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.     

Model weights and the foundations of multimodel inference

Statistical thinking in wildlife biology and ecology has been profoundly influenced by the introduction of AIC (Akaike's information criterion) as a tool for model selection and as a basis for model averaging. In this paper, we advocate the Bayesian paradigm as a broader framework for multimodel inference, one in which model averaging and model selection are naturally linked, and in which the performance of AIC-based tools is naturally evaluated. Prior model weights implicitly associated with the use of AIC are seen to highly favor complex models: in some cases, all but the most highly parameterized models in the model set are virtually ignored a priori. We suggest the usefulness of the weighted BIC (Bayesian information criterion) as a computationally simple alternative to AIC, based on explicit selection of prior model probabilities rather than acceptance of default priors associated with AIC. We note, however, that both procedures are only approximate to the use of exact Bayes factors. We discuss and illustrate technical difficulties associated with Bayes factors, and suggest approaches to avoiding these difficulties in the context of model selection for a logistic regression. Our example highlights the predisposition of AIC weighting to favor complex models and suggests a need for caution in using the BIC for computing approximate posterior model weights.     

Growth and reproduction of Loxechinus albus (Echinodermata: Echinoidea) at the southerly peripheries of their species range, Falkland Islands (South Atlantic)

Sea urchins Loxechinus albus were collected from an unfished population inhabiting inshore areas of the eastern part of the Falkland Islands (Southwest Atlantic) at bimonthly intervals between January 2005 and March 2006. The gonads of 491 specimens were sampled to ascertain GSI, and a subsample was examined microscopically in order to determine sex and maturity stage. Age was determined by ring counts on genital plates of 349 specimens with the annual periodicity of ring deposition validated by marginal increment analysis. This investigation suggests that the study population spawns at the warmest time of the year thus allowing for greater growth in their pelagic larvae and newly settled juveniles. Size at age data were found to be asymptotic and out of three growth functions tested the von Beralanffy growth model best described the data. The relatively short period of optimal water temperatures around the Falkland Islands does not enable L. albus to grow as fast as in other regions with higher seasonal temperatures (central Chilean coasts) resulting in relatively small adult sizes.     

Model selection criteria for overdispersed data and their application to the characterization of a host-parasite relationship

In the statistical modeling of a biological or ecological phenomenon, selecting an optimal model among a collection of candidates is a critical issue. To identify an optimal candidate model, a number of model selection criteria have been developed and investigated based on estimating Kullback’s (Information theory and statistics. Dover, Mineola, 1968) directed or symmetric divergence. Criteria that target the directed divergence include the Akaike (2nd international symposium on information theory. Akadémia Kiadó, Budapest, Hungary, pp 267–281, 1973, IEEE Trans Autom Control AC 19:716–723, 1974) information criterion, AIC, and the “corrected” Akaike information criterion (Hurvich and Tsai in Biometrika 76:297–307, 1989), AICc; criteria that target the symmetric divergence include the Kullback information criterion, KIC, and the “corrected” Kullback information criterion, KICc (Cavanaugh in Stat Probab Lett 42:333–343, 1999; Aust N Z J Stat 46:257–274, 2004). For overdispersed count data, simple modifications of AIC and AICc have been increasingly utilized: specifically, the quasi Akaike information criterion, QAIC, and its corrected version, QAICc (Lebreton et al. in Ecol Monogr 62(1):67–118 1992). In this paper, we propose analogues of QAIC and QAICc based on estimating the symmetric as opposed to the directed divergence: QKIC and QKICc. We evaluate the selection performance of AIC, AICc, QAIC, QAICc, KIC, KICc, QKIC, and QKICc in a simulation study, and illustrate their practical utility in an ecological application. In our application, we use the criteria to formulate statistical models of the tick (Dermacentor variabilis) load on a white-footed mouse (Peromyscus leucopus) in northern Missouri.     

Selecting a binary Markov model for a precipitation process

This paper uses rth-order categorical Markov chains to model the probability of precipitation. Several stationary and non-stationary high-order Markov models are proposed and compared using BIC. The number of parameters increases exponentially by adding the Markov order. Several classes of high-order Markov models are proposed which their increase of number of parameters are modest. For example models that use the number of precipitation days in a period prior to date, temperature of the previous day and sines/cosines periodic functions (to model the seasonality) are considered. The theory of partial likelihood is used to estimate the parameters. Parsimonious non-stationary first order Markov models with few seasonal terms are found optimal using BIC and temperature does not turn out to be a useful covariate. However BIC seems to underestimate the number of seasonal terms. We have also compared the results with AIC in some cases which tends to pick parsimonious models with more seasonal terms and higher order. We also show that ignoring seasonal terms result in picking higher order Markov chains. Finally we apply the methods to build confidence intervals for the probability of periods with no precipitation or low number of precipitation days in Calgary using historical data from 1980 to 2000.     

