Comparison between Hansen-Hurwitz and Horvitz-Thompson estimators for adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design. This sampling design has been shown to be a useful sampling method for parameter estimation of a clustered and scattered population (Roesch, 1993; Smith et al., 1995; Thompson and Seber, 1996). Two estimators, the modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators, are available to estimate the mean or total of a population. Empirical results from previous researches indicate that the modified HT estimator has smaller variance than the modified HH estimator. We analytically compare the properties of these two estimators. Some results are obtained in favor of the modified HT estimator so that practitioners are strongly recommended to use the HT estimator despite easiness of computations for the HH estimator.     

Empirical likelihood confidence intervals for adaptive cluster sampling

Adaptive cluster sampling (ACS) is an efficient sampling design for estimating parameters of rare and clustered populations. It is widely used in ecological research. The modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators based on small samples under ACS have often highly skewed distributions. In such situations, confidence intervals based on traditional normal approximation can lead to unsatisfactory results, with poor coverage properties. Christman and Pontius (Biometrics 56:503–510, 2000) showed that bootstrap percentile methods are appropriate for constructing confidence intervals from the HH estimator. But Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) showed that bootstrap confidence intervals from the HT estimator are even worse than the normal approximation confidence intervals. In this article, we consider two pseudo empirical likelihood functions under the ACS design. One leads to the HH estimator and the other leads to a HT type estimator known as the Hájek estimator. Based on these two empirical likelihood functions, we derive confidence intervals for the population mean. Using a simulation study, we show that the confidence intervals obtained from the first EL function perform as good as the bootstrap confidence intervals from the HH estimator but the confidence intervals obtained from the second EL function perform much better than the bootstrap confidence intervals from the HT estimator, in terms of coverage rate.     

Estimator bias and efficiency for adaptive cluster sampling with order statistics and a stopping rule

Practical problems facing adaptive cluster sampling with order statistics (acsord) are explored using Monte Carlo simulation for three simulated fish populations and two known waterfowl populations. First, properties of an unbiased Hansen-Hurwitz (HH) estimator and a biased alternative Horvitz-Thompson (HT) estimator are evaluated. An increase in the level of population aggregation or the initial sample size increases the efficiencies of the two acsord estimators. For less aggregated fish populations, the efficiencies decrease as the order statistic parameter r (the number of units about which adaptive sampling is carried out) increases; for the highly aggregated fish and waterfowl populations, they increase with r. Acsord is almost always more efficient than simple random sampling for the highly aggregated populations. Positive bias is observed for the HT estimator, with the maximum bias usually occurring at small values of r. Secondly, a stopping rule at the Sth iteration of adaptive sampling beyond the initial sampling unit was applied to the acsord design to limit the otherwise open-ended sampling effort. The stopping rule induces relatively high positive bias to the HH estimator if the level of the population aggregation is high, the stopping level S is small, and r is large. The bias of HT is not very sensitive to the stopping rule and its bias is often reduced by the stopping rule at smaller values of r. For more aggregated populations, the stopping rule often reduces the efficiencies of the estimators compared to the non-stopping-rule scheme, but acsord still remains more efficient than simple random sampling. Despite its bias and lack of theoretical grounding, the HT estimator is usually more efficient than the HH estimator. In the stopping rule case, the HT estimator is preferable, because its bias is less sensitive to the stopping level.     

Comparison of distribution function estimators for forestry applications following variable probability sampling

Practical considerations often motivate employing variable probability sampling designs when estimating characteristics of forest populations. Three distribution function estimators, the Horvitz-Thompson estimator, a difference estimator, and a ratio estimator, are compared following variable probability sampling in which the inclusion probabilities are proportional to an auxiliary variable, X. Relative performance of the estimators is affected by several factors, including the distribution of the inclusion probabilities, the correlation () between X and the response Y, and the position along the distribution function being estimated. Both the ratio and difference estimators are superior to the Horvitz-Thompson estimator. The difference estimator gains better precision than the ratio estimator toward the upper portion of the distribution function, but the ratio estimator is superior toward the lower end of the distribution function. The point along the distribution function at which the difference estimator becomes more precise than the ratio estimator depends on the sampling design, as well as the coefficient of variation of X and . A simple confidence interval procedure provides close to nominal coverage for intervals constructed from both the difference and ratio estimators, with the exception that coverage may be poor for the lower tail of the distribution function when using the ratio estimator.     

