Complete allocation sampling: an efficient and easily implemented adaptive sampling design

Adaptive sampling designs are becoming increasingly popular in environmental science, particularly for surveying rare and aggregated populations. An adaptive sample is one in which the survey design is modified, or adapted, in some way on the basis of information gained during the survey. There are many different adaptive survey designs that can be used to estimate animal and plant abundance. In adaptive cluster sampling, additional sample effort is allocated during the survey to the immediate neighborhood in which the species is found. In adaptive stratified sampling, additional sample effort is allocated during the survey to strata of high abundance. The appealing feature of these adaptive designs is that the field biologist gets to do what innately seems sensible when working with rare and aggregated populations—field effort is targeted around where the species is observed in the first wave of the survey. However, there are logistical challenges of applying this principle of targeted field effort while remaining in the framework of probability-based sampling. We propose a simplified adaptive survey design that incorporates both targeting field effort and being logistically feasible. We show with a case study population of rockfish that complete allocation stratified sampling is a very efficient design.     

Adaptive line transect sampling

Adaptive line transect sampling offers the potential of improved population density estimation efficiency over conventional line transect sampling when populations are spatially clustered. In adaptive sampling, survey effort is increased when areas of high animal density are located, thereby increasing the number of observations. Its disadvantage is that the survey effort required is not known in advance. We develop an adaptive line transect methodology that, by varying the degree of adaptation, allows total effort to be fixed at the design stage. Relative to conventional line transect surveys, it also provides better survey coverage in the event of disruption in survey effort, e.g., due to poor weather. In analysis, sightings from the adaptive sections are downweighted in proportion to the increase in effort. We evaluate the methodology by simulation and report on surveys of harbor porpoise in the Gulf of Maine, in which the approach was compared with conventional line transect sampling.     

Adaptive Cluster Sampling in the Context of Restoration

Abundance is an important population state variable for monitoring restoration progress. Efficient sampling often proves difficult, however, when populations are sparse and patchily distributed, such as early after restoration planting. Adaptive cluster sampling (ACS) can help by concentrating search effort in high density areas, improving the encounter rate and the ability to detect a population change over time. To illustrate the problem, I determined conventional design sample sizes for estimating abundance of 12 natural populations and 24 recently planted populations (divided among two preserves) of Lupinus perennis L. (wild blue lupine). I then determined the variance efficiency of ACS relative to simple random sampling at fixed effort and cost for 10 additional planted populations in two habitats (field vs. shrubland). Conventional design sample sizes to estimate lupine stem density with 10% or 20% margins of error were many times greater than initial sample size and would require sampling at least 90% of the study area. Differences in effort requirements were negligible for the two preserves and natural versus planted populations. At fixed sample size, ACS equaled or outperformed simple random sampling in 40% of populations; this shifted to 50% after correcting for travel time among sample units. ACS appeared to be a better strategy for inter‐seeded shrubland habitat than for planted field habitat. Restoration monitoring programs should consider adaptive sampling designs, especially when reliable abundance estimation under conventional designs proves elusive.     

Temporally Adaptive Sampling: A Case Study in Rare Species Survey Design with Marbled Salamanders (Ambystoma opacum)

Improving detection rates for elusive species with clumped distributions is often accomplished through adaptive sampling designs. This approach can be extended to include species with temporally variable detection probabilities. By concentrating survey effort in years when the focal species are most abundant or visible, overall detection rates can be improved. This requires either long-term monitoring at a few locations where the species are known to occur or models capable of predicting population trends using climatic and demographic data. For marbled salamanders (Ambystoma opacum) in Massachusetts, we demonstrate that annual variation in detection probability of larvae is regionally correlated. In our data, the difference in survey success between years was far more important than the difference among the three survey methods we employed: diurnal surveys, nocturnal surveys, and dipnet surveys. Based on these data, we simulate future surveys to locate unknown populations under a temporally adaptive sampling framework. In the simulations, when pond dynamics are correlated over the focal region, the temporally adaptive design improved mean survey success by as much as 26% over a non-adaptive sampling design. Employing a temporally adaptive strategy costs very little, is simple, and has the potential to substantially improve the efficient use of scarce conservation funds.     

