A Numerical Method for Estimating the Variance of Age at Maximum Growth Rate in Growth Models

Studies on maturation and body composition mention age at peak height velocity (PHV) as an important measure that could predict adulthood outcome. The age at PHV is often derived from growth models such as the triple logistic fitted to the stature (height) data. Theoretically, for a well-behaved growth function, age at PHV could be obtained by setting the second derivative of the growth function to zero and solving for age. Such a solution obviously depends on the parameters of the growth function. Therefore, the uncertainty in the estimation of age at PHV resulting from the uncertainty in the estimation of the growth model, need to be accounted for in the models in which it is used as a predictor. Explicit expressions for the age at PHV and, consequently the variance of the estimate of the age at PHV, do not exist for some of the commonly used nonlinear growth functions, such as the triple logistic function. Once an estimate of this variance is obtained, it could be incorporated in subsequent modeling either through measurement error models or by using the inverse variances as weights. A numerical method for estimating the variance is implemented. The accuracy of this method is demonstrated through comparisons in models where explicit solution for the variance exists. The method of estimating the variance is illustrated by applying to growth data from the Fels study and subsequently used as weights in modeling two adulthood outcomes from the same study.     

Quantification of symmetry for functional data with application to equine lameness classification

This paper presents a study on symmetry of repeated bi-phased data signals, in particular, on quantification of the deviation between the two parts of the signal. Three symmetry scores are defined using functional data techniques such as smoothing and registration. One score is related to the L ₂-distance between the two parts of the signal, whereas the other two are constructed to specifically measure differences in amplitude and phase. Moreover, symmetry scores based on functional principal component analysis (PCA) are examined. The scores are applied to acceleration signals from a study on equine gait. The scores turn out to be highly associated with lameness, and their applicability for lameness quantification and detection is investigated. Four classification approaches turn out to give similar results. The scores describing amplitude and phase variation turn out to outperform the PCA scores when it comes to the classification of lameness.     

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.     

用函数型数据分析网络团购市场的结构与市场发展

以美团网为例,使用函数型数据研究网络团购市场的结构与发展。首先将从网络采集到的团购离散数据,根据分析目的构造成函数化数据,然后对函数化的数据进行描述统计分析和函数主成分分析,并对不同种类的团购与不同地区的团购进行比较。最终得出区别于以往研究团购的结论:当期团购市场仍以美食类为主,购物类团购逐渐被大家认同,休闲类团购集中度下降;上海与北京都属于团购发展较快地区,而北京地区相对变动较大;热门地区团购销售额的集中度随着团购地区的扩张而被稀释。     

Application of ridge regression and factor analysis in design and production of alloy wheels

This study proposes using statistical approaches to help with both the design and manufacture of wheels. The quality of a wheel is represented by the mechanical properties of spokes. Variation in the mechanical properties of different wheels is attributed to two sources, i.e. between-model variation and within-model variation. The between-model variation is due to different shapes of different wheel models. To model the effect of shapes on the mechanical properties, we first specify eight shape variables potentially critical to the mechanical properties, and then we collect relevant data on 18-wheel models and perform ridge regression to find the effects of these variables on the mechanical properties. These results are linked to the solidification theory of the A356 alloy. The within-model variation is due to natural variability and process abnormality. We extract mechanical data of a particular wheel model from the database. Factor analysis is employed to analyze the data with a view to identifying the latent factors behind the mechanical properties. We then look into the microstructure of the alloy to corroborate the fact that these two latent factors are essentially the Si phase and the Mg₂Si phase, respectively. These results can be used to efficiently identify the root cause when the manufacturing process goes wrong.     

Investigating impacts of complex sampling on latent growth curve modelling

We investigate the impacts of complex sampling on point and standard error estimates in latent growth curve modelling of survey data. Methodological issues are illustrated with empirical evidence from the analysis of longitudinal data on life satisfaction trajectories using data from the British Household Panel Survey, a national representative survey in Great Britain. A multi-process second-order latent growth curve model with conditional linear growth is used to study variation in the two perceived life satisfaction latent factors considered. The benefits of accounting for the complex survey design are considered, including obtaining unbiased both point and standard error estimates, and therefore correctly specified confidence intervals and statistical tests. We conclude that, even for the rather elaborated longitudinal data models that were considered, estimation procedures are affected by variance-inflating impacts of complex sampling.     

Frailty effects in networks: comparison and identification of individual heterogeneity versus preferential attachment in evolving networks

Preferential attachment is a proportionate growth process in networks, where nodes receive new links in proportion to their current degree. Preferential attachment is a popular generative mechanism to explain the widespread observation of power-law-distributed networks. An alternative explanation for the phenomenon is a randomly grown network with large individual variation in growth rates among the nodes (frailty). We derive analytically the distribution of individual rates, which will reproduce the connectivity distribution that is obtained from a general preferential attachment process (Yule process), and the structural differences between the two types of graphs are examined by simulations. We present a statistical test to distinguish the two generative mechanisms from each other and we apply the test to both simulated data and two real data sets of scientific citation and sexual partner networks. The findings from the latter analyses argue for frailty effects as an important mechanism underlying the dynamics of complex networks.     

