Horvitz-Thompson Estimator Research Articles

Graph sampling

We synthesise the existing theory of graph sampling. We propose a formal definition of sampling in finite graphs, and provide a classification of potential graph parameters. We develop a general approach of Horvitz–Thompson estimation to T-stage snowball sampling, and present various reformulations of some common network sampling methods in the literature in terms of the outlined graph sampling theory.

District-Level Crop Yield Estimation Using Calibration Approach

Estimation of major crop yield rates at the district level using calibration estimation technique is reported here when auxiliary information is available at the unit level only for the selected villages within each district and when the sampling design under consideration is two-stage equal probability without replacement. An estimator was developed for the complex sampling design under consideration using the calibration approach. Through evaluation using real data collected from a pilot survey, we found that the proposed calibration estimator performs better than the usual design-based Horvitz-Thompson estimator under two-stage sampling design.

Estimation Under Cross-Classified Sampling With Application to a Childhood Survey

ABSTRACT The cross-classified sampling design consists in drawing samples from a two-dimensional population, independently in each dimension. Such design is commonly used in consumer price index surveys and has been recently applied to draw a sample of babies in the French Longitudinal Survey on Childhood, by crossing a sample of maternity units and a sample of days. We propose to derive a general theory of estimation for this sampling design. We consider the Horvitz–Thompson estimator for a total, and show that the cross-classified design will usually result in a loss of efficiency as compared to the widespread two-stage design. We obtain the asymptotic distribution of the Horvitz–Thompson estimator and several unbiased variance estimators. Facing the problem of possibly negative values, we propose simplified nonnegative variance estimators and study their bias under a super-population model. The proposed estimators are compared for totals and ratios on simulated data. An application on real data from the French Longitudinal Survey on Childhood is also presented, and we make some recommendations. Supplementary materials for this article are available online.

Estimating long-term trends in abundance and survival for nesting flatback turtles in Kakadu National Park, Australia

Flatback turtles Natator depressus are endemic to Australia and Papua New Guinea's tropical oceans and, although the species has an extensive distribution around northern Australia, there are few published long-term abundance trends of nesting populations. We conducted a longterm capture-mark-recapture program on nesting flatback turtles on Field Island in Kakadu National Park, a World Heritage Area that is jointly managed by Aboriginal landowners and the Australian Government, from 2002 to 2013 for between 12 and 20 monitoring days per year. We used a Cormack-Jolly-Seber (CJS) model that accounted for transience and recapture heterogeneity to estimate apparent survival and recapture probability, and estimated abundance using a Horvitz-Thompson type estimator. A total of 257 flatback turtles attempted nesting during that period, averaging 3.68 ± 0.28 (mean ± SE) nesting attempts per night of monitoring. Annual apparent survival of nesting flatback turtles was 0.97 (95% CI = 0.94 to 0.98) and increased relative to body size. Recapture probability averaged 0.38 (95% CI = 0.34 to 0.42) and was influenced by inter-annual climatic variability. The size of the Field Island nesting flatback turtle population ranged from 97 (95% CI = 87 to 106) to 183 (95% CI = 165 to 200) and there was a non-significant trend over 12 yr of monitoring. Understanding long-term population trends of nesting marine turtles is fundamental for management and recovery of these at-risk species.

Estimation when the Covariance Structure of the Variable of Interest is Positive Definite

Abstract Generalized regression (GREG) estimation uses a model that assumes that the values of the variable of interest are not correlated. An extension of the GREG estimator to the case where the vector of interest has a positive definite covariance structure is presented in this article. This extension can be translated to the calibration estimators. The key to this extension lies in a generalization of the Horvitz-Thompson estimator which, in some sense, also assumes that the values of the variable of interest are not correlated. The Godambe-Joshi lower bound is another result which assumes a model with no correlation. This is also generalized to a vector of interest with a positive definite covariance structure, and it is shown that the generalized calibration estimator asymptotically attains this generalized lower bound. Properties of the new estimators are given, and they are compared with the Horvitz-Thompson estimator and the usual calibration estimator. The new estimators are applied to the Canadian Reverse Record Check survey and to the problem of variance estimation.

