A new approach to weighting and inference in sample surveys

The validity of design-based inference is not dependent on anymodel assumption. However, it is well known that estimatorsderived through design-based theory may be inefficient for theestimation of population totals when the design weights areweakly related to the variables of interest and have widelydispersed values. We propose estimators that have the potentialto improve the efficiency of any estimator derived under thedesign-based theory. Our main focus is limited to the improvementof the Horvitz–Thompson estimator, but we also discussthe extension to calibration estimators. The new estimatorsare obtained by smoothing design or calibration weights usingan appropriate model. Our approach to inference requires themodelling of only one variable, the weight, and it leads toa single set of smoothed weights in multipurpose surveys. Thisis to be contrasted with other model-based approaches, suchas the prediction approach, in which it is necessary to postulateand validate a model for each variable of interest leading potentiallyto variable-specific sets of weights. Our proposed approachis first justified theoretically and then evaluated througha simulation study.     

Estimation of fish abundance indices based on scientific research trawl surveys

The fish abundance index over an ocean region is defined here to be the integral of expected catch per unit effort (CPUE), approximated by the sum of expected CPUE over grid squares. When trawl surveys are done within grid squares selected according to a probability sampling design, several other sources of variation such as the fish population dynamics and the catching process are also involved. In such situations model-assisted methods for estimating abundance, assessed under both design and model perspectives, have some advantages over purely design-based methods such as the Horvitz-Thompson (HT) estimator or purely model-based prediction approaches. This article develops model-assisted empirical likelihood (EL) methods via loglinear regression and nonparametric smoothing. The methods are applied to grid surveys of the Grand Bank region carried out annually by Fishery Products International from 1996 through 2002. The HT and EL methods produce similar point estimates of abundance indices. Simulation results, however, indicate that the EL estimator under local linear smoothing is associated with smaller standard errors.     

A Jackknife Variance Estimator for Unistage Stratified Samples with Unequal Probabilities

Existing jackknife variance estimators used with sample surveyscan seriously overestimate the true variance under unistagestratified sampling without replacement with unequal probabilities.A novel jackknife variance estimator is proposed which is asnumerically simple as existing jackknife variance estimators.Under certain regularity conditions, the proposed variance estimatoris consistent under stratified sampling without replacementwith unequal probabilities. The high entropy regularity conditionnecessary for consistency is shown to hold for the Rao–Sampforddesign. An empirical study of three unequal probability samplingdesigns supports our findings.     

A practical sampling design for acoustic surveys of bats

Acoustic surveys are widely used for describing bat occurrence and activity patterns and are increasingly important for addressing concerns for habitat management, wind energy, and disease on bat populations. Designing these surveys presents unique challenges, particularly when a probabilistic sample is required for drawing inference to unsampled areas. Sampling frame errors and other logistical constraints often require survey sites to be dropped from the sample and new sites added. Maintaining spatial balance and representativeness of the sample when these changes are made can be problematic. Spatially balanced sampling designs recently developed to support aquatic surveys along rivers provide solutions to a number of practical challenges faced by bat researchers and allow for sample site additions and deletions, support unequal-probability selection of sites, and provide an approximately unbiased local neighborhood-weighted variance estimator that is efficient for spatially structured populations such as is typical for bats. We implemented a spatially balanced design to survey canyon bat (Parastrellus hesperus) activity along a stream network. The spatially balanced design accommodated typical logistical challenges and yielded a 25% smaller estimated standard error for the mean activity level than the usual simple random sampling estimator. Spatially balanced designs have broad application to bat research and monitoring programs and will improve studies relying on model-based inference (e.g., occupancy models) by providing flexibility and protection against violations of the independence assumption, even if design-based estimators are not used. Our approach is scalable and can be used for pre- and post-construction surveys along wind turbine arrays and for regional monitoring programs. © 2011 The Wildlife Society.     

Estimating the encounter rate variance in distance sampling

Summary .  The dominant source of variance in line transect sampling is usually the encounter rate variance. Systematic survey designs are often used to reduce the true variability among different realizations of the design, but estimating the variance is difficult and estimators typically approximate the variance by treating the design as a simple random sample of lines. We explore the properties of different encounter rate variance estimators under random and systematic designs. We show that a design-based variance estimator improves upon the model-based estimator of Buckland et al. (2001, Introduction to Distance Sampling. Oxford: Oxford University Press, p. 79) when transects are positioned at random. However, if populations exhibit strong spatial trends, both estimators can have substantial positive bias under systematic designs. We show that poststratification is effective in reducing this bias.     

