Zero-inflated Poisson models with measurement error in the response

Zero-inflated count data arise frequently from genomics studies. Analysis of such data is often based on a mixture model which facilitates excess zeros in combination with a Poisson distribution, and various inference methods have been proposed under such a model. Those analysis procedures, however, are challenged by the presence of measurement error in count responses. In this article, we propose a new measurement error model to describe error-contaminated count data. We show that ignoring the measurement error effects in the analysis may generally lead to invalid inference results, and meanwhile, we identify situations where ignoring measurement error can still yield consistent estimators. Furthermore, we propose a Bayesian method to address the effects of measurement error under the zero-inflated Poisson model and discuss the identifiability issues. We develop a data-augmentation algorithm that is easy to implement. Simulation studies are conducted to evaluate the performance of the proposed method. We apply our method to analyze the data arising from a prostate adenocarcinoma genomic study.     

Mixtures of Berkson and classical covariate measurement error in the linear mixed model: Bias analysis and application to a study on ultrafine particles

The ultrafine particle measurements in the Augsburger Umweltstudie, a panel study conducted in Augsburg, Germany, exhibit measurement error from various sources. Measurements of mobile devices show classical possibly individual–specific measurement error; Berkson–type error, which may also vary individually, occurs, if measurements of fixed monitoring stations are used. The combination of fixed site and individual exposure measurements results in a mixture of the two error types. We extended existing bias analysis approaches to linear mixed models with a complex error structure including individual–specific error components, autocorrelated errors, and a mixture of classical and Berkson error. Theoretical considerations and simulation results show, that autocorrelation may severely change the attenuation of the effect estimations. Furthermore, unbalanced designs and the inclusion of confounding variables influence the degree of attenuation. Bias correction with the method of moments using data with mixture measurement error partially yielded better results compared to the usage of incomplete data with classical error. Confidence intervals (CIs) based on the delta method achieved better coverage probabilities than those based on Bootstrap samples. Moreover, we present the application of these new methods to heart rate measurements within the Augsburger Umweltstudie: the corrected effect estimates were slightly higher than their naive equivalents. The substantial measurement error of ultrafine particle measurements has little impact on the results. The developed methodology is generally applicable to longitudinal data with measurement error.     

Correction for Covariate Measurement Error in Nonparametric Longitudinal Regression

Summary We introduce a correction for covariate measurement error in nonparametric regression applied to longitudinal binary data arising from a study on human sleep. The data have been surveyed to investigate the association of some hormonal levels and the probability of being asleep. The hormonal effect is modeled flexibly while we account for the error‐prone measurement of its concentration in the blood and the longitudinal character of the data. We present a fully Bayesian treatment utilizing Markov chain Monte Carlo inference techniques, and also introduce block updating to improve sampling and computational performance in the binary case. Our model is partly inspired by the relevance vector machine with radial basis functions, where usually very few basis functions are automatically selected for fitting the data. In the proposed approach, we implement such data‐driven complexity regulation by adopting the idea of Bayesian model averaging. Besides the general theory and the detailed sampling scheme, we also provide a simulation study for the Gaussian and the binary cases by comparing our method to the naive analysis ignoring measurement error. The results demonstrate a clear gain when using the proposed correction method, particularly for the Gaussian case with medium and large measurement error variances, even if the covariate model is misspecified.     

Socially informed random walks: incorporating group dynamics into models of population spread and growth

Simple correlated random walk (CRW) models are rarely sufficient to describe movement of animals over more than the shortest time scales. However, CRW approaches can be used to model more complex animal movement trajectories by assuming individuals move in one of several different behavioural or movement states, each characterized by a different CRW. The spatial and social context an individual experiences may influence the proportion of time spent in different movement states, with subsequent effects on its spatial distribution, survival and fecundity. While methods to study habitat influences on animal movement have been previously developed, social influences have been largely neglected. Here, we fit a 'socially informed' movement model to data from a population of over 100 elk (Cervus canadensis) reintroduced into a new environment, radio-collared and subsequently tracked over a 4-year period. The analysis shows how elk move further when they are solitary than when they are grouped and incur a higher rate of mortality the further they move away from the release area. We use the model to show how the spatial distribution and growth rate of the population depend on the balance of fission and fusion processes governing the group structure of the population. The results are briefly discussed with respect to the design of species reintroduction programmes.     

