Multiplicative Measurement Errors Research Articles

A Multiplicative Model for Gauge R & R Studies

We introduce a multiplicative measurement error model and analyze a gauge R & R study with the new model using data from a sample exchange program. Some aspects of designing a gauge R & R study are considered. Also, we analyze data from a factorial experiment where the measurement error arises from the new model using a simultaneous analysis of experimental and gauge R & R study data. WinBUGS code for these analyses is provided. Copyright © 2014 John Wiley & Sons, Ltd.

Cramer-Rao Lower Bound for a Nonlinear Filtering Problem with Multiplicative Measurement Errors and Forcing Noise

The recurrence algorithms for the Cramer-Rao lower bound for a discrete-time nonlinear filtering problem in the conditions when a forcing noise, measurement errors and initial covariance matrix depend on the state vector to be estimated are derived. It is assumed that the state vector being estimated includes a subvector of time-invariant unknown parameters. Some examples are given to illustrate the applicability of the algorithms obtained.

A Bayesian Measurement Error Model for Misaligned Radiographic Data

An understanding of the inherent variability in micro-computed tomography (micro-CT) data is essential to tasks such as statistical process control and the validation of radiographic simulation tools. These data present unique challenges to variability analysis due to the relatively low resolution of radiographs, and also due to minor variations from run to run which can result in misalignment or magnification changes between repeated measurements of a sample. Such positioning changes artificially inflate the variability of the data in ways that mask true physical phenomena. We present a novel Bayesian nonparametric regression model that incorporates both additive and multiplicative measurement error in addition to heteroscedasticity to address this problem. We use this model to assess the effects of sample thickness and sample position on measurement variability for an aluminum specimen. Supplementary materials for this article are available online.

Long-distance photogrammetric trait estimation in free-ranging animals: A new approach

The acquisition of accurate information on the size of traits in animals is fundamental for the study of animal ecology and evolution and their management. We demonstrate how morphological traits of free-ranging animals can reliably be estimated on very large observation distances of several hundred meters by the use of ordinary digital photographic equipment and simple photogrammetric software. In our study, we estimated the length of horn annuli in free-ranging male Alpine ibex (Capra ibex) by taking already measured horn annuli of conspecifics on the same photographs as scaling units. Comparisons with hand-measured horn annuli lengths and repeatability analyses revealed a high accuracy of the photogrammetric estimates. If length estimations of specific horn annuli are based on multiple photographs measurement errors of <5.5mm can be expected. In the current study the application of the described photogrammetric procedure increased the sample size of animals with known horn annuli length by an additional 104%. The presented photogrammetric procedure is of broad applicability and represents an easy, robust and cost-efficient method for the measuring of individuals in populations where animals are hard to capture or to approach.

Linear Least Squares Approach for Accurate Received Signal Strength Based Source Localization

A conventional approach for passive source localization is to utilize signal strength measurements of the emitted source received at an array of spatially separated sensors. The received signal strength (RSS) information can be converted to distance estimates for constructing a set of circular equations, from which the target position is determined. Nevertheless, a major challenge in this approach lies in the shadow fading effect which corresponds to multiplicative measurement errors. By utilizing the mean and variance of the squared distance estimates, we devise two linear least squares (LLS) estimators for RSS-based positioning in this paper. The first one is a best linear unbiased estimator while the second is its improved version by exploiting the known relation between the parameter estimates. The variances of the position estimates are derived and confirmed by computer simulations. In particular, it is proved that the performance of the improved LLS estimator achieves Cramér-Rao lower bound at sufficiently small noise conditions.

A Bayesian Adjustment for Multiplicative Measurement Errors for a Calibration Problem with Application to a Stem Cell Study

We develop a Bayesian approach to a calibration problem with one interested covariate subject to multiplicative measurement errors. Our work is motivated by a stem cell study with the objective of establishing the recommended minimum doses for stem cell engraftment after a blood transplant. When determining a safe stem cell dose based on the prefreeze samples, the postcryopreservation recovery rate enters in the model as a multiplicative measurement error term, as shown in the model. We examine the impact of ignoring measurement errors in terms of asymptotic bias in the regression coefficient. According to the general structure of data available in practice, we propose a two-stage Bayesian method to perform model estimation via R2WinBUGS (Sturtz, Ligges, and Gelman, 2005, Journal of Statistical Software 12, 1-16). We illustrate this method by the aforementioned motivating example. The results of this study allow routine peripheral blood stem cell processing laboratories to establish recommended minimum stem cell doses for transplant and develop a systematic approach for further deciding whether the postthaw analysis is warranted.

Panel regression with multiplicative measurement errors

The paper explores the effect of multiplicative measurement errors on the estimation of a linear panel data model. Multiplicative errors are often used to minimize disclosure risk of micro data. We use unbiased estimating equations to construct consistent and asymptotically normal estimators.

