An under-appreciated difficulty: sampling of plant populations for analysis using molecular markers

Results of studies using molecular markers for determining demographic and genetical population parameters especially in plants or sessile animals under field conditions are strongly dependent on the sampling strategy adopted. There are two critical decisions to make when determining this strategy: (i) what is the unit to be sampled?, (ii) how should units to be sampled in the field be selected? For the first decision, there are two conceptually different approaches: sampling ramets of clonal plants as units (to get information about within-genet parameters, such as genet sizes or numbers) and sampling genets of clonal or non-clonal plants as units (to get information of the genetic structure of the population). For the second decision, it is critically important to make the goal of the study explicit. We argue that in this case fully random sampling is needed only when an estimate of the true value of the population parameter is needed; if a comparison between populations is the goal, however, other sampling schemes may be adopted. The efficiency of different types of sampling strategies to recover relative values in a spatially extended population is studied by means of a spatially explicit simulation model. The results show that a regular pattern of sampling is best for obtaining information on genet sizes or inbreeding coefficients; in contrast, random or hierarchical sampling strategies are better for obtaining information on parameters that are based on comparison of pairs of individuals, such as distribution of genet sizes or autocorrelation in genetic structure. A set of recommendations is provided for designing a good sampling strategy.     

On a Simple Procedure of Inclusion Probability Proportional to Sizes (IPPS) Sampling

A simple method of inclusion probability proportional to sizes is proposed for samples of size three units. It is shown that the variance of the HORVITZ -THOMPSON estimator based on the proposed sampling scheme is uniformly smaller than that of the customary estimator used in the probability proportional to sizes with replacement sampling. Further, its performance over RAO -HARTLEY -COCHRAN and SAMPFORD sampling schemes has been studied empirically for some of the natural populations.     

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.     

Effect of sampling scale on the assessment of epiphytic bacterial populations

Bacterial populations on above-ground plant surfaces were estimated at three different biological scales, including leaflet disks, entire leaflets, and whole plants. The influence of sample scale on the estimation of mean bacterial population size per unit and per gram and on the variability among sampling units was quantified at each scale. Populations were highly variable among sampling units at every scale examined, suggesting that there is no optimal scale at which sample variance is reduced. The distribution of population sizes among sample units was sometimes, but not consistently, described by the lognormal. Regardless of the sampling scale, expression of population sizes on a per gram basis may not reduce variance, because population size was not generally a function of sample unit weight within any single sampling scale. In addition, the data show that scaling populations on a per gram basis does not provide a useful means of comparing population estimates from samples taken at different scales. The implications of these results for designing sampling strategies to address specific issues in microbial ecology are discussed. Correspondence to: L.L. Kinkel     

Variance and spatial scales in a tropical rain forest: changing the size of sampling units

The size of a sampling unit has a critical effect on our perception of ecological phenomena; it influences the variance and correlation structure estimates of the data. Classical statistical theory works well to predict the changes in variance when there is no autocorrelation structure, but it is not applicable when the data are spatially autocorrelated. Geostatistical theory, on the other hand, uses analytical relationships to predict the variance and autocorrelation structure that would be observed if a survey was conducted using sampling units of a different size. To test the geostatistical predictions, we used information about individual tree locations in the tropical rain forest of the Pasoh Reserve, Malaysia. This allowed us to simulate and compare various sampling designs. The original data were reorganised into three artificial data sets, computing tree densities (number of trees per square meter in each quadrat) corresponding to three quadrat sizes (5×5, 10×10 and 20×20 m⁽²⁾). Based upon the 5×5 m⁽²⁾ data set, the spatial structure was modelled using a random component (nugget effect) plus an exponential model for the spatially structured component. Using the within-quadrat variances inferred from the variogram model, the change of support relationships predicted the spatial autocorrelation structure and new variances corresponding to 10×10 m⁽²⁾ and 20×20 m⁽²⁾ quadrats. The theoretical and empirical results agreed closely, while the classical approach would have largely underestimated the variance. As quadrat size increases, the range of the autocorrelation model increases, while the variance and proportion of noise in the data decrease. Large quadrats filter out the spatial variation occurring at scales smaller than the size of their sampling units, thus increasing the proportion of spatially structured component with range larger than the size of the sampling units.     

A Note on RAO's Inclusion Probability Proportional to size Sampling Scheme

A sufficient condition that the variance of HORVITZ -THOMPSON estimator for RAO 's (1965) inclusion probability proportional to sizes sampling scheme of selecting two units is uniformly smaller than that of RAO , HARTLEY and COCHRAN (1962) estimator has been obtained.     

