Addendum to counting by weighing and the seed testing problem

The subject of counting by weighing as addressed in Ridout & Suntheralingam (1997) is revisited. When the coefficient of variation of the seed weights is known, we show how to modify the seed testing procedure in the aforementioned paper to account for the uncertainty introduced by obtaining a given number of seeds by weight, rather than exact counting. We find that counting by weighing requires a greater number of seeds per sub‐sample than that required by the exact counting method of Ridout & Roberts (1997). Furthermore, the number of seeds per sub‐sample increases with decreasing initial sample size used to estimate mean seed weight.     

Estimation of the seed dispersal kernel from exact identification of source plants

The exact identification of individual seed sources through genetic analysis of seed tissue of maternal origin has recently brought the full analytical potential of parentage analysis to the study of seed dispersal. No specific statistical methodology has been described so far, however, for estimation of the dispersal kernel function from categorical maternity assignment. In this study, we introduce a maximum-likelihood procedure to estimate the seed dispersal kernel from exact identification of seed sources. Using numerical simulations, we show that the proposed method, unlike other approaches, is independent of seed fecundity variation, yielding accurate estimates of the shape and range of the seed dispersal kernel under varied sampling and dispersal conditions. We also demonstrate how an obvious estimator of the dispersal kernel, the maximum-likelihood fit of the observed distribution of dispersal distances to seed traps, can be strongly biased due to the spatial arrangement of seed traps relative to source plants. Finally, we illustrate the use of the proposed method with a previously published empirical example for the animal-dispersed tree species Prunus mahaleb.     

Test-based exact confidence intervals for the difference of two binomial proportions

Confidence intervals are often provided to estimate a treatment difference. When the sample size is small, as is typical in early phases of clinical trials, confidence intervals based on large sample approximations may not be reliable. In this report, we propose test-based methods of constructing exact confidence intervals for the difference in two binomial proportions. These exact confidence intervals are obtained from the unconditional distribution of two binomial responses, and they guarantee the level of coverage. We compare the performance of these confidence intervals to ones based on the observed difference alone. We show that a large improvement can be achieved by using the standardized Z test with a constrained maximum likelihood estimate of the variance.     

Exact analysis of dose response for multiple correlated binary outcomes

The neurotoxicity of a substance is often tested using animal bioassays. In the functional observational battery, animals are exposed to a test agent and multiple outcomes are recorded to assess toxicity, using approximately 40 animals measured on up to 30 different items. This design gives rise to a challenging statistical problem: a large number of outcomes for a small sample of subjects. We propose an exact test for multiple binary outcomes, under the assumption that the correlation among these items is equal. This test is based upon an exponential model described by Molenberghs and Ryan (1999, Environmetrics 10, 279-300) and extends the methods developed by Corcoran et al. (2001, Biometrics 57, 941-948) who developed an exact test for exchangeably correlated binary data for groups (clusters) of correlated observations. We present a method that computes an exact p-value testing for a joint dose-response relationship. An estimate of the parameter for dose response is also determined along with its 95% confidence bound. The method is illustrated using data from a neurotoxicity bioassay for the chemical perchlorethylene.     

Nondestructive, Stereological Estimation of Canopy Surface Area

Summary .  We describe a stereological procedure to estimate the total leaf surface area of a plant canopy in vivo, and address the problem of how to predict the variance of the corresponding estimator. The procedure involves three nested systematic uniform random sampling stages: (i) selection of plants from a canopy using the  smooth fractionator , (ii) sampling of leaves from the selected plants using the  fractionator , and (iii) area estimation of the sampled leaves using  point counting . We apply this procedure to estimate the total area of a chrysanthemum ( Chrysanthemum morifolium L. ) canopy and evaluate both the time required and the precision of the estimator. Furthermore, we compare the precision of point counting for three different grid intensities with that of several standard leaf area measurement techniques. Results showed that the precision of the plant leaf area estimator based on point counting is high. Using a grid intensity of 1.76 cm²/point we estimated plant and canopy surface areas with accuracies similar to or better than those obtained using image analysis and a commercial leaf area meter. For canopy surface areas of approximately 1 m² (10 plants), the fractionator leaf approach with sampling fraction equal to 1/9 followed by point counting using a 4.3 cm²/point grid produced a coefficient of error of less than 7%. The  smooth fractionator  can be used to ensure that the additional contribution to the estimator variance due to between-plant variability is small.     

