Stasinopoulos 2007 - GAMLSS in R - Overview

Summary

Stasinopoulos and Rigby (2007) define GAMLSS (generalized additive models for location, scale and shape): regression where every parameter of a flexible response distribution (mean $\mu$, scale $\sigma$, skewness $\nu$, kurtosis $\tau$) gets its own additive predictor. The paper documents the gamlss R package and works four examples, ending with BMI/head-circumference centile curves from the Fourth Dutch Growth Study. Key takeaway: model the whole conditional distribution, not only the mean.

Overview

Research question. GLMs and GAMs restrict the response to the exponential family and model only the mean (plus a dispersion that is constant or tied to the mean). How can one regress on a response that is skewed, heavy-tailed or overdispersed, with spread and shape that change with covariates?

Contribution. (1) A unified model class (see Stasinopoulos 2007 - GAMLSS Model Framework); (2) a catalogue of >50 continuous and discrete distributions and ten additive-term types (Stasinopoulos 2007 - Distributions and Additive Terms); (3) the RS and CG penalized-likelihood fitting algorithms with GAIC/validation tuning (Stasinopoulos 2007 - Fitting Algorithms and Model Selection); (4) four worked examples (Stasinopoulos 2007 - Lice, CD4 and Claims Examples, Stasinopoulos 2007 - Centile Curves, LMS and Residual Diagnostics).

Key result. On the head-circumference data a Box-Cox-t (BCT) GAMLSS with smooth $\mu ,\sigma$ and linear $\nu ,\tau$ had no worm-plot violations, whereas the earlier LMS fit had 16 violations (van Buuren and Fredriks 2001).

Motivating PyMC examples

This paper underlies the PyMC example “Is my child normal? Distributional regression (GAMLSS/LMS) and growth centile charts” (E41; Fourth Dutch Growth Study BMI of 7,294 boys) at https://redam94.github.io/pymc-challenges/examples.html. E41 compares constant-spread, smooth-spread, skewed and heavy-tailed models with Bayesian P-splines for every distribution parameter; see Stasinopoulos 2007 - Bayesian P-Splines and Distributional Regression in PyMC. The same ideas appear in E83 (quantile regression) and E108 (usual-intake distributions).

Main Content — Source Map

Paper section	Pages	Covered in
1.1 Introduction	1-2	this note; Stasinopoulos 2007 - GAMLSS Model Framework
1.2 The GAMLSS model, Eqs. (1)-(6)	2-4	Stasinopoulos 2007 - GAMLSS Model Framework
1.3 Distributions, Tables 1-2, censoring, truncation, mixtures	4-7	Stasinopoulos 2007 - Distributions and Additive Terms
1.4 Additive terms, Table 3, P-splines	7-11	Stasinopoulos 2007 - Distributions and Additive Terms
1.5 Algorithms (CG, RS)	11	Stasinopoulos 2007 - Fitting Algorithms and Model Selection
2 The R packages and functions	12-13	Stasinopoulos 2007 - Fitting Algorithms and Model Selection
3.2 Lice data (count distributions)	13-16	Stasinopoulos 2007 - Lice, CD4 and Claims Examples
3.3 CD4 data (mean/variance smoothing)	16-25	Stasinopoulos 2007 - Lice, CD4 and Claims Examples
3.4 Third-party claims (term selection)	26-33	Stasinopoulos 2007 - Lice, CD4 and Claims Examples
3.5 Head circumference, BCT, worm plot, Q statistics	33-44	Stasinopoulos 2007 - Centile Curves, LMS and Residual Diagnostics
4 Conclusions	44-45	Stasinopoulos 2007 - Fitting Algorithms and Model Selection
(not in paper) Bayesian reading, PyMC sketch	—	Stasinopoulos 2007 - Bayesian P-Splines and Distributional Regression in PyMC

Examples

See the example note; the compact comparison is: Poisson AIC 29,174.8 vs Sichel 4,646.2 on the lice counts; CD4 with a smooth $\sigma$ lowers the needed $\mu$ complexity (df 3.72 and 1.81 under AIC); head circumference BCT total df $(\mu ,\sigma ,\nu ,\tau )$ = (17.77, 10.05, 2, 2) (extra smoothing df 15.77 and 8.05 plus constant and linear parts; linear $\nu ,\tau$) with age power $\xi =0.28$.

Relevance to marketing measurement

Revenue, spend and conversion outcomes are skewed and heteroscedastic (spread grows with level). GAMLSS-style models let the variance and tail of KPI distributions depend on channel mix or seasonality, giving calibrated predictive intervals rather than a single mean response — useful when an MMM’s residuals are clearly non-Gaussian. Related vault notes: Generalized Linear Models, Quantile Regression, Hilbert Space Gaussian Processes (an alternative smoother).

Connections

Depends on Generalized Linear Models. Contrasts with quantile approaches (Ji 2024 - Valid Standard Errors for Bayesian Quantile Regression - Overview), which leave the distribution unspecified.

See Also

• Overfitting and Information Criteria, Posterior Predictive Checking