Classifying movement behaviour in relation to environmental conditions using hidden Markov models

1.  Linking the movement and behaviour of animals to their environment is a central problem in ecology. Through the use of electronic tagging and tracking (ETT), collection of in situ data from free-roaming animals is now commonplace, yet statistical approaches enabling direct relation of movement observations to environmental conditions are still in development.
2.  In this study, we examine the hidden Markov model (HMM) for behavioural analysis of tracking data. HMMs allow for prediction of latent behavioural states while directly accounting for the serial dependence prevalent in ETT data. Updating the probability of behavioural switches with tag or remote-sensing data provides a statistical method that links environmental data to behaviour in a direct and integrated manner.
3.  It is important to assess the reliability of state categorization over the range of time-series lengths typically collected from field instruments and when movement behaviours are similar between movement states. Simulation with varying lengths of times series data and contrast between average movements within each state was used to test the HMMs ability to estimate movement parameters.
4.  To demonstrate the methods in a realistic setting, the HMMs were used to categorize resident and migratory phases and the relationship between movement behaviour and ocean temperature using electronic tagging data from southern bluefin tuna ( Thunnus maccoyii ). Diagnostic tools to evaluate the suitability of different models and inferential methods for investigating differences in behaviour between individuals are also demonstrated.     

Linear measurement error models with restricted sampling

The relationship between nutrient consumption and chronic disease risk is the focus of a large number of epidemiological studies where food frequency questionnaires (FFQ) and food records are commonly used to assess dietary intake. However, these self-assessment tools are known to involve substantial random error for most nutrients, and probably important systematic error as well. Study subject selection in dietary intervention studies is sometimes conducted in two stages. At the first stage, FFQ-measured dietary intakes are observed and at the second stage another instrument, such as a 4-day food record, is administered only to participants who have fulfilled a prespecified criterion that is based on the baseline FFQ-measured dietary intake (e.g., only those reporting percent energy intake from fat above a prespecified quantity). Performing analysis without adjusting for this truncated sample design and for the measurement error in the nutrient consumption assessments will usually provide biased estimates for the population parameters. In this work we provide a general statistical analysis technique for such data with the classical additive measurement error that corrects for the two sources of bias. The proposed technique is based on multiple imputation for longitudinal data. Results of a simulation study along with a sensitivity analysis are presented, showing the performance of the proposed method under a simple linear regression model.     

Bayesian semiparametric nonlinear mixed-effects joint models for data with skewness, missing responses, and measurement errors in covariates

Summary It is a common practice to analyze complex longitudinal data using semiparametric nonlinear mixed-effects (SNLME) models with a normal distribution. Normality assumption of model errors may unrealistically obscure important features of subject variations. To partially explain between- and within-subject variations, covariates are usually introduced in such models, but some covariates may often be measured with substantial errors. Moreover, the responses may be missing and the missingness may be nonignorable. Inferential procedures can be complicated dramatically when data with skewness, missing values, and measurement error are observed. In the literature, there has been considerable interest in accommodating either skewness, incompleteness or covariate measurement error in such models, but there has been relatively little study concerning all three features simultaneously. In this article, our objective is to address the simultaneous impact of skewness, missingness, and covariate measurement error by jointly modeling the response and covariate processes based on a flexible Bayesian SNLME model. The method is illustrated using a real AIDS data set to compare potential models with various scenarios and different distribution specifications.     

Integrative modelling of animal movement: incorporating in situ habitat and behavioural information for a migratory marine predator

A fundamental goal in animal ecology is to quantify how environmental (and other) factors influence individual movement, as this is key to understanding responsiveness of populations to future change. However, quantitative interpretation of individual-based telemetry data is hampered by the complexity of, and error within, these multi-dimensional data. Here, we present an integrative hierarchical Bayesian state-space modelling approach where, for the first time, the mechanistic process model for the movement state of animals directly incorporates both environmental and other behavioural information, and observation and process model parameters are estimated within a single model. When applied to a migratory marine predator, the southern elephant seal (Mirounga leonina), we find the switch from directed to resident movement state was associated with colder water temperatures, relatively short dive bottom time and rapid descent rates. The approach presented here can have widespread utility for quantifying movement–behaviour (diving or other)–environment relationships across species and systems.     

Reliability of reliability coefficients in the estimation of asymmetry

Although promising to provide insight into the interaction between genotype and environment, investigations into fluctuating asymmetry suffer from a lack of standardization in the reporting of measurement error. In the present paper we show, using both anthropometric and odonto-metric data, that the use of the reliability coefficient calculated for a bilateral measurement provides no indication of the reliability of the corresponding asymmetry estimate, because reliability of asymmetry depends on the relationship between measurement error and the difference between sides. Thus, we suggest that future investigations either provide reliability coefficients for asymmetry estimates specifically, or use methods that account for measurement error. © 1995 Wiley-Liss, Inc.     