Pleistocene megafaunal extinctions and the functional loss of long‐distance seed‐dispersal services

Pleistocene extinctions affected mainly large‐bodied animals, determining the loss or changes in numerous ecological functions. Evidence points to a central role of many extinct megafauna herbivores as seed dispersers. An important step in understanding the legacy of extinct mutualistic interactions is to evaluate the roles and effectiveness of megafauna herbivores in seed dispersal. Here we use morphological and ecophysiological allometries to estimate both quantitative and qualitative aspects of seed‐dispersal services likely provided by extinct megafauna. We developed a mechanistic model that encompasses four stages of seed dispersal – seed ingestion, gut retention, animal movement, and seed deposition. We estimate seed‐dispersal kernels through simulations to infer the role of Pleistocene megafauna in promoting long‐distance dispersal and examine how seed dispersal was affected by extinctions. Simulations suggest extinct large‐bodied frugivores would frequently disperse large seeds over a thousand meters, whereas smaller‐bodied frugivores are more likely to deposit the seeds over a few hundred meters. Moreover, events of long‐distance seed dispersal by the extinct megafauna would be up to ten times longer than long‐distance dispersal by smaller‐sized extant mammals. By estimating the combined distribution of seed dispersal distances considering all large‐bodied mammalian frugivores in specific South American Pleistocene assemblages we found that long‐distance dispersal contracted by at least two thirds after the megafauna died out. The disruption of long‐distance dispersal is expected to have consequences for recruitment, spatial and genetic structure of plant populations, population persistence and community composition. Promoting long‐distance seed dispersal was one among other salient features of extinct Pleistocene megafauna that reveal their influence on natural ecosystems. Modeling the consequences of megafaunal extinctions can offer quantitative predictions on the consequences of ongoing defaunation to plant populations and ecological communities.     

Strategies for the development of a peptide computer

MOTIVATION: We devise a computational model using protein-protein interactions. RESULTS: Peptide-antibody interactions can be used to perform a large number of small logical operations in parallel. We show for example how a sequence of operations can be used to compare the number of occurrences of an element in two sets and how to estimate the number of occurrences of an element in a set. Similar to DNA-computing, these techniques could in principle be extended to solve instances of NP-complete problems. We give as an example a procedure to solve examples of the satisfiability problem.     

Parameter estimation following an adaptive treatment selection trial design

Two-stage, drop-the-losers designs for adaptive treatment selection have been considered by many authors. The distributions of conditional sufficient statistics and the Rao-Blackwell technique were used to obtain an unbiased estimate and to construct an exact confidence interval for the parameter of interest. In this paper, we characterize the selection process from a binomial drop-the-losers design using a truncated binomial distribution. We propose a new estimator and show that it is asymptotically consistent with a large sample size in either the first stage or the second stage. Supported by simulation analyses, we recommend the new estimator over the naive estimator and the Rao-Blackwell-type estimator based on its robustness in the finite-sample setting. We frame the concept as a simple and easily implemented procedure for phase 2 oncology trial design that can be confirmatory in nature, and we use an example to illustrate its application.     

Exact confidence bounds following adaptive group sequential tests

Summary .  We provide a method for obtaining confidence intervals, point estimates, and p-values for the primary effect size parameter at the end of a two-arm group sequential clinical trial in which adaptive changes have been implemented along the way. The method is based on applying the adaptive hypothesis testing procedure of Müller and Schäfer (2001, Biometrics 57, 886–891) to a sequence of dual tests derived from the stage-wise adjusted confidence interval of Tsiatis, Rosner, and Mehta (1984, Biometrics 40, 797–803). In the nonadaptive setting this confidence interval is known to provide exact coverage. In the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage. In general, however, all that can be claimed theoretically is that the coverage is guaranteed to be conservative. Nevertheless, extensive simulation experiments, supported by an empirical characterization of the conditional error function, demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased. No procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting. The methodology is illustrated by an application to a clinical trial of deep brain stimulation for Parkinson's disease.     

