Semiparametric space–time survival modeling of chronic wasting disease in deer

In this paper, we propose a semiparametric survival model to investigate the pattern of spatial and temporal variation in disease prevalence of chronic wasting disease (CWD) in wild deer in Wisconsin over the years 2002 and 2006. The semiparametric survival model we suggested allows to build a more flexible model than the parametric model with fewer parametric assumptions by modeling the baseline hazard using a Gamma process prior. Based on the proposed model, we investigate the geographical distribution of CWD, and assess the effect of sex on disease prevalence. We use a Bayesian hierarchical framework where latent parameters capture temporal and spatial trends in disease incidence, incorporating sex and spatially correlated random effects. We also propose bivariate baseline hazard which change over age and time simultaneously to adopt different effects of age and time on the baseline hazard. Inference is carried out by using MCMC simulation techniques in a fully Bayesian framework. Our results suggest that disease has been spreaded mainly in the disease eradication zone and male deer show a significantly higher infection probability than female deer.     

Best linear unbiased estimators for the simple linear regression model using ranked set sampling

When sample observations are expensive or difficult to obtain, ranked set sampling is known to be an efficient method for estimating the population mean, and in particular to improve on the sample mean estimator. Using best linear unbiased estimators, this paper considers the simple linear regression model with replicated observations. Use of a form of ranked set sampling is shown to be markedly more efficient for normal data when compared with the traditional simple linear regression estimators.     

Flexible modeling for stock-recruitment relationships using Bayesian nonparametric mixtures

The stock and recruitment relationship is fundamental to the management of fishery natural resources. However, inferring stock-recruitment relationships is a challenging problem because of the limited available data, the collection of plausible models, and the biological characteristics that should be reflected in the model. Motivated by limitations of traditional parametric stock-recruitment models, we propose a Bayesian nonparametric approach based on a mixture model for the joint distribution of log-reproductive success and stock biomass. Flexible mixture modeling for this bivariate distribution yields rich inference for the stock-recruitment relationship through the implied conditional distribution of log-reproductive success given stock biomass. The method is illustrated with cod data from six regions of the North Atlantic, including comparison with simpler Bayesian parametric and semiparametric models.     

Optimal model averaging by data perturbation for spatial noisy data

Geostatistical models play an important role in spatial data analysis, in which model selection is inevitable. Model selection methods, such as AIC and BIC, are popular for selecting appropriate models. In recent years, some model averaging methods, such as smoothed AIC and smoothed BIC, are also applied to spatial data models. However, the corresponding averaging estimators are outperformed by optimal model averaging estimators (Hansen in Econometrica 75:1175–1189, 2007) for the ordinary linear models. Therefore, this paper focuses on the optimal model averaging method for geostatistical models. We propose a weight choice criterion for the model averaging estimator on the basis of the generalized degrees of freedom and data perturbation technique. We further theoretically prove the resultant estimator is asymptotically optimal in terms of the mean squared error, and numerically demonstrate its satisfactory performance. Finally, the proposed method is applied to a mercury data set.     

Modeling misidentification errors that result from use of genetic tags in capture–recapture studies

Misidentification of animals is potentially important when naturally existing features (natural tags) such as DNA fingerprints (genetic tags) are used to identify individual animals. For example, when misidentification leads to multiple identities being assigned to an animal, traditional estimators tend to overestimate population size. Accounting for misidentification in capture–recapture models requires detailed understanding of the mechanism. Using genetic tags as an example, we outline a framework for modeling the effect of misidentification in closed population studies when individual identification is based on natural tags that are consistent over time (non-evolving natural tags). We first assume a single sample is obtained per animal for each capture event, and then generalize to the case where multiple samples (such as hair or scat samples) are collected per animal per capture occasion. We introduce methods for estimating population size and, using a simulation study, we show that our new estimators perform well for cases with moderately high capture probabilities or high misidentification rates. In contrast, conventional estimators can seriously overestimate population size when errors due to misidentification are ignored.     

A statistical evaluation of non-ergodic variogram estimators

Geostatistics is a set of statistical techniques that is increasingly used to characterize spatial dependence in spatially referenced ecological data. A common feature of geostatistics is predicting values at unsampled locations from nearby samples using the kriging algorithm. Modeling spatial dependence in sampled data is necessary before kriging and is usually accomplished with the variogram and its traditional estimator. Other types of estimators, known as non-ergodic estimators, have been used in ecological applications. Non-ergodic estimators were originally suggested as a method of choice when sampled data are preferentially located and exhibit a skewed frequency distribution. Preferentially located samples can occur, for example, when areas with high values are sampled more intensely than other areas. In earlier studies the visual appearance of variograms from traditional and non-ergodic estimators were compared. Here we evaluate the estimators' relative performance in prediction. We also show algebraically that a non-ergodic version of the variogram is equivalent to the traditional variogram estimator. Simulations, designed to investigate the effects of data skewness and preferential sampling on variogram estimation and kriging, showed the traditional variogram estimator outperforms the non-ergodic estimators under these conditions. We also analyzed data on carabid beetle abundance, which exhibited large-scale spatial variability (trend) and a skewed frequency distribution. Detrending data followed by robust estimation of the residual variogram is demonstrated to be a successful alternative to the non-ergodic approach.     

