Swap and fill algorithms in null model analysis: rethinking the knight's tour

Community assembly rules are often inferred from patterns in presence-absence matrices. A challenging problem in the analysis of presence-absence matrices has been to devise a null model algorithm to produce random matrices with fixed row and column sums. Previous studies by Roberts and Stone [(1990) Oecologia 83:560-567] and Manly [(1995) Ecology 76:1109-1115] used a "Sequential Swap" algorithm in which submatrices are repeatedly swapped to produce null matrices. Sanderson et al. [(1998) Oecologia 116:275-283] introduced a "Knight's Tour" algorithm that fills an empty matrix one cell at a time. In an analysis of the presence-absence matrix for birds of the Vanuatu islands, Sanderson et al. obtained different results from Roberts and Stone and concluded that "results from previous studies are generally flawed". However, Sanderson et al. did not investigate the statistical properties of their algorithm. Using simple probability calculations, we demonstrate that their Knight's Tour is biased and does not sample all unique matrices with equal frequency. The bias in the Knight's Tour arises because the algorithm samples exhaustively at each step before retreating in sequence. We introduce an unbiased Random Knight's Tour that tests only a small number of cells and retreats by removing a filled cell from anywhere in the matrix. This algorithm appears to sample unique matrices with equal frequency. The Random Knight's Tour and Sequential Swap algorithms generate very similar results for the large Vanuatu matrix, and for other presence-absence matrices we tested. As a further test of the Sequential Swap, we constructed a set of 100 random matrices derived from the Vanuatu matrix, analyzed them with the Sequential Swap, and found no evidence that the algorithm is prone to Type I errors (rejecting the null hypothesis too frequently). These results support the original conclusions of Roberts and Stone and are consistent with Gotelli's [(2000) Ecology 81:2606-2621] Type I and Type II error tests for the Sequential Swap. In summary, Sanderson et al.'s Knight's Tourgenerates large variances and does not sample matrices equiprobably. In contrast, the Sequential Swap generates results that are very similar to those of an unbiased Random Knight's Tour, and is not overly prone to Type I or Type II errors. We suggest that the statistical properties of proposed null model algorithms be examined carefully, and that their performance judged by comparisons with artificial data sets of known structure. In this way, Type I and Type II error frequencies can be quantified, and different algorithms and indices can be compared meaningfully.     

On the methods to assess significance in nestedness analyses

Use of Z values to evaluate nestedness significance is a common procedure. An appealing alternative to the use of Z values is that of using a value of relative nestedness (RN). However, there is no agreement on the preferable procedures to generate the null matrices needed to compute both Z and RN. In general, it is recommended to use restrictive null models that take into account row and column totals. The two most widely used null models of this kind, namely, FF and CE [that generate matrices with row and column sums equal (FF) or proportional (CE) to the row and column totals of the original matrix, respectively], are very different in terms of restrictiveness. We performed a set of comparative analyses on both theoretical and real matrices to investigate the differences between the use of Z and RN values, and between the use of FF and CE null models, when NODF (Nestedness metric based on overlap and decreasing fill) or ρ(A) (i.e., the largest eigenvalue of the adjacency matrix) are used to measure nestedness. We found no difference in the use of Z or RN values. On the other hand, we found that different combinations of nestedness measures and null models may lead to inconsistent outcomes. Our results offer some clarity on a few issues that, despite playing a central role in the practical application of nestedness analysis, have been little explored, and highlight the need for the definition of some commonly accepted standards.     

An evaluation of randomization models for nested species subsets analysis

Randomization models, often termed “null” models, have been widely used since the 1970s in studies of species community and biogeographic patterns. More recently they have been used to test for nested species subset patterns (or nestedness) among assemblages of species occupying spatially subdivided habitats, such as island archipelagoes and terrestrial habitat patches. Nestedness occurs when the species occupying small or species-poor sites have a strong tendency to form proper subsets of richer species assemblages. In this paper, we examine the ability of several published simulation models to detect, in an unbiased way, nested subset patterns from a simple matrix of site-by-species presence-absence data. Each approach attempts to build in biological realism by following the assumption that the ecological processes that generated the patterns observed in nature would, if they could be repeated many times over using the same species and landscape configuration, produce islands with the same number of species and species present on the same number of islands as observed. In mathematical terms, the mean marginal totals (column and row sums) of many simulated matrices would match those of the observed matrix. Results of model simulations suggest that the true probability of a species occupying any given site cannot be estimated unambiguously. Nearly all of the models tested were shown to bias simulation matrices toward low levels of nestedness, increasing the probability of a Type I statistical error. Further, desired marginal totals could be obtained only through ad-hoc manipulation of the calculated probabilities. Paradoxically, when such results are achieved, the model is shown to have little statistical power to detect nestedness. This is because nestedness is determined largely by the marginal totals of the matrix themselves, as suggested earlier by Wright and Reeves. We conclude that at the present time, the best null model for nested subset patterns may be one based on equal probabilities of occurrence for all species. Examples of such models are readily available in the literature. Received: 3 February 1997 / Accepted: 21 September 1997     