Gluttonous predators: how to estimate prey size when there are too many prey

Prey size is an important factor in food consumption. In studies of feeding ecology, prey items are usually measured individually using calipers or ocular micrometers. Among amphibians and reptiles, there are species that feed on large numbers of small prey items (e.g. ants, termites). This high intake makes it difficult to estimate prey size consumed by these animals. We addressed this problem by developing and evaluating a procedure for subsampling the stomach contents of such predators in order to estimate prey size. Specifically, we developed a protocol based on a bootstrap procedure to obtain a subsample with a precision error of at the most 5%, with a confidence level of at least 95%. This guideline should reduce the sampling effort and facilitate future studies on the feeding habits of amphibians and reptiles, and also provide a means of obtaining precise estimates of prey size.     

The role of visual information in numerosity estimation

Mainstream theory suggests that the approximate number system supports our non-symbolic number abilities (e.g. estimating or comparing different sets of items). It is argued that this system can extract number independently of the visual cues present in the stimulus (diameter, aggregate surface, etc.). However, in a recent report we argue that this might not be the case. We showed that participants combined information from different visual cues to derive their answers. While numerosity comparison requires a rough comparison of two sets of items (smaller versus larger), numerosity estimation requires a more precise mechanism. It could therefore be that numerosity estimation, in contrast to numerosity comparison, might rely on the approximate number system. To test this hypothesis, we conducted a numerosity estimation experiment. We controlled for the visual cues according to current standards: each single visual property was not informative about numerosity. Nevertheless, the results reveal that participants were influenced by the visual properties of the dot arrays. They gave a larger estimate when the dot arrays consisted of dots with, on average, a smaller diameter, aggregate surface or density but a larger convex hull. The reliance on visual cues to estimate numerosity suggests that the existence of an approximate number system that can extract numerosity independently of the visual cues is unlikely. Instead, we propose that humans estimate numerosity by weighing the different visual cues present in the stimuli.     

Image correlation spectroscopy of randomly distributed disks

Image correlation spectroscopy (ICS) has been widely used to quantify spatiotemporal distributions of fluorescently labelled cell membrane proteins and receptors. When the membrane proteins are randomly distributed, ICS may be used to estimate protein densities, provided the proteins behave as point-like objects. At high protein area fraction, however, even randomly placed proteins cannot obey Poisson statistics, because of excluded area. The difficulty can arise if the protein effective area is quite large, or if proteins form large complexes or aggregate into clusters. In these cases, there is a need to determine the correct form of the intensity correlation function for hard disks in two dimensions, including the excluded area effects. We present an approximate but highly accurate algorithm for the computation of this correlation function. The correlation function was verified using test images of randomly distributed hard disks of uniform intensity convolved with the microscope point spread function. This algorithm can be readily modified to compute exact intensity correlation functions for any probe geometry, interaction potential, and fluorophore distribution; we show how to apply it to describe a random distribution of large proteins labeled with a single fluorophore.     

Errors in aerial survey count data: Identifying pitfalls and solutions

Accurate estimates of animal abundance are essential for guiding effective management, and poor survey data can produce misleading inferences. Aerial surveys are an efficient survey platform, capable of collecting wildlife data across large spatial extents in short timeframes. However, these surveys can yield unreliable data if not carefully executed. Despite a long history of aerial survey use in ecological research, problems common to aerial surveys have not yet been adequately resolved. Through an extensive review of the aerial survey literature over the last 50 years, we evaluated how common problems encountered in the data (including nondetection, counting error, and species misidentification) can manifest, the potential difficulties conferred, and the history of how these challenges have been addressed. Additionally, we used a double‐observer case study focused on waterbird data collected via aerial surveys and an online group (flock) counting quiz to explore the potential extent of each challenge and possible resolutions. We found that nearly three quarters of the aerial survey methodology literature focused on accounting for nondetection errors, while issues of counting error and misidentification were less commonly addressed. Through our case study, we demonstrated how these challenges can prove problematic by detailing the extent and magnitude of potential errors. Using our online quiz, we showed that aerial observers typically undercount group size and that the magnitude of counting errors increases with group size. Our results illustrate how each issue can act to bias inferences, highlighting the importance of considering individual methods for mitigating potential problems separately during survey design and analysis. We synthesized the information gained from our analyses to evaluate strategies for overcoming the challenges of using aerial survey data to estimate wildlife abundance, such as digital data collection methods, pooling species records by family, and ordinal modeling using binned data. Recognizing conditions that can lead to data collection errors and having reasonable solutions for addressing errors can allow researchers to allocate resources effectively to mitigate the most significant challenges for obtaining reliable aerial survey data.     

