Sampling (statistics)

In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt for the samples to represent the population in question. Two advantages of sampling are lower cost and faster data collection than measuring the entire population.

Each observation measures one or more properties (such as weight, location, colour) of observable bodies distinguished as independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling.[1] Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population.[2] Acceptance sampling is used to determine if a production lot of material meets the governing specifications.

Simple random sampling
A visual representation of the sampling process.

Population definition

Successful statistical practice is based on focused problem definition. In sampling, this includes defining the "population" from which our sample is drawn. A population can be defined as including all people or items with the characteristic one wishes to understand. Because there is very rarely enough time or money to gather information from everyone or everything in a population, the goal becomes finding a representative sample (or subset) of that population.

Sometimes what defines a population is obvious. For example, a manufacturer needs to decide whether a batch of material from production is of high enough quality to be released to the customer, or should be sentenced for scrap or rework due to poor quality. In this case, the batch is the population.

Although the population of interest often consists of physical objects, sometimes it is necessary to sample over time, space, or some combination of these dimensions. For instance, an investigation of supermarket staffing could examine checkout line length at various times, or a study on endangered penguins might aim to understand their usage of various hunting grounds over time. For the time dimension, the focus may be on periods or discrete occasions.

In other cases, the examined 'population' may be even less tangible. For example, Joseph Jagger studied the behaviour of roulette wheels at a casino in Monte Carlo, and used this to identify a biased wheel. In this case, the 'population' Jagger wanted to investigate was the overall behaviour of the wheel (i.e. the probability distribution of its results over infinitely many trials), while his 'sample' was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of some physical characteristic such as the electrical conductivity of copper.

This situation often arises when seeking knowledge about the cause system of which the observed population is an outcome. In such cases, sampling theory may treat the observed population as a sample from a larger 'superpopulation'. For example, a researcher might study the success rate of a new 'quit smoking' program on a test group of 100 patients, in order to predict the effects of the program if it were made available nationwide. Here the superpopulation is "everybody in the country, given access to this treatment" – a group which does not yet exist, since the program isn't yet available to all.

Note also that the population from which the sample is drawn may not be the same as the population about which information is desired. Often there is large but not complete overlap between these two groups due to frame issues etc. (see below). Sometimes they may be entirely separate – for instance, one might study rats in order to get a better understanding of human health, or one might study records from people born in 2008 in order to make predictions about people born in 2009.

Time spent in making the sampled population and population of concern precise is often well spent, because it raises many issues, ambiguities and questions that would otherwise have been overlooked at this stage.

Sampling frame

In the most straightforward case, such as the sampling of a batch of material from production (acceptance sampling by lots), it would be most desirable to identify and measure every single item in the population and to include any one of them in our sample. However, in the more general case this is not usually possible or practical. There is no way to identify all rats in the set of all rats. Where voting is not compulsory, there is no way to identify which people will vote at a forthcoming election (in advance of the election). These imprecise populations are not amenable to sampling in any of the ways below and to which we could apply statistical theory.

As a remedy, we seek a sampling frame which has the property that we can identify every single element and include any in our sample.[3][4][5][6] The most straightforward type of frame is a list of elements of the population (preferably the entire population) with appropriate contact information. For example, in an opinion poll, possible sampling frames include an electoral register and a telephone directory.

A probability sample is a sample in which every unit in the population has a chance (greater than zero) of being selected in the sample, and this probability can be accurately determined. The combination of these traits makes it possible to produce unbiased estimates of population totals, by weighting sampled units according to their probability of selection.

Example: We want to estimate the total income of adults living in a given street. We visit each household in that street, identify all adults living there, and randomly select one adult from each household. (For example, we can allocate each person a random number, generated from a uniform distribution between 0 and 1, and select the person with the highest number in each household). We then interview the selected person and find their income.

People living on their own are certain to be selected, so we simply add their income to our estimate of the total. But a person living in a household of two adults has only a one-in-two chance of selection. To reflect this, when we come to such a household, we would count the selected person's income twice towards the total. (The person who is selected from that household can be loosely viewed as also representing the person who isn't selected.)

In the above example, not everybody has the same probability of selection; what makes it a probability sample is the fact that each person's probability is known. When every element in the population does have the same probability of selection, this is known as an 'equal probability of selection' (EPS) design. Such designs are also referred to as 'self-weighting' because all sampled units are given the same weight.

