Non‐parametric habitat models with automatic interactions

Questions: Can a statistical model be designed to represent more directly the nature of organismal response to multiple interacting factors? Can multiplicative kernel smoothers be used for this purpose? What advantages does this approach have over more traditional habitat modelling methods? Methods: Non‐parametric multiplicative regression (NPMR) was developed from the premises that: the response variable has a minimum of zero and a physiologically‐determined maximum, species respond simultaneously to multiple ecological factors, the response to any one factor is conditioned by the values of other factors, and that if any of the factors is intolerable then the response is zero. Key features of NPMR are interactive effects of predictors, no need to specify an overall model form in advance, and built‐in controls on overfitting. The effectiveness of the method is demonstrated with simulated and real data sets. Results: Empirical and theoretical relationships of species response to multiple interacting predictors can be represented effectively by multiplicative kernel smoothers. NPMR allows us to abandon simplistic assumptions about overall model form, while embracing the ecological truism that habitat factors interact.     

Rotifer species richness along an altitudinal gradient in the Alps

Aim Biodiversity patterns along altitudinal gradients are less studied in aquatic than terrestrial systems, even though aquatic sites provide a more homogeneous environment independent of moisture constraints. We studied the altitudinal species richness pattern for planktonic rotifers in freshwater lakes and identified the environmental predictors for which altitude is a proxy. Location Two hundred and eighteen lakes of Trentino–South Tyrol (Italy) in the eastern Alps; lakes covered 98% (range 65–2960 m above sea level) of the altitudinal gradient in the Alps. Methods We performed: (1) linear regression between species richness and altitude to evaluate the general pattern, (2) multiple linear regression between species richness and environmental predictors excluding altitude to identify the most important predictors, and (3) linear regression between the residuals of the best model of step (2) and altitude to investigate any additional explanatory power of altitude. Selection of environmental predictors was based on limnological importance and non‐parametric Spearman correlations. We applied ordinary least squares regression, generalized linear, and generalized least squares modelling to select the most statistically appropriate model. Results Rotifer species richness showed a monotonic decrease with altitude independent of scale effects. Species richness could be explained (R²= 51%) by lake area as a proxy for habitat diversity, reactive silica and total phosphorus as proxies for productivity, water temperature as a proxy for energy, nitrate as a proxy for human influence and north–south and east–west directions as covariates. These predictors completely accounted for the species richness–altitude pattern, and altitude had no additional effect on species richness. Main conclusions The linear decrease of species richness along the altitudinal gradient was related to the interplay of habitat diversity, productivity, heat content and human influence. These factors are the same in terrestrial and aquatic habitats, but the greater environmental stability of aquatic systems seems to favour a linear pattern.     

Probabilistic modeling and mapping of plant indicator species in a Northeast Oregon industrial forest,USA

Managing forest ecosystems for sustainable, multiple use requires forest resource managers to understand and predict how plant species composition and distribution varies across environmental gradients and responds to landscape scale disturbances. This study demonstrates predictive vegetation modeling and mapping for a Northeast Oregon forest using non-parametric Multiplicative Regression (NPMR) with presence/absence data for the species Clintonia uniflora (CLUN) and a set of stand structural and raster-based predictor variables. NPMR is a flexible probability modeling system that can find the best subset of habitat factors influencing species occurrence. NPMR was compared with logistic regression (LR) by building reduced models from variables selected as best by NPMR and full models from variables identified as significant with a forward stepwise process and further manual testing. log β was used to select models with the highest predictive capability. NPMR models were less complex and had higher predictive capability than LR for all modeling approaches. Spatial coordinates were among the most powerful predictors and the modeling approach with physiographic and stand structural variables together was the most improved relative to the average frequency of occurrence. GIS probability maps produced with the application of the physiographic models showed good spatial congruence between high probability values and plots that contained CLUN. NPMR proved to be a reliable probability modeling and mapping tool that could be used as the analytical link between monitoring and quantifying the status and trends of vegetation resources.     

