Compound Regression and Constrained Regression: Nonparametric Regression Frameworks for EIV Models

Abstract

Errors-in-variable (EIV) regression is often used to gauge linear relationship between two variables both suffering from measurement and other errors, such as, the comparison of two measurement platforms (e.g., RNA sequencing vs. microarray). Scientists are often at a loss as to which EIV regression model to use for there are infinite many choices. We provide sound guidelines toward viable solutions to this dilemma by introducing two general nonparametric EIV regression frameworks: the compound regression and the constrained regression. It is shown that these approaches are equivalent to each other and, to the general parametric structural modeling approach. The advantages of these methods lie in their intuitive geometric representations, their distribution free nature, and their ability to offer candidate solutions with various optimal properties when the ratio of the error variances is unknown. Each includes the classic nonparametric regression methods of ordinary least squares, geometric mean regression (GMR), and orthogonal regression as special cases. Under these general frameworks, one can readily uncover some surprising optimal properties of the GMR, and truly comprehend the benefit of data normalization. Supplementary materials for this article are available online.     

Some remarks on measurement models in the structural equation model: an application for socially responsible food consumption

Considering the structural equation model (SEM), usually the main researches are based on the structural model rather than on the measurement one. So, this context implies some problems: construct misspecification, identification and validation. Starting from the most recent articles in terms of these issues, we achieve – and formalize through two tables – a general framework that could help researchers select and assess both formative and reflective measurement models with special attention on statistical implications. To show this general framework, we present a survey on customer behaviours for socially responsible food consumption. The survey was carried out by delivering a questionnaire administered to a representative sample of 332 families. In order to detect the main aspects impacting consumers’ preferences, a factor analysis has been performed. Then the general framework has been used to select and assess the measurement models in SEM. The estimation of the SEM has been worked out by partial least squares. The significance of the indicators has been tested using bootstrap. As far as we know, it is the first time that a model for the analysis of the consumers’ behaviour for social responsibility is formalized through a SEM.     

Using structural equation modelling to detect measurement bias and response shift in longitudinal data

We propose a three step procedure to investigate measurement bias and response shift, a special case of measurement bias in longitudinal data. Structural equation modelling is used in each of the three steps, which can be described as (1) establishing a measurement model using confirmatory factor analysis, (2) detecting measurement bias by testing the equivalence of model parameters across measurement occasions, (3) detecting measurement bias with respect to additional exogenous variables by testing their direct effects on the indicator variables. The resulting model can be used to investigate true change in the attributes of interest, by testing changes in common factor means. Solutions for the issue of constraint interaction and for chance capitalisation in model specification searches are discussed as part of the procedure. The procedure is illustrated by applying it to longitudinal health-related quality-of-life data of HIV/AIDS patients, collected at four semi-annual measurement occasions.     

Consistency of completely outlier-adjusted simultaneous redescending M-estimators of location and scale

This paper gives conditions for the consistency of simultaneous redescending M-estimators for location and scale. The consistency postulates the uniqueness of the parameters μ and σ, which are defined analogously to the estimations by using the population distribution function instead of the empirical one. The uniqueness of these parameters is no matter of course, because redescending ψ- and χ-functions, which define the parameters, cannot be chosen in a way that the parameters can be considered as the result of a common minimizing problem where the sum of ρ-functions of standardized residuals is to be minimized. The parameters arise from two minimizing problems where the result of one problem is a parameter of the other one. This can give different solutions. Proceeding from a symmetrical unimodal distribution and the usual symmetry assumptions for ψ and χ leads, in most but not in all cases, to the uniqueness of the parameters. Under this and some other assumptions, we can also prove the consistency of the according M-estimators, although these estimators are usually not unique even when the parameters are. The present article also serves as a basis for a forthcoming paper, which is concerned with a completely outlier-adjusted confidence interval for μ. So we introduce a ñ where data points far away from the bulk of the data are not counted at all.     