A capture-recapture model with heterogeneity and behavioural response

We develop the non-parametric maximum likelihood estimator (MLE) of the full M_bh capture-recapture model which utilizes both initial capture and recapture data and permits both heterogeneity (h) between animals and behavioural (b) response to capture. Our MLE procedure utilizes non-parametric maximum likelihood estimation of mixture distributions (Lindsay, 1983; Lindsay and Roeder, 1992) and the EM algorithm (Dempsteret al., 1977). Our MLE estimate provides the first non-parametric estimate of the bivariate capture-recapture distribution.Since non-parametric maximum likelihood estimation exists for submodels M_h (allowing heterogeneity only), Mb (allowing behavioural response only) and M₀ (allowing no changes), we develop maximum likelihood-based model selection, specifically the Akaike information criterion (AIC) (Akaike, 1973). The AIC procedure does well in detecting behavioural response but has difficulty in detecting heterogeneity.     

Proposing an information criterion for individual-based models developed in a pattern-oriented modelling framework

Individual-based models (IBMs) have been improved in quality and reliability in recent years with an approach called pattern-oriented modelling (POM). POM proposes guidelines to develop models reproducing multiple patterns observed on the field and to test systematically how well the IBMs reproduce them. POM studies used generally traditional methods of goodness of fit such as the sum of squares evaluation or ad hoc comparisons of fitting errors and variations. Model selection, however, can be a rigorous statistical approach based on information theory and information criteria such as the Akaike's information criterion (AIC) or the deviance information criterion (DIC). So far, it has not been tried to link POM to these rigorous techniques. The main problems to achieve that are: (a) the difficulty to have likelihood functions for IBMs’ parameters and (b) the possibility to obtain posterior distributions of IBMs’ parameters given the patterns to reproduce. In a first part, this paper answers problem (a) by proposing and explaining how to calculate a deviance measure (POMDEV) for models developed in a context of POM. And while answering the second problem, a second part of the paper proposes an information criterion for model selection in a POM context (the pattern-oriented modelling information criterion: POMIC). This criterion does not yet have the same theoretical foundation as, e.g., AIC, but uses formal analogies to the DIC. In a third part POMIC is tested with a modelling exercise. This exercise shows the potential of POMIC to use multiple patterns for selecting among multiple potential submodels and eventually select the most parsimonious and well fitting model version. We conclude that POMIC, although being a heuristically derived approach, can greatly improve the POM framework.     

Comparison of absolute and relative growth patterns among five <Emphasis Type="Italic">Pinna nobilis</Emphasis> populations along the Tunisian coastline: an information theory approach

The variability in absolute and relative growth of Pinna nobilis along the Tunisian coastline was investigated. Five populations of P. nobilis were sampled, three from northern and two from eastern Tunisia. The specimens were aged and ten morphometric characters were measured on each individual. To test if differences existed in absolute and relative growth patterns among the different populations an information theory approach was followed. For absolute growth, von Bertalanffy, Gompertz, the logistic and the power models were fitted in combination with three assumptions regarding inter-population differences in absolute growth patterns: no differences, differences among all five populations or just between northern and eastern populations. The assumption of common absolute growth parameters among all five populations had the greatest support by the data, whereas the assumption of different growth patterns among all five populations had no support. Von Bertalanffy growth model and the power model were both equally supported by the data (while Gompertz had considerably less support and the logistic model had no support), and thus it may not be definitely concluded whether P. nobilis grows asymptotically or not. The P. nobilis populations of the Tunisian coastline had a slow growth and up to an age of ∼ 9 years their shells were smaller than from all other reported populations in the Mediterranean. For relative growth, apart from the classical allometric model Y = aX ^b , relating the size of a part of a body Y to another reference dimension X, more complicated models were used in combination with the three abovementioned assumptions regarding inter-population differences. Those models, of the form logY = f (logX), either assumed breakpoints in the relative growth trajectories or non-linearities. For most morphometric characters, the classical allometric model had no support by the data and more complicated models were necessary. In most cases, different relative growth either among all five populations or between the northern and eastern population groups was supported by the data. Further investigation is needed to relate the morphological differences observed among different populations of P. nobilis to environmental factors.     

