Modeling combined continuous and ordinal outcomes in a clustered setting

Measurements of both continuous and categorical outcomes appear in many statistical problems. One such example is the study of teratology and developmental toxicity, where both the probability that a live fetus is malformed (ordinal) or of low birth weight (continuous) are important measures in the context of teratogenicity. Although multivariate methods of the analysis of continuous outcomes are well understood, methods for jointly continuous and discrete outcomes are less familiar. We propose a likelihood-based method that is an extension of the Plackett-Dale approach. Specification of the full likelihood will be avoided using pseudo-likelihood methodology. The estimation of safe dose levels as part of quantitative risk assessment will be illustrated based on a developmental toxicity experiment of diethylene glycol dimethyl ether in mice.     

A Measurement Error Model for Heterogeneous Capture Probabilities in Mark-Recapture Experiments: An Estimating Equation Approach

Logistic models for capture probabilities that depend on covariates are effective if the covariates can be measured exactly. If there is measurement error so that a surrogate for the covariate is observed rather than the covariate itself, simple adjustments may be made if the parameters of joint distribution of the covariate and the surrogate are known. Here we consider the case when a surrogate is observed whenever an individual is captured and the parameters must also be estimated from the data. An estimating equation regression calibration approach is developed and it is illustrated on a real dataset where the surrogate is an individual bird’s wing-length, which varies from occasion to occasion.     

ANOVA for Factors With Ordered Levels

In its simplest case, ANOVA can be seen as a generalization of the t-test for comparing the means of a continuous variable in more than two groups defined by the levels of a discrete covariate, a so-called factor. Testing is then typically done by using the standard F-test. Here, we consider the special but frequent case of factor levels that are ordered. We propose an alternative test using mixed models methodology. The new test often outperforms the standard F-test when factor levels are ordered. We illustrate the proposed testing procedure in simulation studies and three typical applications: nonparametric dose response analysis in agriculture, associations between rating scales and a continuous outcome, and testing differentially expressed genes with ordinal phenotypes.     

Risk Assessment for Toxicity Experiments with Discrete and Continuous Outcomes: A Bayesian Nonparametric Approach

We present a Bayesian nonparametric modeling approach to inference and risk assessment for developmental toxicity studies. The primary objective of these studies is to determine the relationship between the level of exposure to a toxic chemical and the probability of a physiological or biochemical response. We consider a general data setting involving clustered categorical responses on the number of prenatal deaths, the number of live pups, and the number of live malformed pups from each laboratory animal, as well as continuous outcomes (e.g., body weight) on each of the live pups. We utilize mixture modeling to provide flexibility in the functional form of both the multivariate response distribution and the various dose–response curves of interest. The nonparametric model is built from a structured mixture kernel and a dose-dependent Dirichlet process prior for the mixing distribution. The modeling framework enables general inference for the implied dose–response relationships and for dose-dependent correlations between the different endpoints, features which provide practical advances relative to traditional parametric models for developmental toxicology. We use data from a toxicity experiment that investigated the toxic effects of an organic solvent (diethylene glycol dimethyl ether) to demonstrate the range of inferences obtained from the nonparametric mixture model, including comparison with a parametric hierarchical model.Supplementary materials accompanying this paper appear on-line.     

Gaussian Copula Mixed Models for Clustered Mixed Outcomes,With Application in Developmental Toxicology

This paper is concerned with the analysis of clustered data from developmental toxicity studies with mixed responses, i.e., where each member of the cluster has binary and continuous outcomes. A copula-based random effects model is proposed that accounts for associations between binary and/or continuous outcomes within clusters, including the intrinsic association between the mixed outcomes for the same subject. The approach allows the adoption of flexible distributions for the mixed outcomes as well as for the random effects. The model includes the correlated probit model of Gueorguieva and Agresti (2001) and the generalized linear mixed models of Faes et al. (2008), and Faes, Geys, and Catalano (2009) as special cases. Maximum likelihood estimation of our model parameters is implemented using standard software such as PROC NLMIXED in SAS. The proposed methodology is motivated by and illustrated using a developmental toxicity study of ethylene glycol in mice. This article has supplementary material online.     

