Summary. A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y, as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9876.2005.00510.x

Generalized additive models for location, scale and shape

Summary A general method for constructing low-rank tensor product smooths for use as components of generalized additive models or generalized additive mixed models is presented. A penalized regression approach is adopted in which tensor product smooths of several variables are constructed from smooths of each variable separately, these “marginal” smooths being represented using a low-rank basis with an associated quadratic wiggliness penalty. The smooths offer several advantages: (i) they have one wiggliness penalty per covariate and are hence invariant to linear rescaling of covariates, making them useful when there is no “natural” way to scale covariates relative to each other; (ii) they have a useful tuneable range of smoothness, unlike single-penalty tensor product smooths that are scale invariant; (iii) the relatively low rank of the smooths means that they are computationally efficient; (iv) the penalties on the smooths are easily interpretable in terms of function shape; (v) the smooths can be generated completely automatically from any marginal smoothing bases and associated quadratic penalties, giving the modeler considerable flexibility to choose the basis penalty combination most appropriate to each modeling task; and (vi) the smooths can easily be written as components of a standard linear or generalized linear mixed model, allowing them to be used as components of the rich family of such models implemented in standard software, and to take advantage of the efficient and stable computational methods that have been developed for such models. A small simulation study shows that the methods can compare favorably with recently developed smoothing spline ANOVA methods.

/pdf/low-rank-scale-invariant-tensor-product-smooths-for-2arz1rpjta.pdf

Low‐Rank Scale‐Invariant Tensor Product Smooths for Generalized Additive Mixed Models

Biochars produced by slow pyrolysis of greenwaste (GW), poultry litter (PL), papermill waste (PS), and biosolids (BS) were shown to reduce N2O emissions from an acidic Ferrosol. Similar reductions were observed for the untreated GW feedstock. Soil was amended with biochar or feedstock giving application rates of 1 and 5%. Following an initial incubation, nitrogen (N) was added at 165 kg/ha as urea. Microcosms were again incubated before being brought to 100% water-filled porosity and held at this water content for a further 47 days. The flooding phase accounted for the majority (<80%) of total N2O emissions. The control soil released 3165 mg N2O-N/m2, or 15.1% of the available N as N2O. Amendment with 1 and 5% GW feedstock significantly reduced emissions to 1470 and 636 mg N2O-N/m2, respectively. This was equivalent to 8.6 and 3.8% of applied N. The GW biochar produced at 350°C was least effective in reducing emissions, resulting in 1625 and 1705 mg N2O-N/m2 for 1 and 5% amendments. Amendment with BS biochar at 5% had the greatest impact, reducing emissions to 518 mg N2O-N/m2, or 2.2% of the applied N over the incubation period. Metabolic activity as measured by CO2 production could not explain the differences in N2O emissions between controls and amendments, nor could NH4+ or NO3– concentrations in biochar-amended soils. A decrease in NH4+ and NO3– following GW feedstock application is likely to have been responsible for reducing N2O emissions from this amendment. Reduction in N2O emissions from the biochar-amended soils was attributed to increased adsorption of NO3–. Small reductions are possible due to improved aeration and porosity leading to lower levels of denitrification and N2O emissions. Alternatively, increased pH was observed, which can drive denitrification through to dinitrogen during soil flooding.

Influence of biochars on flux of N2O and CO2 from Ferrosol.

We consider local polynomial kernel regression with a single covariate for clustered data using estimating equations. We assume that at most m < ∞ observations are available on each cluster. In the case of random regressors, with no measurement error in the predictor, we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent. In the further special case of longitudinal data on individuals with fixed common observation times, we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average. We also consider what happens when the predictor is measured with error. Using the SIMEX approach to correct for measurement error, we construct an asymptotic theory for both the pooled and the weighted average estimators. Surprisingly, for the same amount of smoothing, the weighted a...

https://www.jstor.org/stable/pdfplus/2669396.pdf

Nonparametric Function Estimation for Clustered Data When the Predictor is Measured without/with Error

Summary. We propose a class of double hierarchical generalized linear models in which random effects can be specified for both the mean and dispersion. Heteroscedasticity between clusters can be modelled by introducing random effects in the dispersion model, as is heterogeneity between clusters in the mean model. This class will, among other things, enable models with heavy-tailed distributions to be explored, providing robust estimation against outliers. The h-likelihood provides a unified framework for this new class of models and gives a single algorithm for fitting all members of the class. This algorithm does not require quadrature or prior probabilities.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9876.2006.00538.x

Double hierarchical generalized linear models (with discussion)

In designed experiments and in particular longitudinal studies, the aim may be to assess the effect of a quantitative variable such as time on treatment effects. Modelling treatment effects can be complex in the presence of other sources of variation. Three examples are presented to illustrate an approach to analysis in such cases. The first example is a longitudinal experiment on the growth of cows under a factorial treatment structure where serial correlation and variance heterogeneity complicate the analysis. The second example involves the calibration of optical density and the concentration of a protein DNase in the presence of sampling variation and variance heterogeneity. The final example is a multienvironment agricultural field experiment in which a yield-seeding rate relationship is required for several varieties of lupins. Spatial variation within environments, heterogeneity between environments and variation between varieties all need to be incorporated in the analysis. In this paper, the cubic smoothing spline is used in conjunction with fixed and random effects, random coefficients and variance modelling to provide simultaneous modelling of trends and covariance structure. The key result that allows coherent and flexible empirical model building in complex situations is the linear mixed model representation of the cubic smoothing spline. An extension is proposed in which trend is partitioned into smooth and non-smooth components. Estimation and inference, the analysis of the three examples and a discussion of extensions and unresolved issues are also presented.

A. P. Verbyla

Papers

The analysis of designed experiments and longitudinal data by using smoothing splines. Discussion. Authors' reply