Investigating the complex relationship between in situ Southern Ocean pCO_{2} and its ocean physics and biogeochemical drivers using a nonparametric regression approach

The objective in this paper is to investigate the use of a non-parametric approach to model the relationship between oceanic carbon dioxide \((pCO_2)\) and a range of ocean physics and biogeochemical in situ variables in the Southern Ocean, which influence its in situ variability. The need for this stems from the need to obtain reliable estimates of carbon dioxide concentrations in the Southern Ocean which plays an important role in the global carbon flux cycle. The main challenge involved in this objective is the spatial limitation and seasonal bias of the in situ data. Moreover, studies have also reported that the relationship between \(pCO_2\) and its drivers is complex. As such, in this paper, we use the non-parametric kernel regression approach since it is able to accurately capture the complex relationships between the response and predictor variables. In this analysis we use the in situ data obtained from the SANAE49 return leg journey between Antarctic to Cape Town. To the best of our knowledge, this is the first time this data set has been subjected to such analysis. The model variants were developed on a training data subset, and the ‘goodness’ of the models were assessed on an “unseen” test data subset. Results indicate that the nonparametric approach consistently captures the relationship more accurately in terms of mean square error, root mean square error and mean absolute error, over a standard parametric approach (multiple linear regression). These results provide a platform for using the developed nonparametric regression model based on in situ measurements to predict \(pCO_2\) for a larger spatial region in the Southern Ocean based on satellite biogeochemical measurements of predictor variables, given that satellites do not measure \(pCO_2\) .     

Empirical Bayes calculations of concordance between endpoints in environmental toxicity experiments

Hierarchical models are considered for estimating the probability of agreement between two outcomes or endpoints from an environmental toxicity experiment. Emphasis is placed on generalized regression models, under which the prior mean is related to a linear combination of explanatory variables via a monotone function. This function defines the scale over which the systematic effects are modelled as additive. Specific illustration is provided for the logistic link function. The hierarchical model employs a conjugate beta prior that leads to parametric empirical Bayes estimators of the individual agreement parameters. An example from environmental carcinogenesis illustrates the methods, with motivation derived from estimation of the concordance between two species carcinogenicity outcomes. Based on a large database of carcinogenicity studies, the inter-species concordance is seen to be reasonably informative, i.e. in the range 67–84%. Stratification into pertinent potency-related sub-groups via the logistic model is seen to improve concordance estimation: for environmental stimuli at the extremes of the potency spectrum, concordance can reach well above 90%.     

A diagnostic test for autocorrelation in increment-averaged data with application to soil sampling

Motivated by the problem of detecting spatial autocorrelation in increment- averaged data from soil core samples, we use the Cholesky decomposition of the inverse of an autocovariance matrix to derive a parametric linear regression model for autocovariances. In the absence of autocorrelation, the off-diagonal terms in the lower triangular matrix from the Cholesky decomposition should be identically zero, and so the regression coefficients should be identically zero. The standard F-test of this hypothesis and two bootstrapped versions of the test are evaluated as autocorrelation diagnostics via simulation. Size is assessed for a variety of heteroskedastic null hypotheses. Power is evaluated against autocorrelated alternatives, including increment-averaged Ornstein-Uhlenbeck and Matérn processes. The bootstrapped tests maintain approximately the correct size and have good power against moderately autocorrelated alternatives. The methods are applied to data from a study of carbon sequestration in agricultural soils.     

A copula-based inference to piecewise exponential models under dependent censoring,with application to time to metamorphosis of salamander larvae

In ecology and evolutionary biology, controlled animal experiments are often conducted to measure time to metamorphosis which is possibly censored by the competing risk of death and the follow-up end. This paper considers the problem of estimating the survival function of time-to-event when it is subject to dependent censoring. When the censorship is due to competing risks, the traditional assumption of independent censorship may not be satisfied, and hence, the usual application of the Kaplan–Meier estimator yields a biased estimation for the survival function of the event time. This paper follows an assumed copula approach (Zheng and Klein in Biometrika 82(1):127–138, 1995) to adjust for dependence between the event time of interest and the competing event time. While the literature on an assumed copula approach has mostly focused on semiparametric settings, we alternatively consider a parametric approach with piecewise exponential models for fitting the survival function. We develop maximum likelihood estimation under the piecewise exponential models with an assumed copula. A goodness-of-fit procedure is also developed, which touches upon the identifiability issue of the copula. We conduct simulations to examine the performance of the proposed method and compare it with an existing semiparametric method. The method is applied to real data analysis on time to metamorphosis for salamander larvae living in Hokkaido, Japan (Michimae et al. in Evol Ecol Res 16:617–629, 2014).     

