Generalized Linear Models (GLMs) are a versatile class of models that extend linear regression to handle a variety of response variable distributions and relationships. In R, GLMs are implemented through the glm() function and related packages, providing a powerful framework for analyzing both continuous and categorical data across a wide range of contexts. This tutorial introduces several types of GLMs, as well as related models, and demonstrates how to implement each in R.
The Generalized Linear Model (GLM) is a sophisticated extension of linear regression designed to model relationships between a dependent variable and independent variables when the underlying assumptions of linear regression are unmet. The GLM was first introduced by Sir John Nelder and Robert Wedderburn, both acclaimed statisticians, in 1972.
The GLM is an essential tool in modern data analysis, as it can be used to model a wide range of data types that may not conform to the assumptions of traditional linear regression. It allows for modeling non-normal distributions, non-linear relationships, and correlations between observations. By utilizing maximum likelihood estimation (MLE), the GLM can also handle missing data and provide accurate estimates even when some observations are missing. This makes it a valuable tool in business and academia, where the ability to model complex relationships accurately is essential.
The GLM is a powerful and flexible tool integral to modern data analysis. Its ability to model complex relationships between variables and handle missing data has made it a valuable asset in business and academia.
Note
Maximum Likelihood Estimation (MLE) is a statistical technique used to estimate the parameters of a model by analyzing the observed data. This method involves finding the optimal values for the model parameters by maximizing the likelihood function. The likelihood function measures how well the model can explain the observed data. The higher the likelihood function, the more accurate the model explains the data. MLE is widely used in fields such as finance, economics, and engineering to create models that can predict future outcomes based on the available data.
Key features of Generalized Linear Models
Link Function: GLMs are characterized by a link function that connects the linear predictor, a combination of independent variables, to the mean of the dependent variable. This connection enables the estimation of the relationship between independent and dependent variables in a non-linear fashion.
The selection of a link function in GLMs is contingent upon the nature of the data and the distribution of the response variable. The identity link function is utilized when the continuous response variable follows a normal distribution. The logit link function is employed when the response variable is binary, meaning it can only take on two values and follows a binomial distribution. The log link function is utilized when the response variable is count data and follows a Poisson distribution.
Choosing an appropriate link function is a crucial aspect of modeling, as it impacts the interpretation of the estimated coefficients for independent variables. Therefore, a thorough understanding of the nature of the data and the response variable’s distribution is necessary when selecting a link function.
Distribution Family: Unlike linear regression, which assumes a normal distribution for the residuals, GLMs allow for a variety of probability distributions for the response variable. The choice of distribution is based on the characteristics of the data. Commonly used distributions include:
Normal distribution (Gaussian): For continuous data.
Binomial distribution: For binary or dichotomous data.
Poisson distribution: For count data.
Gamma distribution: For continuous, positive, skewed data.
Variance Function: GLMs accommodate heteroscedasticity (unequal variances across levels of the independent variables) by allowing the variance of the response variable to be a function of the mean.
Deviance: Instead of using the sum of squared residuals as in linear regression, GLMs use deviance to measure lack of fit. Deviance compares the fit of the model to a saturated model (a model that perfectly fits the data).
The mathematical expression of a Generalized Linear Model (GLM) involves the linear predictor, the link function, and the probability distribution of the response variable.
\(\beta_0, \beta_1, \ldots, \beta_k\) are the coefficients,
\(x_1, x_2, \ldots, x_k\) are the independent variables.
Link Function (g):
\[ g(\mu) = \eta \]
The link function connects the linear predictor to the mean of the response variable. It transforms the mean (μ) to the linear predictor (η). Common link functions include:
Identity link (for normal distribution):
\[ g(\mu) = \mu \]
Logit link (for binary data in logistic regression):
\[ g(\mu) = log(\frac{\mu}{1-\mu}) \]
Log link(for Poisson regression):
\[ g(\mu) = \log(\mu )\]
Probability Distribution: The response variable follows a probability distribution from the exponential family. The distribution is chosen based on the nature of the data. Common choices include:
Normal distribution (Gaussian) for continuous data.
Binomial distribution for binary or dichotomous data.
Poisson distribution for count data.
Gamma distribution for continuous, positive, skewed data.
Putting it all together, the probability mass function (PMF) or probability density function (PDF) for the response variable (Y) is expressed as:
The primary difference between linear models (LM) and generalized linear models (GLM) is in their flexibility to handle different types of response variables and error distributions. Here’s a breakdown of the key distinctions:
1. Type of Response Variable
LM (Linear Model): Assumes that the response variable is continuous and normally distributed. For example, predicting a continuous variable like height or weight.
GLM (Generalized Linear Model): Extends linear models to accommodate response variables that are not normally distributed, such as binary outcomes (0 or 1), counts, or proportions. GLMs can handle a variety of distributions (e.g., binomial, Poisson).
