Generalized additive model for location, scale and shape

The generalized additive model for location, scale and shape (GAMLSS) is a statistical model developed by Rigby and Stasinopoulos (and later expanded) to overcome some of the limitations associated with the popular generalized linear models (GLMs) and generalized additive models (GAMs). For an overview of these limitations see Nelder and Wedderburn (1972){{cite journal|last1=Nelder|first1=J.A.|last2=Wedderburn|first2=R.W.M|title=Generalized linear models|journal=J. R. Stat. Soc. A|year=1972|volume=135|issue=3|pages=370–384|doi=10.2307/2344614|jstor=2344614}} and Hastie's and Tibshirani's book.{{cite book|last1=Hastie|first1=TJ|last2=Tibshirani|first2=RJ|title=Generalized additive models|year=1990|publisher=Chapman and Hall|location=London}}

In GAMLSS the exponential family distribution assumption for the response variable, ($y$), (essential in GLMs and GAMs), is relaxed and replaced by a general distribution family, including highly skew and/or kurtotic continuous and discrete distributions.

The systematic part of the model is expanded to allow modeling not only of the mean (or location) but other parameters of the distribution of y as linear and/or nonlinear, parametric and/or additive non-parametric functions of explanatory variables and/or random effects.

GAMLSS is especially suited for modelling a leptokurtic or platykurtic and/or positively or negatively skewed response variable. For count type response variable data it deals with over-dispersion by using proper over-dispersed discrete distributions. Heterogeneity also is dealt with by modeling the scale or shape parameters using explanatory variables. There are several packages written in R related to GAMLSS models,{{cite journal|last2=Rigby|first2=Robert A|date=December 2007|title=Generalized additive models for location scale and shape (GAMLSS) in R|journal=Journal of Statistical Software|volume=23|issue=7|doi=10.18637/jss.v023.i07|last1=Stasinopoulos|first1=D. Mikis|doi-access=free}} and tutorials for using and interpreting GAMLSS.{{cite journal |last1=David |first1=Bann |last2=Liam |first2=Wright |last3=Tim J |first3=Cole |title=Risk factors relate to the variability of health outcomes as well as the mean: A GAMLSS tutorial |journal=eLife |date=2022 |volume=11 |issue=11 |doi=10.7554/eLife.72357 |pmid=34985412 |pmc=8791632 |doi-access=free }}

A GAMLSS model assumes independent observations $y\_i$ for $i\; =\; 1,\; 2,\; \backslash dots\; ,\; n$

with probability (density) function $f\; (y\_i\; |\; \backslash mu\_i\; ,\; \backslash sigma\_i\; ,\; \backslash nu\_i\; ,\; \backslash tau\_i\; )$ conditional on $(\backslash mu\_i\; ,\; \backslash sigma\_i\; ,\; \backslash nu\_i\; ,\; \backslash tau\_i\; )$ a vector of four distribution parameters, each of which can be a function of the explanatory variables. The first two population distribution parameters $\backslash mu\_i$ and $\backslash sigma\_i$ are usually characterized as location and scale parameters, while the remaining parameter(s), if any, are characterized as shape parameters, e.g. skewness and kurtosis parameters, although the model may be applied more generally to the parameters of any population distribution with up to four distribution parameters, and can be generalized to more than four distribution parameters.

: $$

\begin{align}

g_1 (\mu) = \eta_1= X_1 \beta_1 + \sum_{j=1}^{J_1} {h}_{j1}(x_{j1}) \\

g_2(\sigma) = \eta_2= X_2 \beta_2 + \sum_{j=1}^{J_2}{h}_{j2}(x_{j2}) \\

g_3(\nu) = \eta_3 = X_3 \beta_3 + \sum_{j=1}^{J_3}{h}_{j3}(x_{j3}) \\

g_4(\tau)=\eta_4=X_4 \beta_4 + \sum_{j=1}^{J_4}{h}_{j4}(x_{j4})