Silvicolous on a Small Scale: Possibilities and Limitations of Habitat Suitability Models for Small,Elusive Mammals in Conservation Management and Landscape Planning

Species distribution and endangerment can be assessed by habitat-suitability modelling. This study addresses methodical aspects of habitat suitability modelling and includes an application example in actual species conservation and landscape planning. Models using species presence-absence data are preferable to presence-only models. In contrast to species presence data, absences are rarely recorded. Therefore, many studies generate pseudo-absence data for modelling. However, in this study model quality was higher with null samples collected in the field. Next to species data the choice of landscape data is crucial for suitability modelling. Landscape data with high resolution and ecological relevance for the study species improve model reliability and quality for small elusive mammals like Muscardinus avellanarius. For large scale assessment of species distribution, models with low-detailed data are sufficient. For regional site-specific conservation issues like a conflict-free site for new wind turbines, high-detailed regional models are needed. Even though the overlap with optimally suitable habitat for M. avellanarius was low, the installation of wind plants can pose a threat due to habitat loss and fragmentation. To conclude, modellers should clearly state the purpose of their models and choose the according level of detail for species and environmental data.     

Review. Theoretical and empirical evidence for the impact of inductive biases on cultural evolution

The question of how much the outcomes of cultural evolution are shaped by the cognitive capacities of human learners has been explored in several disciplines, including psychology, anthropology and linguistics. We address this question through a detailed investigation of transmission chains, in which each person passes information to another along a chain. We review mathematical and empirical evidence that shows that under general conditions, and across experimental paradigms, the information passed along transmission chains will be affected by the inductive biases of the people involved-the constraints on learning and memory, which influence conclusions from limited data. The mathematical analysis considers the case where each person is a rational Bayesian agent. The empirical work consists of behavioural experiments in which human participants are shown to operate in the manner predicted by the Bayesian framework. Specifically, in situations in which each person's response is used to determine the data seen by the next person, people converge on concepts consistent with their inductive biases irrespective of the information seen by the first member of the chain. We then relate the Bayesian analysis of transmission chains to models of biological evolution, clarifying how chains of individuals correspond to population-level models and how selective forces can be incorporated into our models. Taken together, these results indicate how laboratory studies of transmission chains can provide information about the dynamics of cultural evolution and illustrate that inductive biases can have a significant impact on these dynamics.     

A Modeling Study on How Cell Division Affects Properties of Epithelial Tissues Under Isotropic Growth

Cell proliferation affects both cellular geometry and topology in a growing tissue, and hence rules for cell division are key to understanding multicellular development. Epithelial cell layers have for long times been used to investigate how cell proliferation leads to tissue-scale properties, including organism-independent distributions of cell areas and number of neighbors. We use a cell-based two-dimensional tissue growth model including mechanics to investigate how different cell division rules result in different statistical properties of the cells at the tissue level. We focus on isotropic growth and division rules suggested for plant cells, and compare the models with data from the Arabidopsis shoot. We find that several division rules can lead to the correct distribution of number of neighbors, as seen in recent studies. In addition we find that when also geometrical properties are taken into account other constraints on the cell division rules result. We find that division rules acting in favor of equally sized and symmetrically shaped daughter cells can best describe the statistical tissue properties.     

Cadmium uptake by algal biomass in batch and continuous (CSTR and packed bed column) adsorbers

In this work, the properties of marine algae Gelidium, algal waste from agar extraction industry and a composite material were investigated for cadmium(II) biosorption. Equilibrium experiments were performed at three pH values (4, 5.3 and 6.5). Equilibrium data were well described by the Langmuir and Langmuir–Freundlich isotherms. Two models predicting the pH influence in the cadmium biosorption (discrete and continuous models) have been developed in order to better describe the equilibrium. The continuous model also considers a heterogeneous distribution of carboxylic groups, determined by potentiometric titration. The results of batch kinetic experiments performed at different pH values were well fitted by two mass transfer models and the homogeneous diffusion coefficients for the cadmium ions inside the biosorbent were obtained. Continuous stirred tank reactor (CSTR) and packed bed column configurations were also examined for the biosorption of cadmium ions. A strong acid (0.1 M HNO₃) was used as eluant to regenerate the biosorbents in the column. Several mass transfer models were applied with success to describe the biosorption process in batch mode, CSTR and fixed bed column.     

