Robert Rigby

Generalized Autoregressive Moving Average Models

A class of generalized autoregressive moving average (GARMA) models is developed that extends the univariate Gaussian ARMA time series model to a flexible observation-driven model for non-Gaussian time series data. The dependent variable is assumed to have a conditional exponential family distribution given the past history of the process. The model estimation is carried out using an iteratively reweighted least squares algorithm. Properties of the model, including stationarity and marginal moments, are either derived explicitly or investigated using Monte Carlo simulation. The relationship of the GARMA model to other models is shown, including the autoregressive models of Zeger and Qaqish, the moving average models of Li, and the reparameterized generalized autoregressive conditional heteroscedastic GARCH model (providing the formula for its fourth marginal moment not previously derived). The model is demonstrated by the application of the GARMA model with a negative binomial conditional distribution to a well-known time series dataset of poliomyelitis counts.

Using the Box-Cox t distribution in GAMLSS to model skewness and kurtosis

The Box-Cox t (BCT) distribution is presented as a model for a dependent variable Y exhibiting both skewness and leptokurtosis. The distribution is defined by a power transformation Y v having a shifted and scaled (truncated) t distribution with degrees of freedom parameter τ. The distribution has four parameters and is denoted by BCT(μ, σ,ν, τ). The parameters μ, σ,ν and τ may be interpreted as relating to location (median), scale (centile-based coefficient of variation), skewness (power transformation to symmetry) and kurtosis (degrees of freedom), respectively. The generalized additive model for location, scale and shape (GAMLSS) is extended to allow each of the parameters of the distribution to be modelled as linear and/or non-linear parametric and/or smooth non-parametric functions of explanatory variables. A Fisher scoring algorithm is used to fit the model by maximizing a (penalized) likelihood. The first and expected second and cross derivatives of the likelihood with respect to μ, σ,ν and τ, required for the algorithm, are provided. The use of the BCT distribution is illustrated by two data applications.

Automatic smoothing parameter selection in GAMLSS with an application to centile estimation

A method for automatic selection of the smoothing parameters in a generalised additive model for location, scale and shape (GAMLSS) model is introduced. The method uses a P-spline representation of the smoothing terms to express them as random effect terms with an internal (or local) maximum likelihood estimation on the predictor scale of each distribution parameter to estimate its smoothing parameters. This provides a fast method for estimating multiple smoothing parameters. The method is applied to centile estimation where all four parameters of a distribution for the response variable are modelled as smooth functions of a transformed explanatory variable x. This allows smooth modelling of the location, scale, skewness and kurtosis parameters of the response variable distribution as functions of x.

Modelling Skewness and Kurtosis with the Bcpe Density in Gamlss

This paper illustrates the power of modern statistical modelling in understanding processes characterised by data that are skewed and have heavy tails. Our particular substantive problem concerns film box-office revenues. We are able to show that traditional modelling techniques based on the Pareto–Levy–Mandelbrot distribution led to what is actually a poorly supported conclusion that these data have infinite variance. This in turn led to the dominant paradigm of the movie business that ‘nobody knows anything’ and hence that box-office revenues cannot be predicted. Using the Box–Cox power exponential distribution within the generalized additive models for location, scale and shape framework, we are able to model box-office revenues and develop probabilistic statements about revenues.

Estimating regional centile curves from mixed data sources and countries

Regional or national growth distributions can provide vital information on the health status of populations. In most resource poor countries, however, the required anthropometric data from purpose-designed growth surveys are not readily available. We propose a practical method for estimating regional (multi-country) age-conditional weight distributions based on existing survey data from different countries. We developed a two-step method by which one is able to model data with widely different age ranges and sample sizes. The method produces references both at the country level and at the regional (multi-country) level. The first step models country-specific centile curves by Box-Cox t and Box-Cox power exponential distributions implemented in generalized additive model for location, scale and shape through a common model. Individual countries may vary in location and spread. The second step defines the regional reference from a finite mixture of the country distributions, weighted by population size. To demonstrate the method we fitted the weight-for-age distribution of 12 countries in South East Asia and the Western Pacific, based on 273 270 observations. We modeled both the raw body weight and the corresponding Z score, and obtained a good fit between the final models and the original data for both solutions. We briefly discuss an application of the generated regional references to obtain appropriate, region specific, age-based dosing regimens of drugs used in the tropics. The method is an affordable and efficient strategy to estimate regional growth distributions where the standard costly alternatives are not an option.