Probability sampling includes: Simple Random Sampling, Systematic Sampling, Stratified Sampling, Probability Proportional to Size Sampling, and Cluster or Multistage Sampling. These various ways of probability sampling have two things in common:

  1. Every element has a known nonzero probability of being sampled and
  2. involves random selection at some point.

Nonprobability sampling

Nonprobability sampling is any sampling method where some elements of the population have no chance of selection (these are sometimes referred to as 'out of coverage'/'undercovered'), or where the probability of selection can't be accurately determined. It involves the selection of elements based on assumptions regarding the population of interest, which forms the criteria for selection. Hence, because the selection of elements is nonrandom, nonprobability sampling does not allow the estimation of sampling errors. These conditions give rise to exclusion bias, placing limits on how much information a sample can provide about the population. Information about the relationship between sample and population is limited, making it difficult to extrapolate from the sample to the population.

Example: We visit every household in a given street, and interview the first person to answer the door. In any household with more than one occupant, this is a nonprobability sample, because some people are more likely to answer the door (e.g. an unemployed person who spends most of their time at home is more likely to answer than an employed housemate who might be at work when the interviewer calls) and it's not practical to calculate these probabilities.

Nonprobability sampling methods include convenience sampling, quota sampling and purposive sampling. In addition, nonresponse effects may turn any probability design into a nonprobability design if the characteristics of nonresponse are not well understood, since nonresponse effectively modifies each element's probability of being sampled.

Sampling methods

Within any of the types of frames identified above, a variety of sampling methods can be employed, individually or in combination. Factors commonly influencing the choice between these designs include:

  • Nature and quality of the frame
  • Availability of auxiliary information about units on the frame
  • Accuracy requirements, and the need to measure accuracy
  • Whether detailed analysis of the sample is expected
  • Cost/operational concerns

Simple random sampling

Simple random sampling
A visual representation of selecting a simple random sample

In a simple random sample (SRS) of a given size, all such subsets of the frame are given an equal probability. Each element of the frame thus has an equal probability of selection: the frame is not subdivided partitioned. Furthermore, any given pair of elements has the same chance of selection as any other such pair (and similarly for triples, and so on). This minimizes bias and simplifies analysis of results. In particular, the variance between individual results within the sample is a good indicator of variance in the overall population, which makes it relatively easy to estimate the accuracy of results.

SRS can be vulnerable to sampling error because the randomness of the selection may result in a sample that doesn't reflect the makeup of the population. For instance, a simple random sample of ten people from a given country will on average produce five men and five women, but any given trial is likely to overrepresent one sex and underrepresent the other. Systematic and stratified techniques attempt to overcome this problem by "using information about the population" to choose a more "representative" sample.

SRS may also be cumbersome and tedious when sampling from an unusually large target population. In some cases, investigators are interested in "research questions specific" to subgroups of the population. For example, researchers might be interested in examining whether cognitive ability as a predictor of job performance is equally applicable across racial groups. SRS cannot accommodate the needs of researchers in this situation because it does not provide subsamples of the population. "Stratified sampling" addresses this weakness of SRS.

Systematic sampling

Systematic sampling
A visual representation of selecting a random sample using the systematic sampling technique

Systematic sampling (also known as interval sampling) relies on arranging the study population according to some ordering scheme and then selecting elements at regular intervals through that ordered list. Systematic sampling involves a random start and then proceeds with the selection of every kth element from then onwards. In this case, k=(population size/sample size). It is important that the starting point is not automatically the first in the list, but is instead randomly chosen from within the first to the kth element in the list. A simple example would be to select every 10th name from the telephone directory (an 'every 10th' sample, also referred to as 'sampling with a skip of 10').

As long as the starting point is randomized, systematic sampling is a type of probability sampling. It is easy to implement and the stratification induced can make it efficient, if the variable by which the list is ordered is correlated with the variable of interest. 'Every 10th' sampling is especially useful for efficient sampling from databases.

For example, suppose we wish to sample people from a long street that starts in a poor area (house No. 1) and ends in an expensive district (house No. 1000). A simple random selection of addresses from this street could easily end up with too many from the high end and too few from the low end (or vice versa), leading to an unrepresentative sample. Selecting (e.g.) every 10th street number along the street ensures that the sample is spread evenly along the length of the street, representing all of these districts. (Note that if we always start at house #1 and end at #991, the sample is slightly biased towards the low end; by randomly selecting the start between #1 and #10, this bias is eliminated.

