Sample survey theory and methods: Past, present, and future directions
Section 2. Early landmark contributions: 1920-1960
Kiaer (1897) is perhaps the first to promote sampling (or what was then called the representative method) over complete enumeration (census), although the oldest reference can be traced back to 1000 BC. In the representative method, the objective is for the sample to mirror the parent finite population and this may be achieved either by balanced sampling on known auxiliary totals, through purposive selection or by random sampling leading to equal inclusion probabilities. By the 1920s the representative method was widely used. The International Statistical Institute (ISI) played a vital role by creating an expert committee to report on this method. Bowley’s (1926) contribution to the ISI report includes his fundamental work on stratified random sampling with proportional allocation, leading to equal inclusion probabilities. Bowley (1936) states that the “first application of this principle” of inferring the population from the sample was the 1912 study in Reading. Bowley specified the sampling procedure for that study as a systematic sample from a list of houses. Bowley called the systematic procedure a “pure method of sampling” and stated, “This is literally the method of stratified sampling”. Bowley gives a number of examples where systematic sampling was used after 1912. Bowley (1936) emphasized the importance of a complete frame and equal probabilities of selection. But it was Neyman (1934) who laid the foundations of probability sampling (or design-based approach). He demonstrated that stratified random sampling is preferable to balanced (representative) sampling as it was used then. He also introduced the concept of efficiency and optimal sample allocation, now called Neyman allocation, that minimizes the total size of the sample for a specified precision by relaxing Bowley’s condition of equal inclusion probabilities. In fact, Tchuprow (1923) derived the Neyman allocation ten years earlier, in a paper discovered after the Neyman paper appeared. Neyman (1934) also showed that for large enough samples one could obtain confidence intervals on the population mean of a variable of interest such that the frequency of errors in the confidence statement in repeated sampling does not exceed the limit prescribed in advance, “whatever the unknown properties of the population”. In recent years, balanced sampling, originally advocated by Gini and Galvani, has been refined to incorporate the nice features of both probability sampling and balanced sampling on known auxiliary totals (Deville and Tillé, 2004). The new balanced sampling method is now used in Europe, especially in France, to select samples for establishment surveys. A second method of probability controlled selection is rejective sampling, introduced by Hájek (1964) as a method for controlling the sample size in Poisson sampling. Fuller (2009a) extended the procedure to restrict acceptable samples to the set where estimates of the means of auxiliary variables are close to the population mean.
The 1930s witnessed a rapid growth in demand for socio-economic information, and the advantages of probability sampling in terms of greater scope, reduced cost, and greater speed relative to censuses, were soon recognized worldwide. This led to an increase in number and type of surveys based on probability sampling and covering large populations. Neyman’s probability sampling (or design-based approach) was almost universally accepted and it became a standard tool for empirical research in social sciences and official statistics. It was also recognized that the precision of an estimator is determined largely by the sample size and not by the sampling fraction. The 1940’s saw a number of studies on the properties of systematic sampling for different populations. See Madow and Madow (1944), Cochran (1946), and Yates (1948). Cochran (1977, Chapter 8) is an excellent discussion of systematic sampling, a discussion that makes clear why only model-based estimators of variance are possible. Also see Bellhouse (1988). In the early development of sampling theory, focus was on estimating totals and means and associated sampling errors. Non-sampling errors such as nonresponse, coverage errors, and measurement errors, were largely ignored in theoretical research.
We now list a few important post-Neyman theoretical developments in the design-based approach. Mahalanobis used multi-stage sampling designs for crop surveys in India as early as 1937. His classic 1944 paper (Mahalanobis, 1944) rigorously formulated cost and variance functions for the efficient design of surveys. He was instrumental in creating the National Sample Survey of India, the largest multi-subject continuing survey with full-time staff using personal interviews for socio-economic surveys and physical measurements for crop surveys. Sukhatme, who studied under Neyman, also made pioneering contributions to the design and analysis of large scale agricultural surveys in India, using stratified multi-stage sampling. Classic text books on sampling by Cochran (1953), Deming (1950), Hansen, Hurwitz and Madow (1953), Sukhatme (1954) and Yates (1949) benefited students as well as practitioners.
