Generalized linear models provide a flexible framework that extends ordinary linear regression to handle non-normal outcomes and non-linear relationships. This approach unifies many common statistical models under a single coherent structure.
By linking a systematic component to a probability distribution and a transformation of the mean, generalized linear models enable robust inference for real-world data. The following overview highlights core concepts, model families, diagnostics, and practical guidance for practitioners.
| Model Component | Description | Example Link Function | Typical Distribution |
|---|---|---|---|
| Random Component | Specifies the conditional distribution of the response | Identity, log, logit | Gaussian, Binomial, Poisson |
| Systematic Component | Linear predictor formed from explanatory variables | Linear combination of parameters and predictors | Commonly normal errors in linear case |
| Link Function | Connects expectation of the response to the linear predictor | Log for counts, logit for probabilities | Ensures valid mean range |
| Estimation Approach | Fitting based on maximum likelihood rather than least squares | Iterative reweighted least squares | Assesses uncertainty via likelihood |
Model Specification and Components
Proper specification begins by identifying the outcome type and selecting an appropriate distribution family. The choice of link function determines how predictors influence the expected response scale.
Components Overview
Each generalized linear model consists of a random component, a systematic component, and a link function. Together, these elements define how observed data relate to underlying parameters.
Model Families and Link Functions
Different scientific questions require different model families, such as Gaussian for continuous outcomes or binomial for binary outcomes. Matching the variance structure to the data improves accuracy and interpretation.
Common Link Functions
The identity link suits continuous measurements, while the log link is natural for rates and counts. For probabilities, the logit link maps values between zero and one to the full real line.
Estimation and Inference
Maximum likelihood estimation replaces ordinary least squares, enabling parameter estimation for all exponential family distributions. This framework yields standard errors, confidence intervals, and hypothesis tests.
Model Diagnostics
Diagnostics include residual analysis, influential observations, and goodness-of-fit measures to validate distributional assumptions and link function adequacy.
Interpretation and Prediction
Coefficients represent changes in the link-transformed mean for unit changes in predictors, requiring careful back-transformation for substantive conclusions. Prediction intervals must account for both parameter uncertainty and response variability.
Practical Guidance
Visualization of fitted values against observed data helps assess alignment. Sensitivity checks across alternative link functions and distributions ensure robustness.
Key Recommendations
- Match the distribution family and link function to the outcome type and variance structure.
- Inspect residuals and influence diagnostics to validate model assumptions.
- Interpret coefficients on the link scale and back-transform for practical meaning.
- Consider extensions for correlated data when independence is violated.
- Compare models using formal criteria rather than visual fit alone.
FAQ
Reader questions
How do I choose the correct link function for my data?
Select a link function that ensures valid predictions for the scale of your outcome. Use identity for continuous unbounded responses, log for positive counts, and logit for proportions bounded between zero and one.
What diagnostics should I perform after fitting a generalized linear model?
Examine residuals, deviance, and leverage measures to detect outliers and assess distributional assumptions. Evaluate influence diagnostics and goodness-of-fit statistics to identify poorly fitted regions.
Can generalized linear models handle correlated data such as clustered observations?
Standard generalized linear models assume independence, but extensions like generalized estimating equations and mixed models can account for clustering and repeated measures.
How do I compare nested generalized linear models objectively?
Use likelihood ratio tests, information criteria such as AIC or BIC, and cross-validation to compare models while balancing fit and complexity.