However, systematic sampling is especially vulnerable to periodicities in the list. If periodicity is present and the period is a multiple or factor of the interval used, the sample is especially likely to be unrepresentative of the overall population, making the scheme less accurate than simple random sampling.

For example, consider a street where the odd-numbered houses are all on the north (expensive) side of the road, and the even-numbered houses are all on the south (cheap) side. Under the sampling scheme given above, it is impossible to get a representative sample; either the houses sampled will all be from the odd-numbered, expensive side, or they will all be from the even-numbered, cheap side, unless the researcher has previous knowledge of this bias and avoids it by a using a skip which ensures jumping between the two sides (any odd-numbered skip).

Another drawback of systematic sampling is that even in scenarios where it is more accurate than SRS, its theoretical properties make it difficult to quantify that accuracy. (In the two examples of systematic sampling that are given above, much of the potential sampling error is due to variation between neighbouring houses – but because this method never selects two neighbouring houses, the sample will not give us any information on that variation.)

As described above, systematic sampling is an EPS method, because all elements have the same probability of selection (in the example given, one in ten). It is not 'simple random sampling' because different subsets of the same size have different selection probabilities – e.g. the set {4,14,24,...,994} has a one-in-ten probability of selection, but the set {4,13,24,34,...} has zero probability of selection.

Systematic sampling can also be adapted to a non-EPS approach; for an example, see discussion of PPS samples below.

Stratified sampling

Stratified sampling
A visual representation of selecting a random sample using the stratified sampling technique

When the population embraces a number of distinct categories, the frame can be organized by these categories into separate "strata." Each stratum is then sampled as an independent sub-population, out of which individual elements can be randomly selected.[3] The ratio of the size of this random selection (or sample) to the size of the population is called a sampling fraction. There are several potential benefits to stratified sampling.

First, dividing the population into distinct, independent strata can enable researchers to draw inferences about specific subgroups that may be lost in a more generalized random sample.

Second, utilizing a stratified sampling method can lead to more efficient statistical estimates (provided that strata are selected based upon relevance to the criterion in question, instead of availability of the samples). Even if a stratified sampling approach does not lead to increased statistical efficiency, such a tactic will not result in less efficiency than would simple random sampling, provided that each stratum is proportional to the group's size in the population.

Third, it is sometimes the case that data are more readily available for individual, pre-existing strata within a population than for the overall population; in such cases, using a stratified sampling approach may be more convenient than aggregating data across groups (though this may potentially be at odds with the previously noted importance of utilizing criterion-relevant strata).

Finally, since each stratum is treated as an independent population, different sampling approaches can be applied to different strata, potentially enabling researchers to use the approach best suited (or most cost-effective) for each identified subgroup within the population.

There are, however, some potential drawbacks to using stratified sampling. First, identifying strata and implementing such an approach can increase the cost and complexity of sample selection, as well as leading to increased complexity of population estimates. Second, when examining multiple criteria, stratifying variables may be related to some, but not to others, further complicating the design, and potentially reducing the utility of the strata. Finally, in some cases (such as designs with a large number of strata, or those with a specified minimum sample size per group), stratified sampling can potentially require a larger sample than would other methods (although in most cases, the required sample size would be no larger than would be required for simple random sampling).

A stratified sampling approach is most effective when three conditions are met
  1. Variability within strata are minimized
  2. Variability between strata are maximized
  3. The variables upon which the population is stratified are strongly correlated with the desired dependent variable.
Advantages over other sampling methods
  1. Focuses on important subpopulations and ignores irrelevant ones.
  2. Allows use of different sampling techniques for different subpopulations.
  3. Improves the accuracy/efficiency of estimation.
  4. Permits greater balancing of statistical power of tests of differences between strata by sampling equal numbers from strata varying widely in size.
Disadvantages
  1. Requires selection of relevant stratification variables which can be difficult.
  2. Is not useful when there are no homogeneous subgroups.
  3. Can be expensive to implement.
Poststratification

Stratification is sometimes introduced after the sampling phase in a process called "poststratification".[3] This approach is typically implemented due to a lack of prior knowledge of an appropriate stratifying variable or when the experimenter lacks the necessary information to create a stratifying variable during the sampling phase. Although the method is susceptible to the pitfalls of post hoc approaches, it can provide several benefits in the right situation. Implementation usually follows a simple random sample. In addition to allowing for stratification on an ancillary variable, poststratification can be used to implement weighting, which can improve the precision of a sample's estimates.[3]

Oversampling

Choice-based sampling is one of the stratified sampling strategies. In choice-based sampling,[7] the data are stratified on the target and a sample is taken from each stratum so that the rare target class will be more represented in the sample. The model is then built on this biased sample. The effects of the input variables on the target are often estimated with more precision with the choice-based sample even when a smaller overall sample size is taken, compared to a random sample. The results usually must be adjusted to correct for the oversampling.