On Bootstrap Confidence Intervals Associated with Nonparametric Regression Estimators for A Finite Population Total

The precision of an estimator is at times discussed regarding the variance. Usually, the exact value of the variance is unknown. The discussion relies on unknown populace quantities. When a researcher obtains the survey data, an estimate of the variance can, therefore, be calculated. When survey results are presented, it is good practice to provide variance estimates for the estimator used in the study. The estimator of the variance can further be used to construct confidence interval, assuming that the sampling distribution of estimator is approximately normal. This study proposes estimation of standard error and confidence interval for a nonparametric regression estimator for a finite population using bootstrapping method. The idea behind bootstrapping is to carry out computations on the collected data. Computation activity assists in estimating the disparity of statistics that are themselves computed from the same data. The variance of the Nadaraya-Watson estimator is derived, based on bootstrap procedure. This operation has led to the derivation of confidence interval associated with Nadaraya-Watson estimator of the population total. A simulation study has been carried out. The overall conclusion is that the confidence interval associated with Nadaraya-Watson estimator is tighter than all the other estimators (Horvitz-Thompson estimator, Local linear estimator, and Ratio estimator).

Application of Balanced Sampling, Non-Response and Calibrated Estimator

The aim of this paper is to study the interplay between balanced sampling, non-response and calibratedestimator by simulation. The results of seven strategies, embracing a combination of balanced sampling via the cubemethod, simple random cluster sampling, adjustment for non-response, Horvitz–Thompson estimator of the total andcalibration of design weights, are compared. Auxiliary information is used for all strategies at least at one of the stages(sampling or estimation). This auxiliary information consists of indicator variables for sex, age groups and urban/ruralliving area, and their totals. Real Labour Force Survey data of Statistics Lithuania are used for simulation. Bias, varianceand relative mean squared error are measures of accuracy for the comparison of results.

Estimation of the Population Size by Using the One-Inflated Positive Poisson Model

SummaryIn population size estimation, many capture–recapture-type data exhibit a preponderance of ‘1’-counts. This excess of 1s can arise as subjects gain information from the initial capture that provides a desire and ability to avoid subsequent captures. Existing population size estimators that purport to deal with heterogeneity can be much too large in the presence of 1-inflation, which is a specific form of heterogeneity. To deal with the phenomena of excess 1s, we propose the one-inflated positive Poisson model for use as the truncated count distribution in Horvitz–Thompson estimation of the population size.

A comparison of pivotal sampling and unequal probability sampling with replacement

We prove that any implementation of pivotal sampling is more efficient than multinomial sampling. This yields the weak consistency of the Horvitz–Thompson estimator and the existence of a conservative variance estimator. A small simulation study supports our findings.

One-inflation and unobserved heterogeneity in population size estimation.

We present the one-inflated zero-truncated negative binomial (OIZTNB) model, and propose its use as the truncated count distribution in Horvitz-Thompson estimation of an unknown population size. In the presence of unobserved heterogeneity, the zero-truncated negative binomial (ZTNB) model is a natural choice over the positive Poisson (PP) model; however, when one-inflation is present the ZTNB model either suffers from a boundary problem, or provides extremely biased population size estimates. Monte Carlo evidence suggests that in the presence of one-inflation, the Horvitz-Thompson estimator under the ZTNB model can converge in probability to infinity. The OIZTNB model gives markedly different population size estimates compared to some existing truncated count distributions, when applied to several capture-recapture data that exhibit both one-inflation and unobserved heterogeneity.

An extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe

ABSTRACTIn this paper, an extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe is developed. Main new results are presented in theorems. The primary notions of discrete population are transferred to continuous population. First and second order inclusion probabilities for networks are delivered. Horvitz–Thompson estimator for adaptive cluster sampling in continuous universe is constructed. The unbiasedness of the estimator is proven. Variance and unbiased variance estimator are delivered. Finally, the theory is illustrated with an example.

Estimation of mean on the basis of conditional simple random sample

Estimation of the population mean in a finite and fixed population on the basis of the conditional simple random sampling design dependent on order statistics (quantiles) of an auxiliary variable is considered. Properties of the well-known Horvitz-Thompson and ratio type estimators as well as the sample mean are taken into account under the conditional simple random sampling designs. The considered examples of empirical analysis lead to the conclusion that under some additional conditions the proposed estimation strategies based on the conditional simple random sample are usually more accurate than the mean from the simple random sample drawn without replacement.

Estimation of Mean on the Basis of Conditional Simple Random Sample

Abstract Estimation of the population mean in a finite and fixed population on the basis of the conditional simple random sampling design dependent on order statistics (quantiles) of an auxiliary variable is considered. Properties of the well-known Horvitz-Thompson and ratio type estimators as well as the sample mean are taken into account under the conditional simple random sampling designs. The considered examples of empirical analysis lead to the conclusion that under some additional conditions the proposed estimation strategies based on the conditional simple random sample are usually more accurate than the mean from the simple random sample drawn without replacement.