Genotype‐based association mapping of complex diseases: gene‐environment interactions with multiple genetic markers and measurement error in environmental exposures

With the advent of dense single nucleotide polymorphism genotyping, population-based association studies have become the major tools for identifying human disease genes and for fine gene mapping of complex traits. We develop a genotype-based approach for association analysis of case-control studies of gene-environment interactions in the case when environmental factors are measured with error and genotype data are available on multiple genetic markers. To directly use the observed genotype data, we propose two genotype-based models: genotype effect and additive effect models. Our approach offers several advantages. First, the proposed risk functions can directly incorporate the observed genotype data while modeling the linkage disequilibrium information in the regression coefficients, thus eliminating the need to infer haplotype phase. Compared with the haplotype-based approach, an estimating procedure based on the proposed methods can be much simpler and significantly faster. In addition, there is no potential risk due to haplotype phase estimation. Further, by fitting the proposed models, it is possible to analyze the risk alleles/variants of complex diseases, including their dominant or additive effects. To model measurement error, we adopt the pseudo-likelihood method by Lobach et al. [2008]. Performance of the proposed method is examined using simulation experiments. An application of our method is illustrated using a population-based case-control study of association between calcium intake with the risk of colorectal adenoma development.

Population dynamics of an Arctiid caterpillar–tachinid parasitoid system using state‐space models

1. Population dynamics of insect host-parasitoid systems are important in many natural and managed ecosystems and have inspired much ecological theory. However, ecologists have a limited knowledge about the relative strengths of species interactions, abiotic effects and density dependence in natural host-parasitoid dynamics. Statistical time-series analyses would be more informative by incorporating multiple factors, measurement error and noisy dynamics. 2. We use a novel maximum likelihood and model-selection analysis of a state-space model for host-parasitoid dynamics to examine 21 years of annual census data for woolly bear caterpillars (Platyprepia virginalis) and their locally host-specific tachinid parasitoids (Thelaira americana). 3. Caterpillar densities varied by three orders of magnitude and were driven by density dependence and precipitation from the previous March but not detectably by parasitoids, despite variable and sometimes high (>50%) parasitism. 4. Fly fluctuations, as estimated from per cent parasitism, were affected by density dependence and precipitation from the previous July. There was marginal evidence that host abundance drives fly fluctuations as a generic linear effect but no evidence for classical Nicholson-Bailey coupling. 5. The state-space model analysis includes new methods for likelihood calculation and allows a balanced consideration of effect magnitude and statistical significance in a nonlinear model with multiple alternative explanatory variables.

Linear models with measurement errors arising from mixture distributions

The paper considers the linear model with multiplicative measurement errors. In particular, errors arising from mixture distributions will be analyzed. Such a model has to be used if the micro data have been protected by multiplicative noise. If all (continuous) variables are anonymized jointly by this approach, measurement errors will be correlated which is of special concern if the dependent variable is measured with error, too. The paper presents results for the biased naive least-squares estimator both in case of cross-section data and panel data. Moreover, derivation of consistent estimators is shortly discussed.

Bayesian methods for estimating multi-segment discharge rating curves

This study explores Bayesian methods for handling compound stage–discharge relationships, a problem which arises in many natural rivers. It is assumed: (1) the stage–discharge relationship in each rating curve segment is a power-law with a location parameter, or zero-plane displacement; (2) the segment transitions are abrupt and continuous; and (3) multiplicative measurement errors are of equal variance. The rating curve fitting procedure is then formulated as a piecewise regression problem where the number of segments and the associated changepoints are assumed unknown. Procedures are developed for describing both global and site-specific prior distributions for all rating curve parameters, including the changepoints. Estimation and uncertainty analysis is evaluated using Markov chain Monte Carlo simulation (MCMC) techniques. The first model explored accounts for parameter and model uncertainties in the interpolated area, i.e. within the range of available stage–discharge measurements. A second model is constructed in an attempt to include the uncertainty in extrapolation, which is necessary when the rating curve is used to estimate discharges beyond the highest or lowest measurement. This is done by assuming that the rate of changepoints both inside and outside the measured area follows a Poisson process. The theory is applied to actual data from Norwegian gauging stations. The MCMC solutions give results that appear sensible and useful for inferential purposes, though the latter model needs further efforts in order to obtain a more efficient simulation scheme.

Models for Measured Income

Measurement error can impact estimator precision, obscure estimated relationships between variables, and distort the estimated intertemporal behavior of important economic characteristics. A commonly known model for measurement error assumes that measured income is the product of true income and a multiplicative measurement error, which is distributed independently of the level of true income. Based on this model, we derive a collection of flexible parametric forms for the distribution of measured income. We feel that this work could serve as an important reference for measurement error modeling.