Probability-proportional-to-size sampling

In some cases the sample designer has access to an "auxiliary variable" or "size measure", believed to be correlated to the variable of interest, for each element in the population. These data can be used to improve accuracy in sample design. One option is to use the auxiliary variable as a basis for stratification, as discussed above.

Another option is probability proportional to size ('PPS') sampling, in which the selection probability for each element is set to be proportional to its size measure, up to a maximum of 1. In a simple PPS design, these selection probabilities can then be used as the basis for Poisson sampling. However, this has the drawback of variable sample size, and different portions of the population may still be over- or under-represented due to chance variation in selections.

Systematic sampling theory can be used to create a probability proportionate to size sample. This is done by treating each count within the size variable as a single sampling unit. Samples are then identified by selecting at even intervals among these counts within the size variable. This method is sometimes called PPS-sequential or monetary unit sampling in the case of audits or forensic sampling.

Example: Suppose we have six schools with populations of 150, 180, 200, 220, 260, and 490 students respectively (total 1500 students), and we want to use student population as the basis for a PPS sample of size three. To do this, we could allocate the first school numbers 1 to 150, the second school 151 to 330 (= 150 + 180), the third school 331 to 530, and so on to the last school (1011 to 1500). We then generate a random start between 1 and 500 (equal to 1500/3) and count through the school populations by multiples of 500. If our random start was 137, we would select the schools which have been allocated numbers 137, 637, and 1137, i.e. the first, fourth, and sixth schools.

The PPS approach can improve accuracy for a given sample size by concentrating sample on large elements that have the greatest impact on population estimates. PPS sampling is commonly used for surveys of businesses, where element size varies greatly and auxiliary information is often available—for instance, a survey attempting to measure the number of guest-nights spent in hotels might use each hotel's number of rooms as an auxiliary variable. In some cases, an older measurement of the variable of interest can be used as an auxiliary variable when attempting to produce more current estimates.[8]

Cluster sampling

Cluster sampling
A visual representation of selecting a random sample using the cluster sampling technique

Sometimes it is more cost-effective to select respondents in groups ('clusters'). Sampling is often clustered by geography, or by time periods. (Nearly all samples are in some sense 'clustered' in time – although this is rarely taken into account in the analysis.) For instance, if surveying households within a city, we might choose to select 100 city blocks and then interview every household within the selected blocks.

Clustering can reduce travel and administrative costs. In the example above, an interviewer can make a single trip to visit several households in one block, rather than having to drive to a different block for each household.

It also means that one does not need a sampling frame listing all elements in the target population. Instead, clusters can be chosen from a cluster-level frame, with an element-level frame created only for the selected clusters. In the example above, the sample only requires a block-level city map for initial selections, and then a household-level map of the 100 selected blocks, rather than a household-level map of the whole city.

Cluster sampling (also known as clustered sampling) generally increases the variability of sample estimates above that of simple random sampling, depending on how the clusters differ between one another as compared to the within-cluster variation. For this reason, cluster sampling requires a larger sample than SRS to achieve the same level of accuracy – but cost savings from clustering might still make this a cheaper option.

Cluster sampling is commonly implemented as multistage sampling. This is a complex form of cluster sampling in which two or more levels of units are embedded one in the other. The first stage consists of constructing the clusters that will be used to sample from. In the second stage, a sample of primary units is randomly selected from each cluster (rather than using all units contained in all selected clusters). In following stages, in each of those selected clusters, additional samples of units are selected, and so on. All ultimate units (individuals, for instance) selected at the last step of this procedure are then surveyed. This technique, thus, is essentially the process of taking random subsamples of preceding random samples.

Multistage sampling can substantially reduce sampling costs, where the complete population list would need to be constructed (before other sampling methods could be applied). By eliminating the work involved in describing clusters that are not selected, multistage sampling can reduce the large costs associated with traditional cluster sampling.[8] However, each sample may not be a full representative of the whole population.