Mark-recapture estimators for dual frame population size of prominent nesting structures: the effect of uncertain detection probability

The combined mark-recapture and line transect sampling methodology proposed by Alpizar-Jara and Pollock [Journal of Environmental and Ecological Statistics, 3(4), 311–327, 1996; In Marine Mammal Survey and Assessment Methods Symposium. G.W. Garner, S.C. Amstrup, J.L. Laake, B.F.J. Manly, L.L. McDonald, and D.C. Robertson (Eds.), A.A. Balkema, Rotterdam, Netherlands, pp. 99–114, 1999] is used to illustrate the estimation of population size for populations with prominent nesting structures (i.e., bald eagle nests). In the context of a bald eagle population, the number of nests in a list frame corresponds to a pre-marked sample of nests, and an area frame corresponds to a set of transect strips that could be regularly monitored. Unlike previous methods based on dual frame methodology using the screening estimator [Haines and Pollock (Journal of Environmental and Ecological Statistics, 5, 245–256, 1998a; Survey Methodology, 24(1), 79–88, 1998b)], we no longer need to assume that the area frame is complete (i.e., all the nests in the sampled sites do not need to be seen). One may use line transect sampling to estimate the probability of detection in a sampled area. Combining information from list and area frames provides more efficient estimators than those obtained by using data from only one frame. We derive an estimator for detection probability and generalize the screening estimator. A simulation study is carried out to compare the performance of the Chapman modification of the Lincoln–Petersen estimator to the screening estimator. Simulation results show that although the Chapman estimator is generally less precise than the screening estimator, the latter can be severely biased in presence of uncertain detection. The screening estimator outperforms the Chapman estimator in terms of mean squared error when detection probability is near 1 wheareas the Chapman estimator outperforms the screening estimator when detection probability is lower than a certain threshold value depending on particular scenarios.     

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.     

Regression-type estimators for adaptive two-stage sequential sampling

Adaptive two-stage sequential sampling (ATSSS) design was developed to observe more rare units and gain higher efficiency, in the sense of having a smaller variance estimator, than conventional sampling designs with equal effort for rare and spatially cluster populations. For certain rare populations, incorporating auxiliary variables into a sampling design can further improve the observation of rare units and increase efficiency. In this article, we develop regression-type estimators for ATSSS so that auxiliary variables can be incorporated into the ATSSS design when warranted. Simulation studies on two populations show that the regression-type estimators can significantly increase the efficiency of ATSSS and the detection of more rare units as compared to conventional sampling counterparts. Simulation of sampling of desert shrubs in Inner Mongolia (one of the two populations studied) showed that by incorporating a GIS auxiliary variable into ATSSS with the regression estimators resulted in a gain in efficiency over ATSSS without the auxiliary variable. Further, we found that the use of the GIS auxiliary variable in a conventional two-stage design with a regression estimator did not show a gain in efficiency.     

Ecophysiology of the benthic amphipod Monoporeia affinis in an open-sea area of the northern Baltic Sea: seasonal variations in oxygen consumption and ammonia excretion

Seasonal variations in the oxygen uptake rate ( ) and ammonia excretion rate ( ) of the benthic deposit-feeding amphipod Monoporeia affinis vere studied in an open-sea population of the northern Baltic Sea during three consecutive years (1991 to 1993). In the early summer an increase in the weightspecific by 22% (standard 1-mg individual) was registered; an even more striking change was noted in the , which showed an increase in the early summer of ca. six times the extremely low winter/spring valnes. The metabolic changes observed could not be attributed to temperature, which remained at steadystate throughout the year, nor could seasonal variations in the biochemical composition of the amphipods explain the results. Therefor, it is evident that quality-food abundance following the sedimentation of the spring phytoplankton bloom induces the observed early-summer elevations in and . In turn, the marked decrease in the towards autumn reflects the gradual depletion of quality-food in the benthic environment. Atomic O: N ratios indicated that at the sampling location, M. affinis uses lipid almost exclusively for metabolic energy throughout the year, except in early summer when the use of lipid and protein was observed to be almost equal. Results also indicated a time-lag between sedimentation of the phytoplankton bloom and metabolic response in the amphipod population.     

Sample support and related scale issues in composite sampling

Composite sampling offers the great promise of efficiency and economy in environmental decision making. However, if careful attention is not paid to matching the support of the sample to that required for making the desired decision, the promise is unfulfilled. Obviously this attention must be applied in the design phase of a composite sampling strategy. Less obvious is the potential for alteration of the design sample support during sample collection and assay. The consequences of not paying attention to these aspects of sample design and assay are discussed in this issue paper illustrated with a series of examples taken from the authors consulting experience.     