The simultaneous analysis of mixed discrete and continuous outcomes using nonlinear threshold models

Mixed discrete and continuous outcomes are commonly measured on each experimental unit in dose-response studies in toxicology. The dose-response relationships for these outcomes often have dose thresholds and nonlinear patterns. In addition, the endpoints are typically correlated, and a statistical analysis that incorporates the association may result in improved precision. We propose an extension of the generalized estimating equation (GEE) methodology to simultaneously analyze binary, count, and continuous outcomes with nonlinear threshold models that incorporates the intra-subject correlation. The methodology uses a quasi-likelihood framework and a working correlation matrix, and is appropriate when the marginal expectation of each outcome is of primary interest and the correlation between endpoints is a nuisance parameter. Because the derivatives of threshold models are not continuous at each point of the parameter space, we describe the necessary modifications that result in asymptotically normal and consistent estimators. Using dose-response data from a neurotoxicity experiment, the methodology is illustrated by analyzing five outcomes of mixed type with nonlinear threshold models. In this example, the incorporation of the intra-subject correlation resulted in decreased standard errors for the threshold parameters.     

Analysis of Clustered Binary Data With Unequal Cluster Sizes: A Semiparametric Bayesian Approach

The analysis of clustered binary data is a common task in many areas of application. Parametric approaches to the analysis of such data are numerous, but there has been much recent interest in nonparametric and semiparametric approaches. When cluster sizes are unequal, an assumption is often made of compatibility of marginal distributions in order for semiparametric approaches to be developed when there is little replication for different cluster sizes. Here, we use the marginal compatibility assumption to extend flexible semiparametric Bayesian methods able to shrink towards a “parametric backbone” to the situation where there are few replicated observations for distinct cluster sizes and each distinct value of a covariate. A motivating application is the analysis of developmental toxicology data where pregnant laboratory animals are exposed to a dose of some potentially toxic compound and interest lies in describing the distribution, as a function of the dose level, of the number of fetuses exhibiting some characteristic abnormality. Flexible semiparametric methods are required here, as the data typically exhibit overdispersion and complex structure. We also consider a further extension appropriate to the analysis of clustered binary data in the situation where there is little or no replication for distinct covariate values.     

D-optimal designs for mixed discrete and continuous outcomes analyzed using nonlinear models

Many dose-response experiments in toxicology and other biological sciences are designed to measure multiple outcomes. Unfortunately, most of these studies are powered or designed for a single response, and the inference on the under-powered endpoints is limited. As additional design challenges, the outcomes may have different regions and shapes of activity or have different response types. As a new application to the traditional D-optimality criterion, we have developed optimal designs for mixed discrete and continuous outcomes that are analyzed with nonlinear models. These designs use a numerical algorithm to choose the location of the dose groups and proportion of total sample size allocated to each group that minimize the generalized variance of a model-based covariance matrix that incorporates the correlation between outcomes. Using this methodology, we designed a dose-response experiment with binary, count, and continuous outcomes to evaluate neurotoxicity. In this example, the optimal designs placed dose groups at the predicted dose thresholds and throughout the active range. The designs were generally robust to different correlation structures. In addition, when the expected correlation was moderate or large, we observed a substantial gain in efficiency compared to optimal designs created for each outcome separately.     

Volatilization of ketones from water

The overall mass-transfer coefficients for the volatilization from water of acetone, 2-butanone, 2-pentanone, 3-pentanone, 4-methyl-2-pentanone, 2-heptanone, and 2-octanone were measured simultaneously with the oxygen-absorption coefficient in a laboratory stirred water bath. The liquid-film and gas-film coefficients of the two-film model were determined for the ketones from the overall coefficients, and both film resistances were important for volatilization of the ketones. The liquid-film coefficients for the ketones varied with the 0.719 power of the molecular-diffusion coefficient, in agreement with the literature. The liquid-film coefficients showed a variable dependence on molecular weight, with the dependence ranging from the ?0.263 power for acetone to the ?0.378 power for 2-octanone. This is in contrast with the literature where a constant ?0.500 power dependence on the molecular weight is assumed. The gas-film coefficients for the ketones showed no dependence on molecular weight, in contrast with the literature where a ?0.500 power is assumed.     

