Iterative weighted estimation based on variance modelling in linear regression models

ABSTRACT

The estimation of variance function plays an extremely important role in statistical inference of the regression models. In this paper we propose a variance modelling method for constructing the variance structure via combining the exponential polynomial modelling method and the kernel smoothing technique. A simple estimation method for the parameters in heteroscedastic linear regression models is developed when the covariance matrix is unknown diagonal and the variance function is a positive function of the mean. The consistency and asymptotic normality of the resulting estimators are established under some mild assumptions. In particular, a simple version of bootstrap test is adapted to test misspecification of the variance function. Some Monte Carlo simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by the ozone concentration dataset.     

Assessing switchability for biosimilar products: modelling approaches applied to children's growth

The present paper describes two statistical modelling approaches that have been developed to demonstrate switchability from the original recombinant human growth hormone (rhGH) formulation (Genotropin^®) to a biosimilar product (Omnitrope^®) in children suffering from growth hormone deficiency. Demonstrating switchability between rhGH products is challenging because the process of growth varies with the age of the child and across children. The first modelling approach aims at predicting individual height measured at several time‐points after switching to the biosimilar. The second modelling approach provides an estimate of the deviation from the overall growth rate after switching to the biosimilar, which can be regarded as an estimate of switchability. The results after applying these approaches to data from a randomized clinical trial are presented. The accuracy and precision of the predictions made using the first approach and the small deviation from switchability estimated with the second approach provide sufficient evidence to conclude that switching from Genotropin^® to Omnitrope^® has a very small effect on growth, which is neither statistically significant nor clinically relevant. Copyright © 2015 John Wiley & Sons, Ltd.     

Proposed best practice for projects that involve modelling and simulation

Modelling and simulation has been used in many ways when developing new treatments. To be useful and credible, it is generally agreed that modelling and simulation should be undertaken according to some kind of best practice. A number of authors have suggested elements required for best practice in modelling and simulation. Elements that have been suggested include the pre‐specification of goals, assumptions, methods, and outputs. However, a project that involves modelling and simulation could be simple or complex and could be of relatively low or high importance to the project. It has been argued that the level of detail and the strictness of pre‐specification should be allowed to vary, depending on the complexity and importance of the project. This best practice document does not prescribe how to develop a statistical model. Rather, it describes the elements required for the specification of a project and requires that the practitioner justify in the specification the omission of any of the elements and, in addition, justify the level of detail provided about each element. This document is an initiative of the Special Interest Group for modelling and simulation. The Special Interest Group for modelling and simulation is a body open to members of Statisticians in the Pharmaceutical Industry and the European Federation of Statisticians in the Pharmaceutical Industry. Examples of a very detailed specification and a less detailed specification are included as appendices.     

Hierarchical spatially varying coefficient and temporal dynamic process models using spTDyn

Bayesian hierarchical spatio-temporal models are becoming increasingly important due to the increasing availability of space-time data in various domains. In this paper we develop a user friendly R package, spTDyn, for spatio-temporal modelling. It can be used to fit models with spatially varying and temporally dynamic coefficients. The former is used for modelling the spatially varying impact of explanatory variables on the response caused by spatial misalignment. This issue can arise when the covariates only vary over time, or when they are measured over a grid and hence do not match the locations of the response point-level data. The latter is to examine the temporally varying impact of explanatory variables in space-time data due, for example, to seasonality or other time-varying effects. The spTDyn package uses Markov chain Monte Carlo sampling written in C, which makes computations highly efficient, and the interface is written in R making these sophisticated modelling techniques easily accessible to statistical analysts. The models and software, and their advantages, are illustrated using temperature and ozone space-time data.     

Specification and testing of multiplicative time-varying GARCH models with applications

In this article, we develop a specification technique for building multiplicative time-varying GARCH models of Amado and Teräsvirta (2008, 2013). The variance is decomposed into an unconditional and a conditional component such that the unconditional variance component is allowed to evolve smoothly over time. This nonstationary component is defined as a linear combination of logistic transition functions with time as the transition variable. The appropriate number of transition functions is determined by a sequence of specification tests. For that purpose, a coherent modelling strategy based on statistical inference is presented. It is heavily dependent on Lagrange multiplier type misspecification tests. The tests are easily implemented as they are entirely based on auxiliary regressions. Finite-sample properties of the strategy and tests are examined by simulation. The modelling strategy is illustrated in practice with two real examples: an empirical application to daily exchange rate returns and another one to daily coffee futures returns.     