Filtering and interpreting location errors in satellite telemetry of marine animals

Argos telemetry offers a powerful means of tracking wild animals in their habitat, yet the delivered locations are subject to complex errors and random coverage. Bayesian filters and statistical models allow for objective trajectory estimates and inference on movement rates. As an alternative to Monte-Carlo methods, we investigate here how classic time series technique, such as the Kalman Filter, can be made robust to uncover patterns in the data. Our approach relies on a composite measurement model to account for outliers, and makes use of all the Location Classes to smooth observations and regularize the track to a regular time grid. Two application examples are presented. Using data from freely-swimming leatherback turtles, we confirm that locations of class A (LCA) are more accurate on average than class 0, and we recommend their use in tracking studies. We further show how measurement errors (and their geometry) interact with the assumed movement model, further modulating the final location error and the discriminating ability of the filter. The choice of the movement model appears important, since a model with no velocity constraint may fit observational errors at the expense of trajectory smoothness, while a speed-based model is better behaved but less forgiving for data fitting and outlier identification. Varying sea surface temperatures also appear to degrade the quality of locations and increase the occurrence of outliers, possibly in relation to thermal stratification and depth behavior. These results have important implications when inferring changes in behavior from long-term movements.     

A comparison of normalization techniques for microRNA microarray data

Normalization of expression levels applied to microarray data can help in reducing measurement error. Different methods, including cyclic loess, quantile normalization and median or mean normalization, have been utilized to normalize microarray data. Although there is considerable literature regarding normalization techniques for mRNA microarray data, there are no publications comparing normalization techniques for microRNA (miRNA) microarray data, which are subject to similar sources of measurement error. In this paper, we compare the performance of cyclic loess, quantile normalization, median normalization and no normalization for a single-color microRNA microarray dataset. We show that the quantile normalization method works best in reducing differences in miRNA expression values for replicate tissue samples. By showing that the total mean squared error are lowest across almost all 36 investigated tissue samples, we are assured that the bias correction provided by quantile normalization is not outweighed by additional error variance that can arise from a more complex normalization method. Furthermore, we show that quantile normalization does not achieve these results by compression of scale.     

New method for analyzing the molecular weights of proteins by sodium dodecyl sulfate-polyacrylamide gel electrophoresis

Previously the method for determining protein molecular weights from SDS-PAGE depended on the accidental, only partial linearity of protein movement with the logarithm of its molecular weight. A new, mathematically rigorous method with supporting data is now described demonstrating that such movement is dependent upon the reciprocal of protein size. Experimental data, therefore, follow most closely a hyperbolic curve when plotted directly; it becomes linear and passes through the origin when movement is plotted vs the reciprocal of protein molecular weight. In the earlier method determination of the error of a measurement of molecular weight is very complex and never determined. In the method presented here such error is easily estimated and it is identical in both the hyperbolic and linear forms of data presentation. This method may eventually also allow other less-significant forces controlling movement such as protein charge to be analyzed and understood.     

A hidden Markov movement model for rapidly identifying behavioral states from animal tracks

Electronic telemetry is frequently used to document animal movement through time. Methods that can identify underlying behaviors driving specific movement patterns can help us understand how and why animals use available space, thereby aiding conservation and management efforts. For aquatic animal tracking data with significant measurement error, a Bayesian state‐space model called the first‐Difference Correlated Random Walk with Switching (DCRWS) has often been used for this purpose. However, for aquatic animals, highly accurate tracking data are now becoming more common. We developed a new hidden Markov model (HMM) for identifying behavioral states from animal tracks with negligible error, called the hidden Markov movement model (HMMM). We implemented as the basis for the HMMM the process equation of the DCRWS, but we used the method of maximum likelihood and the R package TMB for rapid model fitting. The HMMM was compared to a modified version of the DCRWS for highly accurate tracks, the DCRWS, and to a common HMM for animal tracks fitted with the R package moveHMM. We show that the HMMM is both accurate and suitable for multiple species by fitting it to real tracks from a grey seal, lake trout, and blue shark, as well as to simulated data. The HMMM is a fast and reliable tool for making meaningful inference from animal movement data that is ideally suited for ecologists who want to use the popular DCRWS implementation and have highly accurate tracking data. It additionally provides a groundwork for development of more complex modeling of animal movement with TMB. To facilitate its uptake, we make it available through the R package swim.     