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.     

Robustness of ANCOVA in randomized trials with unequal randomization

Randomized trials with continuous outcomes are often analyzed using analysis of covariance (ANCOVA), with adjustment for prognostic baseline covariates. The ANCOVA estimator of the treatment effect is consistent under arbitrary model misspecification. In an article recently published in the journal, Wang et al proved the model-based variance estimator for the treatment effect is also consistent under outcome model misspecification, assuming the probability of randomization to each treatment is 1/2. In this reader reaction, we derive explicit expressions which show that when randomization is unequal, the model-based variance estimator can be biased upwards or downwards. In contrast, robust sandwich variance estimators can provide asymptotically valid inferences under arbitrary misspecification, even when randomization probabilities are not equal.     

Accounting for imperfect detection of groups and individuals when estimating abundance

If animals are independently detected during surveys, many methods exist for estimating animal abundance despite detection probabilities <1. Common estimators include double‐observer models, distance sampling models and combined double‐observer and distance sampling models (known as mark‐recapture‐distance‐sampling models; MRDS). When animals reside in groups, however, the assumption of independent detection is violated. In this case, the standard approach is to account for imperfect detection of groups, while assuming that individuals within groups are detected perfectly. However, this assumption is often unsupported. We introduce an abundance estimator for grouped animals when detection of groups is imperfect and group size may be under‐counted, but not over‐counted. The estimator combines an MRDS model with an N‐mixture model to account for imperfect detection of individuals. The new MRDS‐Nmix model requires the same data as an MRDS model (independent detection histories, an estimate of distance to transect, and an estimate of group size), plus a second estimate of group size provided by the second observer. We extend the model to situations in which detection of individuals within groups declines with distance. We simulated 12 data sets and used Bayesian methods to compare the performance of the new MRDS‐Nmix model to an MRDS model. Abundance estimates generated by the MRDS‐Nmix model exhibited minimal bias and nominal coverage levels. In contrast, MRDS abundance estimates were biased low and exhibited poor coverage. Many species of conservation interest reside in groups and could benefit from an estimator that better accounts for imperfect detection. Furthermore, the ability to relax the assumption of perfect detection of individuals within detected groups may allow surveyors to re‐allocate resources toward detection of new groups instead of extensive surveys of known groups. We believe the proposed estimator is feasible because the only additional field data required are a second estimate of group size.     

Ranked set sampling for replicated sampling designs

In practical ecological sampling studies, a certain design (such as plot sampling or line-intercept sampling) is usually replicated more than once. For each replication, the Horvitz-Thompson estimation of the objective parameter is considered. Finally, an overall estimator is achieved by averaging the single Horvitz-Thompson estimators. Because the design replications are drawn independently and under the same conditions, the overall estimator is simply the sample mean of the Horvitz-Thompson estimators under simple random sampling. This procedure may be wisely improved by using ranked set sampling. Hence, we propose the replicated protocol under ranked set sampling, which gives rise to a more accurate estimation than the replicated protocol under simple random sampling.     

Using Inverse Probability Bootstrap Sampling to Eliminate Sample Induced Bias in Model Based Analysis of Unequal Probability Samples

In ecology, as in other research fields, efficient sampling for population estimation often drives sample designs toward unequal probability sampling, such as in stratified sampling. Design based statistical analysis tools are appropriate for seamless integration of sample design into the statistical analysis. However, it is also common and necessary, after a sampling design has been implemented, to use datasets to address questions that, in many cases, were not considered during the sampling design phase. Questions may arise requiring the use of model based statistical tools such as multiple regression, quantile regression, or regression tree analysis. However, such model based tools may require, for ensuring unbiased estimation, data from simple random samples, which can be problematic when analyzing data from unequal probability designs. Despite numerous method specific tools available to properly account for sampling design, too often in the analysis of ecological data, sample design is ignored and consequences are not properly considered. We demonstrate here that violation of this assumption can lead to biased parameter estimates in ecological research. In addition, to the set of tools available for researchers to properly account for sampling design in model based analysis, we introduce inverse probability bootstrapping (IPB). Inverse probability bootstrapping is an easily implemented method for obtaining equal probability re-samples from a probability sample, from which unbiased model based estimates can be made. We demonstrate the potential for bias in model-based analyses that ignore sample inclusion probabilities, and the effectiveness of IPB sampling in eliminating this bias, using both simulated and actual ecological data. For illustration, we considered three model based analysis tools—linear regression, quantile regression, and boosted regression tree analysis. In all models, using both simulated and actual ecological data, we found inferences to be biased, sometimes severely, when sample inclusion probabilities were ignored, while IPB sampling effectively produced unbiased parameter estimates.     