Atlantic bluefin tuna in the Gulf of Maine,II: precision of sampling designs in estimating seasonal abundance accounting for tuna behaviour

A primary challenge of animal surveys is to understand how to reliably sample populations exhibiting strong spatial heterogeneity. Building upon recent findings from survey, tracking and tagging data, we investigate spatial sampling of a seasonally resident population of Atlantic bluefin tuna in the Gulf of Maine, Northwestern Atlantic Ocean. We incorporate empirical estimates to parameterize a stochastic population model and simulate measurement designs to examine survey efficiency and precision under variation in tuna behaviour. We compare results for random, systematic, stratified, adaptive and spotter-search survey designs, with spotter-search comprising irregular transects that target surfacing schools and known aggregation locations (i.e., areas of expected high population density) based on a priori knowledge. Results obtained show how survey precision is expected to vary on average with sampling effort, in agreement with general sampling theory and provide uncertainty ranges based on simulated variance in tuna behaviour. Simulation results indicate that spotter-search provides the highest level of precision, however, measurable bias in observer-school encounter rate contributes substantial uncertainty. Considering survey bias, precision, efficiency and anticipated operational costs, we propose that an adaptive-stratified sampling alone or a combination of adaptive-stratification and spotter-search (a mixed-layer design whereby a priori information on the location and size of school aggregations is provided by sequential spotter-search sampling) may provide the best approach for reducing uncertainty in seasonal abundance estimates.

Searching for weeds – how to search where the weeds are?

Sampling for rare events, such as a new weed incursion, is not easy. At most of the sample points the species of interest is absent and only occasionally the species is recorded. Very often surveillance and monitoring for rare events (e.g. new weed incursions) is done using local knowledge and a statistical sampling design is not used. The stated reason for this approach is usually because the biodiversity managers knew where to look and didn’t need statistics. Adaptive, unequal probability survey designs can be used in these situations, ensuring both sample effort is focused on locations where there is a high likelihood of a weed being present. Time in the field is spent within locations where weeds are present and minimal time spent where weeds are absent. Any relevant information on where weeds are likely to be found (e.g. local knowledge and expertise) can be used to target survey effort in unequal probability survey designs. The advantage of an adaptive, unequal probability survey design is that not only can field effort be focused on areas where the weeds are thought to be. In addition, important weed parameters can be estimated and reported along with estimates of uncertainty. Weed parameters include the proportion of the total area that weeds are present, the diversity of weed species, the total abundance of weeds, and the total area covered by weeds. With reliable and consistent estimates of these weed parameters (e.g. weed cover or abundance) the efficacy of weed management can be tracked. Over time, with regular reporting of weed cover or abundance, the success (or otherwise) of weed management strategies can be measured.     

Self-designing two-stage trials to minimize expected costs

In the design of clinical trials, the sample size for the trial is traditionally calculated from estimates of parameters of interest, such as the mean treatment effect, which can often be inaccurate. However, recalculation of the sample size based on an estimate of the parameter of interest that uses accumulating data from the trial can lead to inflation of the overall Type I error rate of the trial. The self-designing method of Fisher, also known as the variance-spending method, allows the use of all accumulating data in a sequential trial (including the estimated treatment effect) in determining the sample size for the next stage of the trial without inflating the Type I error rate. We propose a self-designing group sequential procedure to minimize the expected total cost of a trial. Cost is an important parameter to consider in the statistical design of clinical trials due to limited financial resources. Using Bayesian decision theory on the accumulating data, the design specifies sequentially the optimal sample size and proportion of the test statistic's variance needed for each stage of a trial to minimize the expected cost of the trial. The optimality is with respect to a prior distribution on the parameter of interest. Results are presented for a simple two-stage trial. This method can extend to nonmonetary costs, such as ethical costs or quality-adjusted life years.     