Developing optimum sample size and multistage sampling plans for Lobesia botrana (Lepidoptera: Tortricidae) larval infestation and injury in northern Greece

The purpose of this research was to quantify the spatial pattern and develop a sampling program for larvae of Lobesia botrana Denis and Schiffermüller (Lepidoptera: Tortricidae), an important vineyard pest in northern Greece. Taylor's power law and Iwao's patchiness regression were used to model the relationship between the mean and the variance of larval counts. Analysis of covariance was carried out, separately for infestation and injury, with combined second and third generation data, for vine and half-vine sample units. Common regression coefficients were estimated to permit use of the sampling plan over a wide range of conditions. Optimum sample sizes for infestation and injury, at three levels of precision, were developed. An investigation of a multistage sampling plan with a nested analysis of variance showed that if the goal of sampling is focusing on larval infestation, three grape clusters should be sampled in a half-vine; if the goal of sampling is focusing on injury, then two grape clusters per half-vine are recommended.     

A nonparametric mean estimator for judgment poststratified data

Summary .   MacEachern, Stasny, and Wolfe (2004, Biometrics 60 , 207–215) introduced a data collection method, called judgment poststratification (JPS), based on ideas similar to those in ranked set sampling, and proposed methods for mean estimation from JPS samples. In this article, we propose an improvement to their methods, which exploits the fact that the distributions of the judgment poststrata are often stochastically ordered, so as to form a mean estimator using isotonized sample means of the poststrata. This new estimator is strongly consistent with similar asymptotic properties to those in MacEachern et al. (2004) . It is shown to be more efficient for small sample sizes, which appears to be attractive in applications requiring cost efficiency. Further, we extend our method to JPS samples with imprecise ranking or multiple rankers. The performance of the proposed estimators is examined on three data examples through simulation.     

Measures of geographic range size: the effects of sample size

A number of methods have been used for quantifying the sizes of the geographic ranges of species. The consequences of different levels of sampling (the proportion of actual spatial occurrences) are explored for eight of these, using data on the occurrences of butterfly species on a 10 × 10 km grid across Britain. For all methods, the percentage error of estimation (PEE) decreases with the number of 10 × 10 km squares which a species occupies, most rapidly for extent measures, and more rapidly for area measures than for measures of numbers of units occupied. The rate of decline in PEE itself falls as sampling effort increases. At a given sampling level, rank correlations between range sizes measured by different methods are generally high, but there is no consistent change in the magnitude of these correlations as the level of sampling increases. The composition of the set of species with the smallest range sizes changes with the level of sampling.     

Ranked set sampling: cost and optimal set size

McIntyre (1952, Australian Journal of Agricultural Research 3, 385-390) introduced ranked set sampling (RSS) as a method for improving estimation of a population mean in settings where sampling and ranking of units from the population are inexpensive when compared with actual measurement of the units. Two of the major factors in the usefulness of RSS are the set size and the relative costs of the various operations of sampling, ranking, and measurement. In this article, we consider ranking error models and cost models that enable us to assess the effect of different cost structures on the optimal set size for RSS. For reasonable cost structures, we find that the optimal RSS set sizes are generally larger than had been anticipated previously. These results will provide a useful tool for determining whether RSS is likely to lead to an improvement over simple random sampling in a given setting and, if so, what RSS set size is best to use in this case.     

Effects of sample size on estimates of population growth rates calculated with matrix models

Background

Matrix models are widely used to study the dynamics and demography of populations. An important but overlooked issue is how the number of individuals sampled influences estimates of the population growth rate (λ) calculated with matrix models. Even unbiased estimates of vital rates do not ensure unbiased estimates of λ–Jensen''s Inequality implies that even when the estimates of the vital rates are accurate, small sample sizes lead to biased estimates of λ due to increased sampling variance. We investigated if sampling variability and the distribution of sampling effort among size classes lead to biases in estimates of λ.

Methodology/Principal Findings

Using data from a long-term field study of plant demography, we simulated the effects of sampling variance by drawing vital rates and calculating λ for increasingly larger populations drawn from a total population of 3842 plants. We then compared these estimates of λ with those based on the entire population and calculated the resulting bias. Finally, we conducted a review of the literature to determine the sample sizes typically used when parameterizing matrix models used to study plant demography.