Statistical inference on accumulation curves for inventorying forest diversity: A design-based critical look

Abstract

Statistical inference on accumulation curves is considered from a design-based perspective. Preliminaries on probabilistic sampling of plants and species are given, emphasizing the fundamental role of independent replications of the sampling scheme. The role of rarefaction curves as a tool for making inference on the effectiveness of the sampling procedures to compile accurate species lists is outlined. Design-based and model-based inference are discussed and compared. Some future developments for design-based inference are considered.     

Proportional hazards regression for cancer studies

Summary.   There has been some recent work in the statistical literature for modeling the relationship between the size of cancers and probability of detecting metastasis, i.e., aggressive disease. Methods for assessing covariate effects in these studies are limited. In this article, we formulate the problem as assessing covariate effects on a right-censored variable subject to two types of sampling bias. The first is the length-biased sampling that is inherent in screening studies; the second is the two-phase design in which a fraction of tumors are measured. We construct estimation procedures for the proportional hazards model that account for these two sampling issues. In addition, a Nelson–Aalen type estimator is proposed as a summary statistic. Asymptotic results for the regression methodology are provided. The methods are illustrated by application to data from an observational cancer study as well as to simulated data.     

Variance estimation for systematic designs in spatial surveys

Summary In spatial surveys for estimating the density of objects in a survey region, systematic designs will generally yield lower variance than random designs. However, estimating the systematic variance is well known to be a difficult problem. Existing methods tend to overestimate the variance, so although the variance is genuinely reduced, it is over‐reported, and the gain from the more efficient design is lost. The current approaches to estimating a systematic variance for spatial surveys are to approximate the systematic design by a random design, or approximate it by a stratified design. Previous work has shown that approximation by a random design can perform very poorly, while approximation by a stratified design is an improvement but can still be severely biased in some situations. We develop a new estimator based on modeling the encounter process over space. The new “striplet” estimator has negligible bias and excellent precision in a wide range of simulation scenarios, including strip‐sampling, distance‐sampling, and quadrat‐sampling surveys, and including populations that are highly trended or have strong aggregation of objects. We apply the new estimator to survey data for the spotted hyena (Crocuta crocuta) in the Serengeti National Park, Tanzania, and find that the reported coefficient of variation for estimated density is 20% using approximation by a random design, 17% using approximation by a stratified design, and 11% using the new striplet estimator. This large reduction in reported variance is verified by simulation.     

A Note on Balanced Sampling and Robust Genetic Predictors

The concept of balanced sampling is applied to prediction in finite samples using model based inference procedures. Necessary and sufficient conditions are derived for a general linear model with arbitrary covariance structure to yield the expansion estimator as the best linear unbiased predictor for the mean. The analysis is extended to produce a robust estimator for the mean squared error under balanced sampling and the results are discussed in the context of statistical genetics where appropriate sampling produces simple efficient and robust genetic predictors free from unnecessary genetic assumptions.     

Logistic regression analysis of two-phase studies using generalized method of moments

Two-phase designs can reduce the cost of epidemiological studies by limiting the ascertainment of expensive covariates or/and exposures to an efficiently selected subset (phase-II) of a larger (phase-I) study. Efficient analysis of the resulting data set combining disparate information from phase-I and phase-II, however, can be complex. Most of the existing methods, including semiparametric maximum-likelihood estimator, require the information in phase-I to be summarized into a fixed number of strata. In this paper, we describe a novel method for the analysis of two-phase studies where information from phase-I is summarized by parameters associated with a reduced logistic regression model of the disease outcome on available covariates. We then setup estimating equations for parameters associated with the desired extended logistic regression model, based on information on the reduced model parameters from phase-I and complete data available at phase-II after accounting for nonrandom sampling design. We use generalized method of moments to solve overly identified estimating equations and develop the resulting asymptotic theory for the proposed estimator. Simulation studies show that the use of reduced parametric models, as opposed to summarizing data into strata, can lead to more efficient utilization of phase-I data. An application of the proposed method is illustrated using the data from the U.S. National Wilms Tumor Study.     