Continuous Spatial Process Models for Spatial Extreme Values

We propose a hierarchical modeling approach for explaining a collection of point-referenced extreme values. In particular, annual maxima over space and time are assumed to follow generalized extreme value (GEV) distributions, with parameters μ, σ, and ξ specified in the latent stage to reflect underlying spatio-temporal structure. The novelty here is that we relax the conditionally independence assumption in the first stage of the hierarchial model, an assumption which has been adopted in previous work. This assumption implies that realizations of the the surface of spatial maxima will be everywhere discontinuous. For many phenomena including, e.g., temperature and precipitation, this behavior is inappropriate. Instead, we offer a spatial process model for extreme values that provides mean square continuous realizations, where the behavior of the surface is driven by the spatial dependence which is unexplained under the latent spatio-temporal specification for the GEV parameters. In this sense, the first stage smoothing is viewed as fine scale or short range smoothing while the larger scale smoothing will be captured in the second stage of the modeling. In addition, as would be desired, we are able to implement spatial interpolation for extreme values based on this model. A simulation study and a study on actual annual maximum rainfall for a region in South Africa are used to illustrate the performance of the model.     

Continuous-Time Proportional Hazards Regression for Ecological Monitoring Data

We consider a continuous-time proportional hazards model for the analysis of ecological monitoring data where subjects are monitored at discrete times and fixed sites across space. Since the exact time of event occurrence is not directly observed, we rely on dichotomous event indicators observed at monitoring times to make inference about the model parameters. We use autoregression on the response at neighboring sites from a previous time point to take into account spatial dependence. The interesting fact is utilized that the probability of observing an event at a monitoring time when the underlying hazards is proportional falls under the class of generalized linear models with binary responses and complementary log-log link functions. Thus, a maximum likelihood approach can be taken for inference and the computation can be carried out using standard statistical software packages. This approach has significant computational advantages over some of the existing methods that rely on Monte Carlo simulations. Simulation experiments are conducted and demonstrate that our method has sound finite-sample properties. A real dataset from an ecological study that monitored bark beetle colonization of red pines in Wisconsin is analyzed using the proposed models and inference. Supplementary materials that contain technical details are available online.     

Small Area Estimation of Proportions with Constraint for National Resources Inventory Survey

Motivated by the need to produce small area estimates for the National Resources Inventory survey, we develop a spatial hierarchical model based on the generalized Dirichlet distribution to construct small area estimators of compositional proportions in several mutually exclusive and exhaustive landcover categories. At the observation level, the standard design-based estimators of the proportions are assumed to follow the generalized Dirichlet distribution. After proper transformation of the design-based estimators, beta regression is applicable. We consider a logit mixed model for the expectation of the beta distribution, which incorporates covariates through fixed effects and spatial effect through a conditionally autoregressive process. In a design-based evaluation study, the proposed model-based estimators are shown to have smaller root-mean-square error and relative root-mean-square error than design-based estimators and multinomial model-based estimators. Supplementary materials accompanying this paper appear online.     

Estimating Disease Prevalence Using Inverse Binomial Pooled Testing

Monitoring populations of hosts as well as insect vectors is an important part of agricultural and public health risk assessment. In applications where pathogen prevalence is likely low, it is common to test pools of subjects for the presence of infection, rather than to test subjects individually. This technique is known as pooled (group) testing. In this paper, we revisit the problem of estimating the population prevalence p from pooled testing, but we consider applications where inverse binomial sampling is used. Our work is unlike previous research in pooled testing, which has largely assumed a binomial model. Inverse sampling is natural to implement when there is a need to report estimates early on in the data collection process and has been used in individual testing applications when disease incidence is low. We consider point and interval estimation procedures for p in this new pooled testing setting, and we use example data sets from the literature to describe and to illustrate our methods.     