Modelling long-term pan-European population change from 1870 to 2000 by using geographical information systems

Summary.  The paper presents work that creates a geographical information system database of European census data from 1870 to 2000. The database is integrated over space and time. Spatially it consists of regional level data for most of Europe; temporally it covers every decade from 1870 to 2000. Crucially the data have been interpolated onto the administrative units that were available in 2000, thus allowing contemporary population patterns to be understood in the light of the changes that have occurred since the late 19th century. The effect of interpolation error on the resulting estimates is explored. This database will provide a framework for much future analysis on long-term Europewide demographic processes over space and time.     

Improvements to the Escalation with Overdose Control design and a comparison with the restricted Continual Reassessment Method

The Escalation with Overdose Control (EWOC) design for cancer dose finding clinical trials is a variation of the Continual Reassessment Method (CRM) that was proposed to overcome the limitation of the original CRM of exposing patients to high toxic doses. The properties of EWOC have been studied to some extent, but some aspects of the design are not well studied, and its performance is not fully understood. Comparisons of the EWOC design to the most commonly used modified CRM designs have not yet been performed, and the advantages of EWOC over the modified CRM designs are unclear. In this paper, we assess the properties of the EWOC design and of the restricted CRM and some variations of these designs. We show that EWOC has several weaknesses that CRM does not have that make it impractical to use in its original formulation. We propose modified EWOC designs that address some of the weaknesses and that have some desirable statistical properties compared with the original EWOC design, the restricted CRM design, and the 3 + 3 design. However, their statistical properties are sensitive to correct specification of the prior distribution of their parameters and hence nevertheless will need to be used with some caution. The restricted CRM design is shown to have more stable performance across a wider family of dose‐toxicity curves than EWOC and therefore may be a preferable general choice in cancer clinical research.     

An Exact Corrected Log-Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract.  This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [ Biometrika 77 (1990) 127] methodology of corrected log-likelihood will be applied to the so-called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [ Scand. J. Statist. 25 (1998) 573] and Kong & Gu [ Statistica Sinica 9 (1999) 953] are re-established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.     

Estimators based on trimmed Kendall’s tau in multivariate copula models

A common method of estimating the parameters of dependency in multivariate copula models is by maximum likelihood principle, termed as Inference From Marginals (IFM); see Joe (1997)  [13]. To avoid possible misspecification of the marginal distributions, some authors suggest rank-based procedures for estimating the parameters of dependency in a multivariate copula model. A standard approach for this problem is through maximization of the pseudolikelihood, as discussed in Genest et al. (1995)  [9] and Shih and Louis (1995)  [23]. Alternative estimators based on the inversion of two multivariate extensions of Kendall’s tau, due to Kendall and Babington Smith (1940)  [14] and Joe (1990)  [12], were used in Genest et al. (2011)  [10]. In the literature, dependency of data was considered in the whole data space. However, it may be better to divide the data set into two distinct sets, lower and higher than a threshold, and then evaluate the dependency parameters in these sets. In this way, we may have different dependency parameters in these sets which may shed additional light. For example, in drought analysis, precipitation and minimum temperature may be modeled using copulas in which case we can infer that dependency between precipitation and minimum temperature are severe when they are less than a certain threshold. In this paper, after introducing trimmed Kendall’s tau when such a threshold is imposed, we consider modeling dependency using it as a measure. Asymptotic distribution of trimmed Kendall’s tau is also investigated, and a test for the null hypothesis of equality between Kendall’s tau and trimmed Kendall’s tau is constructed. We can use this hypothesis testing procedure for testing the hypothesis that data are dependent before a threshold value and are independent after the threshold. An explicit form of the asymptotic distribution of trimmed Kendall’s tau and of the mentioned test statistic are also derived for some special families of copulas. Finally, the results of a simulation study and an illustrative example are provided.     