Information-theory approach to allometric growth of marine organisms

Allometric growth investigations are usually conducted by fitting the allometric model (L) (y, x are morphometric characters and b the allometric exponent), which is quite simple both conceptually and mathematically, and its parameters are easy to estimate by linear regression. However b is not necessarily constant and it may change either continuously or abruptly at specific breakpoints; thus, the simple L model quite often fails to describe allometric growth successfully. In the current context, a better alternative is proposed, based on Kullback–Leibler (K-L) information theory and multi-model inference (MMI). Allometric growth was investigated in eight marine species: the bivalves Pecten jacobaeus and Pinna nobilis, the squids Todarodes sagittatus and Todaropsis eblanae, the crab Pachygrapsus marmoratus (females), the ghost shrimp Pestarella tyrrhena (males), and the fishes Trachurus trachurus and Sparus aurata. In each of the eight species, a pair of body parts was measured and the allometric growth of one body part in relation to the other (reference dimension) was studied, by fitting five different candidate models including: the simple allometric model, two models assuming that b changed continuously and two other assuming that b had a breakpoint. For each species, the ‘best’ model was selected by minimizing the small-sample, bias-corrected form of the Akaike Information Criterion. To quantify the plausibility of each model, given the data and the set of five models, the ‘Akaike weight’ w _i of each model was calculated; based on w _i the average model was estimated for each case. MMI is beneficial, more robust, and may reveal more information than the classical approach. As demonstrated with the given examples, estimation of b from the linear model, when it was not supported by the data, revealed some characteristic pitfalls, such as concluding positive allometry when there is actually negative or vice versa, or reporting allometry when the data in reality support isometric growth or vice versa.     

Growth of Chaetodon larvatus (Chaetodontidae: Pisces) in the southern Red Sea

Growth and age of Chaetodon larvatus were studied using growth bands in otoliths and length-frequency analyses. Otoliths of 180 C. larvatus were extracted and measured. Polished sections of sagittae revealed alternating opaque and translucent bands corresponding with a seasonal growth pattern. Both mass and size of the otoliths continue to grow steadily throughout life. Length-at-age data revealed very fast growth during the first year. Growth proceeded at a decreasing rate during the second and the third year; fishes older than 3 years did not grow noticeably. No difference in growth patterns between males and females could be detected. The growth parameters obtained for the whole population are: the asymptotic length (L _∞)=10.64 cm, growth constant (K)=1.14 year⁻¹ and the theoretical age at length zero (t ₀)=−0.30 year. The maximum age recorded was 14 years. Length frequency data collected at a recruitment site confirmed the fast growth of juveniles.     

Comparative growth and mortality of Bonamia ostreae -resistant and wild flat oysters, Ostrea edulis, in an intensive system. I. First year of experiment

Since the early nineteen-seventies, the parasitic protozoans Marteilia refringens and Bonamia ostreae have consistently deleteriously affected the French production of Ostrea edulis (Linné). Purification and inoculation of B. ostreae that were achieved in the late nineteen-eighties allowed a selection program to be initiated. Two O. edulis strains, S85 and S89, were raised that displayed significant resistance to B. ostreae in the field. Growth was monitored to ensure that the resistant strains grew at a rate comparable to that of controls, using two crosses between the first generation of S89 and the second generation of S85 (G1G2), and between the first generation of S89 and wild oysters (G0G1). Growth and mortality were monitored for seven consecutive months in an intensive open circulating system at the IFREMER station of Bouin (Vendée, France). A Chapman-Richards model of growth revealed that the two “resistant” crosses grew significantly better than two controls from Quiberon Bay (Brittany), an area in which B. ostreae is endemic, and Palavas (Mediterranean Sea), where oysters are less infested by this parasite. The asymptotic values for growth curves ranged between 27.2 and 28.2 g for G1G2 and G0G1 vs 21.0 and 22.8 g for the controls. There was a similar trend in mortality, with G1G2 (11.8%) surviving better than the three other populations (from 28.7 to 57.5%). B. ostreae was not detected during the experiments and mortalities were attributed to feeding conditions that were not optimal, reinforced by gametogenesis and high temperatures during the summer. It is suggested that the better performance of G1G2 and G0G1 resulted from their increased resistance to stress. Received: 10 March 1997 / Accepted: 4 August 1997     

Diet of some Antarctic nudibranchs (Gastropoda,Opisthobranchia, Nudibranchia)

Austrodoris kerguelenensis feeds on sponges of the genera Rosella and Scolymastra; Aegires albus, on Hexactinellida (Porifera); and Tritoniella belli, probably on non-determined synascidians. Bathydoris clavigera and B. hodgsoni are omnivorous.     