Modeling carcass removal time for avian mortality assessment in wind farms using survival analysis

In monitoring studies at wind farms, the estimation of bird and bat mortality caused by collision must take into account carcass removal by scavengers or decomposition. In this paper we propose the use of survival analysis techniques to model the time of carcass removal. The proposed method is applied to data collected in ten Portuguese wind farms. We present and compare results obtained from semiparametric and parametric models assuming four main competing lifetime distributions (exponential, Weibull, log-logistic and log-normal). Both homogeneous parametric models and accelerated failure time models were used. The fitted models enabled the estimation of the carcass persistence rates and the calculation of a scavenging correction factor for avian mortality estimation. Additionally, we discuss the impact that the distributional assumption can have on parameter estimation. The proposed methodology integrates the survival probability estimation problem with the analysis of covariate effects. Estimation is based on the most suitable model while simultaneously accounting for censored observations, diminishing scavenging rate estimation bias. Additionally, the method establishes a standardized statistical procedure for the analysis of carcass removal time in subsequent studies.     

An application of Bayesian variable selection to spatial concurrent linear models

Spatial concurrent linear models, in which the model coefficients are spatial processes varying at a local level, are flexible and useful tools for analyzing spatial data. One approach places stationary Gaussian process priors on the spatial processes, but in applications the data may display strong nonstationary patterns. In this article, we propose a Bayesian variable selection approach based on wavelet tools to address this problem. The proposed approach does not involve any stationarity assumptions on the priors, and instead we impose a mixture prior directly on each wavelet coefficient. We introduce an option to control the priors such that high resolution coefficients are more likely to be zero. Computationally efficient MCMC procedures are provided to address posterior sampling, and uncertainty in the estimation is assessed through posterior means and standard deviations. Examples based on simulated data demonstrate the estimation accuracy and advantages of the proposed method. We also illustrate the performance of the proposed method for real data obtained through remote sensing.     

Local spatial modeling of white-tailed deer distribution

Complex spatial heterogeneity of ecological systems is difficult to capture and interpret using global models alone. For this reason, recent attention has been paid to local spatial modeling techniques. We used one local modeling approach, geographically weighted regression (GWR), to investigate the effects of local spatial heterogeneity on multivariate relationships of white-tailed deer distribution using land cover patch metrics and climate factors. The results of these analyses quantify differences in the contributions of model parameters to estimates of deer density over space. A GWR model with local kernel bandwidth was compared to a GWR model with global kernel bandwidth and an ordinary least-squares regression (OLS) model with the same parameters to evaluate their relative abilities in modeling deer distributions. The results indicated that the GWR models predicted deer density better than the traditional ordinary least-squares model and also provided useful information regarding local environmental processes affecting deer distribution. GWR model comparisons showed that the local kernel bandwidth GWR model was more realistic than the global kernel bandwidth GWR model, as the latter exaggerated local spatial variation. The parameter estimates and model statistics (e.g., model R²) of the GWR models were mapped using geographic information systems (GIS) to illustrate local spatial variation in the regression relationship and to identify causes of large-scale model misspecifications and low estimation efficiencies.     

Spatial smoothing techniques for the assessment of habitat suitability

Precise knowledge about factors influencing the habitat suitability of a certain species forms the basis for the implementation of effective programs to conserve biological diversity. Such knowledge is frequently gathered from studies relating abundance data to a set of influential variables in a regression setup. In particular, generalised linear models are used to analyse binary presence/absence data or counts of a certain species at locations within an observation area. However, one of the key assumptions of generalised linear models, the independence of observations is often violated in practice since the points at which the observations are collected are spatially aligned. In this paper, we describe a general framework for semiparametric spatial generalised linear models that allows for the routine analysis of non-normal spatially aligned regression data. The approach is utilised for the analysis of a data set of synthetic bird species in beech forests, revealing that ignorance of spatial dependence actually may lead to false conclusions in a number of situations.