Mapping Land Suitability for Agriculture in Malawi

Population growth, climate sensitivity, and edaphic properties are important factors that influence decision‐making and risk mitigation for agricultural production. Within the agricultural sector in Malawi, continuous cropping without the use of long‐term sustainable strategies and frequent cultivation on marginal lands have resulted in continually declining soil fertility. Improving soil quality of marginal lands using innovative technologies is imperative for increasing agricultural productivity and improving food security. Here, we propose an ensemble approach to map agricultural land suitability and identify the distribution of marginal land in Malawi. Quantitative data available for eight soil and terrain factors were rated individually, and five distinct models were applied to generate a spatial distribution map of land suitability. The results indicate that highly suitable, moderately suitable, marginally suitable, and unsuitable agricultural areas account for 8·2%, 24·1%, 28·0%, and 39·7% of the total land area, respectively. The majority of suitable lands are currently used for agriculture, but more than half (57·4%) of Malawi's total cropland exists on marginally suitable or unsuitable land categories and is likely a candidate for rehabilitation through sustainable agricultural practices. The methods and products herein will be valuable resources for effectively managing and improving Malawi's agricultural lands for increasing food security. Copyright © 2017 John Wiley & Sons, Ltd.     

Analysing flow patterns from dye tracer experiments in a forest soil using extreme value statistics

Preferential flow of water in soil is now recognized as a common phenomenon. It results in complex flow patterns that can be visualized by dye tracers and increases the risk of pollutants reaching greater depths. We analysed the behaviour of a risk index for vertical solute propagation based on extreme value theory. This risk index can be calculated from binary images of dye‐stained soil profiles and is defined as the form parameter of the generalized Pareto distribution. We did five tracer experiments with Brilliant Blue and iodide under changing initial (variable initial soil moisture) and experimental conditions (different irrigation rates). Our results indicate some persistence of the risk index against small changes of experimental conditions such as the irrigation rate. On the other hand, it seems to be affected by initial soil moisture. Comparisons of Brilliant Blue and iodide patterns show that the form parameter alone is not sufficient to estimate the risk of vertical solute propagation. Therefore we propose to combine the risk index with the scale parameter of the generalized Pareto distribution.     

Multilevel Latent Gaussian Process Model for Mixed Discrete and Continuous Multivariate Response Data

We propose a Bayesian model for mixed ordinal and continuous multivariate data to evaluate a latent spatial Gaussian process. Our proposed model can be used in many contexts where mixed continuous and discrete multivariate responses are observed in an effort to quantify an unobservable continuous measurement. In our example, the latent, or unobservable measurement is wetland condition. While predicted values of the latent wetland condition variable produced by the model at each location do not hold any intrinsic value, the relative magnitudes of the wetland condition values are of interest. In addition, by including point-referenced covariates in the model, we are able to make predictions at new locations for both the latent random variable and the multivariate response. Lastly, the model produces ranks of the multivariate responses in relation to the unobserved latent random field. This is an important result as it allows us to determine which response variables are most closely correlated with the latent variable. Our approach offers an alternative to traditional indices based on best professional judgment that are frequently used in ecology. We apply our model to assess wetland condition in the North Platte and Rio Grande River Basins in Colorado. The model facilitates a comparison of wetland condition at multiple locations and ranks the importance of in-field measurements.     

Using portfolio theory to assess tradeoffs between return from natural capital and social equity across space

Spatial variance in returns from natural resources, driven by resource dynamics and regulations, can have strong consequences for equitable delivery of value to individuals and communities. Yet resource management models implicitly weight returns equally across space, even when space is explicitly included in model dynamics and policy. Here we translate financial portfolio theory from the temporal to spatial realm and use it to quantify the inherent tradeoff between resource returns and social equity, defined as a more uniform distribution of resource value across space. We illustrate this approach with a marine case study of the Channel Islands, California, USA. Depending on the spatial distribution of resources, increasing spatial equity requires nonlinear reductions in resource returns. Realistic management options, such as effort-based fisheries regulations or marine protected areas, increase or reduce this tradeoff, respectively. We also quantify two critical advantages of portfolio approaches to management: they improve outcomes by avoiding false expectations and increase either resource return or social equity while maintaining the other.     