2. Link Function
LM: The relationship between the predictor variables and the response is assumed to be linear, with an identity link function (i.e., (\(Y = X \beta + \epsilon\)), where (\(\epsilon\)) is normally distributed).
GLM: Uses a link function to transform the linear predictor to accommodate different types of response variables. Common link functions include:
Logit link for binary data (logistic regression)
Log link for count data (Poisson regression)
Identity link for normal data (same as in LM)
3. Error Distribution
LM: Assumes errors are normally distributed with constant variance (homoscedasticity).
GLM: Allows for different error distributions (e.g., binomial, Poisson, gamma) to better suit the data.
4. Use Cases
LM: Used when the response variable is continuous, normally distributed, and has a linear relationship with predictors.
GLM: Used when the response variable does not fit these assumptions, such as binary outcomes (yes/no), counts, or proportions.
5. Examples
LM: Simple linear regression, multiple linear regression
GLM: Logistic regression, Poisson regression, negative binomial regression, etc.
In summary, GLMs generalize LMs by allowing for non-normal distributions and providing flexibility with link functions, making them more suitable for a wider range of data types and applications.
In summary, the GLM combines the linear predictor, link function, and probability distribution to model the relationship between the mean of the response variable and the predictors, allowing for flexibility in handling various data types. The specific form of the GLM will depend on the chosen link function and distribution.
GLM Models in R
Before starting, ensure you have R and the necessary packages installed. Key packages include stats (for basic GLMs) and mgcv (for Generalized Additive Models). You may also need packages such as MASS for ordinal models and betareg for Beta regression.
# Install necessary packages if you haven't alreadyinstall.packages(c("MASS", "mgcv", "betareg"))
Family objects are a convenient way to specify the models used by functions like glm(). See help(family) for other allowable link functions for each family.
binomial(link = "logit")
gaussian(link = "identity")
Gamma(link = "inverse")
inverse.gaussian(link = "1/mu\^2")
poisson(link = "log")
quasi(link = "identity", variance = "constant")
quasibinomial(link = "logit")
quasipoisson(link = "log")
There are several GLM model families depending on the make-up of the response variable.
Probit regression is similar to logistic regression but uses the probit link function. It’s useful for modeling binary outcomes when the probit function is a better fit than the logit.
# Example of probit regressionmodel_probit <-glm(y ~ x1 + x2, data = data, family =binomial(link ="probit"))summary(model_probit)
For nominal categorical responses with more than two levels, multinomial logistic regression can be used. The nnet package provides multinom for fitting such models.
# Example of multinomial logistic regressionlibrary(nnet)model_multinom <-multinom(y ~ x1 + x2, data = data)summary(model_multinom)
GAMs allow for flexible relationships between predictors and the response by using smooth functions. The mgcv package’s gam function is used for GAMs.
# Example of GAMlibrary(mgcv)model_gam <-gam(y ~s(x1) +s(x2), data = data, family = gaussian)summary(model_gam)
Required R Packages
The following R packages are required for running the code examples in this tutorial:
Here’s the organized list of R packages grouped by primary use, with duplicates removed: Here’s the comprehensive list with descriptions and references for all packages, organized by category:
Data Wrangling & Visualization
tidyverse (Wickham): Meta-package for data science workflows (includes dplyr, ggplot2, tidyr). Streamlines data manipulation and visualization. tidyverse.org | CRAN
plyr (Wickham): Split-apply-combine workflows (predecessor to dplyr). CRAN
patchwork (Pedersen): Combine multiple ggplot2 plots into unified layouts. CRAN
RColorBrewer (Neuwirth): Color palettes for thematic maps and statistical graphics. CRAN
GGally (Schloerke): Extend ggplot2 with correlation matrices and multivariate plots. CRAN
Exploratory Data Analysis (EDA)
dlookr (Ruy C): Automate data quality checks, outlier detection, and EDA reports. CRAN
DataExplorer (Peters): Quickly profile datasets with automatic visualizations. CRAN
Statistical Tests & Diagnostics
rstatix (Kassambara): Tidy-friendly interface for t-tests, ANOVA, and non-parametric tests. CRAN
lmtest (Zeileis & Hothorn): Diagnostic tests for linear models (e.g., Breusch-Pagan). CRAN
generalhoslem (Kassambara): Goodness-of-fit tests for logistic regression models. CRAN
moments (Komsta & Novomestky): Calculate skewness, kurtosis, and distribution moments. CRAN
Modeling & Regression
MASS (Venables & Ripley): Robust regression, LDA, and negative binomial GLMs. CRAN
This tutorial covered various GLMs, each suited to different types of response data. Use summary() to inspect model results and diagnostics for each type of GLM. These models allow for a range of data structures and distributions, making them a versatile toolset in R for real-world applications.