Making control charts more effective by time series analysis: three illustrative applications

Control charts contribute to the monitoring and improvement of process quality by helping to separate out special cause variation from common cause variation. By common cause variation we mean the usual variation in an in-control process. Special causes can be thought of as disturbances, possibly transitory, impacting a process that is in a state of statistical control. However, there is no clear place in this scheme of special causes and common causes for systematic non-iid variation, such as trend, seasonal, autoregression variation, and intervention effects from efforts to improve the proess. When systematic non-iid variation is present, time series modeling and fitting can fill in this picture. In the time series framework, observations influenced by special causes can be treated as outliers from the currently-entertained time-series model and can be detected by outlier detection methods. We discuss three data sets that illustrate how this can be done in order to make control charts more effective. We show also how a standard control-chart supplement called "pattern analysis" can be useful in time-series work.     

Trend,Growth Rate,and Change Point Analysis—A Data Driven Approach

A data-driven technique is proposed to estimate the trend and relative growth rate of time series data. The method is based on the local linear regression smoother and the only assumption about the form of the trend and growth rate function is that they are smooth functions of time. We also extended the method for handling sudden shifts or changes in the trend or growth rate functions by adding dummy variables for the jumps. Simulation studies are carried out to see the performance of the proposed procedure. The method is applied to study the trend and growth rate of wheat production in India from 1951–2005.     

A Spatio-Temporal Model for Functional Magnetic Resonance Imaging Data – with a View to Resting State Networks

Abstract.  Functional magnetic resonance imaging (fMRI) is a technique for studying the active human brain. During the fMRI experiment, a sequence of MR images is obtained, where the brain is represented as a set of voxels. The data obtained are a realization of a complex spatio-temporal process with many sources of variation, both biological and technical. We present a spatio-temporal point process model approach for fMRI data where the temporal and spatial activation are modelled simultaneously. It is possible to analyse other characteristics of the data than just the locations of active brain regions, such as the interaction between the active regions. We discuss both classical statistical inference and Bayesian inference in the model. We analyse simulated data without repeated stimuli both for location of the activated regions and for interactions between the activated regions. An example of analysis of fMRI data, using this approach, is presented.     

Functional Analysis of Variance,Discriminant Analysis,and Clustering in a Manifold of Elastic Curves

We develop functional data analysis techniques using the differential geometry of a manifold of smooth elastic functions on an interval in which the functions are represented by a log-speed function and an angle function. The manifold's geometry provides a method for computing a sample mean function and principal components on tangent spaces. Using tangent principal component analysis, we estimate probability models for functional data and apply them to functional analysis of variance, discriminant analysis, and clustering. We demonstrate these tasks using a collection of growth curves from children from ages 1–18.     

Conservative prior distributions for variance parameters in hierarchical models

Bayesian hierarchical models typically involve specifying prior distributions for one or more variance components. This is rather removed from the observed data, so specification based on expert knowledge can be difficult. While there are suggestions for “default” priors in the literature, often a conditionally conjugate inverse‐gamma specification is used, despite documented drawbacks of this choice. The authors suggest “conservative” prior distributions for variance components, which deliberately give more weight to smaller values. These are appropriate for investigators who are skeptical about the presence of variability in the second‐stage parameters (random effects) and want to particularly guard against inferring more structure than is really present. The suggested priors readily adapt to various hierarchical modelling settings, such as fitting smooth curves, modelling spatial variation and combining data from multiple sites.     

A Non-Gaussian Spatial Process Model for Opacity of Flocculated Paper

ABSTRACT.  Product quality in the paper-making industry can be assessed by opacity of a linear trace through continuous production sheets, summarized in spectral form. We adopt a class of non-Gaussian stochastic models for continuous spatial variation to describe data of this type. The model has flexible covariance structure, physically interpretable parameters and allows several scales of variation for the underlying process. We derive the spectral properties of the model, and develop methods of parameter estimation based on maximum likelihood in the frequency domain. The methods are illustrated using sample data from a UK mill.     

Fledging size and survival in snow geese: Timing is everything (or is it?)