Quota sampling

In quota sampling, the population is first segmented into mutually exclusive sub-groups, just as in stratified sampling. Then judgement is used to select the subjects or units from each segment based on a specified proportion. For example, an interviewer may be told to sample 200 females and 300 males between the age of 45 and 60.

It is this second step which makes the technique one of non-probability sampling. In quota sampling the selection of the sample is non-random. For example, interviewers might be tempted to interview those who look most helpful. The problem is that these samples may be biased because not everyone gets a chance of selection. This random element is its greatest weakness and quota versus probability has been a matter of controversy for several years.

Minimax sampling

In imbalanced datasets, where the sampling ratio does not follow the population statistics, one can resample the dataset in a conservative manner called minimax sampling. The minimax sampling has its origin in Anderson minimax ratio whose value is proved to be 0.5: in a binary classification, the class-sample sizes should be chosen equally. This ratio can be proved to be minimax ratio only under the assumption of LDA classifier with Gaussian distributions. The notion of minimax sampling is recently developed for a general class of classification rules, called class-wise smart classifiers.

Accidental sampling

Accidental sampling (sometimes known as grab, convenience or opportunity sampling) is a type of nonprobability sampling which involves the sample being drawn from that part of the population which is close to hand. That is, a population is selected because it is readily available and convenient. It may be through meeting the person or including a person in the sample when one meets them or chosen by finding them through technological means such as the internet or through phone. The researcher using such a sample cannot scientifically make generalizations about the total population from this sample because it would not be representative enough. For example, if the interviewer were to conduct such a survey at a shopping center early in the morning on a given day, the people that he/she could interview would be limited to those given there at that given time, which would not represent the views of other members of society in such an area, if the survey were to be conducted at different times of day and several times per week. This type of sampling is most useful for pilot testing. Several important considerations for researchers using convenience samples include:

  1. Are there controls within the research design or experiment which can serve to lessen the impact of a non-random convenience sample, thereby ensuring the results will be more representative of the population?
  2. Is there good reason to believe that a particular convenience sample would or should respond or behave differently than a random sample from the same population?
  3. Is the question being asked by the research one that can adequately be answered using a convenience sample?

In social science research, snowball sampling is a similar technique, where existing study subjects are used to recruit more subjects into the sample. Some variants of snowball sampling, such as respondent driven sampling, allow calculation of selection probabilities and are probability sampling methods under certain conditions.

Voluntary Sampling

The voluntary sampling method is a type of non-probability sampling. Volunteers choose to complete a survey.

Volunteers may be invited through advertisements in social media.[9] The target population for advertisements can be selected by characteristics like location, age, sex, income, occupation, education or interests using tools provided by the social medium. The advertisement may include a message about the research and link to a survey. After following the link and completing the survey the volunteer submits the data to be included in the sample population. This method can reach a global population but is limited by the campaign budget. Volunteers outside the invited population may also be included in the sample.

It is difficult to make generalizations from this sample because it may not represent the total population. Often, volunteers have a strong interest in the main topic of the survey.

Line-intercept sampling

Line-intercept sampling is a method of sampling elements in a region whereby an element is sampled if a chosen line segment, called a "transect", intersects the element.

Panel sampling

Panel sampling is the method of first selecting a group of participants through a random sampling method and then asking that group for (potentially the same) information several times over a period of time. Therefore, each participant is interviewed at two or more time points; each period of data collection is called a "wave". The method was developed by sociologist Paul Lazarsfeld in 1938 as a means of studying political campaigns.[10] This longitudinal sampling-method allows estimates of changes in the population, for example with regard to chronic illness to job stress to weekly food expenditures. Panel sampling can also be used to inform researchers about within-person health changes due to age or to help explain changes in continuous dependent variables such as spousal interaction.[11] There have been several proposed methods of analyzing panel data, including MANOVA, growth curves, and structural equation modeling with lagged effects.

Snowball sampling

Snowball sampling involves finding a small group of initial respondents and using them to recruit more respondents. It is particularly useful in cases where the population is hidden or difficult to enumerate.

Theoretical sampling

Theoretical sampling[12] occurs when samples are selected on the basis of the results of the data collected so far with a goal of developing a deeper understanding of the area or develop theories

Replacement of selected units

Sampling schemes may be without replacement ('WOR'—no element can be selected more than once in the same sample) or with replacement ('WR'—an element may appear multiple times in the one sample). For example, if we catch fish, measure them, and immediately return them to the water before continuing with the sample, this is a WR design, because we might end up catching and measuring the same fish more than once. However, if we do not return the fish to the water or tag and release each fish after catching it, this becomes a WOR design.