Theoretic and empirical data-inclusive process characterization

Summary.  In process characterization the quality of information that is obtained depends directly on the quality of process model. The current quality revolution is now providing a strong stimulus for rethinking and re-evaluating many statistical ideas. Among these are the role of theoretic knowledge and data in statistical inference and some issues in theoretic–empirical modelling. With this concern the paper takes a broad, pragmatic view of statistical inference to include all aspects of model formulation. The estimation of model parameters traditionally assumes that a model has a prespecified known form and takes no account of possible uncertainty regarding model structure. But in practice model structural uncertainty is a fact of life and is likely to be more serious than other sources of uncertainty which have received far more attention. This is true whether the model is specified on subject-matter grounds or when a model is formulated, fitted and checked on the same data set in an iterative interactive way. For that reason novel modelling techniques have been fashioned for reducing model uncertainty. Using available knowledge for theoretic model elaboration the techniques that have been created approximate the exact unknown process model concurrently by accessible theoretic and polynomial empirical functions. The paper examines the effects of uncertainty for hybrid theoretic–empirical models and, for reducing uncertainty, additive and multiplicative methods of model formulation are fashioned. Such modelling techniques have been successfully applied to perfect a steady flow model for an air gauge sensor. Validation of the models elaborated has revealed that the multiplicative modelling approach allows us to attain a satisfactory model with small discrepancy from empirical evidence.     

Analysis of longitudinal data with drop-out: objectives, assumptions and a proposal

Summary. The problem of analysing longitudinal data that are complicated by possibly informative drop-out has received considerable attention in the statistical literature. Most researchers have concentrated on either methodology or application, but we begin this paper by arguing that more attention could be given to study objectives and to the relevant targets for inference. Next we summarize a variety of approaches that have been suggested for dealing with drop-out. A long-standing concern in this subject area is that all methods require untestable assumptions. We discuss circumstances in which we are willing to make such assumptions and we propose a new and computationally efficient modelling and analysis procedure for these situations. We assume a dynamic linear model for the expected increments of a constructed variable, under which subject-specific random effects follow a martingale process in the absence of drop-out. Informal diagnostic procedures to assess the tenability of the assumption are proposed. The paper is completed by simulations and a comparison of our method and several alternatives in the analysis of data from a trial into the treatment of schizophrenia, in which approximately 50% of recruited subjects dropped out before the final scheduled measurement time.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.     

Imputation of missing variance data using non-linear mixed effects modelling to enable an inverse variance weighted meta-analysis of summary-level longitudinal data: a case study

Missing variances, on the basis of the summary-level data, can be a problem when an inverse variance weighted meta-analysis is undertaken. A wide range of approaches in dealing with this issue exist, such as excluding data without a variance measure, using a function of sample size as a weight and imputing the missing standard errors/deviations. A non-linear mixed effects modelling approach was taken to describe the time-course of standard deviations across 14 studies. The model was then used to make predictions of the missing standard deviations, thus, enabling a precision weighted model-based meta-analysis of a mean pain endpoint over time. Maximum likelihood and Bayesian approaches were implemented with example code to illustrate how this imputation can be carried out and to compare the output from each method. The resultant imputations were nearly identical for the two approaches. This modelling approach acknowledges the fact that standard deviations are not necessarily constant over time and can differ between treatments and across studies in a predictable way.     

Simplicity out of complexity in environmental modelling: Occam's razor revisited

While large models based on a deterministic-reductionist philosophy have an important part to play in environmental research, it is advantageous to consider alternative modelling methodologies which overtly acknowledge the poorly defined and uncertain nature of most environmental systems. The paper discusses this topic and presents an integrated statistical modelling procedure which involves three main methodological tools: uncertainty and sensitivity studies based on Monte Carlo simulation techniques; dominant mode analysis using a new method of combined linearization and model-order reduction; and data-based mechanistic modelling. This novel approach is illustrated by two practical examples: modelling the global carbon cycle in relation to possible climate change; and modelling a horticultural glasshouse for the purposes of automatic climate control system design.     

Simplicity out of complexity in environmental modelling: Occam's razor revisited

While large models based on a deterministic-reductionist philosophy have an important part to play in environmental research, it is advantageous to consider alternative modelling methodologies which overtly acknowledge the poorly defined and uncertain nature of most environmental systems. The paper discusses this topic and presents an integrated statistical modelling procedure which involves three main methodological tools: uncertainty and sensitivity studies based on Monte Carlo simulation techniques; dominant mode analysis using a new method of combined linearization and model-order reduction; and data-based mechanistic modelling. This novel approach is illustrated by two practical examples: modelling the global carbon cycle in relation to possible climate change; and modelling a horticultural glasshouse for the purposes of automatic climate control system design.     

Reliability properties of record values from non-identically distributed random variables

In reliability theory, order statistics and record values are used for statistical modelling. The r-th order statistic in a sample of size n represents the life—length of a (n?r+l)-out-of-n system, and record values are used in shock models. In recent years, reliability properties of order statistics and record values have been investigated. The two models are included in Pfeifer's concept of record values from non-identically distributed random variables. Here, some results on the transmission of distributional properties, such as increasing failure rate, are shown for such records, which contain the results for order statistics and ordinary record values as particular cases.     

