Stasinopoulos 2007 - Distributions and Additive Terms

Summary

The gamlss.family catalogue spans 1-4 parameter continuous and discrete distributions with default links; ten additive-term functions (cubic splines, P-splines, loess, fractional polynomials, random effects, …) build the smooth predictors. The P-spline term is a B-spline basis with an $r$-th order difference penalty on coefficients — the same construction used in Bayesian distributional regression.

Overview

Section 1.3 lists the distributions (Tables 1-2); Section 1.4 the smoothers (Table 3). Every family exposes d, p, q, r functions (e.g. dGA, pGA, qGA, rGA for the gamma) and a fitting function that supplies links, first and (approximate) expected second derivatives and starting values.

Main Content

Continuous families (Table 1, selected, with default links $\mu ,\sigma ,\nu ,\tau$)

Distribution	Name	$\mu$	$\sigma$	$\nu$	$\tau$
normal	NO	identity	log	-	-
gamma	GA	log	log	-	-
inverse Gaussian	IG	log	log	-	-
log-normal	LOGNO	log	log	-	-
Weibull	WEI	log	log	-	-
Gumbel / reverse Gumbel	GU, RG	identity	log	-	-
beta	BE	logit	logit	-	-
zero/one-inflated beta	BEINF	logit	logit	log	log
Box-Cox Cole-Green (BCCG)	BCCG	identity	log	identity	-
Box-Cox power exponential	BCPE	identity	log	identity	log
Box-Cox $t$	BCT	identity	log	identity	log
Johnson SU	JSU	identity	log	identity	log
skew $t$ (types 1-5)	ST1-ST5	identity	log	identity or log	log
skew power exponential (1-4)	SEP1-SEP4	identity	log	identity or log	log
sinh-arcsinh	SHASH	identity	log	log	log
generalized beta type 2	GB2	log	identity	log	log

Table 1 also includes exponential, exponential Gaussian, generalized gamma, generalized inverse Gaussian, logistic, power exponential, NET, $t$ family and zero-adjusted inverse Gaussian.

BCCG = LMS

The BCCG distribution is the Box-Cox transformation model of Cole and Green (1992), also known as the LMS method of centile estimation. BCPE (Rigby and Stasinopoulos 2004) and BCT (2006) generalise it to model both skewness and kurtosis. See Stasinopoulos 2007 - Centile Curves, LMS and Residual Diagnostics.

Discrete families (Table 2)

Poisson PO (log), binomial BI (logit), negative binomial type I/II NBI/NBII (log, log), Poisson inverse Gaussian PIG, Sichel SI/SICHEL (log, log, identity), Delaporte DEL, beta binomial BB (logit, log), zero-inflated Poisson ZIP/ZIP2 (log, logit).

Parameterisations. Several have alternatives, e.g. Weibull: WEI $f=\sigma y^{\sigma -1}{\mu }^{-\sigma }\exp \{-(y/\mu {)}^{\sigma }\}$; WEI2 $f=\sigma \mu y^{\sigma -1}{e}^{-\mu y^{\sigma }}$ (proportional hazards); WEI3 has $\mu$ equal to the mean via $\beta =\mu /[\Gamma (1/\sigma )+1]$. Choice affects optimisation: near-orthogonal parameters fit more easily.

Extending the families: (i) write a new gamlss.family (needs the pdf), (ii) truncate (gamlss.tr, gen.trun()), (iii) censor (gamlss.cens, gen.cens()), (iv) mix (gamlss.mx, gamlssMX(), gamlssNP()).

Additive terms (Table 3)

Function	Term
cs()	cubic smoothing spline (every distinct $x$ is a knot; penalty $\lambda \int h^{\prime \prime }(t{)}^{2}dt$; solution is a natural cubic spline)
vc()	varying coefficient $\beta (r)x$ (Hastie and Tibshirani 1993)
ps()	penalized (P-)spline (Eilers and Marx 1996)
lo()	loess (only term allowing smoothing in several variables)
fp()	fractional polynomials (Royston and Altman 1994), powers from $\{-2,-1,-0.5,0,0.5,1,2,3\}$
pp()	power polynomials, powers estimated non-linearly (experimental)
nl()	non-linear fit (gamlss.nl)
random(), ra(), rc()	random effects, augmented-least-squares version, random coefficient

P-spline term

For a distribution parameter $\mathbf{\theta }$, $\mathbf{\theta }=\mathbf{Z}(x)\mathbf{\gamma }$, where $\mathbf{Z}(x)$ is the $n\times q$ B-spline basis (degree degree) on $q$ knots (ps.intervals, equally spaced) and $\mathbf{\gamma }\in {\mathbb{R}}^q$ satisfies $\mathbf{D}\mathbf{\gamma }\sim N(0,{\lambda }^{-1}\mathbf{I})$, equivalently $\mathbf{\gamma }\sim N(0,{\lambda }^{-1}{\mathbf{K}}^{-})$ with $\mathbf{K}={\mathbf{D}}^{\top }\mathbf{D}$. $\mathbf{D}$ is the $(q-r)\times q$ $r$-th order difference matrix (order). Smoothness is set by df (equivalent degrees of freedom) or lambda.

Degrees of freedom of smoothers are counted beyond the constant and linear part: cs(x, df=3) gives 5 total df; this differs from gam::s(x, df=4).

Backfitting convention. The linear part of an additive term is placed in the linear design matrix and the smoother is fitted as a deviation (modified backfitting). Output components per parameter: mu.s (smoother fits), mu.var, mu.df, mu.lambda, mu.coefSmo. Random-effect terms estimate $\mathbf{\gamma }$ given $\lambda$; hyperparameters come from Rigby and Stasinopoulos (2005, App. A) or from gamlssNP() with Gaussian quadrature.

Examples

Why not a polynomial? (CD4 data)

A degree-7 polynomial minimised both AIC and SBC for the CD4 mean but looked “wobbly at the end of the age range” (p. 19) — typical polynomial behaviour. The penalized/smoothing-spline alternatives are in Stasinopoulos 2007 - Lice, CD4 and Claims Examples.

Connections

• Hilbert-space/GP alternatives to P-splines: Hilbert Space Gaussian Processes.
• Count families (NBI, PIG, Sichel, ZIP) vs Generalized Linear Models.

See Also

Stasinopoulos 2007 - GAMLSS Model Framework, Stasinopoulos 2007 - Bayesian P-Splines and Distributional Regression in PyMC