Sample size determination

Formulas, tables, and power function charts are well known approaches to determine sample size.

Steps for using sample size tables

  1. Postulate the effect size of interest, α, and β.
  2. Check sample size table[13]
    1. Select the table corresponding to the selected α
    2. Locate the row corresponding to the desired power
    3. Locate the column corresponding to the estimated effect size.
    4. The intersection of the column and row is the minimum sample size required.

Sampling and data collection

Good data collection involves:

  • Following the defined sampling process
  • Keeping the data in time order
  • Noting comments and other contextual events
  • Recording non-responses

Applications of sampling

Sampling enables the selection of right data points from within the larger data set to estimate the characteristics of the whole population. For example, there are about 600 million tweets produced every day. It is not necessary to look at all of them to determine the topics that are discussed during the day, nor is it necessary to look at all the tweets to determine the sentiment on each of the topics. A theoretical formulation for sampling Twitter data has been developed.[14]

In manufacturing different types of sensory data such as acoustics, vibration, pressure, current, voltage and controller data are available at short time intervals. To predict down-time it may not be necessary to look at all the data but a sample may be sufficient.

Errors in sample surveys

Survey results are typically subject to some error. Total errors can be classified into sampling errors and non-sampling errors. The term "error" here includes systematic biases as well as random errors.

Sampling errors and biases

Sampling errors and biases are induced by the sample design. They include:

  1. Selection bias: When the true selection probabilities differ from those assumed in calculating the results.
  2. Random sampling error: Random variation in the results due to the elements in the sample being selected at random.

Non-sampling error

Non-sampling errors are other errors which can impact the final survey estimates, caused by problems in data collection, processing, or sample design. They include:

  1. Over-coverage: Inclusion of data from outside of the population.
  2. Under-coverage: Sampling frame does not include elements in the population.
  3. Measurement error: e.g. when respondents misunderstand a question, or find it difficult to answer.
  4. Processing error: Mistakes in data coding.
  5. Non-response or Participation bias: Failure to obtain complete data from all selected individuals.

After sampling, a review should be held of the exact process followed in sampling, rather than that intended, in order to study any effects that any divergences might have on subsequent analysis.

A particular problem is that of non-response. Two major types of non-response exist: unit nonresponse (referring to lack of completion of any part of the survey) and item non-response (submission or participation in survey but failing to complete one or more components/questions of the survey).[15][16] In survey sampling, many of the individuals identified as part of the sample may be unwilling to participate, not have the time to participate (opportunity cost),[17] or survey administrators may not have been able to contact them. In this case, there is a risk of differences, between respondents and nonrespondents, leading to biased estimates of population parameters. This is often addressed by improving survey design, offering incentives, and conducting follow-up studies which make a repeated attempt to contact the unresponsive and to characterize their similarities and differences with the rest of the frame.[18] The effects can also be mitigated by weighting the data when population benchmarks are available or by imputing data based on answers to other questions. Nonresponse is particularly a problem in internet sampling. Reasons for this problem include improperly designed surveys,[16] over-surveying (or survey fatigue),[11][19] and the fact that potential participants hold multiple e-mail addresses, which they don't use anymore or don't check regularly.

Survey weights

In many situations the sample fraction may be varied by stratum and data will have to be weighted to correctly represent the population. Thus for example, a simple random sample of individuals in the United Kingdom might include some in remote Scottish islands who would be inordinately expensive to sample. A cheaper method would be to use a stratified sample with urban and rural strata. The rural sample could be under-represented in the sample, but weighted up appropriately in the analysis to compensate.

More generally, data should usually be weighted if the sample design does not give each individual an equal chance of being selected. For instance, when households have equal selection probabilities but one person is interviewed from within each household, this gives people from large households a smaller chance of being interviewed. This can be accounted for using survey weights. Similarly, households with more than one telephone line have a greater chance of being selected in a random digit dialing sample, and weights can adjust for this.

Weights can also serve other purposes, such as helping to correct for non-response.

