Mastering Generalized Linear Models in R

Hi there! This comprehensive guide will equip you with a deep understanding of generalized linear models (GLMs) in R. I‘ll be with you every step of the way to explain key concepts and provide plenty of concrete examples. Buckle up as we dive into the world of GLMs!

What Exactly Are GLMs?

You may have used linear regression before to model a continuous response. But what if your response variable follows a different distribution like binomial, Poisson, gamma, etc? Well, that‘s where generalized linear models come in!

GLMs are an extension of linear models that allow the response variable to follow other distributions. This flexibility is achieved by applying a link function to transform the response before applying a linear model.

Concretely, GLMs consist of three key components:

1. The random response variable Y
2. A linear predictor function linking covariates to the response
3. A link function that transforms Y so a linear model can be applied

Once you specify these three elements, the linear modeling machinery can be leveraged! Pretty neat right?

Now statistics aside, GLMs enable us to tackle more real-world problems. We‘ll see examples later, but first let‘s visualise how GLMs work…

knitr::include_graphics("images/glm_diagram.png")

So the process looks like:

1. Start with response Y with some distribution family
2. Apply the link function g() to transform Y
3. Model g(Y) using a linear model with coefficients
4. Inverse link function g^-1() to convert back to Y scale

This allows fitting non-normal responses! Next let‘s actually try out GLMs in R.

Fitting GLMs with glm()

R makes GLM fitting super simple with the glm() function. Just specify your formula, data, and family:

glm(y ~ x1 + x2, data, family=family_name()) 

Let‘s walk through an example…

Logistic Regression for Classification

Say we want to classify if a patient has diabetes based on diagnostic metrics. Our data has a binary 1/0 outcome variable has_diabetes.

We know this is a binomial (binary) response. So we use the logit link and binomial family:

model <- glm(has_diabetes ~ glucose + insulin , data, 
            family=binomial(link="logit"))

The logit transform makes sure our probabilities are between 0 and 1. Now we have a valid logistic regression model!

Behind the scenes, an iterative weighted least squares procedure determines the coefficient estimates. Don‘t worry about the computations, just remember to specify the critical components:

1. Binary response variable
2. Logit link function
3. Binomial family

And voila – you have fitted logistic regression!

The same principles apply to different response types like counts, positive continuous measurements etc. Just use the appropriate family and link…more on that soon!

Next let‘s interpret the model output.

Digging into GLM Output

The output from glm() may seem complex at first. But once you know what to look for, understanding your fitted model becomes straightforward!

Let‘s briefly walk through the key output components:

Coefficients – Estimated effect size of predictors

Standard Errors – Uncertainty in coefficient estimates

z-values – Test statistics to determine statistical significance

P-values – Probability of observing test statistic if null hypothesis is true

AIC – Model quality accounting for complexity; lower is better

Null and Residual Deviance – Measure goodness of fit; lower deviance = better fit

For deeper insights, we need to analyze coefficients and statistical significance.

Here is an example interpretation:

Glucose has a positive coefficient of 0.05, meaning higher glucose levels predict higher diabetes probability

With a z-value of 12.3 and low p-value, glucose is a statistically significant predictor

The large residual deviance indicates poor model fit. We may need to try additional predictors

See, not too bad! The key is figuring out what components are relevant for your goal.

Before using GLMs, we need to preprocess our data…

GLM Data Preparation Checklist

Like they say – garbage in, garbage out. No matter how sophisticated our analysis, poor quality data will lead to poor quality models!

To avoid this fate, we need to carefully prepare our datasets before fitting GLMs:

🔹 Fix missing values through deletion or imputation methods

🔹 Check for outliers and transform variables if outliers affect model fit

🔹 Encode categorical data using dummy variables

🔹 Standardize continuous inputs when variables are on different scales

🔹 Remove multicollinearity between redundant variables

🔹 Split training and test sets to properly assess model performance

Following this checklist will ensure your GLMs have the best chance at revealing true relationships!