Searching for weeds – a GIS tool

Sampling for rare events, such as a new weed incursion, can be surprisingly efficient when adaptive, unequal probability survey designs are used. Spatially explicit habitat models and expert knowledge of weedy species can be used to identify areas of varied survey intensity. We introduce a GIS-based tool that can be used for designing such a survey. The user-friendly tool interfaces (behind the scenes) with the US Environmental Protection Agency’s spatially balanced sampling design functions in R. The functions ensure that the location of the sample points are spatially balanced while at the same time, allowing the user to specify survey intensity in area of special interest (preferred habitats, areas of high conservation value, areas of high public use, etc). We discuss the use of the GIS tool in a case study where we designed a 5-year weed monitoring plan for a local authority in New Zealand. The plan includes ‘over sample’ sites to replace any original sample sites that were impractical or costly to visit. Initial results include estimates of what proportion of the total region has weeds present and an estimate of weed density. More detailed results are produced for specified known weed hot spots, such as areas adjacent to roads and rivers. These estimates are available for all weed species, and for individual species. Because the system is GIS-based, spatial information is stored. Over time, as the weed surveillance and monitoring progresses, regional changes in weed distribution can be tracked, and species and locations that require more targeted weed management can be identified. Further results of such a probability-based design can be used to develop habitat models for predicting future distributions.     

Temporal Patterns of Species Accumulation in a Survey of Lepidoptera in a Beech-Maple Forest

One of the most significant challenges to insect conservation is lack of information concerning species diversity and distribution. Because a complete inventory of all species in an area is virtually impossible, interest has turned to developing statistical techniques to guide sampling design and to estimate total species richness within a site. We used two such techniques, diversity partitioning and non-parametric richness estimation, to determine how variation in sampling effort over time affected species accumulation for a survey of Lepidoptera in an old-growth beech-maple forest. Temporal scaling of sampling effort had significant effects on two measures of species diversity. Increases in species richness were primarily driven by changes in species occurrences with season, while Shannon diversity was largely determined at the scale of individual sampling units (i.e. by spatial effects). Variation in sampling effort affected the values of the two most widely regarded richness estimators (ICE and Chao 2); neither diversity estimator achieved stable values across a range of sampling efforts. Even after 52 trap-nights and accounting for seasonality, rare species (singletons and uniques) remained a significant component of the moth community. To the extent that moth communities in other forest systems are similarly comprised of many rare species, non-parametric richness estimators should be expected to yield variable estimates with increased effort and should only be used to provide a minimum benchmark for predicting the number of species remaining to be sampled. Our results suggest the best strategy for a short-term survey of forest Lepidoptera should emphasize spreading sampling intervals throughout a given year rather than focusing on intensive sampling during a short time period or prolonged sampling over many years.     

Monitoring temporal trends in spatially structured populations: how should sampling effort be allocated between space and time?

Estimating temporal trends in spatially structured populations has a critical role to play in understanding regional changes in biological populations and developing management strategies. Designing effective monitoring programmes to estimate these trends requires important decisions to be made about how to allocate sampling effort among spatial replicates (i.e. number of sites) and temporal replicates (i.e. how often to survey) to minimise uncertainty in trend estimates. In particular, the optimal mix of spatial and temporal replicates is likely to depend upon the spatial and temporal correlations in population dynamics. Although there has been considerable interest in the ecological literature on understanding spatial and temporal correlations in species’ population dynamics, little attention has been paid to its consequences for monitoring design. We address this issue using model‐based survey design to identify the optimal allocation of sampling effort among spatial and temporal replicates for estimating population trends under different levels of spatial and temporal correlation. Based on linear trends, we show that how we should allocate sampling effort among spatial and temporal replicates depends crucially on the spatial and temporal correlations in population dynamics, environmental variation, observation error and the spatial variation in temporal trends. When spatial correlation is low and temporal correlation is high, the best option is likely to be to sample many sites infrequently, particularly when observation error and/or spatial variation in temporal trends are high. When spatial correlation is high and temporal correlation is low, the best option is likely to be to sample few sites frequently, particularly when observation error and/or spatial variation in temporal trends are low. When abundances are spatially independent, it is always preferable to maximise spatial replication. This provides important insights into how spatio‐temporal monitoring programmes should be designed to estimate temporal trends in spatially structured populations.     