A consumer's guide to nestedness analysis

Nestedness analysis has become increasingly popular in the study of biogeographic patterns of species occurrence. Nested patterns are those in which the species composition of small assemblages is a nested subset of larger assemblages. For species interaction networks such as plant–pollinator webs, nestedness analysis has also proven a valuable tool for revealing ecological and evolutionary constraints. Despite this popularity, there has been substantial controversy in the literature over the best methods to define and quantify nestedness, and how to test for patterns of nestedness against an appropriate statistical null hypothesis. Here we review this rapidly developing literature and provide suggestions and guidelines for proper analyses. We focus on the logic and the performance of different metrics and the proper choice of null models for statistical inference. We observe that traditional 'gap-counting' metrics are biased towards species loss among columns (occupied sites) and that many metrics are not invariant to basic matrix properties. The study of nestedness should be combined with an appropriate gradient analysis to infer possible causes of the observed presence–absence sequence. In our view, statistical inference should be based on a null model in which row and columns sums are fixed. Under this model, only a relatively small number of published empirical matrices are significantly nested. We call for a critical reassessment of previous studies that have used biased metrics and unconstrained null models for statistical inference.     

Occupancy modelling as a new approach to assess supranational trends using opportunistic data: a pilot study for the damselfly Calopteryx splendens

There is limited information available on changes in biodiversity at the European scale, because there is a lack of data from standardised monitoring for most species groups. However, a great number of observations made without a standardised field protocol is available in many countries for many species. Such opportunistic data offer an alternative source of information, but unfortunately such data suffer from non-standardised observation effort and geographical bias. Here we describe a new approach to compiling supranational trends using opportunistic data which adjusts for these two major imperfections. The non-standardised observation effort is dealt with by occupancy modelling, and the unequal geographical distribution of sites by a weighting procedure. The damselfly Calopteryx splendens was chosen as our test species. The data were collected from five countries (Ireland, Great Britain, the Netherlands, Belgium and France), covering the period 1990–2008. We used occupancy models to estimate the annual number of occupied 1 × 1 km sites per country. Occupancy models use presence-absence data, account for imperfect detection of species, and thereby correct for between-year variability in observation effort. The occupancy models were run per country in a Bayesian mode of inference using JAGS. The occupancy estimates per country were then aggregated to assess the supranational trend in the number of occupied 1 × 1 km². To adjust for the unequal geographical distribution of surveyed sites, we weighted the countries according to the number of sites surveyed and the range of the species per country. The distribution of C. splendens has increased significantly in the combined five countries. Our trial demonstrated that a supranational trend in distribution can be derived from opportunistic data, while adjusting for observation effort and geographical bias. This opens new perspectives for international monitoring of biodiversity.     

Linear Diagonals‐Parameter Symmetry and Quasi‐Symmetry Models for Cumulative Probabilities in Square Contingency Tables with Ordered Categories

For the analysis of square contingency tables with ordered categories, Caussinus (1965) considered the quasi‐symmetry (QS) model, Goodman (1979) considered the diagonals‐parameter symmetry (DPS) model, and Agresti (1983) considered the linear diagonals‐parameter symmetry (LDPS) model. These models show the structures of symmetry for cell probabilities. Tomizawa (1993) proposed another DPS model which has a similar multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (>i) or above. This paper proposes another LDPS and QS models that have the corresponding similar multiplicative forms for cumulative probabilities instead of cell probabilities. Special cases of the proposed models include symmetry. Two kinds of unaided distance vision data and endometrial cancer data are analyzed using these models. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ecologists should not use statistical significance tests to interpret simulation model results

Simulation models are widely used to represent the dynamics of ecological systems. A common question with such models is how changes to a parameter value or functional form in the model alter the results. Some authors have chosen to answer that question using frequentist statistical hypothesis tests (e.g. ANOVA). This is inappropriate for two reasons. First, p‐values are determined by statistical power (i.e. replication), which can be arbitrarily high in a simulation context, producing minuscule p‐values regardless of the effect size. Second, the null hypothesis of no difference between treatments (e.g. parameter values) is known a priori to be false, invalidating the premise of the test. Use of p‐values is troublesome (rather than simply irrelevant) because small p‐values lend a false sense of importance to observed differences. We argue that modelers should abandon this practice and focus on evaluating the magnitude of differences between simulations. Synthesis Researchers analyzing field or lab data often test ecological hypotheses using frequentist statistics (t‐tests, ANOVA, etc.) that focus on p‐values. Field and lab data usually have limited sample sizes, and p‐values are valuable for quantifying the probability of making incorrect inferences in that situation. However, modern ecologists increasingly rely on simulation models to address complex questions, and those who were trained in frequentist statistics often apply the hypothesis‐testing approach inappropriately to their simulation results. Our paper explains why p‐values are not informative for interpreting simulation models, and suggests better ways to evaluate the ecological significance of model results.     