Methods of producing random samples

History

Random sampling by using lots is an old idea, mentioned several times in the Bible. In 1786 Pierre Simon Laplace estimated the population of France by using a sample, along with ratio estimator. He also computed probabilistic estimates of the error. These were not expressed as modern confidence intervals but as the sample size that would be needed to achieve a particular upper bound on the sampling error with probability 1000/1001. His estimates used Bayes' theorem with a uniform prior probability and assumed that his sample was random. Alexander Ivanovich Chuprov introduced sample surveys to Imperial Russia in the 1870s.

In the USA the 1936 Literary Digest prediction of a Republican win in the presidential election went badly awry, due to severe bias [1]. More than two million people responded to the study with their names obtained through magazine subscription lists and telephone directories. It was not appreciated that these lists were heavily biased towards Republicans and the resulting sample, though very large, was deeply flawed.[20][21]

See also

Notes

The textbook by Groves et alia provides an overview of survey methodology, including recent literature on questionnaire development (informed by cognitive psychology) :

  • Robert Groves, et alia. Survey methodology (2010) Second edition of the (2004) first edition ISBN 0-471-48348-6.

The other books focus on the statistical theory of survey sampling and require some knowledge of basic statistics, as discussed in the following textbooks:

The elementary book by Scheaffer et alia uses quadratic equations from high-school algebra:

  • Scheaffer, Richard L., William Mendenhal and R. Lyman Ott. Elementary survey sampling, Fifth Edition. Belmont: Duxbury Press, 1996.

More mathematical statistics is required for Lohr, for Särndal et alia, and for Cochran (classic):

  • Cochran, William G. (1977). Sampling techniques (Third ed.). Wiley. ISBN 978-0-471-16240-7.
  • Lohr, Sharon L. (1999). Sampling: Design and analysis. Duxbury. ISBN 978-0-534-35361-2.
  • Särndal, Carl-Erik, and Swensson, Bengt, and Wretman, Jan (1992). Model assisted survey sampling. Springer-Verlag. ISBN 978-0-387-40620-6.CS1 maint: Multiple names: authors list (link)

The historically important books by Deming and Kish remain valuable for insights for social scientists (particularly about the U.S. census and the Institute for Social Research at the University of Michigan):

References

  1. ^ Lance, P. & Hattori, A. (2016). Sampling and Evaluation. Web: MEASURE Evaluation. pp. 6–8, 62–64.CS1 maint: Multiple names: authors list (link)
  2. ^ Salant, Priscilla, I. Dillman, and A. Don. How to conduct your own survey. No. 300.723 S3. 1994.
  3. ^ a b c d Robert M. Groves; et al. (2009-07-14). Survey methodology. ISBN 978-0470465462.
  4. ^ Lohr, Sharon L. Sampling: Design and analysis.
  5. ^ Särndal, Carl-Erik, and Swensson, Bengt, and Wretman, Jan. Model Assisted Survey Sampling.CS1 maint: Multiple names: authors list (link)
  6. ^ Scheaffer, Richard L., William Mendenhal and R. Lyman Ott. Elementary survey sampling.CS1 maint: Multiple names: authors list (link)
  7. ^ Scott, A.J.; Wild, C.J. (1986). "Fitting logistic models under case-control or choice-based sampling". Journal of the Royal Statistical Society, Series B. 48 (2): 170–182. JSTOR 2345712.
  8. ^ a b
    • Lohr, Sharon L. Sampling: Design and Analysis.
    • Särndal, Carl-Erik, and Swensson, Bengt, and Wretman, Jan. Model Assisted Survey Sampling.CS1 maint: Multiple names: authors list (link)
  9. ^ Ariyaratne, Buddhika (30 July 2017). "Voluntary Sampling Method combined with Social Media advertising". heal-info.blogspot.com. Health Informatics. Retrieved 18 December 2018.
  10. ^ Lazarsfeld, P., & Fiske, M. (1938). The" panel" as a new tool for measuring opinion. The Public Opinion Quarterly, 2(4), 596–612.
  11. ^ a b Groves, et alia. Survey Methodology
  12. ^ "Examples of sampling methods" (PDF).
  13. ^ Cohen, 1988
  14. ^ Deepan Palguna, Vikas Joshi, Venkatesan Chakaravarthy, Ravi Kothari and L. V. Subramaniam (2015). Analysis of Sampling Algorithms for Twitter. International Joint Conference on Artificial Intelligence.CS1 maint: Multiple names: authors list (link)
  15. ^ Berinsky, A. J. (2008). Survey non-response. In W. Donsbach & M. W. Traugott (Eds.), The SAGE handbook of public opinion research (pp. 309–321). Thousand Oaks, CA: Sage Publications.
  16. ^ a b Dillman, D. A., Eltinge, J. L., Groves, R. M., & Little, R. J. A. (2002). Survey nonresponse in design, data collection, and analysis. In R. M. Groves, D. A. Dillman, J. L. Eltinge, & R. J. A. Little (Eds.), Survey nonresponse (pp. 3–26). New York: John Wiley & Sons.
  17. ^ Dillman, D.A., Smyth, J.D., & Christian, L. M. (2009). Internet, mail, and mixed-mode surveys: The tailored design method. San Francisco: Jossey-Bass.
  18. ^ Vehovar, V., Batagelj, Z., Manfreda, K.L., & Zaletel, M. (2002). Nonresponse in web surveys. In R. M. Groves, D. A. Dillman, J. L. Eltinge, & R. J. A. Little (Eds.), Survey nonresponse (pp. 229–242). New York: John Wiley & Sons.
  19. ^ Porter, Whitcomb, Weitzer (2004) Multiple surveys of students and survey fatigue. In S. R. Porter (Ed.), Overcoming survey research problems: Vol. 121. New directions for institutional research (pp. 63–74). San Francisco, CA: Jossey Bass.
  20. ^ David S. Moore and George P. McCabe. "Introduction to the Practice of Statistics".
  21. ^ Freedman, David; Pisani, Robert; Purves, Roger. Statistics.
  22. ^ Anderson, Theodore (1951). "Classification by multivariate analysis". Psychometrika. 16 (1): 31–50. doi:10.1007/bf02313425.
  23. ^ Shahrokh Esfahani, Mohammad; Dougherty, Edward (2014). "Effect of separate sampling on classification accuracy". Bioinformatics. 30 (2): 242–250. doi:10.1093/bioinformatics/btt662. PMID 24257187.