Sensitivity analysis for causal inference using inverse probability weighting

Evaluation of impact of potential uncontrolled confounding is an important component for causal inference based on observational studies. In this article, we introduce a general framework of sensitivity analysis that is based on inverse probability weighting. We propose a general methodology that allows both non‐parametric and parametric analyses, which are driven by two parameters that govern the magnitude of the variation of the multiplicative errors of the propensity score and their correlations with the potential outcomes. We also introduce a specific parametric model that offers a mechanistic view on how the uncontrolled confounding may bias the inference through these parameters. Our method can be readily applied to both binary and continuous outcomes and depends on the covariates only through the propensity score that can be estimated by any parametric or non‐parametric method. We illustrate our method with two medical data sets.     

Information‐theoretic model selection affects home‐range estimation and habitat preference inference: a case study of male Reeves’s Pheasants Syrmaticus reevesii

Reeves’s Pheasant Syrmaticus reevesii is a vulnerable forest bird inhabiting broadleaved habitats dominated by oaks Quercus spp. in central China. Identifying home‐ranges and habitat associations is important for understanding the biology of this species and developing effective management and conservation plans. We used information‐theoretic criteria to evaluate the relative performance of four parametric (exponential power, one‐mode bivariate normal, two‐mode bivariate normal and two‐mode bivariate circle) and two non‐parametric models (adaptive and fixed kernel) for estimating home‐ranges and habitat associations of Reeves’s Pheasants. For parametric models, Akaike’s information criterion (AIC_c) and the likelihood cross‐validation criterion (CVC) were relatively consistent in ranking the bivariate exponential power model the least acceptable, whereas the two‐mode bivariate models performed better. The CVC suggested that kernel models, particularly the adaptive kernel, performed best among all six models evaluated. The average core area and 95% contour area based on the model with greatest support were 6.1 and 54.9 ha, respectively, and were larger than those estimated from other models. The discrepancy in estimates between models with highest and the lowest support decreased as the contour size increased; however, home‐range shapes differed between models. Minimum convex polygons that removed 5% of extreme data points (MCP95) were roughly half the size of home‐ranges based on kernel models. Estimates of home‐range and model evaluation were not affected by sample size (> 50 observations for each bird). Inference about habitat preference based on composition analysis and home‐range overlap varied between models. That with strongest support suggested that Reeves’s Pheasants selected mature fir and mixed forest, avoided farmland, and had mean among‐individual home‐range overlaps of 20%. We recommend non‐parametric methods, particularly the adaptive kernel method, for estimating home‐ranges and core areas for species with complex multi‐polar habitat preferences in heterogeneous environments with large habitat patches. However, we caution against the traditional convenience of using a single model to estimate home‐ranges and recommend exploration of multiple models for describing and understanding the ecological processes underlying space use and habitat associations.     

Iterative partial least squares with right-censored data analysis: a comparison to other dimension reduction techniques

In the linear model with right-censored responses and many potential explanatory variables, regression parameter estimates may be unstable or, when the covariates outnumber the uncensored observations, not estimable. We propose an iterative algorithm for partial least squares, based on the Buckley-James estimating equation, to estimate the covariate effect and predict the response for a future subject with a given set of covariates. We use a leave-two-out cross-validation method for empirically selecting the number of components in the partial least-squares fit that approximately minimizes the error in estimating the covariate effect of a future observation. Simulation studies compare the methods discussed here with other dimension reduction techniques. Data from the AIDS Clinical Trials Group protocol 333 are used to motivate the methodology.     

Spatial relationships in a dendritic network: the herpetofaunal metacommunity of the Mattole River catchment of northwest California

We investigated the aquatic and riparian herpetofauna in a 789 km² river catchment in northwest California to examine competing theories of biotic community structuring in catchment stream networks. Research in fluvial geomorphology has resulted in multi‐scale models of dynamic processes that cyclically create, maintain, and destroy environments in stream networks of mountain catchments. These models have been applied to understanding distributions of invertebrates, algae, fishes and their habitats across entire basin networks, but similar approaches with herpetofauna are rare. We examined multi‐scale spatial patterns of multiple species as they related to variation in channel types, channel settings, and within‐channel attributes that result from these processes. From 83 reaches distributed randomly throughout the watershed, we distinguished four channel types: 1) high gradient with cascade structure; 2) 2–4% gradient with step‐pool structure controlled by moderately steep valleys; 3) slightly entrenched, lower gradient, plane‐bed structure; and 4) low gradient, shallow, unconfined, multiple or migrating pool/riffle channels in broad alluvial valleys. The composition of herpetofauna differed in five of six pair‐wise comparisons among these channel types, indicating a minimum of three distinct mesoscale assemblages. We used non‐parametric multiple regression (NPMR) to examine relationships at multiple spatial scales. NPMR revealed species‐specific associations with channel settings and within‐channel environments among species sharing the same sets of channel types. Morphological adaptations, biophysical limits and natural histories of each species best explained their associations with distinct sets of attributes surrounding and within channel types. While each set of species has similarly adapted to fluvial and geomorphic disturbance processes structuring channels at the mesoscale, species within each set have adapted to a unique set of attributes that are best discerned when their spatial relationships are examined across multiple spatial scales. We evaluated the various spatial patterns against hypotheses of stream community organization and metacommunity perspectives.     