Empirical study of GARCH models with leverage effect in an environmental application

Atmospheric carbon dioxide concentration (ACDC) level is an important factor for predicting temperature and climate changes. We analyze the conditional variance of a function of ACDC level known as ACDC level growth rate (ACDCGR) using the generalised autoregressive conditional heteroskedasticity (GARCH) and GARCH models with leverage effect. The data are a subset of the well known Mauna Loa atmosphere carbon dioxide record. We test for the presence of stylized facts in the ACDCGR time series. The performance of GARCH models are compared to EGARCH, TGARCH and PGARCH models. Model fit measures AIC, BIC and likelihood is calculated for each fitted model. The results do confirm the presence of some of important stylized facts in the ACDCGR time series, but the presence of leverage effect is not significant . The out of sample one step ahead forecasting performances of the models based on RMSE and MAE metrics are evaluated. EGARCH model with student $t$ disturbances showed the best fit and a valid forecasting performance. A bootstrap algorithm is employed to calculate confidence intervals for future values of ACDCGR time series and its volatility. The constructed bootstrap confidence intervals showed a reasonable performance.     

Model averaging,missing data and multiple imputation: a case study for behavioural ecology

Model averaging, specifically information theoretic approaches based on Akaike’s information criterion (IT-AIC approaches), has had a major influence on statistical practices in the field of ecology and evolution. However, a neglected issue is that in common with most other model fitting approaches, IT-AIC methods are sensitive to the presence of missing observations. The commonest way of handling missing data is the complete-case analysis (the complete deletion from the dataset of cases containing any missing values). It is well-known that this results in reduced estimation precision (or reduced statistical power), biased parameter estimates; however, the implications for model selection have not been explored. Here we employ an example from behavioural ecology to illustrate how missing data can affect the conclusions drawn from model selection or based on hypothesis testing. We show how missing observations can be recovered to give accurate estimates for IT-related indices (e.g. AIC and Akaike weight) as well as parameters (and their standard errors) by utilizing ‘multiple imputation’. We use this paper to illustrate key concepts from missing data theory and as a basis for discussing available methods for handling missing data. The example is intended to serve as a practically oriented case study for behavioural ecologists deciding on how to handle missing data in their own datasets and also as a first attempt to consider the problems of conducting model selection and averaging in the presence of missing observations.     

A Matrix‐Calibrated Species‐Area Model for Predicting Biodiversity Losses Due to Land‐Use Change

Abstract: Application of island biogeography theory to prediction of species extinctions resulting from habitat loss is based on the assumption that the transformed landscape matrix is completely inhospitable to the taxa considered, despite evidence demonstrating the nontrivial influence of matrix on populations within habitat remnants. The island biogeography paradigm therefore needs refining to account for specific responses of taxa to the area of habitat “islands” and to the quality of the surrounding matrix. We incorporated matrix effects into island theory by partitioning the slope (z value) of species–area relationships into two components: γ, a constant, and σ, a measure of taxon‐specific responses to each component of a heterogeneous matrix. We used our matrix‐calibrated model to predict extinction and endangerment of bird species resulting from land‐use change in 20 biodiversity hotspots and compared these predictions with observed numbers of extinct and threatened bird species. We repeated this analysis with the conventional species–area model and the countryside species–area model, considering alternative z values of 0.35 (island) or 0.22 (continental). We evaluated the relative strength of support for each of the five candidate models with Akaike's information criterion (AIC). The matrix‐calibrated model had the highest AIC weight (w_i = 89.21%), which means the weight of evidence in support of this model was the optimal model given the set of candidate models and the data. In addition to being a valuable heuristic tool for assessing extinction risk, our matrix‐calibrated model also allows quantitative assessment of biodiversity benefits (and trade‐offs) of land‐management options in human‐dominated landscapes. Given that processes of secondary regeneration have become more widespread across tropical regions and are predicted to increase, our matrix‐calibrated model will be increasingly appropriate for practical conservation in tropical landscapes.     