Conclusions/Significance

We found significant bias at small sample sizes when survival was low (survival = 0.5), and that sampling with a more-realistic inverse J-shaped population structure exacerbated this bias. However our simulations also demonstrate that these biases rapidly become negligible with increasing sample sizes or as survival increases. For many of the sample sizes used in demographic studies, matrix models are probably robust to the biases resulting from sampling variance of vital rates. However, this conclusion may depend on the structure of populations or the distribution of sampling effort in ways that are unexplored. We suggest more intensive sampling of populations when individual survival is low and greater sampling of stages with high elasticities.     

Notes in Case-Control Studies with Matched Pairs under Inverse Sampling

In case-control studies with matched pairs, the traditional point estimator of odds ratio (OR) is well-known to be biased with no exact finite variance under binomial sampling. In this paper, we consider use of inverse sampling in which we continue to sample subjects to form matched pairs until we obtain a pre-determined number (>0) of index pairs with the case unexposed but the control exposed. In contrast to use of binomial sampling, we show that the uniformly minimum variance unbiased estimator (UMVUE) of OR does exist under inverse sampling. We further derive an exact confidence interval of OR in closed form. Finally, we develop an exact test and an asymptotic test for testing the null hypothesis H₀: OR = 1, as well as discuss sample size determination on the minimum required number of index pairs for a desired power at α-level.     

Judgement post-stratification for designed experiments

Summary .   In many scientific studies, information that is not easily translated into covariates is ignored in the analysis. However, this type of information may significantly improve inference. In this research, we apply the idea of judgment post-stratification to utilize such information. Specifically, we consider experiments that are conducted under a completely randomized design. Sets of experimental units are formed, and the units in a set are ranked. Estimation is performed conditional on the sets and ranks. We propose a new estimator for a treatment contrast. We improve the new estimator by Rao–Blackwellization. Asymptotic distribution theory and corresponding inferential procedures for both estimators are developed. Simulation studies quantify the superiority of the new estimators and show their desirable properties for small and moderate sample sizes. The impact of the new techniques is illustrated with data from a clinical trial.     

Bias-corrected variance estimation and hypothesis testing for spatial point and marked point processes using subsampling

Summary We introduce novel regression extrapolation based methods to correct the often large bias in subsampling variance estimation as well as hypothesis testing for spatial point and marked point processes. For variance estimation, our proposed estimators are linear combinations of the usual subsampling variance estimator based on subblock sizes in a continuous interval. We show that they can achieve better rates in mean squared error than the usual subsampling variance estimator. In particular, for n×n observation windows, the optimal rate of n^?2 can be achieved if the data have a finite dependence range. For hypothesis testing, we apply the proposed regression extrapolation directly to the test statistics based on different subblock sizes, and therefore avoid the need to conduct bias correction for each element in the covariance matrix used to set up the test statistics. We assess the numerical performance of the proposed methods through simulation, and apply them to analyze a tropical forest data set.     

Monitoring population size of mammals using a spotlight-count-based abundance index: How to relate the number of counts to the precision?

Abundance indices are widely used to study changes in population size in wildlife management. However, a truly appropriate measure of precision is often lacking in such studies. Statistically, the two crucial issues regarding the use of an abundance index are sampling and observability, which lead one to consider two kinds of errors, namely sampling and observation errors. The purpose of this methodological paper is to relate the number of counts to the precision of an abundance index by introducing the Hansen–Hurwitz–Bershad model which takes into account both sampling and observation errors. We illustrate this statistical approach in the case of a European rabbit (Oryctolagus cuniculus) abundance index based on spotlight counts, for two fixed spatial sampling units located in different ecological contexts. We show (i) that the usual sampling variance estimator is a downward-biased estimator of the total variance of the abundance index, (ii) that the bias of the usual variance estimator does not decrease when increasing the sampling size, (iii) that correlated observation errors may have a dramatic impact on the total variance, especially when the sampling size increases. The acknowledgement that the (pure) sampling variance underestimates the total variance because of observation errors is a statistical result that is neither widely known nor appreciated by most wildlife ecologists. The magnitude of this underestimation may be important and, therefore, observation errors cannot be always considered as a priori negligible in assessing the precision of a count-based abundance index.     

The quantitative LOD score: test statistic and sample size for exclusion and linkage of quantitative traits in human sibships.