Log-normal variation belts for growth curves

Prediction (confidence) or tolerance belts compound the uncertainty of sample estimates with the estimated extent of individual variation. The latter is therefore better described by variation belts, in which sample estimates are simply substituted for population parameters. Variation belts can provide valuable graphical indications concerning the goodness of fit of postulated error models. While multiplicative least-squares (MLS) methods appear appropriate in principle for biological growth, they are unsatisfactory in practice when logarithmically transformed data are heteroscedastic. Heteroscedastic multiplicative error models can be fitted by iteratively reweighted multiplicative least squares (IRMLS), but unacceptable negative or infinite residual variance estimates and unreasonably wide variation belts are occasionally obtained. These difficulties can be prevented by constrained iteratively reweighted multiplicative least squares (CIRMLS). Examples are presented concerning the metabolic allometry of white rats, the somatic growth of male elephant seals, and the growth of an experimental population of Paramecium caudatum.     

Kriging with nonparametric variance function estimation

A method for fitting regression models to data that exhibit spatial correlation and heteroskedasticity is proposed. It is well known that ignoring a nonconstant variance does not bias least-squares estimates of regression parameters; thus, data analysts are easily lead to the false belief that moderate heteroskedasticity can generally be ignored. Unfortunately, ignoring nonconstant variance when fitting variograms can seriously bias estimated correlation functions. By modeling heteroskedasticity and standardizing by estimated standard deviations, our approach eliminates this bias in the correlations. A combination of parametric and nonparametric regression techniques is used to iteratively estimate the various components of the model. The approach is demonstrated on a large data set of predicted nitrogen runoff from agricultural lands in the Midwest and Northern Plains regions of the U.S.A. For this data set, the model comprises three main components: (1) the mean function, which includes farming practice variables, local soil and climate characteristics, and the nitrogen application treatment, is assumed to be linear in the parameters and is fitted by generalized least squares; (2) the variance function, which contains a local and a spatial component whose shapes are left unspecified, is estimated by local linear regression; and (3) the spatial correlation function is estimated by fitting a parametric variogram model to the standardized residuals, with the standardization adjusting the variogram for the presence of heteroskedasticity. The fitting of these three components is iterated until convergence. The model provides an improved fit to the data compared with a previous model that ignored the heteroskedasticity and the spatial correlation.     

Nonparametrically Weighted Least Squares Estimation in Heteroscedastic Linear Regression

The weights used in iterative weighted least squares (IWLS) regression are usually estimated parametrically using a working model for the error variance. When the variance function is misspecified, the IWLS estimates of the regression coefficients β are still asymptotically consistent but there is some loss in efficiency. Since second moments can be quite hard to model, it makes sense to estimate the error variances nonparametrically and to employ weights inversely proportional to the estimated variances in computing the WLS estimate for β. Surprisingly, this approach had not received much attention in the literature. The aim of this note is to demonstrate that such a procedure can be implemented easily in S-plus using standard functions with default options making it suitable for routine applications. The particular smoothing method that we use is local polynomial regression applied to the logarithm of the squared residuals but other smoothers can be tried as well. The proposed procedure is applied to data on the use of two different assay methods for a hormone. Efficiency calculations based on the estimated model show that the nonparametric IWLS estimates are more efficient than the parametric IWLS estimates based on three different plausible working models for the variance function. The proposed estimators also perform well in a simulation study using both parametric and nonparametric variance functions as well as normal and gamma errors.     