Bayesian approach to the inverse problem in a light scattering application

In this article, static light scattering (SLS) measurements are processed to estimate the particle size distribution of particle systems incorporating prior information obtained from an alternative experimental technique: scanning electron microscopy (SEM). For this purpose we propose two Bayesian schemes (one parametric and another non-parametric) to solve the stated light scattering problem and take advantage of the obtained results to summarize some features of the Bayesian approach within the context of inverse problems. The features presented in this article include the improvement of the results when some useful prior information from an alternative experiment is considered instead of a non-informative prior as it occurs in a deterministic maximum likelihood estimation. This improvement will be shown in terms of accuracy and precision in the corresponding results and also in terms of minimizing the effect of multiple minima by including significant information in the optimization. Both Bayesian schemes are implemented using Markov Chain Monte Carlo methods. They have been developed on the basis of the Metropolis–Hastings (MH) algorithm using Matlab^® and are tested with the analysis of simulated and experimental examples of concentrated and semi-concentrated particles. In the simulated examples, SLS measurements were generated using a rigorous model, while the inversion stage was solved using an approximate model in both schemes and also using the rigorous model in the parametric scheme. Priors from SEM micrographs were also simulated and experimented, where the simulated ones were obtained using a Monte Carlo routine. In addition to the presentation of these features of the Bayesian approach, some other topics will be discussed, such as regularization and some implementation issues of the proposed schemes, among which we remark the selection of the parameters used in the MH algorithm.     

Modeling population and subject-specific growth in a latent trait measured by multiple instruments over time using a hierarchical Bayesian framework

Psychometric growth curve modeling techniques are used to describe a person’s latent ability and how that ability changes over time based on a specific measurement instrument. However, the same instrument cannot always be used over a period of time to measure that latent ability. This is often the case when measuring traits longitudinally in children. Reasons may be that over time some measurement tools that were difficult for young children become too easy as they age resulting in floor effects or ceiling effects or both. We propose a Bayesian hierarchical model for such a scenario. Within the Bayesian model we combine information from multiple instruments used at different age ranges and having different scoring schemes to examine growth in latent ability over time. The model includes between-subject variance and within-subject variance and does not require linking item specific difficulty between the measurement tools. The model’s utility is demonstrated on a study of language ability in children from ages one to ten who are hard of hearing where measurement tool specific growth and subject-specific growth are shown in addition to a group level latent growth curve comparing the hard of hearing children to children with normal hearing.KEYWORDS: Bayesian hierarchical models, psychometric modeling, language ability, growth curve modeling, longitudinal analysis     

Making predictions from complex longitudinal data, with application to planning monitoring intervals in a national screening programme

When biological or physiological variables change over time, we are often interested in making predictions either of future measurements or of the time taken to reach some threshold value. On the basis of longitudinal data for multiple individuals, we develop Bayesian hierarchical models for making these predictions together with their associated uncertainty. Particular aspects addressed, which include some novel components, are handling curvature in individuals' trends over time, making predictions for both underlying and measured levels, making predictions from a single baseline measurement, making predictions from a series of measurements, allowing flexibility in the error and random-effects distributions, and including covariates. In the context of data on the expansion of abdominal aortic aneurysms over time, where reaching a certain threshold leads to referral for surgery, we discuss the practical application of these models to the planning of monitoring intervals in a national screening programme. Prediction of the time to reach a threshold was too imprecise to be practically useful, and we focus instead on limiting the probability of exceeding the threshold after given time intervals. Although more complex models can be shown to fit the data better, we find that relatively simple models seem to be adequate for planning monitoring intervals.     