Spatial residual analysis of six modeling techniques

In recent years alternative modeling techniques have been used to account for spatial autocorrelations among data observations. They include linear mixed model (LMM), generalized additive model (GAM), multi-layer perceptron (MLP) neural network, radial basis function (RBF) neural network, and geographically weighted regression (GWR). Previous studies show these models are robust to the violation of model assumptions and flexible to nonlinear relationships among variables. However, many of them are non-spatial in nature. In this study, we utilize a local spatial analysis method (i.e., local Moran coefficient) to investigate spatial distribution and heterogeneity in model residuals from those modeling techniques with ordinary least-squares (OLS) as the benchmark. The regression model used in this study has tree crown area as the response variable, and tree diameter and the coordinates of tree locations as the predictor variables. The results indicate that LMM, GAM, MLP and RBF may improve model fitting to the data and provide better predictions for the response variable, but they generate spatial patterns for model residuals similar to OLS. The OLS, LMM, GAM, MLP and RBF models yield more residual clusters of similar values, indicating that trees in some sub-areas are either all underestimated or all overestimated for the response variable. In contrast, GWR estimates model coefficients at each location in the study area, and produces more accurate predictions for the response variable. Furthermore, the residuals of the GWR model have more desirable spatial distributions than the ones derived from the OLS, LMM, GAM, MLP and RBF models.     

Removal process estimation of population size for a population with a known sex ratio

A removal model for estimating population size which uses the known population sex ratio is studied. A maximum likelihood estimate and an optimal martingale estimate of the population size are proposed. Their standard errors and large sample properties are obtained. Simulation studies are reported, and the performance of the proposed estimators are compared with the standard maximum likelihood estimator which ignores the sex ratio information. An example on a capture study of deer mice is given.     

The effect of species response form on species distribution model prediction and inference

Ecological theory and current evidence support the validity of various species response curves according to a variety of environmental gradients. Various methods have been developed for building species distribution models but it is not well known how these methods perform under various assumptions about the form of the underlying species response. It is also not well known how spatial correlation in species occurrence affects model performance. These effects were investigated by applying an environmental envelope method (BIOCLIM) and three regression-based methods: logistic regression (LR), generalized additive modelling (GAM), and classification and regression tree (CART) to simulated species occurrence data. Each simulated species was constructed as a sum of responses with varying weights. Three basic species response curves were assumed: Gaussian (bell-shaped), Beta (skew) and linear. The two non-linear responses conform to standard ecological niche theory. All three responses were applied in turn to three simulated environmental variables, each with varying degrees of spatial autocorrelation. GAM produced the most consistent model performance over all forms of simulated species response. BIOCLIM and CART were inclined to underrate the performance of variables with a linear response. BIOCLIM was less sensitive to data density. LR was susceptible to model misspecification. The use of a linear function in LR underestimated the performance of variables with non-linear species response and contributed to increased spatial autocorrelation in model residuals. Omission of important environmental variables with non-linear species response also contributed to increased spatial autocorrelation in model residuals. Adding a spatial autocovariate term to the LR model (autologistic model) reduced the spatial autocorrelation and improved model performance, but did not correct the misidentification of the dominant environmental determinant. This is to be expected since the autologistic approach was designed primarily for prediction and not for inference. Given that various forms of species response to environmental determinants arise commonly in nature: (1) higher order functions should always be tested when applying LR in modelling species distribution; (2) spatial autocorrelation in species distribution model residuals can indicate that environmental determinants with non-linear response are missing from the model; and (3) deficiencies in LR model performance due to model misspecification can be addressed by adding a spatial autocovariate to the model, but care should be taken when interpreting the coefficients of the model parameters.     

Neglected biological patterns in the residuals

One of the fundamental assumptions underlying linear regression models is that the errors have a constant variance (i.e., homoscedastic). When this assumption is violated, standard errors from a regression can be biased and inconsistent, meaning that the associated p values and 95% confidence intervals cannot be trusted. The assumption of homoscedasticity is made for statistical reasons rather than biological reasons; in most real datasets, some form of heteroscedasticity is likely to exist. However, a survey of the behavioural ecology literature showed that only about 5% of articles explicitly mentioned heteroscedasticity, leaving 95% of articles in which heteroscedasticity was apparently absent. These results strongly indicate that the prevalence of heteroscedasticity is widely under-reported within behavioural ecology. The aim of this article is to raise awareness of heteroscedasticity amongst behavioural ecologists. Using topical examples from fields in behavioural ecology such as sexual dimorphism and animal personality, we highlight the biological importance of considering heteroscedasticity. We also emphasize that researchers should pay closer attention to the variance in their data and consider what factors could cause heteroscedasticity. In addition, we introduce some simple methods of dealing with heteroscedasticity. The two methods we focus on are: (1) incorporating variance functions within a generalised least squares (GLS) framework to model the functional form of heteroscedasticity and; (2) heteroscedasticity-consistent standard error (HCSE) estimators, which can be used when the functional form of heteroscedasticity is unknown. Using case studies, we show how both methods can influence the output from linear regression models. Finally, we hope that more researchers will consider heteroscedasticity as an important source of additional information about the particular biological process being studied, rather than an impediment to statistical analysis.