Modelling of connected processes

The problem of comparing, contrasting and combining information from different sets of data is an enduring one in many practical applications of statistics. A specific problem of combining information from different sources arose in integrating information from three different sets of data generated by three different sampling campaigns at the input stage as well as at the output stage of a grey-water treatment process. For each stage, a common process trend function needs to be estimated to describe the input and output material process behaviours. Once the common input and output process models are established, it is required to estimate the efficiency of the grey-water treatment method. A synthesized tool for modelling different sets of process data is created by assembling and organizing a number of existing techniques: (i) a mixed model of fixed and random effects, extended to allow for a nonlinear fixed effect, (ii) variogram modelling, a geostatistical technique, (iii) a weighted least squares regression embedded in an iterative maximum-likelihood technique to handle linear/nonlinear fixed and random effects and (iv) a formulation of a transfer-function model for the input and output processes together with a corresponding nonlinear maximum-likelihood method for estimation of a transfer function. The synthesized tool is demonstrated, in a new case study, to contrast and combine information from connected process models and to determine the change in one quality characteristic, namely pH, of the input and output materials of a grey-water filtering process.     

A MULTIVARIATE NORMAL MODEL FOR PEDIGREE AND LONGITUDINAL DATA AND THE SOFTWARE 'FISHER'

In 1918 R.A. Fisher published an interpretation of covariation between relatives in terms of Mendelian inheritance, which has allowed inference on genetic and environmental components of variation from plant, animal and human pedigree data. Fisher had introduced maximum likelihood six years earlier. His 1918 paper abo contained the basics of linear regression and decomposition of variance. These concepts have now been united to allow flexible modelling of the mean and covariance structure of non-independent data on continuous traits, using maximum likelihood under a multivariate normal assumption. FISHER is a software package, designed for pedigree analysis and easily adapted for repeated measures and longitudinal data analysis. A range of applications illustrate FISHER as a useful statistical tool. Issues related to assumptions, tests-of-fit, and robustness of inference are discussed.     

A Bayesian hierarchical model for photometric red shifts

Summary.  The Sloan digital sky survey is an extremely large astronomical survey that is conducted with the intention of mapping more than a quarter of the sky. Among the data that it is generating are spectroscopic and photometric measurements, both containing information about the red shift of galaxies. The former are precise and easy to interpret but expensive to gather; the latter are far cheaper but correspondingly more difficult to interpret. Recently, Csabai and co-workers have described various calibration techniques aiming to predict red shift from photometric measurements. We investigate what a structured Bayesian approach to the problem can add. In particular, we are interested in providing uncertainty bounds that are associated with the underlying red shifts and the classifications of the galaxies. We find that quite a generic statistical modelling approach, using for the most part standard model ingredients, can compete with much more specific custom-made and highly tuned techniques that are already available in the astronomical literature.     

Multilevel modelling of the geographical distributions of diseases

Multilevel modelling is used on problems arising from the analysis of spatially distributed health data. We use three applications to demonstrate the use of multilevel modelling in this area. The first concerns small area all-cause mortality rates from Glasgow where spatial autocorrelation between residuals is examined. The second analysis is of prostate cancer cases in Scottish counties where we use a range of models to examine whether the incidence is higher in more rural areas. The third develops a multiple-cause model in which deaths from cancer and cardiovascular disease in Glasgow are examined simultaneously in a spatial model. We discuss some of the issues surrounding the use of complex spatial models and the potential for future developments.     

Estimated estimating equations: semiparametric inference for clustered and longitudinal data

Summary.  We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.     

Data analytic methods for latent partially ordered classification models

Summary. A general framework is presented for data analysis of latent finite partially ordered classification models. When the latent models are complex, data analytic validation of model fits and of the analysis of the statistical properties of the experiments is essential for obtaining reliable and accurate results. Empirical results are analysed from an application to cognitive modelling in educational testing. It is demonstrated that sequential analytic methods can dramatically reduce the amount of testing that is needed to make accurate classifications.     

A Composite Likelihood Approach to Multivariate Survival Data

This paper is about the statistical analysis of multivariate survival data. We discuss the additive and multiplicative frailty models which have been the most popular models for multivariate survival data. As an alternative to the additive and multiplicative frailty models, we propose basing inference on a composite likelihood function that only requires modelling of the marginal distribution of pairs of failure times. Each marginal distribution of a pair of failure times is here assumed to follow a shared frailty model. The method is illustrated with a real-life example.     

Simulating from irregular data: Kernel Carlo Simulation

In this paper, we address the problem of simulating from a data-generating process for which the observed data do not follow a regular probability distribution. One existing method for doing this is bootstrapping, but it is incapable of interpolating between observed data. For univariate or bivariate data, in which a mixture structure can easily be identified, we could instead simulate from a Gaussian mixture model. In general, though, we would have the problem of identifying and estimating the mixture model. Instead of these, we introduce a non-parametric method for simulating datasets like this: Kernel Carlo Simulation. Our algorithm begins by using kernel density estimation to build a target probability distribution. Then, an envelope function that is guaranteed to be higher than the target distribution is created. We then use simple accept–reject sampling. Our approach is more flexible than others, can simulate intelligently across gaps in the data, and requires no subjective modelling decisions. With several univariate and multivariate examples, we show that our method returns simulated datasets that, compared with the observed data, retain the covariance structures and have distributional characteristics that are remarkably similar.