Estimating ungulate abundance while accounting for multiple sources of observation error

ABSTRACTWhether a species is rare or overabundant, accurate estimates of population abundance are essential for the development and assessment of conservation plans and management goals. Aerial surveys are commonly used to estimate population abundance and a variety of methods have been used to account for recognized biases associated with imperfect detection. Rarely addressed, however, is the possibility of recording duplicate observations and the influence of duplicate observations on estimates of abundance. Using data provided by Global Positioning System (GPS)‐collared bison (Bison bison) and helicopter survey paths conducted during 2011–2013 in the Henry Mountains, Utah, USA, we determined whether GPS‐collared bison were observed, missed, or observed multiple times. Using these data, covariates hypothesized to influence observation errors, and generalized linear models, we found the probability of detecting a GPS‐collared bison (study‐wide mean of 95%) was consistently greater for individuals in larger groups and less rugged terrain. We also found that the probability of recording a duplicate observation (study‐wide mean of 4.4%) increased with time since the prior survey observation. By incorporating both types of observation error into a modified Horvitz–Thompson estimator, more accurate estimates of bison abundance can be attained, particularly in years when animals segregate into small groups, utilize terrain that makes them difficult to see, or have ample time to move into areas not yet surveyed and be counted again. Our approach of accounting for multiple sources of observation error can easily be applied to aerial surveys for other species in other systems, and has the potential to improve the rigor of wildlife abundance estimation from aerial surveys. © 2016 The Wildlife Society.

Revisiting Basu's Circus Example: Another Look at the Horvitz-Thompson Estimator

The objective of this article is a critical appraisal of the classical Horvitz-Thompson (HT) estimator used in survey sampling, and examine when and where it is effective. For illustration, we have brought in the hilarious circus example of Basu's [1] where the HT estimator led to a disastrous result. We have pointed out what went wrong with this example and, in the process, have also discussed what one needs for a successful application of the HT estimator. We also provide a model-based interpretation of the HT estimator, and again discuss the success or failure of the HT estimator from a model-based perspective.

Probability-Sampling Approach to Editing

Editing for measurement errors is always part of data processing. In traditional editing, all data records are checked for errors and inconsistencies. In a new way of editing, only the subset with the most important erroneous responses is considered for editing. This approach is applied in selective editing procedures, which have been shown to save resources considerably. However, selective editing lacks a probabilistic basis and the properties of estimators cannot be established using standard methods. In particular,bias properties of the estimator are unknown except for level estimates based on historical data. This paper proposes combining selective editing with an editing procedure based on the traditional probability-sampling framework. The variance of a bias-corrected Horvitz-Thompson estimator is derived and a variance estimator is proposed. The results of a simulation study support the use of the combined editing procedure.

A comparison of small area and calibration estimators via simulation

Domain estimates are typically obtained using calibration estimators that are direct or modified direct. They are direct if they strictly use data within the domain of interest. They are modified direct if they use both data within and outside the domain of interest. An alternative way of producing these estimates is through small area procedures. In this article, we compare the performance of these two approaches via a simulation. The population is generated using a hierarchical model that includes both area effects and unit level random errors. The population is made up of mutually exclusive domains of different sizes, ranging from a small number of units to a large number of units. We select many independent simple random samples of fixed size from the population and compute various estimates for each sample using the available auxiliary information. The estimates computed for the simulation included the Horvitz-Thompson estimator, the synthetic estimator (indirect estimate), calibration estimators, and unit level based estimators (small area estimate). The performance of these estimators is summarized based on their design- based properties.

A Comparison of Small Area and Calibration Estimators Via Simulation

Abstract Domain estimates are typically obtained using calibration estimators that are direct or modified direct. They are direct if they strictly use data within the domain of interest. They are modified direct if they use both data within and outside the domain of interest. An alternative way of producing these estimates is through small area procedures. In this article, we compare the performance of these two approaches via a simulation. The population is generated using a hierarchical model that includes both area effects and unit level random errors. The population is made up of mutually exclusive domains of different sizes, ranging from a small number of units to a large number of units. We select many independent simple random samples of fixed size from the population and compute various estimates for each sample using the available auxiliary information. The estimates computed for the simulation included the Horvitz-Thompson estimator, the synthetic estimator (indirect estimate), calibration estimators, and unit level based estimators (small area estimate). The performance of these estimators is summarized based on their design-based properties.