Multiplicative Measurement Error and the Simulation Extrapolation Method

Whereas the literature on additive measurement error has known a considerable treatment, less work has been done for multiplicative noise. In this paper we concentrate on multiplicative measurement error in the covariates, which contrary to additive error not only modifies proportionally the original value, but also conserves the structural zeros. This paper compares three variants to specify the multiplicative measurement error model in the simulation step of the Simulation-Extrapolation (SIMEX) method originally proposed by Cook and Stefanski (1994): i) as an additive one without using a logarithmic transformation, ii) as the well-known logarithmic transformation of the multiplicative error model, and iii) as an approach using the multiplicative measurement error model as such. The aim of the paper is to analyze how well these three approaches reduce the bias caused by the multiplicative measurement error. We apply three variants to the case of data masking by multiplicative measurement error, in order to obtain parameter estimates of the true data generating process. We produce Monte Carlo evidence on how the reduction of data quality can be minimized.

Make Assurance Double Sure: Combination of Two Disclosure Limitation Methods and Estimation of General Regression Models

In order to guarantee confidentiality and privacy of firm-level data, statistical offices apply various disclosure limitation techniques. However, each anonymization technique has its protection limits, such that the probability of disclosing the individual information for some observations is not minimized. To overcome this problem, we propose to combine two separate disclosure limitation techniques blanking and multiplication of independent noise in order to protect the original dataset. The proposed approach yields a decrease in the probability of reidentifying/disclosing the individual information, and can be applied to linear as well as nonlinear regression models. We show how to combine the blanking method with the multiplicative measurement error method, and how to estimate the model by the combination of the multiplicative Simulation-Extrapolation (M-SIMEX) approach applied by Nolte (2007) and the Inverse Probability Weighting (IPW) approach going back to Horwitz and Thompson (1952). Based on Monte-Carlo simulations, we show that multiplicative measurement error combined with blanking as a masking procedure leads not necessarily to a severe reduction of the estimation quality, provided that its effects on the data generating process are known.

Laboratory longitudinal diffusion tests: 2. Parameter estimation by inverse analysis

This study focuses on the verification of test interpretations for different state analyses of diffusion experiments. Part 1 of this study identified that steady, quasi-steady and equilibrium state analyses for the through- and in-diffusion tests with solution reservoirs are generally feasible where the tracer is not highly sorptive. In Part 2 we investigate parameter identifiability in transient-state analysis of reservoir concentration variation using a numerical approach. For increased generality, the analytical models, objective functions and Jacobian matrix necessary for inverse analysis of transient-state data are reformulated using unified dimensionless parameters. In these dimensionless forms, the number of unknown parameters is reduced and a single dimensionless parameter represents the sorption property. The dimensionless objective functions are evaluated for individual test methods and parameter identifiability is discussed in relation to the sorption property. The effects of multiple minima and measurement error on parameter identifiability are also investigated. The main findings are that inverse problems for inlet and outlet reservoir concentration analyses are generally unstable and well-posed, respectively. Where the tracer is sorptive, the inverse problem for the inlet reservoir concentration analysis may have multiple minima. When insufficient measurement data is collected, multiple solutions may result and this should be taken into consideration when inversely analyzing data including that of inlet reservoir concentration. Verification of test interpretation by cross-checking different state analyses is feasible where the tracer is not highly sorptive. In an actual experiment, test interpretation validity is demonstrated through consistency between theory and practice for different state analyses.

Effect of Berkson measurement error on parameter estimates in Cox regression models

We study the effect of additive and multiplicative Berkson measurement error in Cox proportional hazard model. By plotting the true and the observed survivor function and the true and the observed hazard function dependent on the exposure one can get ideas about the effect of this type of error on the estimation of the slope parameter corresponding to the variable measured with error. As an example, we analyze the measurement error in the situation of the German Uranium Miners Cohort Study both with graphical methods and with a simulation study. We do not see a substantial bias in the presence of small measurement error and in the rare disease case. Even the effect of a Berkson measurement error with high variance, which is not unrealistic in our example, is a negligible attenuation of the observed effect. However, this effect is more pronounced for multiplicative measurement error.

The Multiplicative Simulation-Extrapolation Approach

We develop a new general approach for handling multiplicative measurement error in continuous covariates in linear and nonlinear regression models. We apply the Simulation-Extrapolation (SIMEX) approach, which is a simulation based method of estimating and reducing the bias due to additive measurement error, to the case of multiplicative measurement error. We do not apply a logarithmic transformation, so that the multiplicative measurement error model becomes an additive one, but we show how to modify the SIMEX approach, in order to use the multiplicative measurement error model as such. Multiplying the measurement error by additional measurement error allows us to infer in which way the estimation bias is affected by the increase of variance of the measurement error. In the extrapolation step, the estimated parameters are modelled as a function of the magnitude of the variance of the measurement error and extrapolated to the case of no measurement error. We apply our method to the case of data masking, in order to obtain parameter estimates of the true data generating process, if the data are multiplied by an additional measurement error. We produce Monte-Carlo evidence on how the reduction of data quality can be minimized by masking.