Estimation of the Allele Number in a Natural Population by the Method of Isofemale Lines

The number of alleles present in a natural population of unknown structure is estimated using a sequential sampling procedure applied to isofemale lines. Two questions are raised: how many individuals per isofemale line must be assayed and how many isofemale lines must be sampled to get an adequate sample to estimate the number of alleles, at a given risk, of the natural population? On the one hand, we show that when wild females are inseminated once, only two individuals per line are required. On the other hand, the number of isofemale lines that must be sampled depends on the risk chosen of losing an allele, on the number of alleles present in the population and on their drawing probabilities. When the population structure is known, an accurate answer can be provided. For an unknown population structure, one general sequential sampling previously described by J. Rouault and P. Capy is proposed to estimate the number of alleles in the population from data on isofemale lines.     

Effects of burrow condition and seed handling time on hoarding strategies of Edward's long-tailed rat (Leopoldamys edwardsi)

Many hoarding rodents use burrows not only for dwelling and protection from natural enemies, but also for food storage. However, little is known how burrows used by scatter-hoarding animals influence their foraging behaviors. In addition, handling time for a given food item has a fundamental impact on hoarding strategies of these hoarding animals: food items with longer handling time are more likely to be hoarded due to increasing predation risk because the animals spend more time outside their burrows if they consumed such food. By providing with two types of artificial burrows (aboveground vs. underground) and two types of food items (i.e. seeds) with contrasting handling times, we investigated how burrow condition and handling time co-influence hoarding strategies of a key scatter-hoarding rodent, Edward's long-tailed rat (Leopoldamys edwardsi) in large enclosures in southwest China. We found that only a few animals larder-hoarded fewer seeds when only aboveground burrows were available, while over 80% of the animals preferred to use the underground burrows and hoard significantly more seeds in the burrows when both aboveground and underground burrows were provided simultaneously. We also found that seed handling time significantly affected hoarding strategies of the animals: they consumed and/or scatter-hoarded more Camellia oleifera seeds with shorter handling time outside the burrow, but consumed and larder-hoarded more Lithocarpus harlandii seeds with longer handling time in underground burrows. Our study indicates that both burrow types and seed handling time have important impacts on hoarding strategies of scatter-hoarding animals.     

Aging and visual counting

Background

Much previous work on how normal aging affects visual enumeration has been focused on the response time required to enumerate, with unlimited stimulus duration. There is a fundamental question, not yet addressed, of how many visual items the aging visual system can enumerate in a “single glance”, without the confounding influence of eye movements.

Methodology/Principal Findings

We recruited 104 observers with normal vision across the age span (age 21–85). They were briefly (200 ms) presented with a number of well- separated black dots against a gray background on a monitor screen, and were asked to judge the number of dots. By limiting the stimulus presentation time, we can determine the maximum number of visual items an observer can correctly enumerate at a criterion level of performance (counting threshold, defined as the number of visual items at which ≈63% correct rate on a psychometric curve), without confounding by eye movements. Our findings reveal a 30% decrease in the mean counting threshold of the oldest group (age 61–85: ∼5 dots) when compared with the youngest groups (age 21–40: 7 dots). Surprisingly, despite decreased counting threshold, on average counting accuracy function (defined as the mean number of dots reported for each number tested) is largely unaffected by age, reflecting that the threshold loss can be primarily attributed to increased random errors. We further expanded this interesting finding to show that both young and old adults tend to over-count small numbers, but older observers over-count more.

Conclusion/Significance

Here we show that age reduces the ability to correctly enumerate in a glance, but the accuracy (veridicality), on average, remains unchanged with advancing age. Control experiments indicate that the degraded performance cannot be explained by optical, retinal or other perceptual factors, but is cortical in origin.     

