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.     

Two latent variable risk assessment approaches for mixed continuous and discrete outcomes from developmental toxicity data

Measurements of both continuous and discrete outcomes are encountered in many statistical problems. Here we consider the particular context of teratology studies, where quantitative risk asessment is aimed at determining the effect of dose on the probability that an individual fetus is malformed or of low birth weight, both being important measures of teratogenicity. We will introduce two different joint marginal mean models for outcomes of a mixed nature. First, we propose the Plackett-Dale approach, where for each binary outcome it is assumed that there exists an underlying glatent variable. The latent malformation outcomes are then assumed to follow a Plackett distribution. The second approach we consider is a probit approach. Here it is assumed that there exists an underlying continuous variable for each binary outcome, so the joint distribution for weight and malformation can be assumed to follow a multivariate normal distribution. In both cases, specification of the full distribution will be avoided using pseudolikelihood and generalized estimating equations methodology, respectively. Quantitative risk assessment is illustrated using data from two developmental toxicology experiments.     

Penalized locally optimal experimental designs for nonlinear models

Classical optimal design theory may produce experimental designs that are biologically or characteristically inappropriate. Often, there is a particular study goal along with many practical experimental concerns that a researcher may wish to include in the optimal design process. This article provides a technique that allows a researcher to incorporate desired experimental characteristics into an adjusted optimal design criterion. This technique uses a weighted overall desirability function to penalize the optimal design criterion. A researcher may define an overall desirability function using any number of individual desirability functions to influence the properties of an optimal experimental design. The methodology is illustrated with two dose-response examples.     

Statistical inference for calibration points in nonlinear mixed effects models

Many disciplines conduct studies in which the primary objectives depend on inference based on a nonlinear relationship between the treatment and response. In particular, interest often focuses on calibration—that is, estimating the best treatment level to achieve a particular result. Often, data for such calibration come from experiments with split-plots or other features that result in multiple error terms or other nontrivial error structures. One such example is the time-of-weed-removal study in weed science, designed to estimate the critical period of weed control. Calibration, or inverse prediction, is not a trivial problem with simple linear regression, and the complexities of experiments such as the time-of-weed-removal study further complicate the procedure. In this article, we extend existing calibration techniques to nonlinear mixed effects models, and illustrate the procedure using data from a time-of-weed-removal study.     

Smooth additive mixed models for predicting aboveground biomass

Aboveground biomass estimation in short-rotation forestry plantations is an essential step in the development of crop management strategies as well as allowing the economic viability of the crop to be determined prior to harvesting. Hence, it is important to develop new methodologies that improve the accuracy of predictions, using only a minimum set of easily obtainable information i.e., diameter and height. Many existing models base their predictions only on diameter (mainly due to the complexity of including further covariates), or rely on complicated equations to obtain biomass predictions. However, in tree species, it is important to include height when estimating aboveground biomass because this will vary from one genotype to another. This work proposes the use of a more flexible and easy to implement model for predicting aboveground biomass (stem, branches and total) as a smooth function of height and diameter using smooth additive mixed models which preserve the additive property necessary to model the relationship within wood fractions, and allows the inclusion of random effects and interaction terms. The model is applied to the analysis of three trials carried out in Spain, where nine clones at three different sites are compared. Also, an analysis of slash pine data is carried out in order to compare with the approach proposed by Parresol (Can J For Res 31:865–878, 2001).Supplementary materials accompanying this paper appear on-line     

Temperature dependence of organic matter decomposition: a critical review using literature data analyzed with different models

 The literature was reviewed regarding laboratory incubation studies where C mineralization was measured. Experiments were selected in which the same substrate was incubated at least at two different temperatures and where time-series were available with at least four measurements for each substrate and temperature. A first-order one-component model and a parallel first-order two-component model were fitted to the CO₂–C evolution data in each experiment using a least-squares procedure. After normalising for a reference temperature, the temperature coefficient (Q ₁₀ ) function and three other temperature response functions were fitted to the estimated rate constants. The two-component model could describe the dynamics of the 25 experiments much more adequately than the one-component model (higher R², adjusted for the number of parameters), even when the rate constants for both were assumed to be equally affected by temperature. The goodness-of-fit did not differ between the temperature response models, but was affected by the choice of the reference temperature. For the whole data set, a Q ₁₀ of 2 was found to be adequate for describing the temperature dependence of decomposition in the intermediate temperature range (about 5–35  °C). However, for individual experiments, Q ₁₀ values deviated greatly from 2. At least at temperatures below 5  °C, functions not based on Q ₁₀ are probably more adequate. However, due to the paucity of data from low-temperature incubations, this conclusion is only tentative, and more experimental work is called for. Received: 1 December 1997     