Estimation of Parameters in Logistic Regression Models with Multiplicative Measurement Error

Estimation of Parameters in Logistic Regression Models with Multiplicative Measurement Error

The Effect of Random Error in Exposure Measurement upon the Shape of the Exposure Response

Although statistical analyses of epidemiological data usually treat the exposure variable as being known without error, estimated exposures in epidemiological studies often involve considerable uncertainty. This paper investigates the theoretical effect of random errors in exposure measurement upon the observed shape of the exposure response. The model utilized assumes that true exposures are log-normally distributed, and multiplicative measurement errors are also log-normally distributed and independent of the true exposures. Under these conditions it is shown that whenever the true exposure response is proportional to exposure to a power r, the observed exposure response is proportional to exposure to a power K, where K < r. This implies that the observed exposure response exaggerates risk, and by arbitrarily large amounts, at sufficiently small exposures. It also follows that a truly linear exposure response will appear to be supra-linear-i.e., a linear function of exposure raised to the K-th power, where K is less than 1.0. These conclusions hold generally under the stated log-normal assumptions whenever there is any amount of measurement error, including, in particular, when the measurement error is unbiased either in the natural or log scales. Equations are provided that express the observed exposure response in terms of the parameters of the underlying log-normal distribution. A limited investigation suggests that these conclusions do not depend upon the log-normal assumptions, but hold more widely. Because of this problem, in addition to other problems in exposure measurement, shapes of exposure responses derived empirically from epidemiological data should be treated very cautiously. In particular, one should be cautious in concluding that the true exposure response is supra-linear on the basis of an observed supra-linear form.

Feedback control of surface roughness using stochastic PDEs

The surface roughness of thin films deposited from gas phase precursors is an important variable to control, because it strongly affects the quality of such films. The modern integrated circuit technology depends strongly on the uniformity and microstructure of deposited thin films (Granneman, 1993). Due to the increasingly stringent requirements on the quality of such films, on-line estimation and control of thin film deposition becomes important. In a thin film growth process, the film is directly formed from microscopic random processes (for example, particle adsorption, desorption, migration and surface reaction). Therefore, the stochastic nature of thin film growth processes must be fully considered in the modeling and control of the surface roughness of thin films. The desire to understand and control the thin film micro-structure has motivated extensive research on fundamental mathematical models describing the deposition processes, which include (1) kinetic Monte-Carlo methods (for example, Gillespie, 1976; Fichthorn and Weinberg, 1991; Lam and Vlachos, 2001), and (2) stochastic partial-differential equations (PDEs) (for example, Edwards and Wilkinson, 1982; Villain, 1991; Vvedensky et al., 1993). The kinetic Monte-Carlo simulation method can be used to predict average properties of the thin film (which are of interest from a control point of view, for example, surface roughness), by explicitly accounting for the microprocesses that directly shape thin film microstructure. Recently, a methodology for feedback control of thin film growth using kinetic Monte-Carlo models has been developed in (Lou and Christofides, 2003a,b). The methodology leads to the design of (a) real-time roughness estimators by using multiple small lattice kinetic Monte-Carlo simulators, adaptive filters and measurement error compensators, and (b) feedback controllers based on the real-time roughness estimators. The method was successfully applied to control surface roughness in a GaAs deposition process using an experimentally determined kinetic Monte-Carlo process model (Lou and Christofides, 2004b). Other approaches have also been developed to: (a) identify linear models from outputs of kinetic Monte-Carlo simulators and perform controller design by using linear control theory (Siettos et al., 2003), and (b) construct reduced-order approximations of the master equation (Gallivan and Murray, 2003). However, the fact that kinetic Monte-Carlo models are not available in closed-form makes very difficult to perform model-based controller design directly on the basis of kinetic Monte-Carlo models. To achieve better closed-loop performance, it is desirable to design feedback controllers on the basis of deposition process models. This motivates research on feedback control of deposition processes based on stochastic PDE models of thin film growth. Stochastic PDE models have been developed to describe the evolution of the height profile for surfaces in a variety of physical and chemical processes. Examples include deposition processes including adsorption and surface relaxation (Edwards and Wilkinson, 1982; Vvedensky, 2003), crystal growth from atomic-beams with and without desorption (Villain, Correspondence concerning this article should be addressed to P. D. Christofides; e-mail: pdc@seas.ucla.edu.