Spatial autocorrelation and red herrings in geographical ecology

Aim Spatial autocorrelation in ecological data can inflate Type I errors in statistical analyses. There has also been a recent claim that spatial autocorrelation generates ‘red herrings’, such that virtually all past analyses are flawed. We consider the origins of this phenomenon, the implications of spatial autocorrelation for macro‐scale patterns of species diversity and set out a clarification of the statistical problems generated by its presence. Location To illustrate the issues involved, we analyse the species richness of the birds of western/central Europe, north Africa and the Middle East. Methods Spatial correlograms for richness and five environmental variables were generated using Moran's I coefficients. Multiple regression, using both ordinary least‐squares (OLS) and generalized least squares (GLS) assuming a spatial structure in the residuals, were used to identify the strongest predictors of richness. Autocorrelation analyses of the residuals obtained after stepwise OLS regression were undertaken, and the ranks of variables in the full OLS and GLS models were compared. Results Bird richness is characterized by a quadratic north–south gradient. Spatial correlograms usually had positive autocorrelation up to c. 1600 km. Including the environmental variables successively in the OLS model reduced spatial autocorrelation in the residuals to non‐detectable levels, indicating that the variables explained all spatial structure in the data. In principle, if residuals are not autocorrelated then OLS is a special case of GLS. However, our comparison between OLS and GLS models including all environmental variables revealed that GLS de‐emphasized predictors with strong autocorrelation and long‐distance clinal structures, giving more importance to variables acting at smaller geographical scales. Conclusion Although spatial autocorrelation should always be investigated, it does not necessarily generate bias. Rather, it can be a useful tool to investigate mechanisms operating on richness at different spatial scales. Claims that analyses that do not take into account spatial autocorrelation are flawed are without foundation.     

Pteridophyte richness, climate and topography in the Iberian Peninsula: comparing spatial and nonspatial models of richness patterns

Aim To describe the spatial variation in pteridophyte species richness; evaluate the importance of macroclimate, topography and within‐grid cell range variables; assess the influence of spatial autocorrelation on the significance of the variables; and to test the prediction of the mid‐domain effect. Location The Iberian Peninsula. Methods We estimated pteridophyte richness on a grid map with c. 2500 km² cell size, using published geocoded data of the individual species. Environmental data were obtained by superimposing the grid system over isoline maps of precipitation, temperature, and altitude. Mean and range values were calculated for each cell. Pteridophyte richness was related to the environmental variables by means of nonspatial and spatial generalized least squares models. We also used ordinary least squares regression, where a variance partitioning was performed to partial out the spatial component, i.e. latitude and longitude. Coastal and central cells were compared to test the mid‐domain effect. Results Both spatial and nonspatial models showed that pteridophyte richness was best explained by a second‐order polynomial of mean annual precipitation and a quadratic elevation‐range term, although the relative importance of these two variables varied when spatial autocorrelation was accounted for. Precipitation range was weakly significant in a nonspatial multiple model (i.e. ordinary regression), and did not remain significant in spatial models. Richness is significantly higher along the coast than in the centre of the peninsula. Main conclusions Spatial autocorrelation affects the statistical significance of explanatory variables, but this did not change the biological interpretation of precipitation and elevation range as the main predictors of pteridophyte richness. Spatial and nonspatial models gave very similar results, which reinforce the idea that water availability and topographic relief control species richness in relatively high‐energy regions. The prediction of the mid‐domain effect is falsified.     

Using functional traits to predict species growth trajectories,and cross‐validation to evaluate these models for ecological prediction

Modeling plant growth using functional traits is important for understanding the mechanisms that underpin growth and for predicting new situations. We use three data sets on plant height over time and two validation methods—in‐sample model fit and leave‐one‐species‐out cross‐validation—to evaluate non‐linear growth model predictive performance based on functional traits. In‐sample measures of model fit differed substantially from out‐of‐sample model predictive performance; the best fitting models were rarely the best predictive models. Careful selection of predictor variables reduced the bias in parameter estimates, and there was no single best model across our three data sets. Testing and comparing multiple model forms is important. We developed an R package with a formula interface for straightforward fitting and validation of hierarchical, non‐linear growth models. Our intent is to encourage thorough testing of multiple growth model forms and an increased emphasis on assessing model fit relative to a model's purpose.     