We present a test statistic, the quantitative LOD (QLOD) score, for the testing of both linkage and exclusion of quantitative-trait loci in randomly selected human sibships. As with the traditional LOD score, the boundary values of 3, for linkage, and -2, for exclusion, can be used for the QLOD score. We investigated the sample sizes required for inferring exclusion and linkage, for various combinations of linked genetic variance, total heritability, recombination distance, and sibship size, using fixed-size sampling. The sample sizes required for both linkage and exclusion were not qualitatively different and depended on the percentage of variance being linked or excluded and on the total genetic variance. Information regarding linkage and exclusion in sibships larger than size 2 increased as approximately all possible pairs n(n-1)/2 up to sibships of size 6. Increasing the recombination (theta) distance between the marker and the trait loci reduced empirically the power for both linkage and exclusion, as a function of approximately (1-2theta)4.     

Sequential sampling protocols for Spodoptera frugiperda (Lepidoptera: Noctuidae), on Zea mays fields: influence of sampling unit size

Spodoptera frugiperda (J.E. Smith) is a major pest in maize fields in Argentina. However, a sampling method that accounts for spatial pattern and allows reliable pest density estimations is still lacking. This paper addresses the issue of how sampling unit size can influence the characterization of S. frugiperda spatial pattern and the performance of sampling plans. An intensive sampling programme for S. frugiperda larvae was carried out in maize fields from October until March in four growing seasons. On each sampling date, 12 to 20 sampling units were taken at random. Each unit consisted of 1, 2, 4, 6, 8, 10 or 12 consecutive plants along a row. The linearized version of the Taylor's power law (TPL) was fitted to mean and variance estimated for each sampling date and data set through least squares regression. In general, b values from TPL were significantly >1 (P<0.05), indicating an aggregated sampling distribution. Regression slopes (b) ranged from 1.28 to 1.48 in small larvae and from 1.06 to 1.24 in large larvae, indicating stronger clumping in the former. TPL parameters were used to develop constant precision sampling plans. The efficiency of these plans in terms of expected total cost (searching and handling sampling units) was very sensitive to the size of the sampling units with small larvae but not with large larvae. The influence of sampling unit size and spatial pattern on TPL parameters and sampling costs is discussed.     

Realistic global scouting for pests and diseases on cut rose crops

Scouting is considered an essential component of integrated pest management strategies, but most of the techniques, which involve visual assessment, remain too time-consuming for application on a commercial scale. The global scouting method proposed here for greenhouse rose (Rosa spp.) crops combines several rapid visual methods for common pests and diseases, in a single sampling process. A 2-min observation time per sampling unit is required, with two observers. The sampling unit consists of a single stem with its flower and the corresponding basal foliage. A 90-unit regular grid (1 U/6.4 m2) was used, with weekly assessments, including a spatial distribution approach, for pest monitoring. Different grid sizes were simulated and tested with reference data, to determine whether to decrease the number of sampling units. A grid size of 1 U/21 m2 was found to be acceptable, with no significant loss of information. A more realistic and cheaper sampling strategy of this type is more likely to be accepted by growers, increasing the efficiency of crop monitoring and leading to more rational decisions.     

The Curious Anomaly of Skewed Judgment Distributions and Systematic Error in the Wisdom of Crowds

Judgment distributions are often skewed and we know little about why. This paper explains the phenomenon of skewed judgment distributions by introducing the augmented quincunx (AQ) model of sequential and probabilistic cue categorization by neurons of judges. In the process of developing inferences about true values, when neurons categorize cues better than chance, and when the particular true value is extreme compared to what is typical and anchored upon, then populations of judges form skewed judgment distributions with high probability. Moreover, the collective error made by these people can be inferred from how skewed their judgment distributions are, and in what direction they tilt. This implies not just that judgment distributions are shaped by cues, but that judgment distributions are cues themselves for the wisdom of crowds. The AQ model also predicts that judgment variance correlates positively with collective error, thereby challenging what is commonly believed about how diversity and collective intelligence relate. Data from 3053 judgment surveys about US macroeconomic variables obtained from the Federal Reserve Bank of Philadelphia and the Wall Street Journal provide strong support, and implications are discussed with reference to three central ideas on collective intelligence, these being Galton''s conjecture on the distribution of judgments, Muth''s rational expectations hypothesis, and Page''s diversity prediction theorem.     

Bayesian Statistics-Based Procedure for Sampling of Contaminated Sites

A variable density sampling pattern based on Bayesian statistics is presented and compared to a uniform density statistical pattern and a judgmental approach in a real case study. The Bayesian statistics, supported by a software tool, supplied a soil sampling plan similar to the judgmental one, especially for the number of sampling points and their location. It allowed statistical goals to be set and expert judgment to be included in the sampling strategy in a transparent and systematic procedure. For these reasons, it appears quite suitable for inclusion into Quality Assurance Quality Control plans.