\end{align}

where μ, σ, ν, τ and $\backslash eta\_k$ are vectors of length $n$, $\backslash beta^\{T\}\_k\; =\; (\backslash beta\_\{1k\},\backslash beta\_\{2k\},\backslash ldots,\backslash beta\_\{J\text{'}\_\{k\}$

k}) is a parameter vector of length $J\text{'}\_k$, $X\_k$ is a fixed known design matrix of order $n\; \backslash times\; J\text{'}\_k$ and $h\_\{jk\}$ is a smooth non-parametric function of explanatory variable $x\_\{jk\}$, $j=1,2,\backslash ldots,\; J\_\{k\}$ and $k=1,2,3,4$.

$g\_i$ are link functions.

For centile estimation the [https://www.who.int/childgrowth/en WHO Multicentre Growth Reference Study Group] have recommended GAMLSS and the Box–Cox power exponential (BCPE) distributions{{cite journal|last2=Stasinopoulos|first2=D. Mikis|date=February 2004|title=Smooth Centile Curves for Skew and Kurtotic data Modelled Using the Box–Cox Power Exponential Distribution|journal=Statistics in Medicine|volume=23|issue=19|pages=3053–3076|doi=10.1002/sim.1861|pmid=15351960|last1=Rigby|first1=Robert}} for the construction of the WHO Child Growth Standards.{{Cite journal | last1 = Borghi | first1 = E. | last2 = De Onis | first2 = M. | last3 = Garza | first3 = C. | last4 = Van Den Broeck | first4 = J. | last5 = Frongillo | first5 = E. A. | last6 = Grummer-Strawn | first6 = L. | last7 = Van Buuren | first7 = S. | last8 = Pan | first8 = H. | last9 = Molinari | first9 = L. | doi = 10.1002/sim.2227 | last10 = Martorell | first10 = R. | last11 = Onyango | first11 = A. W. | last12 = Martines | first12 = J. C. | author13 = WHO Multicentre Growth Reference Study Group | title = Construction of the World Health Organization child growth standards: Selection of methods for attained growth curves | journal = Statistics in Medicine | volume = 25 | issue = 2 | pages = 247–265 | year = 2006 | pmid = 16143968}}WHO Multicentre Growth Reference Study Group (2006) WHO Child Growth Standards: Length/height-for-age, weight-for-age, weight-for-length, weight-for-height and body mass index-for-age: Methods and development. Geneva: World Health Organization.

The form of the distribution assumed for the response variable y, is very general. For example, an implementation of GAMLSS in R{{Cite web|url=https://www.gamlss.com/distributions/|title=The R packages {{!}} gamlss|website=The R packages {{!}} gamlss|access-date=2020-05-04}} has around 100 different distributions available. Such implementations also allow use of truncated distributions and censored (or interval) response variables.

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• [http://www.gamlss.org/ GAMLSS official website gamlss.org]
• [http://manual.gamlss.org/ GAMLSS manual (downloadable)]{{Dead link|date=December 2019 |bot=InternetArchiveBot |fix-attempted=yes }}
• [http://gamlss.org/images/stories/papers/Distributions-2010-onlyThetable.pdf Distribution tables in GAMLSS]{{Dead link|date=December 2019 |bot=InternetArchiveBot |fix-attempted=yes }}
• [http://gamlss.org/images/stories/papers/gamlssreferencecard.pdf The GAMLSS packages reference card (downloadable)]{{Dead link|date=December 2019 |bot=InternetArchiveBot |fix-attempted=yes }}
• [http://www.gamlss.org/images/stories/papers/book-2008-27-6-08.pdf The booklet for the Utrecht short course on GAMLSS (downloadable)]{{Dead link|date=December 2019 |bot=InternetArchiveBot |fix-attempted=yes }}
• [https://archive.today/20130414200233/http://packages.gamlss.org/ R packages for GAMLSS on CRAN]

{{DEFAULTSORT:Generalized Additive Model For Location, Scale And Shape}}

Additive model for location, scale and shape

Category:Semi-parametric models