In many birds, body size at fledging is assumed to predict accurately the probability of subsequent survival, and size at fledging is often used as a proxy variable in analyses attempting to assess the pattern of natural selection on body size. However, in some species, size at fledging can vary significantly as a function of variation in the environmental component of growth. Such developmental plasticity has been demonstrated in several species of Arctic-breeding geese. In many cases, slower growth and reduced size at fledging has been suggested as the most parsimonious explanation for reduced post-fledging survival in goslings reared under poor environmental conditions. However, simply quantifying a relationship between mean size at fledging and mean survival rate (Francis et al ., 1992) may obscure the pattern of selection on the interaction of the genetic and environmental components of growth. The hypothesis that selection operates on the environmental component of body size at fledging, rather than the genetic component of size per se, was tested using data from the long-term study of Lesser Snow Geese ( Anser c. caerulescens ) breeding at La Pérouse Bay, Manitoba, Canada. Using data from female goslings measured at fledging, post-fledging survival rates were estimated using combined live encounter and dead recovery data (Burnham, 1993). To control for the covariation between growth and environmental factors, survival rates were constrained to be functions of individual covariation of size at fledging, and various measures of the timing of hatch; in all Arctic-breeding geese studied to date, late hatching goslings grow significantly more slowly than do early hatching goslings. The slower growth of late-hatching goslings has been demonstrated to reflect systematic changes in the environmental component of growth, and thus controlling for hatch date controls for a significant proportion of variation in the environmental component of growth. The relationship between size at fledging, hatch date and survival was found to be significantly non-linear; among early hatching goslings, there was little indication of significant differences in survival rate among large and small goslings. However, with increasingly later hatch dates, there was progressively greater mortality selection against smaller, slower growing goslings in most years. This would appear to suggest that body size matters, but not absolutely; small size leads to reduced survival for late-hatching goslings only at La Pe´rouse Bay. Since at least some of the variation in size among goslings for a given hatch date reflects genetic differences, this suggests selection may favour larger size at fledging, albeit only among late-hatching goslings.     

Overcoming biases and misconceptions in ecological studies

The aggregate data study design provides an alternative group level analysis to ecological studies in the estimation of individual level health risks. An aggregate model is derived by aggregating a plausible individual level relative rate model within groups, such that population-based disease rates are modelled as functions of individual level covariate data. We apply an aggregate data method to a series of fictitious examples from a review paper by Greenland and Robins which illustrated the problems that can arise when using the results of ecological studies to make inference about individual health risks. We use simulated data based on their examples to demonstrate that the aggregate data approach can address many of the sources of bias that are inherent in typical ecological analyses, even though the limited between-region covariate variation in these examples reduces the efficiency of the aggregate study. The aggregate method has the potential to estimate exposure effects of interest in the presence of non-linearity, confounding at individual and group levels, effect modification, classical measurement error in the exposure and non-differential misclassification in the confounder.     

THE REDUCED-RANK GROWTH CURVE MODEL FOR DISCRIMINANT ANALYSIS OF LONGITUDINAL DATA

This paper presents a method of discriminant analysis especially suited to longitudinal data. The approach is in the spirit of canonical variate analysis (CVA) and is similarly intended to reduce the dimensionality of multivariate data while retaining information about group differences. A drawback of CVA is that it does not take advantage of special structures that may be anticipated in certain types of data. For longitudinal data, it is often appropriate to specify a growth curve structure (as given, for example, in the model of Potthoff & Roy, 1964). The present paper focuses on this growth curve structure, utilizing it in a model-based approach to discriminant analysis. For this purpose the paper presents an extension of the reduced-rank regression model, referred to as the reduced-rank growth curve (RRGC) model. It estimates discriminant functions via maximum likelihood and gives a procedure for determining dimensionality. This methodology is exploratory only, and is illustrated by a well-known dataset from Grizzle & Allen (1969).     

Statistical analysis of accelerated temperature cycling test based on Coffin-Manson model

Abstract

This paper investigates the statistical analysis of grouped accelerated temperature cycling test data when the product lifetime follows a Weibull distribution. A log-linear acceleration equation is derived from the Coffin-Manson model. The problem is transformed to a constant-stress accelerated life test with grouped data and multiple acceleration variables. The Jeffreys prior and reference priors are derived. Maximum likelihood estimation and Bayesian estimation with objective priors are obtained by applying the technique of data augmentation. A simulation study shows that both of these two methods perform well when sample size is large, and the Bayesian method gives better performance under small sample sizes.     

A MODIFIED GROWTH CURVE MODEL AND ITS APPLICATION TO CLINICAL STUDIES

This paper considers a technique for applying growth curve analysis to data which are correlated across different groups. The method is illustrated using a study comparing three methods of suctioning an endotracheal tube.     

Multilevel models for longitudinal data

Summary.  Repeated measures and repeated events data have a hierarchical structure which can be analysed by using multilevel models. A growth curve model is an example of a multilevel random-coefficients model, whereas a discrete time event history model for recurrent events can be fitted as a multilevel logistic regression model. The paper describes extensions to the basic growth curve model to handle auto-correlated residuals, multiple-indicator latent variables and correlated growth processes, and event history models for correlated event processes. The multilevel approach to the analysis of repeated measures data is contrasted with structural equation modelling. The methods are illustrated in analyses of children's growth, changes in social and political attitudes, and the interrelationship between partnership transitions and childbearing.