On estimation and local influence analysis for measurement errors models under heavy-tailed distributions

Scale mixtures of normal distributions form a class of symmetric thick-tailed distributions that includes the normal one as a special case. In this paper we consider local influence analysis for measurement error models (MEM) when the random error and the unobserved value of the covariates jointly follow scale mixtures of normal distributions, providing an appealing robust alternative to the usual Gaussian process in measurement error models. In order to avoid difficulties in estimating the parameter of the mixing variable, we fixed it previously, as recommended by Lange et al. (J Am Stat Assoc 84:881–896, 1989) and Berkane et al. (Comput Stat Data Anal 18:255–267, 1994). The local influence method is used to assess the robustness aspects of the parameter estimates under some usual perturbation schemes. However, as the observed log-likelihood associated with this model involves some integrals, Cook’s well–known approach may be hard to apply to obtain measures of local influence. Instead, we develop local influence measures following the approach of Zhu and Lee (J R Stat Soc Ser B 63:121–126, 2001), which is based on the EM algorithm. Results obtained from a real data set are reported, illustrating the usefulness of the proposed methodology, its relative simplicity, adaptability and practical usage.     

Approaches for the direct estimation of u,and demographic contributions to u,using capture-recapture data

We first consider the estimation of the finite rate of population increase or population growth rate, u i , using capture-recapture data from open populations. We review estimation and modelling of u i under three main approaches to modelling openpopulation data: the classic approach of Jolly (1965) and Seber (1965), the superpopulation approach of Crosbie & Manly (1985) and Schwarz & Arnason (1996), and the temporal symmetry approach of Pradel (1996). Next, we consider the contributions of different demographic components to u i using a probabilistic approach based on the composition of the population at time i + 1 (Nichols et al., 2000b). The parameters of interest are identical to the seniority parameters, n i , of Pradel (1996). We review estimation of n i under the classic, superpopulation, and temporal symmetry approaches. We then compare these direct estimation approaches for u i and n i with analogues computed using projection matrix asymptotics. We also discuss various extensions of the estimation approaches to multistate applications and to joint likelihoods involving multiple data types.     

A hierarchical modelling approach to combining environmental data at different scales

Summary. Long-transported air pollution in Europe is monitored by a combination of a highly complex mathematical model and a limited number of measurement stations. The model predicts deposition on a 150 km × 150 km square grid covering the whole of the continent. These predictions can be regarded as spatial averages, with some spatially correlated model error. The measurement stations give a limited number of point estimates, regarded as error free. We combine these two sources of data by assuming that both are observations of an underlying true process. This true deposition is made up of a smooth deterministic trend, due to gradual changes in emissions over space and time, and two stochastic components. One is non- stationary and correlated over long distances; the other describes variation within a grid square. Our approach is through hierarchical modelling with predictions and measurements being independent conditioned on the underlying non-stationary true deposition. We assume Gaussian processes and calculate maximum likelihood estimates through numerical optimization. We find that the variation within a grid square is by far the largest component of the variation in the true deposition. We assume that the mathematical model produces estimates of the mean over an area that is approximately equal to a grid square, and we find that it has an error that is similar to the long-range stochastic component of the true deposition, in addition to a large bias.     

Empirical Performance and Asset Pricing in Hidden Markov Models

Abstract

To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address leptokurtic feature, volatility smile, and volatility clustering effects of the asset return distributions. However, analytical tractability remains a problem for most alternative models. In this article, we study a class of hidden Markov models including Markov switching models and stochastic volatility models, that can incorporate leptokurtic feature, volatility clustering effects, as well as provide analytical solutions to option pricing. We show that these models can generate long memory phenomena when the transition probabilities depend on the time scale. We also provide an explicit analytic formula for the arbitrage-free price of the European options under these models. The issues of statistical estimation and errors in option pricing are also discussed in the Markov switching models.     

Statistics development: statistical methods meeting the user’s needs

Statistical methods have the potential of being effectively used by industrial practitioners if they satisfied two criteria: functionality and usability. Statistical methods are usually the product of statistical research activities of universities and other research organizations. Some already satisfy these criteria; however, many do not. The effect is that potentially relevant methods are not used in practice as often as they could be. In this paper we will present an approach regarding ‘statistics development,’ in which the end-user is given a central position, so that the results from statistical research aim to meet the needs and requirements of the practitioner. Examples of known and new methods will be presented, and we will discuss issues such as education in statistics, the link with statistical consultancy and publication of methods through various channels.     