Survey statisticians at the U.S. Census Bureau, under the leadership of Morris Hansen, made fundamental contributions to sample survey theory and methodology, during the period 1940-1960. This period is regarded as the golden era of the Census Bureau. Hansen and Hurwitz (1943) developed the basic theory of stratified two-stage cluster sampling with one cluster (or primary sampling unit) within each stratum drawn with probability proportional to size (PPS) and then subsampled at a rate to ensure a self-weighting sample (equal overall probabilities of selection). Unequal probability selection of clusters can lead to significant variance reduction by controlling the variability arising from unequal cluster sizes. Another major contribution from the U.S. Census Bureau is the use of rotation sampling with partial replacement of households to handle response burden in surveys repeated over time, such as the monthly U.S. Current Population Survey for measuring unemployment rates. Hansen, Hurwitz, Nisselson and Steinberg (1955) developed simple but efficient composite estimators under rotation sampling. Rotation sampling and composite estimation are widely used in large-scale continuing surveys.
Prior to the 1950s, the primary focus was on estimating population totals and means. Woodruff (1952) of the U.S. Census Bureau developed a unified approach for constructing confidence intervals for quantiles (in particular, the median), applicable to general sampling designs. The procedure remains a cornerstone for quantile estimation (Francisco and Fuller, 1991).
After the consolidation of the basic design-based sampling theory, Hansen, Hurwitz, Marks and Mauldin (1951) and others paid attention to measurement or response errors in survey data. Under additive measurement error models with minimal model assumptions on the observed responses treated as random variables, total variance of an estimator can be decomposed into sampling variance, simple response variance and correlated response variance (CRV) due to interviewers.
Mahalanobis (1946) had developed the method of interpenetrating subsamples for assessing both sampling and interviewer errors. By assigning the subsamples at random to interviewers, both the total variance and the interviewer component can be estimated. The interviewer component can dominate total variance when the number of interviewers is small. To remove the CRV component due to interviewers, self-enumeration by mail was introduced in the 1960 U.S. Census.
Nonresponse in surveys was also addressed in early survey sampling development. Hansen and Hurwitz (1946) proposed two-phase sampling in which the sample is contacted by mail in the first phase and a subsample of nonrespondents is then subjected to personal interview, assuming complete response or negligible nonresponse at the second phase. This method was used recently in Canada when the compulsory long form sample census was replaced by a voluntary National Household Survey. After the change of Government in 2015, the Prime Minister of Canada reinstated the long form census. Two phase sampling is retained but to a lesser extent. The Hansen-Hurwitz two phase sampling method has also been used in other surveys including the American Community Survey.
Attention was also given to inferences for unplanned subpopulations (called domains) such as age-sex groups within a state. Hartley (1959) and Durbin (1958) developed a unified theory for domain estimation applicable to general designs and yet requiring only existing formulae for population totals and means.
Most of the survey sampling theory in the early period was developed by official statisticians while academic researchers, especially in USA, paid little attention to survey sampling. An exception was Iowa State University, where faculty played a leading role from the early stages under the leadership of Cochran, Jessen and Hartley. Another institution making early contribution to survey practice and research is the Survey Research Center at the University of Michigan established in 1947, with Leslie Kish as one of its first members.
In the 1950s formal theoretical frameworks for design-based inference on totals and means were proposed by regarding the sample data as a set of sample labels together with the associated variables of interest. Horvitz and Thompson (1952) derived the well-known estimator with weight inversely proportional to the inclusion probability. Narain (1951) also proposed this estimator. Godambe (1955) developed a general class of linear estimators by letting the sample weight of a unit depend on the label as well as on the labels of the other units in the sample. He then showed that the best linear unbiased estimator does not exist in this general class even under simple random sampling.
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