Repeatability for Gaussian and non‐Gaussian data: a practical guide for biologists

Repeatability (more precisely the common measure of repeatability, the intra‐class correlation coefficient, ICC) is an important index for quantifying the accuracy of measurements and the constancy of phenotypes. It is the proportion of phenotypic variation that can be attributed to between‐subject (or between‐group) variation. As a consequence, the non‐repeatable fraction of phenotypic variation is the sum of measurement error and phenotypic flexibility. There are several ways to estimate repeatability for Gaussian data, but there are no formal agreements on how repeatability should be calculated for non‐Gaussian data (e.g. binary, proportion and count data). In addition to point estimates, appropriate uncertainty estimates (standard errors and confidence intervals) and statistical significance for repeatability estimates are required regardless of the types of data. We review the methods for calculating repeatability and the associated statistics for Gaussian and non‐Gaussian data. For Gaussian data, we present three common approaches for estimating repeatability: correlation‐based, analysis of variance (ANOVA)‐based and linear mixed‐effects model (LMM)‐based methods, while for non‐Gaussian data, we focus on generalised linear mixed‐effects models (GLMM) that allow the estimation of repeatability on the original and on the underlying latent scale. We also address a number of methods for calculating standard errors, confidence intervals and statistical significance; the most accurate and recommended methods are parametric bootstrapping, randomisation tests and Bayesian approaches. We advocate the use of LMM‐ and GLMM‐based approaches mainly because of the ease with which confounding variables can be controlled for. Furthermore, we compare two types of repeatability (ordinary repeatability and extrapolated repeatability) in relation to narrow‐sense heritability. This review serves as a collection of guidelines and recommendations for biologists to calculate repeatability and heritability from both Gaussian and non‐Gaussian data.     

Multiscale resistant kernel surfaces derived from inferred gene flow: An application with vernal pool breeding salamanders

The importance of assessing spatial data at multiple scales when modelling species–environment relationships has been highlighted by several empirical studies. However, no landscape genetics studies have optimized landscape resistance surfaces by evaluating relevant spatial predictors at multiple spatial scales. Here, we model multiscale/layer landscape resistance surfaces to estimate resistance to inferred gene flow for two vernal pool breeding salamander species, spotted (Ambystoma maculatum) and marbled (A. opacum) salamanders. Multiscale resistance surface models outperformed spatial layers modelled at their original spatial scale. A resistance surface with forest land cover at a 500‐m Gaussian kernel bandwidth and normalized vegetation index at a 100‐m Gaussian kernel bandwidth was the top optimized resistance surface for A. maculatum, while a resistance surface with traffic rate and topographic curvature, both at a 500‐m Gaussian kernel bandwidth, was the top optimized resistance surface for A. opacum. Species‐specific resistant kernels were fit at all vernal pools in our study area with the optimized multiscale/layer resistance surface controlling kernel spread. Vernal pools were then evaluated and scored based on surrounding upland habitat (local score) and connectivity with other vernal pools on the landscape, with resistant kernels driving vernal pool connectivity scores. As expected, vernal pools that scored highest were in areas within forested habitats and with high vernal pool densities and low species‐specific landscape resistance. Our findings highlight the success of using a novel analytical approach in a multiscale framework with applications beyond vernal pool amphibian conservation.     

The net result: evaluating species richness extrapolation techniques for littoral pond invertebrates

1. Total species richness for an assemblage or site is a valuable measure in conservation monitoring and assessment, but protocols for sampling and species richness determination in wetland habitats such as ponds, bogs or mires remain largely unrefined. 2. Techniques for estimation of total richness of an assemblage based upon replicated sampling offer the opportunity to derive useful estimates of total richness based upon small numbers of samples, and limit sampling‐derived disturbance which can be particularly problematic in small aquatic habitats. 3. We quantified the performance of eight of the most commonly encountered estimators of species richness for a variety of littoral zone macrofauna from ponds, comparing estimated richness to maximum richness derived from sampling. 4. Estimates using non‐parametric techniques based on species incidence provided the most accurate and precise estimates. The estimators Chao 2 and incidence‐based coverage estimator (ICE) from this category were reliable and consistent slight over‐estimators; the abundance‐based estimator Chao1 also performed well. 5. Species inventory based on relatively small numbers of samples might be significantly improved by use of non‐parametric estimators for quantification of species richness. 6. Use of non‐parametric estimators of species richness can assist biodiversity inventory by preventing erroneous rankings of habitat richness based upon observed species numbers from limited sampling.     