Within-species variation and measurement error in phylogenetic comparative methods

Most phylogenetically based statistical methods for the analysis of quantitative or continuously varying phenotypic traits assume that variation within species is absent or at least negligible, which is unrealistic for many traits. Within-species variation has several components. Differences among populations of the same species may represent either phylogenetic divergence or direct effects of environmental factors that differ among populations (phenotypic plasticity). Within-population variation also contributes to within-species variation and includes sampling variation, instrument-related error, low repeatability caused by fluctuations in behavioral or physiological state, variation related to age, sex, season, or time of day, and individual variation within such categories. Here we develop techniques for analyzing phylogenetically correlated data to include within-species variation, or "measurement error" as it is often termed in the statistical literature. We derive methods for (i) univariate analyses, including measurement of "phylogenetic signal," (ii) correlation and principal components analysis for multiple traits, (iii) multiple regression, and (iv) inference of "functional relations," such as reduced major axis (RMA) regression. The methods are capable of incorporating measurement error that differs for each data point (mean value for a species or population), but they can be modified for special cases in which less is known about measurement error (e.g., when one is willing to assume something about the ratio of measurement error in two traits). We show that failure to incorporate measurement error can lead to both biased and imprecise (unduly uncertain) parameter estimates. Even previous methods that are thought to account for measurement error, such as conventional RMA regression, can be improved by explicitly incorporating measurement error and phylogenetic correlation. We illustrate these methods with examples and simulations and provide Matlab programs.     

Leveraging multidimensional heterogeneity in resource selection to define movement tactics of animals

Increasing interest in the complexity, variation and drivers of movement‐related behaviours promise new insight into fundamental components of ecology. Resolving the multidimensionality of spatially explicit behaviour remains a challenge for investigating tactics and their relation to niche construction, but high‐resolution movement data are providing unprecedented understanding of the diversity of spatially explicit behaviours. We introduce a framework for investigating individual variation in movement‐defined resource selection that integrates the behavioural and ecological niche concepts. We apply it to long‐term tracking data of 115 African elephants (Loxodonta africana), illustrating how a behavioural hypervolume can be defined based on differences between individuals and their ecological settings, and applied to explore population heterogeneity. While normative movement behaviour is frequently used to characterise population behaviour, we demonstrate the value of leveraging heterogeneity in the behaviour to gain greater insight into population structure and the mechanisms driving space‐use tactics.     

Regression Calibration in Semiparametric Accelerated Failure Time Models

Summary In large cohort studies, it often happens that some covariates are expensive to measure and hence only measured on a validation set. On the other hand, relatively cheap but error‐prone measurements of the covariates are available for all subjects. Regression calibration (RC) estimation method ( Prentice, 1982 , Biometrika 69 , 331–342) is a popular method for analyzing such data and has been applied to the Cox model by Wang et al. (1997, Biometrics 53 , 131–145) under normal measurement error and rare disease assumptions. In this article, we consider the RC estimation method for the semiparametric accelerated failure time model with covariates subject to measurement error. Asymptotic properties of the proposed method are investigated under a two‐phase sampling scheme for validation data that are selected via stratified random sampling, resulting in neither independent nor identically distributed observations. We show that the estimates converge to some well‐defined parameters. In particular, unbiased estimation is feasible under additive normal measurement error models for normal covariates and under Berkson error models. The proposed method performs well in finite‐sample simulation studies. We also apply the proposed method to a depression mortality study.     

Combining animal movements and behavioural data to detect behavioural states

Animal movement paths show variation in space caused by qualitative shifts in behaviours. I present a method that (1) uses both movement path data and ancillary sensor data to detect natural breakpoints in animal behaviour and (2) groups these segments into different behavioural states. The method can also combine analyses of different path segments or paths from different individuals. It does not assume any underlying movement mechanism. I give an example with simulated data. I also show the effects of random variation, # of states and # of segments on this method. I present a case study of a fisher movement path spanning 8 days, which shows four distinct behavioural states divided into 28 path segments when only turning angles and speed were considered. When accelerometer data were added, the analysis shows seven distinct behavioural states divided into 41 path segments.     

A New Method of Regression of Time-series Based on a Mathematical Connection Between Path and Compartmental Analyses

In the regression analysis of time-series the error terms may be serially correlated with the results. In this case the possibility arises that the expectations and adjustment processes are themselves mis-specifications of the correct behavioural relationships. In this paper an analogy is pointed out between the compartmental-analysis and the path analysis by the author. It will be argued that the time-series of the standardized partial regression coefficients (path coefficients or beta weights) computed from the cross-sectional data, have to be analysed instead of applying ordinary least squares directly to the timeseries. An application of the new method is briefly discussed.