Natural growth bands and growth variability in the sea urchin Echinus esculentus: results from tetracycline tagging

Growth of the European edible sea urchin Echinus esculentus L. was studied in a population held for 2 yr in cages on the sea bed, after labelling with the skeletal growth marker tetracycline. The final position of the tetracycline tag on the genital plates agreed with an annual periodicity in natural growth zones; two such growth zones appeared beyond the position of the tag on the ground surface of the plate as light-reflecting bands separated by narrow dark lines in the middle layer. Individual and group (pooled data) growth parameters were estimated from the growth increment shown in the genital plate, whose lateral growth displayed a linear relationship to the diameter of the urchin test within the size range of these measurements. Von Bertalanffy growth parameters (asymptotic size and growth-rate function K) fitted to the growth increment on each individual were highly significantly correlated to those fitted to the natural growth lines, assuming an annual periodicity. The inferred growth pattern agrees well with conclusions based on H. B. Moore's growth-band data. The caged urchins can be assumed to have experienced exactly similar conditions, yet the growth curves fitted to individuals showed considerable variability. The good agreement between estimated growth function parameters of individuals obtained by the two methods indicate that this reflects real variability in growth between individuals that probably is genetically rather than environmentally determined. The growth of E. esculentus, and the adaptational significance of high growth variability in the population is briefly discussed.     

Estimating demographic parameters for capture–recapture data in the presence of multiple mark types

In mark-recapture studies, various techniques can be used to uniquely identify individual animals, such as ringing, tagging or photo-identification using natural markings. In some long-term studies more than one type of marking procedure may be implemented during the study period. In these circumstances, ignoring the different mark types can produce biased survival estimates since the assumption that the different mark types are equally catchable (homogeneous capture probability across mark types) may be incorrect. We implement an integrated approach where we simultaneously analyse data obtained using three different marking techniques, assuming that animals can be cross-classified across the different mark types. We discriminate between competing models using the AIC statistic. This technique also allows us to estimate both relative mark-loss probabilities and relative recapture efficiency rates for the different marking methods. We initially perform a simulation study to explore the different biases that can be introduced if we assume a homogeneous recapture probability over mark type, before applying the method to a real dataset. We make use of data obtained from an intensive long-term observational study of UK female grey seals (Halichoerus grypus) at a single breeding colony, where three different methods are used to identify individuals within a single study: branding, tagging and photo-identification based on seal coat pattern or pelage.     

Geographical differences in early growth of walleye pollock Theragra chalcogramma,estimated by back-calculation of otolith daily growth increments

The relationship between otolith length and body length (total length) was examined in walleye pollock Theragra chalcogramma (Pallas), collected from Pacific Ocean waters adjacent to Hokkaido, Japan, and in larvae reared from fertilized eggs. A linear relationship was found between log-transformed body length and otolith length data with two inflection points at ca. 11 mm and 100 mm total length. This relationship was found to be applicable also to samples from the Japan Sea, Bering Sea and Okhotsk Sea. The early growth pattern estimated by back-calculation of otolith increments of 1-yr fish from Funka (Uchiura) Bay (Hokkaido) accorded with that obtained from size-at-age data of 0 yr fish collected in the preceding year. Differences were found in comparisons of the back-calculated early growth pattern between samples from waters adjacent to Hokkaido, the Pacific Ocean, the Japan Sea and the Okhotsk Sea. The growth curve of the Okhotsk samples was markedly different from others, showing rapid initial growth up to about 100 d after hatching and attaining a small body size in the first year.     

Growth,mortality and recruitment of the yellow clam Mesodesma mactroides on Uruguayan beaches

We studied some aspects of the population dynamics of the yellow clam Mesodesma mactroides (Deshayes, 1854) which inhabits the eastern sandy beaches of Uruguay. Observations were made from March 1983 through March 1985 (harvesting season), and additional data from January 1988 to December 1989 (closed season) were also included in the analyses. Parameters of growth and mortality were estimated by methods that make use of age or length information. An age/length relationship was constructed by counting growth rings. Results were consistent with those obtained with modal class progression analysis and other length/frequency methods. The growth curve obtained from the age/length relationship did not account for seasonality in growth rate, and a seasonal growth model explained growth oscillations in a better way. The instantaneous coefficient of natural mortality (M) estimated for the closed season was 1.64, whereas an increase in fishing mortality (F) was observed during 1984. Age composition did not change during the 2 yr of study, and the first 6 mm class accounted for over 50% of the population. The main contribution of recruits was observed between late summer and early fall (February to April), and the recruitment pattern suggested one main settlement period per year. Growth and recruitment success appeared to be regulated by density-dependence processes. Some implications for management are considered in this context. Methodological aspects concerning the reliability of empirical equations and programs for estimating growth and mortality are discussed.