A General Framework for Longitudinal Data through Marked Point Processes

This paper reviews a general framework for the modelling of longitudinal data with random measurement times based on marked point processes and presents a worked example. We construct a quite general regression models for longitudinal data, which may in particular include censoring that only depend on the past and outside random variation, and dependencies between measurement times and measurements. The modelling also generalises statistical counting process models. We review a non-parametric Nadarya-Watson kernel estimator of the regression function, and a parametric analysis that is based on a conditional least squares (CLS) criterion. The parametric analysis presented, is a conditional version of the generalised estimation equations of LIANG and ZEGER (1986). We conclude that the usual nonparametric and parametric regression modelling can be applied to this general set-up, with some modifications. The presented framework provides an easily implemented and powerful tool for model building for repeated measurements.     

Short-term effect of oxic to anoxic transition on benthic microbial activity and solute fluxes in organic-rich phytotreatment ponds

Macroinvertebrates are one of the key components of lake ecosystems and are required to be monitored alongside other biological groups to define ecological status according to European Union legislation. Macroinvertebrate communities are highly variable and complex and respond to a diverse series of environmental conditions. The purpose of this study was to examine the relative importance of environmental variables in explaining macroinvertebrate abundance. A total of 45 sub-alpine lakes were sampled for macroinvertebrates in the shallow sublittoral. Environmental variables were grouped into four types: (1) aquatic physical and chemical parameters, (2) littoral and riparian habitat, (3) lake morphometric parameters and (4) sediment chemical characteristics. Nonparametric multiplicative regression (NPMR) was used to model the abundance of individual macroinvertebrate taxa. Significant models were produced for nine out of the 24 taxa examined. Sediment characteristics were the group most frequently included in models and also the factors to which taxa abundance was the most sensitive. Aquatic physical and chemical variables were the next group most frequently included in models although chlorophyll a was not included in any of the models and total phosphorus in only one. This indicates that many taxa may not show a direct easily interpretable response to eutrophication pressure. Lake morphometric factors were included in several of the models although the sensitivity of macroinvertebrate abundance tended to be lower than for sediment and aquatic physical and chemical factors. Habitat factors were only included in three models although riparian vegetation was found to have a significant influence on the abundance of Ephemera danica indicating that ecotone integrity is likely to play a role in its ecology. Overall, the models tended to be specific for species with limited commonality across taxa. Models produced by NPMR indicate that the response of macroinvertebrates to environmental variables can be successfully described but further research is required focussing in more detail on the response of key taxa to relevant environmental parameters and anthropogenic pressures.     

Environmental factors influencing spatial patterns of shrub diversity in chaparral,Santa Ynez Mountains,California

We examined patterns of shrub species diversity relative to landscape‐scale variability in environmental factors within two watersheds on the coastal flank of the Santa Ynez Mountains, California. Shrub species richness and dominance was sampled at a hierarchy of spatial units using a high‐powered telescope from remote vantage points. Explanatory variables included field estimates of total canopy cover and percentage rock cover, and modeled distributions of slope, elevation, photosynthetically active radiation, topographic moisture index, and local topographic variability. Correlation, multiple regression, and regression tree analyses showed consistent relationships between field‐based measurements of species richness and dominance, and topographically‐mediated environmental variables. In general, higher richness and lower dominance occurred where environmental conditions indicated greater levels of resource limitation with respect to soil moisture and substrate availability. Maximum richness in shrub species occurred on high elevation sites with low topographic moisture index, rocky substrate, and steep slopes. Maximum dominance occurred at low elevation sites with low topographic variability, high potential solar insolation, and high total shrub canopy cover. The observed patterns are evaluated with respect to studies on species‐environment relations, resource use, and regeneration of shrubs in chaparral and coastal sage scrub. The results are discussed in the context of existing species‐diversity hypotheses that hinge on reduced competitive dominance and increased resource heterogeneity under conditions of resource limitation.