Formulating mixed models for experiments, including longitudinal experiments

Mixed models have become important in analyzing the results of experiments, particularly those that require more complicated models (e.g., those that involve longitudinal data). This article describes a method for deriving the terms in a mixed model. Our approach extends an earlier method by Brien and Bailey to explicitly identify terms for which autocorrelation and smooth trend arising from longitudinal observations need to be incorporated in the model. At the same time we retain the principle that the model used should include, at least, all the terms that are justified by the randomization. This is done by dividing the factors into sets, called tiers, based on the randomization and determining the crossing and nesting relationships between factors. The method is applied to formulate mixed models for a wide range of examples. We also describe the mixed model analysis of data from a three-phase experiment to investigate the effect of time of refinement on Eucalyptus pulp from four different sources. Cubic smoothing splines are used to describe differences in the trend over time and unstructured covariance matrices between times are found to be necessary.     

Identification of model components for a class of continuous spatiotemporal models

Environmental data routinely are collected at irregularly spaced monitoring stations and at intermittent times, times which may differ by location. This article introduces a class of continuous-time, continuous-space statistical models that can accommodate many of these more complex environmental processes. This class of models in corporates temporal and spatial variability in a cohesive manner and is broad enough to include temporal processes that are assumed to be generated by stochastic differential equations with possibly temporally and spatially correlated errors. A wide range of ARIMA temporal models and geostatistical spatial models are included in the class of models investigated. Techniques for identifying the structure of the temporal and spatial components of this class of models are detailed. Point estimates of model parameters, asymptotic distributions, and Kalman-filter prediction methods are discussed.     

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.     

Optimizing wetland restoration and management for avian communities using a mixed integer programming approach

Conservation planning and management decisions often present trade-offs among habitats and species, generating uncertainty about the composition and configuration of habitat that will best meet management goals. The public acquisition of 5471 ha of salt ponds in San Francisco Bay for tidal-marsh restoration presents just such a challenge. Because the existing ponds support large numbers of waterbirds, restoring the entire area to tidal marsh could cause undesirable local declines for many species. To identify management strategies that simultaneously maximize abundances of marsh- and pond-associated species, we applied an integer programming approach to maximize avian abundance, comparing across two objectives, two models, and five species weightings (20 runs total). For each pond, we asked: should it be restored to a tidal marsh or kept as a managed pond, and with what salinity and depth? We used habitat relationship models as inputs to non-linear integer programs to find optimal or near-optimal solutions. We found that a simple linear objective, based on maximizing a weighted sum of standardized species’ abundance, led to homogeneous solutions (all-pond or all-marsh). Maximizing a log-linear objective yielded more heterogeneous configurations that benefit more species. Including landscape terms in the models resulted in slightly greater habitat aggregation, but generally favored pond-associated species. It also led to the placement of certain habitats near the bay’s edge. Using the log-linear objective, optimal restoration configurations ranged from 9% to 60% tidal marsh, depending on the species weighting, highlighting the importance of thoughtful a priori consideration of priority species.     

Local linear mixed effect models—Model specification and interpretation in a biological context

Clustered data, either as an explicit part of the study design or due to the natural distribution of habitats, populations, and so on, are frequently encountered by biologists. Mixed effect models provide a framework that can handle clustered data by estimating cluster-specific random effects and introducing correlated residual structures. General parametric models have been shown not to suit all biological problems, resulting in an increased popularity for local regression procedures, such as LOESS and splines. To evaluate similar biological problems for clustered data with cluster-specific random effects and potential dependencies between within-cluster residuals, we suggest a local linear mixed model (LLMM). The LLMM approach is a local version of a linear mixed-effect model (LME), and the LLMM approach produces: (1) local shared predictions, (2) local cluster-specific predictions, and (3) estimates of cluster-specific random effects conditioned on the covariates. Thus, in addition to the local estimates of the expected response, we obtain information about how the cluster-specific random variability depends on the values of the covariate. Ovary data are used to illustrate the flexibility and potential of this procedure in biological contexts.     