A systematic statistical linear modeling approach to oligonucleotide array experiments

We outline and describe steps for a statistically rigorous approach to analyzing probe-level Affymetrix GeneChip data. The approach employs classical linear mixed models and operates on a gene-by-gene basis. Forgoing any attempts at gene presence or absence calls, the method simultaneously considers the data across all chips in an experiment. Primary output includes precise estimates of fold change (some as low as 1.1), their statistical significance, and measures of array and probe variability. The method can accommodate complex experiments involving many kinds of treatments and can test for their effects at the probe level. Furthermore, mismatch probe data can be incorporated in different ways or ignored altogether. Data from an ionizing radiation experiment on human cell lines illustrate the key concepts.     

Comparison of model selection technique performance in predicting the spread of newly invasive species: a case study with <Emphasis Type="Italic">Batrachochytrium salamandrivorans</Emphasis>

Species distribution models (SDMs) increasingly have been used to anticipate the spread of invasive species. However, these models are powerful conservation tools only if they are biologically relevant, and thus validation of these models is essential. Here, we evaluate four model selection frameworks for their ability to identify a best fit model of spread under low data conditions early in an invasion, specifically testing the efficacy of methods that utilize absence data in addition to presence data in evaluating models. We test this question using a simulation where we generated data with varying confidence in the accuracy of the absence data, as absences in early invasions may become presences in the future, and increasing quantity of observation data to test the models. We create these simulations based on a real-world example of a newly emergent, invasive fungal pathogen, Batrachochytrium salamandrivorans (Bsal). The simulation demonstrated that AIC and Likelihood consistently outperform both Kappa and AUC in selecting the true model as the best model when data are limited and absence data are low quality, with AIC providing the most conservative results due to penalties for overparameterization. With these results, we then used these techniques to compare five candidate models for predicting the spread of Bsal. Consistent with the simulation, the best fit model of the candidate models for Bsal was inconsistent across the four metrics. However, AIC, which performed best in the simulation study, suggested that the spread of Bsal into Western Europe was best predicted by a combination of bioclimatic suitability, salamander richness, and number of salamander imports. Our results highlight the difficulty in evaluating predictive models when data are limited and of low quality, as with a newly invasive species, but show that these challenges can be partially addressed with the appropriate model selection approach. Use of this approach is critical as SDMs of invasive species are often used to inform conservation policy and efforts before the invasion spreads, when limited data are available.     

The NATURA 2000 database as a tool in the analysis of habitat selection at large scales: factors affecting the occurrence of pine and stone martens in Southern Europe

Most studies on habitat selection among animals are conducted at local scales, whereas reliable determination of species requirements at larger spatial scales can be problematic. We used data available for NATURA 2000 sites to determine the habitat requirements of two relatively widespread and common species—pine marten Martes martes and stone marten Martes foina—in Southern Europe. Using presence-absence data, we applied statistical models at two spatial scales. At the local scale (within the dispersal distances of the species), sites occupied by martens were compared with unoccupied sites using buffers of different sizes, whereas at the regional scale, unoccupied sites were selected randomly. To adjust for spatial autocorrelation of data, penalised quasi-likelihood approximations were used. Both species of martens demonstrated preferences for areas with higher proportions of forest cover and steeper terrain. At the local, but not at the regional, scale, pine martens occurred at lower elevations, whilst stone martens occurred at higher elevations. We found that climatic variables (mean temperature, precipitation) had no significant effect on the presence of the martens. The results of our analyses generally confirmed findings of previous studies on habitat selection of both marten species in Southern Europe. This demonstrates the utility of data collected for NATURA 2000 sites for use in various analyses such as conservation planning and evaluating the impact of climate change on the distribution of animal species.     