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.     

A Note on the Estimation of the Population Mean in Stratified Random Sampling Using Two Auxiliary Variables

In this paper, a two‐phase sampling estimator for a stratified population mean using two auxiliary variables x and z is considered when the stratum mean of x is unknown but that of z is known. The suggested estimator under its optimal condition is found to be more efficient than the one using only x.     

On Double Sampling Ratio-Cum-Product-Type Estimator with Independent Samples

There are two cases in double sampling; case(i) when the second sample is a sub-sample from preliminary large sample, and case(ii) when the second sample is not a sub-sample from the preliminary large sample. Recently SISODIA and DWIVEDI (1981) proposed a ratio cum product-type estimator in double sampling in which they have studied the properties of this estimator under case (i). In this paper, we have made an attempt to study the properties of the same estimator under case (ii). It is found that the estimator is superior than double sampling linear regression estimator, usual ratio estimator, product estimator and among others. The estimator is also compared with simple mean per unit for a given cost of the survey.     

Species Richness and Invasion Vectors: Sampling Techniques and Biases

During a European Union Concerted Action study on species introductions, an intercalibration workshop on ship ballast water sampling techniques considered various phytoplankton and zooplankton sampling methods. For the first time, all the techniques presently in use worldwide were compared using a plankton tower as a model ballast tank spiked with the brine shrimp and oyster larvae while phytoplankton samples were taken simultaneously in the field (Helgoland Harbour, Germany). Three cone-shaped and 11 non-cone shaped plankton nets of different sizes and designs were employed. Net lengths varied from 50 to 300 cm, diameters 9.7–50 cm, and mesh sizes 10–100 μm. Three pumps, a Ruttner sampler, and a bucket previously used in ballast water sampling studies were also compared. This first assessment indicates that for sampling ballast water a wide range of techniques may be needed. Each method showed different results in efficiency and it is unlikely that any of the methods will sample all taxa. Although several methods proved to be valid elements of a hypothetical `tool box' of effective ship sampling techniques. The Ruttner water sampler and the pump P30 provide suitable means for the quantitative phytoplankton sampling, whereas other pumps prevailed during the qualitative trial. Pump P15 and cone-shaped nets were the best methods used for quantitative zooplankton sampling. It is recommended that a further exercise involving a wider range of taxa be examined in a larger series of mesocosms in conjunction with promising treatment measures for managing ballast water. This revised version was published online in July 2006 with corrections to the Cover Date.     

Branch Length Heterogeneity Leads to Nonindependent Branch Length Estimates and Can Decrease the Efficiency of Methods of Phylogenetic Inference

Branch length estimates play a central role in maximum-likelihood (ML) and minimum-evolution (ME) methods of phylogenetic inference. For various reasons, branch length estimates are not statistically independent under ML or ME. We studied the response of correlations among branch length estimates to the degree of among-branch length heterogeneity (BLH) in the model (true) tree. The frequency and magnitude of (especially negative) correlations among branch length estimates were both shown to increase as BLH increases under simulation and analytically. For ML, we used the correct model (Jukes–Cantor). For ME, we employed ordinary least-squares (OLS) branch lengths estimated under both simple p-distances and Jukes–Cantor distances, analyzed with and without an among-site rate heterogeneity parameter. The efficiency of ME and ML was also shown to decrease in response to increased BLH. We note that the shape of the true tree will in part determine BLH and represents a critical factor in the probability of recovering the correct topology. An important finding suggests that researchers cannot expect that different branches that were in fact the same length will have the same probability of being accurately reconstructed when BLH exists in the overall tree. We conclude that methods designed to minimize the interdependencies of branch length estimates (BLEs) may (1) reduce both the variance and the covariance associated with the estimates and (2) increase the efficiency of model-based optimality criteria. We speculate on possible ways to reduce the nonindependence of BLEs under OLS and ML. Received: 9 March 1999 / Accepted: 4 May 1999