Modeling spatial-temporal binary data using Markov random fields

An autologistic regression model consists of a logistic regression of a response variable on explanatory variables and an autoregression on responses at neighboring locations on a lattice. It is a Markov random field with pairwise spatial dependence and is a popular tool for modeling spatial binary responses. In this article, we add a temporal component to the autologistic regression model for spatial-temporal binary data. The spatial-temporal autologistic regression model captures the relationship between a binary response and potential explanatory variables, and adjusts for both spatial dependence and temporal dependence simultaneously by a space-time Markov random field. We estimate the model parameters by maximum pseudo-likelihood and obtain optimal prediction of future responses on the lattice by a Gibbs sampler. For illustration, the method is applied to study the outbreaks of southern pine bettle in North Carolina. We also discuss the generality of our approach for modeling other types of spatial-temporal lattice data.     

Approximate and Pseudo-Likelihood Analysis for Logistic Regression Using External Validation Data to Model Log Exposure

A common goal in environmental epidemiologic studies is to undertake logistic regression modeling to associate a continuous measure of exposure with binary disease status, adjusting for covariates. A frequent complication is that exposure may only be measurable indirectly, through a collection of subject-specific variables assumed associated with it. Motivated by a specific study to investigate the association between lung function and exposure to metal working fluids, we focus on a multiplicative-lognormal structural measurement error scenario and approaches to address it when external validation data are available. Conceptually, we emphasize the case in which true untransformed exposure is of interest in modeling disease status, but measurement error is additive on the log scale and thus multiplicative on the raw scale. Methodologically, we favor a pseudo-likelihood (PL) approach that exhibits fewer computational problems than direct full maximum likelihood (ML) yet maintains consistency under the assumed models without necessitating small exposure effects and/or small measurement error assumptions. Such assumptions are required by computationally convenient alternative methods like regression calibration (RC) and ML based on probit approximations. We summarize simulations demonstrating considerable potential for bias in the latter two approaches, while supporting the use of PL across a variety of scenarios. We also provide accessible strategies for obtaining adjusted standard errors to accompany RC and PL estimates.     

When the species is also a habitat: Comparing the predictively modelled distributions of Lophelia pertusa and the reef habitat it forms

Internationally there is political momentum to establish networks of marine protected areas for the conservation of threatened species and habitats. Practical implementation of such networks requires an understanding of the distribution of these species and habitats. Predictive modelling provides a method by which continuous distribution maps can be produced from limited sample data. This method is particularly useful in the deep sea where a number of biological communities have been identified as vulnerable ‘habitats’, including Lophelia pertusa reefs. Recent modelling efforts have focused on predicting the distribution of this species. However the species is widely distributed where as reef habitat is not. This study uses Maxent predictive modelling to investigate whether the distribution of the species acts as a suitable proxy for the reef habitat. Models of both species and habitat distribution across Hatton Bank and George Bligh Bank are constructed using multibeam bathymetry, interpreted substrate and geomorphology layers, and derived layers of bathymetric position index (BPI), rugosity, slope and aspect. Species and reef presence records were obtained from video observations. For both models performance is fair to excellent assessed using AUC and additional threshold dependant metrics. 7.17% of the study area is predicted as highly suitable for the species presence while only 0.56% is suitable for reef presence, using the sensitivity–specificity sum maximisation approach to determine the appropriate threshold. Substrate is the most important variable in the both models followed by geomorphology in the RD model and fine scale BPI in the SD model. The difference in the distributions of reef and species suggest that mapping efforts should focus on the habitat rather than the species at fine (100 m) scales.