The clustering of regression models method with applications in gene expression data

Identification of differentially expressed genes and clustering of genes are two important and complementary objectives addressed with gene expression data. For the differential expression question, many "per-gene" analytic methods have been proposed. These methods can generally be characterized as using a regression function to independently model the observations for each gene; various adjustments for multiplicity are then used to interpret the statistical significance of these per-gene regression models over the collection of genes analyzed. Motivated by this common structure of per-gene models, we proposed a new model-based clustering method--the clustering of regression models method, which groups genes that share a similar relationship to the covariate(s). This method provides a unified approach for a family of clustering procedures and can be applied for data collected with various experimental designs. In addition, when combined with per-gene methods for assessing differential expression that employ the same regression modeling structure, an integrated framework for the analysis of microarray data is obtained. The proposed methodology was applied to two microarray data sets, one from a breast cancer study and the other from a yeast cell cycle study.     

Probing single-stranded DNA conformational flexibility using fluorescence spectroscopy

Single-stranded DNA (ssDNA) is an essential intermediate in various DNA metabolic processes and interacts with a large number of proteins. Due to its flexibility, the conformations of ssDNA in solution can only be described using statistical approaches, such as flexibly jointed or worm-like chain models. However, there is limited data available to assess such models quantitatively, especially for describing the flexibility of short ssDNA and RNA. To address this issue, we performed FRET studies of a series of oligodeoxythymidylates, (dT)N, over a wide range of salt concentrations and chain lengths (10 < or = N < or = 70 nucleotides), which provide systematic constraints for testing theoretical models. Unlike in mechanical studies where available ssDNA conformations are averaged out during the time it takes to perform measurements, fluorescence lifetimes may act here as an internal clock that influences fluorescence signals depending on how fast the ssDNA conformations fluctuate. A reasonably good agreement could be obtained between our data and the worm-like chain model provided that limited relaxations of the ssDNA conformations occur within the fluorescence lifetime of the donor. The persistence length thus estimated ranges from 1.5 nm in 2 M NaCl to 3 nm in 25 mM NaCl.     

Sequential hypothesis testing with spatially correlated presence-absence data

A pest management decision to initiate a control treatment depends upon an accurate estimate of mean pest density. Presence-absence sampling plans significantly reduce sampling efforts to make treatment decisions by using the proportion of infested leaves to estimate mean pest density in lieu of counting individual pests. The use of sequential hypothesis testing procedures can significantly reduce the number of samples required to make a treatment decision. Here we construct a mean-proportion relationship for Oligonychus perseae Tuttle, Baker, and Abatiello, a mite pest of avocados, from empirical data, and develop a sequential presence-absence sampling plan using Bartlett's sequential test procedure. Bartlett's test can accommodate pest population models that contain nuisance parameters that are not of primary interest. However, it requires that population measurements be independent, which may not be realistic because of spatial correlation of pest densities across trees within an orchard. We propose to mitigate the effect of spatial correlation in a sequential sampling procedure by using a tree-selection rule (i.e., maximin) that sequentially selects each newly sampled tree to be maximally spaced from all other previously sampled trees. Our proposed presence-absence sampling methodology applies Bartlett's test to a hypothesis test developed using an empirical mean-proportion relationship coupled with a spatial, statistical model of pest populations, with spatial correlation mitigated via the aforementioned tree-selection rule. We demonstrate the effectiveness of our proposed methodology over a range of parameter estimates appropriate for densities of O. perseae that would be observed in avocado orchards in California.     

Functional constraints on tooth morphology in carnivorous mammals

ABSTRACT: BACKGROUND: The range of potential morphologies resulting from evolution is limited by complex interacting processes, ranging from development to function. Quantifying these interactions is important for understanding adaptation and convergent evolution. Using three-dimensional reconstructions of carnivoran and dasyuromorph tooth rows, we compared statistical models of the relationship between tooth row shape and the opposing tooth row, a static feature, as well as measures of mandibular motion during chewing (occlusion), which are kinetic features. This is a new approach to quantifying functional integration because we use measures of movement and displacement, such as the amount the mandible translates laterally during occlusion, as opposed to conventional morphological measures, such as mandible length and geometric landmarks. By sampling two distantly related groups of ecologically similar mammals, we study carnivorous mammals in general rather than a specific group of mammals. RESULTS: Statistical model comparisons demonstrate that the best performing models always include some measure of mandibular motion, indicating that functional and statistical models of tooth shape as purely a function of the opposing tooth row are too simple and that increased model complexity provides a better understanding of tooth form. The predictors of the best performing models always included the opposing tooth row shape and a relative linear measure of mandibular motion. CONCLUSIONS: Our results provide quantitative support of long-standing hypotheses of tooth row shape as being influenced by mandibular motion in addition to the opposing tooth row. Additionally, this study illustrates the utility and necessity of including kinetic features in analyses of morphological integration.