A balanced sampling and calibration

A balanced sampling design is a design in which Horvitz-Thompson estimators of population totals for a set of auxiliary variables equal the known totals of these variables. On the other hand, calibration is a technique where the modification of design weights occurs in such a way that the new weights, when applied to auxiliary variables, reproduce, i.e. estimate withouterror, the known totals for these variables. The general idea behind balanced sampling and calibration is thus the same — both techniques tend to reproduce known totals of the auxiliary variables. The purpose of the paper is to describe and compare both techniques, considering them as alternatives in achieving the same goal. More attention was devoted to balanced sampling. The algorithm for selecting a sample was illustrated with two numerical examples. The comparison between balanced sampling and calibration, as alternatives, indicates calibration, but the best strategy is to use both methods simultaneously.

Abstract B113: Prevalence of germline BRCA and mismatch repair (MMR) gene mutations in pancreatic cancer

Abstract Background: Germline mutations of BRCA1, BRCA2, MLH1, MSH2, and MSH6 are known to increase risk of pancreatic cancer. However, the prevalence of pathogenic germline mutations of these genes among pancreatic cancer patients is unknown. Accurately characterizing the frequencies of mutations of these genes will inform patient selection for future gene discovery studies and, clinically, foster screening strategies and tailored treatments. Objective: To determine the prevalence of pathogenic germline mutations of BRCA1, BRCA2, MLH1, MSH2, and MSH6 in a cohort of pancreatic cancer patients, and to determine the association between mutation carrier status and personal and family history of cancer. Methods: The Ontario Pancreas Cancer Study (OPCS) enrolls all consenting participants with a diagnosis of pancreatic cancer into a province-wide electronic pathology database. A cross-sectional sample of 300 patients from 715 OPCS probands enrolled between April 2003 and August 2012 were randomly selected from three strata based on a family history of pancreatic, breast, and ovarian cancer. Targeted next-generation sequencing was successfully performed on germline DNA from 291 of the 300 selected probands. Mutations were classified as benign, of unknown significance, or pathogenic by literature and database review. Pathogenic mutations were confirmed with Sanger sequencing. Prevalence estimates representing the whole OPCS were estimated using the Horvitz-Thompson estimator. Univariate analysis determined association between carrier status and clinical covariates, and regression analysis for overall survival. Results: A total of 7 pathogenic germline mutations were identified: 1 BRCA1, 2 BRCA2, 1 MLH1, 2 MSH2, and 1 MSH6. The estimated prevalence of pathogenic mutations in BRCA1 and BRCA2 among probands in this OPCS series was 1.1% (95% confidence interval: 0.1-2.0%); in the mismatch repair genes MLH1, MSH2, and MSH6 it was 1.5% (0.3-2.6%). Both a personal history of colorectal cancer and a first-degree relative with colorectal cancer, breast cancer, or melanoma were significantly associated (p<0.001 and p<0.01, respectively) with MMR mutations. There were no significant differences in tumor location within the pancreas (head/uncinate versus body/tail), clinical nodal status, rates of resection, age at diagnosis, BMI, history of smoking, history of pancreatitis, family history of pancreatic cancer, family history of colorectal cancer, or differences in survival between the MMR or sporadic cohorts. Conclusion: This study is the first to quantify population-based prevalences of germline mutations of BRCA1, BRCA2, MLH1, MSH2, and MSH6 in pancreatic cancer. Surprisingly, the prevalence of pathogenic germline mutations of the MMR genes in pancreatic cancer is higher than expected, and is comparable to that of BRCA1 and BRCA2. The prevalence of the MMR germline mutations is also comparable to the prevalence of MMR germline mutations in colorectal cancer cohorts. Relatives who carry MMR gene mutations can benefit from tailored cancer prevention strategies; thus mutational analysis of MLH1, MSH2, and MSH6 should be included in molecular genetic testing and counseling strategies for pancreatic cancer patients, especially those with a family history of malignancy. Translational studies will follow to determine if stratifying pancreatic cancer risk by familial susceptibility genes, as in colorectal cancer, fosters tailored and cost-effective primary and secondary prevention strategies for carriers. Moreover, unique chemotherapy sensitivity and resistance patterns would be expected for MMR-associated pancreatic cancer, as is emerging for BRCA-associated pancreatic cancer. Citation Format: Iris Selander, Robert Grant, Ashton Connor, Shamini Selvarajah, Ayelet Borgida, Laurent Briollais, Jordan Lerner-Ellis, Spring Holter, Steven Gallinger. Prevalence of germline BRCA and mismatch repair (MMR) gene mutations in pancreatic cancer. [abstract]. In: Proceedings of the AACR Special Conference on Pancreatic Cancer: Innovations in Research and Treatment; May 18-21, 2014; New Orleans, LA. Philadelphia (PA): AACR; Cancer Res 2015;75(13 Suppl):Abstract nr B113.