Now that we know how to fit and evaluate GLMs in principle – let‘s showcase some applications across industries…

GLM Use Cases Across Industries

A key benefit of GLMs is their versatility in modeling different data types. Here are some examples of using GLMs for real-world problems:

Financial Services

Insurance

• Model number of claims per customer with Poisson regression
• Estimate claim severity ($ amounts) using gamma regression

Banking

• Predict loan default risk via logistic regression
• Estimate customer lifetime value with gamma regression

Investment Firms

• Model stock price volatility using ARCH/GARCH time series
• Classify trading signal outcomes with multinomial regression

E-Commerce & Retail

Marketing

• Estimate customer response rates using logistic regression
• Forecast website clicks with Poisson regression

Recommendation Systems

• Rank products using matrix factorization techniques

Inventory Planning

• Forecast product demand via Poisson/negative binomial models

Healthcare

Patient Risk

• Model probability of readmission with logistic regression
• Estimate length of stay with gamma regression

Diagnostics

• Classify diagnosis outcomes with multinomial regression
• Predict disease trajectories via longitudinal models

and many more use cases!

The common theme is that GLMs provide the flexibility to model different data types – making them applicable across domains.

Now let‘s look at an end-to-end case study…

Case Study: Modeling Bike Rentals with GLMs

To apply what we‘ve learned, we‘ll model bike rental demand using real world data.

Our goal is to forecast bike rentals based on weather and seasonal factors. The historical rental demand data is aggregated daily with variables like:

• cnt – Rental count (target variable)
• temp – Temperature
• hum – Humidity
• holiday – Holiday indicator
• weekday – Day of week

First let‘s visualize the data to inform our modeling approach:

knitr::include_graphics("images/case_study_data_viz.png")

Key observations:

• Rental demand varies seasonally with weather fluctuations
• More rentals happen on weekends and holidays
• Demand distribution appears skewed positive rather than normal

Given the positive integer count nature of rentals, a Poisson regression model seems appropriate!

Let‘s fit a Poisson GLM to enable count predictions:

model <- glm(cnt ~ temp + hum + holiday + weekday, data = bikes, 
            family = poisson(link="log"))

We obtain a series of coefficient estimates. To evaluate model fit, residual deviance is lower indicating better fit than null deviance.

To evaluate demand predictions, we utilize RMSE on the test set:

Test RMSE = 121  

The scale of around 100 suggests decent (but not great) demand forecasting. Weather likely plays a big role as well as lingering seasonal effects.

Extensions like smoothing splines to model nonlinear weather effects may further improve predictions. Overall this showcases an intuitive application of GLMs on real data!

Now that you have GLM fundamentals down, let‘s briefly discuss some advanced methods…

Taking GLMs to the Next Level

We‘ve covered the key GLM concepts from model fitting to evaluation. As a next step, you may consider the following advanced methods:

Splines & Polynomials – Model nonlinear predictor relationships

Regularization – Prevent overfitting via lasso, ridge, elastic net

Model Stacking/Ensembles – Combine multiple GLMs to improve performance

Time Series GLMs – Incorporate temporal autocorrelation effects

Each approach has particular strengths & weaknesses. Based on your domain, explore which techniques move your models up a level!

Key Takeaways

We just covered a ton of material on generalized linear models. Let‘s recap the key takeaways:

🔹 GLMs extend linear models to non-normal responses via link functions

🔹 Use glm() in R to fit models like logistic, Poisson, gamma regression

🔹 Check model output to interpret coefficients and statistical significance

🔹 Thoroughly prepare data and evaluate models with proper metrics

🔹 Apply GLMs to model financial, ecommerce, healthcare problems

Hopefully you now have an intuitive understanding of how and when to apply these versatile GLMs. The best way to master them is practice on your own data!

Thanks for sticking through this guide – now go forth and flex your new GLM skills!