Estimation of rare and clustered population mean using stratified adaptive cluster sampling

For many clustered populations, the prior information on an initial stratification exists but the exact pattern of the population concentration may not be predicted. Under this situation, the stratified adaptive cluster sampling (SACS) may provide more efficient estimates than the other conventional sampling designs for the estimation of rare and clustered population parameters. For practical interest, we propose a generalized ratio estimator with the single auxiliary variable under the SACS design. The expressions of approximate bias and mean squared error (MSE) for the proposed estimator are derived. Numerical studies are carried out to compare the performances of the proposed generalized estimator over the usual mean and combined ratio estimators under the conventional stratified random sampling (StRS) using a real population of redwood trees in California and generating an artificial population by the Poisson cluster process. Simulation results show that the proposed class of estimators may provide more efficient results than the other estimators considered in this article for the estimation of highly clumped population.

     

The use of composite sampling in contaminated sites—a case study

Simulated composite sampling was carried out using data from a contaminated site. The values obtained by composite sampling were compared with the results obtained using discrete (individual) samples. It is appropriate to use a modified investigation level (MIL) when using composite samples. The MIL is lower than the standard investigation level, (IL). Various MILs were considered in this study. Too low an MIL will indicate that some composite samples require further investigation, when none of the discrete samples comprising the composite would have exceeded the IL. Too high an MIL will result in some discrete samples that exceed the IL being missed. A suggested MIL is IL/ where n is the number of discrete samples in the composite sample. This MIL was found to give few false negatives but many fewer false positives than the IL/n rule. Although this MIL was effective on the test data it could be site specific. Some local areas of high concentration may be missed with composite samples if a lower investigation level is used. These however do not make a large contribution to the health risk because they will have a contaminant level only slightly higher than the IL, and the neighboring samples must have a low concentration of the contaminant. The increased risk due this cause may be more than offset by the higher sampling density made possible through the economies of composite sampling When composite sampling is used as the first phase of an adaptive cluster-sampling scheme, it must be augmented by additional samples to delineate the contaminated area to be cleaned up. Composite sampling can also be effectively used in a clean up unit technique, where a clean up unit is represented by one or more composite samples. Suggestions are given for when composite sampling can be used effectively.     

Adaptive cluster double sampling with post stratification with application to an epiphytic lichen community

The implementation of an adaptive cluster sampling design often becomes logistically challenging because variation in the final sampling effort introduces uncertainty in survey planning. To overcome this drawback, an inexpensive and easy to measure auxiliary variable could be used in a two-phase survey strategy, called adaptive cluster double sampling (Félix-Medina and Thompson in Biometrika 91:877–891, 2004). In this paper, a two-phase sampling strategy is proposed which combines the idea of adaptive cluster double sampling with the principle of post-stratification. In the first-phase an adaptive cluster sample is selected by means of an inexpensive auxiliary variable. Networks from the first phase sampling are then post-stratified according to their size. In the second-phase, the network structure is used to select a subsample of units by means of stratified random sampling. The proposed sampling strategy employs stratification without requiring an a priori delineation of the strata. Indeed, the strata sizes are estimated in the course of the two-phase sampling process. Therefore, it is suitable for situations where stratification is suspected to be efficient but strata cannot be easily delineated in advance. In this framework, a new type of estimator for the population mean which mimics the stratified sampling mean estimator and an estimator of the sampling variance are proposed. The results of a simulation study confirm, as expected, that the use of post-stratification leads to gain in precision for the estimator. The proposed sampling strategy is applied for targeting an epiphytic lichen community Lobarion pulmonariae in a forest area of the Northern Apennines (N-Italy), characterized by several species of conservation concern.     

Improved strategies for coverage estimation by using replicated line-intercept sampling

Coverage, i.e., the area covered by the target attribute in the study region, is a key parameter in many surveys. Coverage estimation is usually performed by adopting a replicated protocol based on line-intercept sampling coupled with a suitable linear homogeneous estimator. Since coverage is a parameter which may be interestingly represented as the integral of a suitable function, improved Monte Carlo strategies for implementing the replicated protocol are introduced in order to achieve estimators with small variance rates. In addition, new specific theoretical results on Monte Carlo integration methods are given to deal with the integrand functions arising in the special coverage estimation setting.