Optimal conditional error functions for the control of conditional power

Ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance. If at an interim analysis the available data are used to decide whether the trial is promising enough to be continued, investigators and sponsors often wish to have a high conditional power, which is the probability to reject the null hypothesis given the interim data and the alternative of interest. Under this requirement a design with interim sample size recalculation, which keeps the overall and conditional power at a prespecified value and preserves the overall type I error rate, is a reasonable alternative to a classical group sequential design, in which the conditional power is often too small. In this article two-stage designs with control of overall and conditional power are constructed that minimize the expected sample size, either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter. The presented optimality result applies to trials with and without an interim hypothesis test; in addition, one can account for constraints such as a minimal sample size for the second stage. The optimal designs will be illustrated with an example, and will be compared to the frequently considered method of using the conditional type I error level of a group sequential design.     

Sequential Analysis of Longitudinal Data in a Prospective Nested Case–Control Study

Summary The nested case–control design is a relatively new type of observational study whereby a case–control approach is employed within an established cohort. In this design, we observe cases and controls longitudinally by sampling all cases whenever they occur but controls at certain time points. Controls can be obtained at time points randomly scheduled or prefixed for operational convenience. This design with longitudinal observations is efficient in terms of cost and duration, especially when the disease is rare and the assessment of exposure levels is difficult. In our design, we propose sequential sampling methods and study both (group) sequential testing and estimation methods so that the study can be stopped as soon as the stopping rule is satisfied. To make such a longitudinal sampling more efficient in terms of both numbers of subjects and replications, we propose applying sequential sampling methods to subjects and replications, simultaneously, until the information criterion is fulfilled. This simultaneous sequential sampling on subjects and replicates is more flexible for practitioners designing their sampling schemes, and is different from the classical approaches used in longitudinal studies. We newly define the σ‐field to accommodate our proposed sampling scheme, which contains mixtures of independent and correlated observations, and prove the asymptotic optimality of sequential estimation based on the martingale theories. We also prove that the independent increment structure is retained so that the group sequential method is applicable. Finally, we present results by employing sequential estimation and group sequential testing on both simulated data and real data on children's diarrhea.     

Sampling strategy optimization to increase statistical power in landscape genomics: A simulation‐based approach

An increasing number of studies are using landscape genomics to investigate local adaptation in wild and domestic populations. Implementation of this approach requires the sampling phase to consider the complexity of environmental settings and the burden of logistical constraints. These important aspects are often underestimated in the literature dedicated to sampling strategies. In this study, we computed simulated genomic data sets to run against actual environmental data in order to trial landscape genomics experiments under distinct sampling strategies. These strategies differed by design approach (to enhance environmental and/or geographical representativeness at study sites), number of sampling locations and sample sizes. We then evaluated how these elements affected statistical performances (power and false discoveries) under two antithetical demographic scenarios. Our results highlight the importance of selecting an appropriate sample size, which should be modified based on the demographic characteristics of the studied population. For species with limited dispersal, sample sizes above 200 units are generally sufficient to detect most adaptive signals, while in random mating populations this threshold should be increased to 400 units. Furthermore, we describe a design approach that maximizes both environmental and geographical representativeness of sampling sites and show how it systematically outperforms random or regular sampling schemes. Finally, we show that although having more sampling locations (between 40 and 50 sites) increase statistical power and reduce false discovery rate, similar results can be achieved with a moderate number of sites (20 sites). Overall, this study provides valuable guidelines for optimizing sampling strategies for landscape genomics experiments.     