Species accumulation curves analysed by a class of null models discovered by Arrhenius

Arrhenius is mainly given credit for his discovery of the log-log relationship between number of species and area, but in 1921 he invented the technique of analysing the data with a null model. This achievement, which Arrhenius explicitly stated was more important than the empirical species–area relationship, has been overlooked in the scientific literature. We suggest here that analysis of large communities should be done by a successive addition of predefined subsets of samples. Plotting the resulting accumulation curves in a common diagram usually will reveal heterogeneity between subsamples. We believe that this is a general feature of a counting process, (over space or time), that keeps record of new objects being generated by an unobservable process. We show this by two nonbiological time series: examination scores at the University of Oslo and stock prices on the Oslo stock market. We then apply these ideas to the analysis of the nested accumulation plots obtained from two unusually large community samples, the benthos of the Norwegian continental shelf and of marine areas of Hong Kong. Discontinuities in the accumulation curves can be explained by changes in habitat heterogeneity. These changes may be interpreted as variations in the parameter values in the class of null models discovered by Arrhenius. By selecting appropriate parameter values, the observed accumulation curves (calculated by an analytical expression giving exact values) can be approximated by null curves. This procedure makes it possible to identifying the major factors responsible for the shape of the curves in the accumulation plots.     

The efficiency of pooling mRNA in microarray experiments

In a microarray experiment, messenger RNA samples are oftentimes pooled across subjects out of necessity, or in an effort to reduce the effect of biological variation. A basic problem in such experiments is to estimate the nominal expression levels of a large number of genes. Pooling samples will affect expression estimation, but the exact effects are not yet known as the approach has not been systematically studied in this context. We consider how mRNA pooling affects expression estimates by assessing the finite-sample performance of different estimators for designs with and without pooling. Conditions under which it is advantageous to pool mRNA are defined; and general properties of estimates from both pooled and non-pooled designs are derived under these conditions. A formula is given for the total number of subjects and arrays required in a pooled experiment to obtain gene expression estimates and confidence intervals comparable to those obtained from the no-pooling case. The formula demonstrates that by pooling a perhaps increased number of subjects, one can decrease the number of arrays required in an experiment without a loss of precision. The assumptions that facilitate derivation of this formula are considered using data from a quantitative real-time PCR experiment. The calculations are not specific to one particular method of quantifying gene expression as they assume only that a single, normalized, estimate of expression is obtained for each gene. As such, the results should be generally applicable to a number of technologies provided sufficient pre-processing and normalization methods are available and applied.     

Counting labeled transitions in continuous-time Markov models of evolution

Counting processes that keep track of labeled changes to discrete evolutionary traits play critical roles in evolutionary hypothesis testing. If we assume that trait evolution can be described by a continuous-time Markov chain, then it suffices to study the process that counts labeled transitions of the chain. For a binary trait, we demonstrate that it is possible to obtain closed-form analytic solutions for the probability mass and probability generating functions of this evolutionary counting process. In the general, multi-state case we show how to compute moments of the counting process using an eigen decomposition of the infinitesimal generator, provided the latter is a diagonalizable matrix. We conclude with two examples that demonstrate the utility of our results. V.N.M. was supported by a Dissertation Year Fellowship from the UCLA Graduate Division. M.A.S. is an Alfred P. Sloan Research Fellow.     

Evaluation of three techniques for the study of harvester ant (Pogonomyrmex spp.) diet

The estimation of an ant's diet is crucial in many ecological studies. Different techniques, which involve different assumptions and field procedures, have been used to estimate the composition of harvester ant diet. In this study, three techniques are compared for the estimation of the diet of Pogonomyrmex rastratus (Mayr), Pogonomyrmex pronotalis (Santschi), and Pogonomyrmex inermis (Forel) in the central Monte desert, Argentina: (1) hand collection of items brought back to the nest by foragers, (2) collection of items with a semiautomated device with pitfall traps, and (3) collection of the discarded material accumulated in middens. The hand collection technique and the collection of middens provided the lowest and the highest number of items, respectively. Midden samples and pitfall traps contained a higher proportion of nonseed items, probably coming from sources other than ants, than hand-collected items. The three techniques provided similar estimations of species richness but a bias against small seeds was detected for P. pronotalis and P. inermis with the hand collection technique, possibly because of the difficulty of collecting small items by hand. The percentage of seed species in the diet obtained with different techniques was positively correlated in the great majority of colonies. Overall, despite their intrinsic differences, the three techniques proved consistent, which constitutes a robustness test for the estimations obtained. In comparative ecological studies, the awareness that results depend on the techniques and their assumptions is particularly important.