Mean and dispersion modelling for policy claims costs

A model for the statistical analysis of the total amount of insurance paid out on a policy is developed and applied. The model simultaneously deals with the number of claims (zero or more) and the amount of each claim. The number of claims is from a Poisson-based discrete distribution. Individual claim sizes are from a continuous right skewed distribution. The resulting distribution of total claim size is a mixed discrete-continuous model, with positive probability of a zero claim. The means and dispersions of the claim frequency and claim size distribution are modeled in terms of risk factors. The model is applied to a car insurance data set.

Developing regional weight-for-age growth references for malaria-endemic countries to optimize age-based dosing of antimalarials

Abstract Objective To derive regional weight-for-age growth references to help optimize age-based dosing of antimalarials in Africa, the Americas, South-East Asia and the Western Pacific. Methods A weight-for-age database was constructed from pre-existing population-based anthropometric data obtained from household surveys and research groups. It contained data collected between 1995 and 2012 on 1 263 119 individuals (909 368 female, 353 751 male) older than 14 days and younger than 50 years in 64 malaria-endemic countries. Regional growth references were generated using a generalized additive model for location, scale and shape by combining data with varying distributions from a range of sources. Countries were weighted by their population at risk of malaria to enable references to be used in optimizing the dosing of antimalarials. Findings Large differences in weight-for-age distributions existed between the regions and between the regions and global growth standards. For example, the average adult male from the Americas weighed 68.1 kg – 6.0 kg more than males in South-East Asia and the Western Pacific (average: 62.1 kg). For adult women, the difference was over 10.4 kg: the average was 60.4 kg in the Americas and 50.0 kg in South-East Asia and the Western Pacific. Conclusion There were substantial variations in weight-for-age growth curves between malaria-endemic areas. The growth reference charts derived here can be used to guide the evidence-based optimization of aged-based dosing regimens for antimalarials and other drugs often prescribed by age.

Flexible Regression and Smoothing: Using GAMLSS in R

This book is about learning from data using the Generalized Additive Models for Location, Scale and Shape (GAMLSS). GAMLSS extends the Generalized Linear Models (GLMs) and Generalized Additive Models (GAMs) to accommodate large complex datasets, which are increasingly prevalent. In particular, the GAMLSS statistical framework enables flexible regression and smoothing models to be fitted to the data. The GAMLSS model assumes that the response variable has any parametric (continuous, discrete or mixed) distribution which might be heavy- or light-tailed, and positively or negatively skewed. In addition, all the parameters of the distribution (location, scale, shape) can be modelled as linear or smooth functions of explanatory variables.

Discussion: A comparison of GAMLSS with quantile regression:

A discussion on the relative merits of quantile, expectile and GAMLSS regression models is given. We contrast the ‘complete distribution models’ provided by GAMLSS to the ‘distribution free models’ provided by quantile (and expectile) regression. We argue that in general, a flexibility parametric distribution assumption has several advantages allowing possible focusing on specific aspects of the data, model comparison and model diagnostics. A new method for concentrating only on the tail of the distributions is suggested combining quantile regression and GAMLSS.

Centile estimation for a proportion response variable.

This paper introduces two general models for computing centiles when the response variable Y can take values between 0 and 1, inclusive of 0 or 1. The models developed are more flexible alternatives to the beta inflated distribution. The first proposed model employs a flexible four parameter logit skew Student t (logitSST) distribution to model the response variable Y on the unit interval (0, 1), excluding 0 and 1. This model is then extended to the inflated logitSST distribution for Y on the unit interval, including 1. The second model developed in this paper is a generalised Tobit model for Y on the unit interval, including 1. Applying these two models to (1-Y) rather than Y enables modelling of Y on the unit interval including 0 rather than 1. An application of the new models to real data shows that they can provide superior fits.