Using habitat suitability models to sample rare species in high-altitude ecosystems: a case study with Tibetan argali

Models of the distribution of rare and endangered species are important tools for their monitoring and management. Presence data used to build up distribution models can be based on simple random sampling, but this for patchy distributed species results in small number of presences and therefore low precision. Convenience sampling, either based on easily accessible units or a priori knowledge of the species habitat but with no known probability of sampling each unit, is likely to result in biased estimates. Stratified random sampling, with strata defined using habitat suitability models [estimated in the resource selection functions (RSFs) framework] is a promising approach for improving the precision of model parameters. We used this approach to sample the Tibetan argali (Ovis ammon hodgsoni) in Indian Transhimalaya in order to estimate their distribution and to test if it can lead to a significant reduction in survey effort compared to random sampling. We first used an initial sample of argali feeding sites in 2005 and 2006 based on a priori selected vantage points and survey transects. This initial sample was used to build up an initial distribution model. The spatial predictions based on estimated RSFs were then used to define three strata of the study area. The strata were randomly sampled in 2007. As expected, much more presences per hour were obtained in the high quality strata compared to the low quality strata—1.33 obs/h vs. 0.080/h. Furthermore the best models selected on the basis of the prospective sample differed from those using the first a priori sample, suggesting bias in the initial sampling effort. The method therefore has significant implications for decreasing sampling effort in terms of sampling time in the field, especially when dealing with rare species, and removing initial sampling bias.     

Sampling strategies for assessing the overall density of the cassava green mite (Mononychellus tanajoa)

Six different sampling methods to estimate the density of the cassava green mite, Mononychellus tanajoa, are categorized according to whether leaves or leaflets are used as secondary sampling units and whether the number of leaves on the sampled plants are enumerated, estimated from an independent plant sample, or not censused at all. In the last case, sampling can provide information only on the average number of mites per leaf and its variance, while information on stratum sizes is necessary to estimate the mean number of mites per plant as well. It is shown that leaflet-sampling is as reliable as leaf-sampling for the same number of sampling units. When stratum sizes are estimated from a separate plant sample, sampling time may also be reduced, but the estimated mean density and its variance may be biased if mite density and plant size are correlated. Sampling data show that the within-plant variance contributes relatively little to the overall variance of the population density estimates. It points at a sampling strategy in which the number of primary units (plants) is as large as possible at the expense of secondary units (leaflets) per plant. Mean-variance relationships may be applied to estimate sample variances and can be used even when only one leaflet is taken per plant per stratum. An unequal allocation of primary units among strata can increase precision, but the gain is small compared with an equal allocation. Leaf area can be predicted from the length of the longest leaflet and the number of leaflets.     

Evaluating Survey Methods for Monitoring a Rare Vertebrate,the Sonoran Desert Tortoise

Abstract: Effective conservation requires strategies to monitor populations efficiently, which can be especially difficult for rare or elusive species where field surveys require high effort and considerable cost. Populations of many reptiles, including Sonoran desert tortoises (Gopherus agassizii), are challenging to monitor effectively because they are cryptic, they occur at low densities, and their activity is limited both seasonally and daily. We compared efficiency and statistical power of 2 survey methods appropriate for tortoises and other rare vertebrates, line-transect distance sampling and site occupancy. In 2005 and 2006 combined, we surveyed 120 1-km transects to estimate density and 40 3-ha plots 5 times each to estimate occupancy of Sonoran desert tortoises in 2 mountain ranges in southern Arizona, USA. For both mountain ranges combined, we estimated density to be 0.30 adult tortoises/ha (95% CI = 0.17–0.43) and occupancy to be 0.72 (95% CI = 0.56–0.89). For the sampling designs we evaluated, monitoring efforts based on occupancy were 8–36% more efficient than those based on density, when contrasting only survey effort, and 17–30% more efficient when contrasting total effort (surveying, hiking to and from survey locations, and radiotracking). Occupancy had greater statistical power to detect annual declines in the proportion of area occupied than did distance sampling to detect annual declines in density. For example, we estimated that power to detect a 5% annual decline with 10 years of annual sampling was 0.92 (95% CI = 0.75–0.98) for occupancy and 0.43 (95% CI = 0.35–0.52) for distance sampling. Although all sampling methods have limitations, occupancy estimation offers a promising alternative for monitoring populations of rare vertebrates, including tortoises in the Sonoran Desert.     