A simulation study on tests of hypotheses and confidence intervals for fixed effects in mixed models for blocked experiments with missing data

This article considers the analysis of experiments with missing data from various experimental designs frequently used in agricultural research (randomized complete blocks, split plots, strip plots). We investigate the small sample properties of REML-based Wald-type F tests using linear mixed models. Several methods for approximating the denominator degrees of freedom are employed, all of which are available with the MIXED procedure of the SAS System (8.02). The simulation results show that the Kenward-Roger method provides the best control of the Type I error rate and is not inferior to other methods in terms of power.     

Developing effective sampling designs for monitoring natural resources in Alaskan national parks: An example using simulations and vegetation data

Monitoring natural resources in Alaskan national parks is challenging because of their remoteness, limited accessibility, and high sampling costs. We describe an iterative, three-phased process for developing sampling designs based on our efforts to establish a vegetation monitoring program in southwest Alaska. In the first phase, we defined a sampling frame based on land ownership and specific vegetated habitats within the park boundaries and used Path Distance analysis tools to create a GIS layer that delineated portions of each park that could be feasibly accessed for ground sampling. In the second phase, we used simulations based on landcover maps to identify size and configuration of the ground sampling units (single plots or grids of plots) and to refine areas to be potentially sampled. In the third phase, we used a second set of simulations to estimate sample size and sampling frequency required to have a reasonable chance of detecting a minimum trend in vegetation cover for a specified time period and level of statistical confidence. Results of the first set of simulations indicated that a spatially balanced random sample of single plots from the most common landcover types yielded the most efficient sampling scheme. Results of the second set of simulations were compared with field data and indicated that we should be able to detect at least a 25% change in vegetation attributes over 31 years by sampling 8 or more plots per year every five years in focal landcover types. This approach would be especially useful in situations where ground sampling is restricted by access.     

Evaluation of linear and nonlinear sediment transport equations using hillslope morphology

Although some simple erosive processes like soil creep or tillage redistribution may be satisfactorily described by linear diffusive equations, the complexity of erosion phenomena requires the use of more complete nonlinear equations. Roering et al. [Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res., 35 853–870.] have proposed a single test, based on the relationship between the curvature and gradient of a hillslope, rather then using a linear diffusion equation. Nevertheless this test, based on steady state conditions, is not complete as it is shown in this work with a counter-example. The hillslope profile used by Roering et al. [Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res., 35 853–870.], (Fig. 2) can be generated either by a linear diffusion equation under transient conditions or a nonlinear diffusion equation under steady state conditions. Additional information on the soil profile changes would give a more complete interpretation of the hillslope evolution.     

Planning for success: Identifying effective and efficient survey designs for monitoring

Selecting a survey design to detect change through time in an ecological resource requires balancing the speed with which a given level of change can be detected against the cost of monitoring. Planning studies allow one to assess these tradeoffs and identify the optimal design choices for a specific scenario of change. However, such studies seldom are conducted. Even worse, they seem least likely to be undertaken when they offer the most insight – when survey methods and monitoring designs are complex and not well captured by simple statistical models. This may be due to limited technical capacity within management agencies. Without such planning, managers risk a potentially severe waste of monitoring resources on ineffective and inefficient monitoring, and institutions will remain ignorant of the true costs of information and the potential efficiency gains afforded by a moderate increase in technical capacity. We discuss the importance of planning studies, outline their main components, and illustrate the process through an investigation of competing designs for monitoring for declining brown bear (Ursus arctos) densities in southwestern Alaska. The results provide guidance on how long monitoring must be sustained before any change is likely to be detected (under a scenario of rather strong true decline), the optimal designs for detecting a change, and a tradeoff where accepting a delay of 2 years in detecting the change could reduce the monitoring cost by almost 50%. This report emphasizes the importance of planning studies for guiding monitoring decisions.     

A mixed model for investigating a population of asymptotic growth curves using restricted B-splines

We introduce a new method for modeling a population of growth curves with B-splines, adapting the usual regression spline basis to ensure a horizontal upper asymptote in all fitted curves. The new method is easily implemented in standard statistical software. We motivate and illustrate our method using data on the growth of Brown Kiwi (Apteryx mantelli) in the North Island of New Zealand, including a time-dependent covariate to investigate the effects of different rearing environments on patterns of weight increase.     