One-sided Continuous Tolerance Limits and their Accompanying Sample Size Problem

Tolerance limits are those limits that contain a certain proportion of the distribution of a characteristic with a given probability. 'They are used to make sure that the production will not be outside of specifications' (Amin & Lee, 1999). Usually, tolerance limits are constructed at the beginning of the monitoring of the process. Since they are calculated just one time, these tolerance limits cannot reflect changes of tolerance level over the lifetime of the process. This research proposes an algorithm to construct tolerance limits continuously over time for any given distribution. This algorithm makes use of the exponentially weighted moving average (EWMA) technique. It can be observed that the sample size required by this method is reduced over time.     

Spectral estimation for locally stationary time series with missing observations

Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled time series, or require modifications to cope with irregular or ‘gappy’ data. Additionally, many techniques also assume that the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally stationary wavelet process in the sense introduced by Nason et al. (J. R. Stat. Soc. B 62(2):271–292, 2000) and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the centre of our construction are second generation wavelets built by means of the lifting scheme (Sweldens, Wavelet Applications in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68–79, 1995), designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate our method with real data and simulated examples.     

The correlation structure of longitudinal measurements of vision in patients with macular degeneration

Background: In age‐related macular degeneration (ARMD) trials, the FDA‐approved endpoint is the loss (or gain) of at least three lines of vision as compared to baseline. The use of such a response endpoint entails a potentially severe loss of information. A more efficient strategy could be obtained by using longitudinal measures of the change in visual acuity. In this paper we investigate, by using data from two randomized clinical trials, the mean and variance–covariance structures of the longitudinal measurements of the change in visual acuity. Methods: Individual patient data were collected in 234 patients in a randomized trial comparing interferon‐ α with placebo and in 1181 patients in a randomized trial comparing three active doses of pegaptanib with sham. A linear model for longitudinal data was used to analyze the repeated measurements of the change in visual acuity. Results: For both trials, the data were adequately summarized by a model that assumed a quadratic trend for the mean change in visual acuity over time, a power variance function, and an antedependence correlation structure. The power variance function was remarkably similar for the two datasets and involved the square root of the measurement time. Conclusions: The similarity of the estimated variance functions and correlation structures for both datasets indicates that these aspects may be a genuine feature of the measurements of changes in visual acuity in patients with ARMD. The feature can be used in the planning and analysis of trials that use visual acuity as the clinical endpoint of interest. Copyright © 2010 John Wiley & Sons, Ltd.     

Trajectory inference and parameter estimation in stochastic models with temporally aggregated data

Stochastic models are of fundamental importance in many scientific and engineering applications. For example, stochastic models provide valuable insights into the causes and consequences of intra-cellular fluctuations and inter-cellular heterogeneity in molecular biology. The chemical master equation can be used to model intra-cellular stochasticity in living cells, but analytical solutions are rare and numerical simulations are computationally expensive. Inference of system trajectories and estimation of model parameters from observed data are important tasks and are even more challenging. Here, we consider the case where the observed data are aggregated over time. Aggregation of data over time is required in studies of single cell gene expression using a luciferase reporter, where the emitted light can be very faint and is therefore collected for several minutes for each observation. We show how an existing approach to inference based on the linear noise approximation (LNA) can be generalised to the case of temporally aggregated data. We provide a Kalman filter (KF) algorithm which can be combined with the LNA to carry out inference of system variable trajectories and estimation of model parameters. We apply and evaluate our method on both synthetic and real data scenarios and show that it is able to accurately infer the posterior distribution of model parameters in these examples. We demonstrate how applying standard KF inference to aggregated data without accounting for aggregation will tend to underestimate the process noise and can lead to biased parameter estimates.