenda

01:46

Linear Regression Exercise Solutions (part 1)

07:31

Linear Regression Exercise Solutions (part 2)

07:29

GLM Exercise Solutions (part 3)

09:30

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.

Example: Poisson Model with Count Data (part 1)

08:15

Example: Poisson Model with Count Data (part 2)

09:29

Exercise: GLM to GAM

01:40

**Log-linear analysis** is a technique used in statistics to examine the relationship between more than two categorical variables.

Example: Log-Linear Model for Categorical Data

05:55

More on Deviance and Overdispersion (slides)

03:11

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Generalized Additive Models Explained
22 Lectures
02:38:16

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.

What are GAMS? (Crawley, slides, part 1)

07:41

What are GAMs? (Crawley, slides, part 2)

06:02

Demonstrate GAM Ozone Data (part 1)

09:40

Demonstrate GAM Ozone Data (part 2)

09:42

What are GAMs? (Wood, slides, part 1)

11:34

Univariate Polynomial GAMs (Wood, slides, part 2)

07:27

Univariate Polynomial GAMs (Wood, slides, part 3)

05:52

GAMs as 4th Order Polynomials (slides, part 1)

06:21

GAMs as 4th Order Polynomials (slides, part 2)

04:29

GAMs as Regression Splines (slides)

03:38

Cubic Splines (slides, part 1)

08:45

Cubic Splines (slides, part 2)

04:21

Function to Establish Basis for Spline (slides)

07:33

Build-a-GAM (slides, part 1)

07:46

Build-a-GAM (slides, part 2)

10:16

Build-a-GAM (slides, part 3)

06:17

Build-a-GAM Cross Validation

08:13

Bivariate GAMs with 2 Explanatory Independent Variables (slides, part 1)

09:17

Bivariate GAMs with 2 Explanatory Independent Variables (slides, part 2)

07:31

Exercises

01:33

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Detailed GAM Examples
15 Lectures
01:55:37

Current Agenda (slides)

05:23

Cherry Trees and Finer Control (slides, part 1)

08:10

Finer Control of GAM (slides, part 2)

10:52

Using Smoothers with More than One Predictor (slides)

07:04

More on Alternative Smoothing Bases (slides)

08:06

Parametric Model Terms (slides)

08:29

Example: Brain Imaging (part 2)

08:09

Example: Brain Imaging (part 3)

07:38

Example: Brain Imaging (part 4)

07:03

Example: Brain Imaging (part 5)

07:41

Example: Air Pollution in Chicago (part 1)

09:33

Example: Air Pollution in Chicago (part 2)

09:17

Air Pollution in Chicago (part 3)

04:40

More Exercises

05:41

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