Some comments on design-based line-intersect sampling with segmented transects

Line-intersect sampling based on segmented transects is adopted in many forest inventories to quantify important ecological indicators such as coarse woody debris attributes. By assuming a design-based approach, Affleck, Gregoire and Valentine (2005, Environ Ecol Stat 12:139–154) have recently proposed a sampling protocol for this line-intersect setting and have suggested an estimation method based on linear homogeneous estimators. However, their proposal does not encompass the estimation procedure currently adopted in some national forest inventories. Hence, the present paper aims to introduce a unifying perspective for both methods. Moreover, it is shown that the two procedures give rise to coincident estimators for almost all the usual field applications. Finally, some strategies for efficient segmented-transect replications are considered.

A robust weighted EM algorithm for use-availability data

Wildlife sampling for habitat selection often combines a random background sample with a random sample of used sites, because the background sample could contain too few used sites to be informative for rare species. This approach is referred to as use-availability sampling. Two variants are considered where there is: (1) a random background sample including used and unused sites augmented with a sample of used sites, and (2) a sample of used sites augmented with a contaminated background sample, i.e. use is not recorded. A weighted estimator first proposed by Manski and Lerman (Econometrica 45(8):1977?C1988, 1977) forms the basis for our suggested approach. The weighted estimator has been shown to perform better than the usual unweighted approach with uncontaminated data and mis-specified logit models (Xie and Manski in Sociol Methods Res 17(3):283?C302, 1989). A weighted EM algorithm is developed for use with contaminated background data. We show that the weighted estimator continues to perform well with contaminated data and maintains its robustness to model mis-specification. The weighted estimator has not been previously used for use-availability sampling due to reliance on the assumption that only the intercept is biased, which is valid for a correct logit model. We show that adjusting the intercept may not eliminate the bias with an incorrect logit model. In this case, the weighted estimator is a relatively simple and effective alternative.     

Abundance and biomass of macrobenthos in the vicinity of Carson Submarine Canyon,northwest Atlantic Ocean

The abundance, biomass and mean weight of macrofauna in the Carson Canyon region (Lat. 45°30N, Long. 48°40W) of the Grand Banks were estimated from 40 quantitative infaunal samples taken at 15 stations in June 1980. In contrast to other areas, there were no significant differences in these values between samples taken in and outside of the canyon. Abundance ( = 716 individuals m^-2), biomass ( = 10.9 gm^-2), and mean weight ( = 0.03 g individual^-1) displayed no consistent relationship with depth, but stations grouped according to sediment type had differing dominant taxa. Biomass and abundance at the edge of the Grand Banks are unexpectedly low. Larger animals, including fishes, that forage there must depend more on pelagic than on benthic food sources.     

Physiological responses of Mytilus edulis and Cardium edule to aerial exposure

Physiological responses of two bivalves (Mytilus edulis L. and Cardium edule L.) to intertidal conditions were studied. Specimens were collected from S. W. England in autumn/winter, 1980, and acclimatized to either intertidal or subtidal regimes before measurement of rates of heat dissipation and oxygen uptake during 5 h of air exposure, and rates of , particle clearance, ammonia excretion, and food-absorption efficiency during 7 h of reimmersion. Subtidal individuals were either intermittently or continuously fed in order to distinguish the effects of periodic food supply from the effects of air exposure. Specimens of M. edulis had low aerial rates of (14 to 20% of aquatic rate), and was greater than the energy equivalent of , indicating that they were largely anaerobic. In contrast, C. edule air-gaped and had higher aerial rates of and (50 to 75% of aquatic rate). There were behavioural and metabolic differences in the responses of intertidally and subtidally acclimatized mussels and cockles to air exposure. Intertidal individuals of both species were more quiescent, had lower aerial rates of and , and showed a conditioned response at the expected time of reimmersion. The reduction in aerial rate of was an energy-saving mechanism and the payment of the oxygen-debt within 2 h of reimmersion represented a significant cost. The heat increment associated with feeding and digestion was estimated as 15 to 17% of the oxygen uptake by M. edulis during all stages of recovery. M. edulis adapted to the intertidal regime by reducing its time-averaged aerial and aquatic rates of ammonia excretion. In contrast, C. edule maintained a high aerial and aquatic rate of ammonia excretion. The clearance rates of M. edulis recovered rapidly (0.5 to 1.5 h) following reimmersion, whereas those of C. edule recovered more slowly, particularly for the subtidal individuals following acute exposure (>4 h). There was no evidence of increased clearance rate or absorption efficiency by intertidal individuals to compensate for the loss of feeding time. Intertidally acclimatized individuals of M. edulis and C. edule had more energy available for growth (scope for growth) integrated over a 12 h period and higher growth efficiencies than subtidal individuals subjected to acute air exposure and intermittent feeding regimes. This was the result of reduced aerial and aquatic rates of energy expenditure, a relatively small cost in terms of the payment of oxygen-debt and a rapid recovery of clearance rate following reimmersion.     

