We give a brief overview of common sampling designs used in a survey setting, and introduce the principal inferential paradigms under which data from complex surveys may be analyzed. In particular, we distinguish between design-based, model-based and model-assisted approaches. Simple examples highlight the key differences between the approaches. We discuss the interplay between inferential approaches and targets of inference and the important issue of variance estimation.

The aim of this paper is twofold. First, three theoretical principles are formalized: randomization, overrepresentation and restriction. We develop these principles and give a rationale for their use in choosing the sampling design in a systematic way. In the model-assisted framework, knowledge of the population is formalized by modelling the population and the sampling design is chosen accordingly. We show how the principles of overrepresentation and of restriction naturally arise from the modelling of the population. The balanced sampling then appears as a consequence of the modelling. Second, a review of probability balanced sampling is presented through the model-assisted framework. For some basic models, balanced sampling can be shown to be an optimal sampling design. Emphasis is placed on new spatial sampling methods and their related models. An illustrative example shows the advantages of the different methods. Throughout the paper, various examples illustrate how the three principles can be applied in order to improve inference.

This paper reviews the design-based, model-assisted approach to using data from a complex survey together with auxiliary information to estimate finite population parameters. A general recipe for deriving model-assisted estimators is presented and design-based asymptotic analysis for such estimators is reviewed. The recipe allows for a very broad class of prediction methods, with examples from the literature including linear models, linear mixed models, nonparametric regression and machine learning techniques.

Weighting is one of the central steps in surveys. The typical weighting process involves three major stages. At the first stage, each unit is assigned a base weight, which is defined as the inverse of its inclusion probability. The base weights are then modified to account for unit nonresponse. At the last stage, the nonresponse-adjusted weights are further modified to ensure consistency between survey estimates and known population totals. When needed, the weights undergo a last modification through weight trimming or weight smoothing methods in order to improve the efficiency of survey estimates. This article provides an overview of the various stages involved in the typical weighting process used by national statistical offices.

In sample surveys, the sample units are typically chosen using a complex design. This may lead to a selection effect and, if uncorrected in the analysis, may lead to biased inferences. To mitigate the effect on inferences of deviations from a simple random sample a common technique is to use survey weights in the analysis. This article reviews approaches to address possible inefficiency in estimation resulting from such weighting.

To improve inferences we emphasize modifications of the basic design-based weight, that is, the inverse of a unit’s inclusion probability. These techniques include weight trimming, weight modelling and incorporating weights via models for survey variables. We start with an introduction to survey weighting, including methods derived from both the design and model-based perspectives. Then we present the rationale and a taxonomy of methods for modifying the weights. We next describe an extensive numerical study to compare these methods. Using as the criteria relative bias, relative mean square error, confidence or credible interval width and coverage probability, we compare the alternative methods and summarize our findings. To supplement this numerical study we use Texas school data to compare the distributions of the weights for several methods. We also make general recommendations, describe limitations of our numerical study and make suggestions for further investigation.

Although selecting a probability sample has been the standard for decades when making inferences from a sample to a finite population, incentives are increasing to use nonprobability samples. In a world of “big data”, large amounts of data are available that are faster and easier to collect than are probability samples. Design-based inference, in which the distribution for inference is generated by the random mechanism used by the sampler, cannot be used for nonprobability samples. One alternative is quasi-randomization in which pseudo-inclusion probabilities are estimated based on covariates available for samples and nonsample units. Another is superpopulation modeling for the analytic variables collected on the sample units in which the model is used to predict values for the nonsample units. We discuss the pros and cons of each approach.

Data from complex surveys are being used increasingly to build the same sort of explanatory and predictive models used in the rest of statistics. Although the assumptions underlying standard statistical methods are not even approximately valid for most survey data, analogues of most of the features of standard regression packages are now available for use with survey data. We review recent developments in the field and illustrate their use on data from NHANES.

Web surveys can be conducted relatively fast and at relatively low cost. However, Web surveys are often conducted with nonprobability samples and, therefore, a major concern is generalizability. There are two main approaches to address this concern: One, find a way to conduct Web surveys on probability samples without losing most of the cost and speed advantages (e.g., by using mixed-mode approaches or probability-based panel surveys). Two, make adjustments (e.g., propensity scoring, post-stratification, GREG) to nonprobability samples using auxiliary variables. We review both of these approaches as well as lesser-known ones such as respondent-driven sampling. There are many different ways Web surveys can solve the challenge of generalizability. Rather than adopting a one-size-fits-all approach, we conclude that the choice of approach should be commensurate with the purpose of the study.

Collecting data using probability samples can be expensive, and response rates for many household surveys are decreasing. The increasing availability of large data sources opens new opportunities for statisticians to use the information in survey data more efficiently by combining survey data with information from these other sources. We review some of the work done to date on statistical methods for combining information from multiple data sources, discuss the limitations and challenges for different methods that have been proposed, and describe research that is needed for combining survey estimates.

This article brings attention to some historical developments that gave rise to the Bayes factor for testing a point null hypothesis against a composite alternative. In line with current thinking, we find that the conceptual innovation—to assign prior mass to a general law—is due to a series of three articles by Dorothy Wrinch and Sir Harold Jeffreys (1919, 1921, 1923a). However, our historical investigation also suggests that in 1932, J. B. S. Haldane made an important contribution to the development of the Bayes factor by proposing the use of a mixture prior comprising a point mass and a continuous probability density. Jeffreys was aware of Haldane’s work and it may have inspired him to pursue a more concrete statistical implementation for his conceptual ideas. It thus appears that Haldane may have played a much bigger role in the statistical development of the Bayes factor than has hitherto been assumed.