Testing aluminum-chelate equilibria models using sorghum root growth as a bioassay for aluminum

Aluminum toxicity is an important limitation to crop yields in the acid soils of southeastern U. S. and other parts of the world due to its detrimental effects on roots. Soluble organics in soil solution ameliorate Al toxicity, a phenomenon which can be studied employing synthetic chelates. Theoretical models predict that Al will be complexed on nearly a one to one molar basis by nitrilotriacetic acid (NTA) at toxic pH levels found in acid soils (4.0 to 4.5). A series of growth chamber experiments were conducted with NTA at various Al and pH levels to test equilibrium models using sorghum [Sorghum bicolor (L.) Moench] as a bioassay for the uncomplexed Al. At pH 3.5 neither Al nor NTA affected root growth which was very poor, probably because of H⁺ toxicity. At pH 4.0 and 4.5 root growth was reduced by Al levels, and NTA ameliorated toxicity as predicted by the theoretical model. Root length was reduced at pH 4.0 relative to pH 4.5 indicating that H⁺ ion was toxic at pH 4.0. The bioassay method was successful in confirming the model for Al-chelate equilibria for NTA. Differential pulse polarography was found to be sensitive to the uncomplexed Al and may have potential in determining toxic Al in soil solution.     

Evaluating the relationship between ecological and habitat conditions using hierarchical models

Understanding the relationship between an index of biological community state and habitat is important for policy makers and environmental managers. A common approach to modeling this relationship is to use regression. However, this simple method becomes complicated when the data are clustered and have both within-cluster and between-cluster spatial correlation. This article proposes a Bayesian hierarchical model that incorporates both types of spatial correlation. This model yields both an understanding of the within-cluster relationships as well as an overall relationship between these variables. We apply this method to evaluate the relationship between the index of biotic integrity (a common measure of fish condition) and the qualitative habitat evaluation index (a common measure of habitat quality). This method allows us to show that there is a relationship between the biological community state and habitat and that this relationship varies across river basins, while accounting for the within- and between-spatial correlations.     

Power analysis based on spatial effects mixed models: A tool for comparing design and analysis strategies in the presence of spatial variability

Spatial variability among experimental units is known to exist in many designed experiments. Agronomic field trials are a particularly well-known example, but there are others. Historically, spatial variability has been dealt with in one of two ways: either though design, by blocking to account for spatial effects, or though analysis, by nearest neighbor adjustment. More recently, mixed models with spatial covariance structures such as those used in geostatistics have been proposed. These mixed model procedures have tempted some to conclude—to the dismay of many consulting statisticians—that design principles may be bypassed, since spatial covariance models can recover any lost information. Although design principles clearly should not be ignored, spatial procedures do raise questions. Are traditional notions of appropriate design affected? If so, how? How do spatial effects mixed models compare to conventional analysis of variance used in conjunction with blocked designs? This article presents mixed model methods to assess power and precision of proposed designs in the presence of spatial variability and to compare competing design and analysis strategies. The main conclusion is that, if anything, spatial models reinforce the need for sound design principles, particularly the use of incomplete block designs.     

Comparison between discontinuous and continuous lactose conversion processes for the production of prebiotic galacto-oligosaccharides using beta-galactosidase from Lactobacillus reuteri

Galacto-oligosaccharide (GOS) formation from lactose in discontinuous and continuous modes of conversion was investigated using beta-galactosidase (beta-gal) from Lactobacillus reuteri. A continuous stirred tank reactor (CSTR) with an external crossflow membrane was set up, and continuous GOS production was analyzed and compared to the batchwise formed GOS product. Marked differences were detected for the two reactor setups. Above 65% lactose conversion, the GOS yield was lower for the CSTR due to a lower content of tri- and tetrasaccharides in the reaction mixture. In the CSTR, beta-gal from L. reuteri showed up to 2-fold higher specificity toward the formation of beta-(1-->6)-linked GOS, with beta-D-Galp-(1-->6)-D-Glc and beta-D-Galp-(1-->6)-D-Gal being the main GOS components formed under these conditions. This could be used to synthesize more defined GOS products.