Temperature and metabolic rate in sedentary fish from the Antarctic,North Sea and Indo-West Pacific Ocean

Resting metabolic rate was measured in demersal stages of the teleostNotothenia neglecta Nybelin from the South Orkney Islands, Antarctica, from 1985 to 1987. The relationship between and body mass (Mb) conformed to the general relationship , wherea is a proportionality constant andb is the scaling exponent. (mg O₂ h^–1) was found to scale toMb ^{(0.82±0.011)} in the summer (November to April, 1.6 to 1 850 g,n=56) and toMb ^{(0.76±0.013)} in the winter (May to October, 0.9 to 1 850 g,n=57) (values ofb are means ± SD). Although the scaling exponents were significantly different (P<0.01), was similar in the juvenile stages of summer- and winter-caught fish matched for body mass. The effects of activity on oxygen consumption was studied using a Brett respirometer. Adult stages had a factorial aerobic scope for activity of 5.7, which is similar to that reported for demersal fish from temperate latitudes. The effects of temperature on resting metabolism was investigated in fish with similar sedentary lifestyles from the North Sea (Agonus cataphractus andMyoxocephalus scorpius) and the Indo-West Pacific (Paracirrhites forsteri, P. arcatus, Neocirrhites armatus andExallias brevis). Extrapolated values of for the tropical species approached zero at 5 to 10°C. For a standard 50 g fish, for the tropical species at 25°C was in the range 3.4 to 4.4 mg O₂ h^–1, compared with 1.3 mg O₂ h^–1 forNotothenia neglecta at its acclimation temperature. Thus, the maximum metabolic rate of sedentary tropical species at 24°C is likely to be 2 to 4 times higher than inN. neglecta at 0°C. This suggests that the energy available for sustained activity is significantly lower in cold- than in warm-water fish.     

Oxygen consumption inNeomysis americana (Crustacea: Mysidacea), and the effects of naphthalene exposure

Oxygen uptake ( ) of individual mysids was measured in a novel continuous flow respirometer for 24-h periods, and in a sealed chamber respirometer for several hours. Mysids were acclimated 30–100d under conditions which allowed complete life-cycle cultivation. was normalized for mysid size using an unconventional linear regression: log ( ) vs log (body water). Compared to the conventional regression, log ( ) vs log (body mass), the new approach showed two advantages: significantly greater coefficient of determination,r ², and removal of differences in between sexes. Minimum rates of oxygen uptake measured in the continuous flow respirometer were lower than sealed chamber measurements, and may more accurately reflect standard metabolic rate. Hourly data also showed a daily cycle when adjusted to represent a standard size mysid. Exposure to ca 1.1 mg l^-1 naphthalene increased and the range of spontaneous shown by the mysids; exposure to ca 0.1 mgl^-1 naphthalene depressed the range of . Both exposures caused aberrations in the daily cycle.     

Adaptive two-stage one-per-stratum sampling

We briefly describe adaptive cluster sampling designs in which the initial sample is taken according to a Markov chain one-per-stratum design (Breidt, 1995) and one or more secondary samples are taken within strata if units in the initial sample satisfy a given condition C. An empirical study of the behavior of the estimation procedure is conducted for three small artificial populations for which adaptive sampling is appropriate. The specific sampling strategy used in the empirical study was a single random-start systematic sample with predefined systematic samples within strata when the initially sampled unit in that stratum satisfies C. The bias of the Horvitz-Thompson estimator for this design is usually very small when adaptive sampling is conducted in a population for which it is suited. In addition, we compare the behavior of several alternative estimators of the standard error of the Horvitz-Thompson estimator of the population total. The best estimator of the standard error is population-dependent but it is not unreasonable to use the Horvitz-Thompson estimator of the variance. Unfortunately, the distribution of the estimator is highly skewed hence the usual approach of constructing confidence intervals assuming normality cannot be used here.