Evaluating sampling designs by computer simulation: a case study with the Missouri bladderpod

To effectively manage rare populations, accurate monitoring data are critical. Yet many monitoring programs are initiated without careful consideration of whether chosen sampling designs will provide accurate estimates of population parameters. Obtaining accurate estimates is especially difficult when natural variability is high, or limited budgets determine that only a small fraction of the population can be sampled. The Missouri bladderpod, Lesquerella filiformis Rollins, is a federally threatened winter annual that has an aggregated distribution pattern and exhibits dramatic interannual population fluctuations. Using the simulation program SAMPLE, we evaluated five candidate sampling designs appropriate for rare populations, based on 4 years of field data: (1) simple random sampling, (2) adaptive simple random sampling, (3) grid-based systematic sampling, (4) adaptive grid-based systematic sampling, and (5) GIS-based adaptive sampling. We compared the designs based on the precision of density estimates for fixed sample size, cost, and distance traveled. Sampling fraction and cost were the most important factors determining precision of density estimates, and relative design performance changed across the range of sampling fractions. Adaptive designs did not provide uniformly more precise estimates than conventional designs, in part because the spatial distribution of L. filiformis was relatively widespread within the study site. Adaptive designs tended to perform better as sampling fraction increased and when sampling costs, particularly distance traveled, were taken into account. The rate that units occupied by L. filiformis were encountered was higher for adaptive than for conventional designs. Overall, grid-based systematic designs were more efficient and practically implemented than the others. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Accounting for cluster sampling in constructing enumerative sequential sampling plans

Green's sequential sampling plan is widely used in applied entomology. Green's equation can be used to construct sampling stop charts, and a crop can then be surveyed using a simple random sampling (SRS) approach. In practice, however, crops are rarely surveyed according to SRS. Rather, some type of hierarchical design is usually used, such as cluster sampling, where sampling units form distinct groups. This article explains how to make adjustments to sampling plans that intend to use cluster sampling, a commonly used hierarchical design, rather than SRS. The methodologies are illustrated using diamondback moth, Plutella xylostella (L.), a pest of Brassica crops, as an example.     

Pellet group count methods to estimate red deer densities: Precision,potential accuracy and efficiency

Pellet group count methods are key tools for wildlife conservation and management. Therefore, it is essential to clarify the performance and suitability of the different pellet group count methods at distinct realities. This issue has been discussed by other researchers but with inconsistent conclusions, leaving open the necessity of additional studies. Aiming to fill this need, we used a combination of field data and simulation models to evaluate the density estimates, precision and potential accuracy of the results obtained through each pellet group count method, and also the sampling effort and efficiency associated with them. The methods evaluated were standing crop plot counts (FSCP), clearance plot counts (FAR), standing crop strip transect counts (ST) and standing crop line transect counts (LT). Deer density estimates by the four methods were statistically similar at all effort levels simulated. The analyses of CV trends reveal a better precision supplied by FSCP than by FAR for the same effort, while ST and LT yielded comparable values in analogous situations. The time required to make a survey is a key factor in the choice of a field technique. LT with distance sampling was the most efficient method to count pellet groups, while FAR seems to be the less proficient method. Attending to the limitations usually inherent to field surveys, like time, technicians and budget, LT appeared more efficient than the other methods, providing great precision and accuracy in less time. Nevertheless, at high densities and pronounced habitat heterogeneity, FAR became more efficient than FSC, do not requiring the decay rates, allowing accurate estimates in a few months when applied with the proper effort and environmental conditions. This study highlights the importance of carrying out pilot studies and simulation models to evaluate the sampling design prior to its implementation in the field.