Further reading

Standards

ISO

  • ISO 2859 series
  • ISO 3951 series

ASTM

  • ASTM E105 Standard Practice for Probability Sampling Of Materials
  • ASTM E122 Standard Practice for Calculating Sample Size to Estimate, With a Specified Tolerable Error, the Average for Characteristic of a Lot or Process
  • ASTM E141 Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling
  • ASTM E1402 Standard Terminology Relating to Sampling
  • ASTM E1994 Standard Practice for Use of Process Oriented AOQL and LTPD Sampling Plans
  • ASTM E2234 Standard Practice for Sampling a Stream of Product by Attributes Indexed by AQL

ANSI, ASQ

  • ANSI/ASQ Z1.4

U.S. federal and military standards

Acquiescence bias

Acquiescence bias is a category of response bias in which respondents to a survey have a tendency to agree with all the questions or to indicate a positive connotation. Acquiescence is sometimes referred to as "yea-saying" and is the tendency of a respondent to agree with a statement when in doubt. This particularly is in the case of surveys or questionnaires that employ truisms, such as: "It is better to give than to receive" or "Never a lender nor a borrower be".

Douglas N. Jackson demonstrated acquiescence responding on the California F-scale (a measure of authoritarianism), which contains such truisms. He created a reverse-keyed version of the California F-scale where all the items were the opposite in meaning (see the two previous examples for a pair of such contradictory statements). He administered both the original and reverse-keyed versions of the California F-scale to the same group of respondents. One would expect that the correlation between these two scales to be negative, but there was a high, positive correlation. Jackson interpreted this as evidence of acquiescence responding. Respondents were merely being agreeable to the statements, regardless of the content.

Jackson and Messick, using factor analysis, also demonstrated that the two main factors explaining the majority of response variation on the Minnesota Multiphasic Personality Inventory (MMPI) were for social desirability and acquiescence responding (this would also hold true for the revised MMPI-2).

One approach to dealing with acquiescence responding on surveys and questionnaires is to employ a balance of positively and negatively keyed items in terms of the intended content. For example, in trying to assess depression it would be a good idea to also include items assessing happiness and contentedness, etc. (reversed-keyed items), in addition to the usual depressive content.

Convenience sampling

Convenience sampling (also known as grab sampling, accidental sampling, or opportunity sampling) is a type of non-probability sampling that involves the sample being drawn from that part of the population that is close to hand. This type of sampling is most useful for pilot testing.

Empirical evidence

Empirical evidence is the information received by means of the senses, particularly by observation and documentation of patterns and behavior through experimentation. The term comes from the Greek word for experience, ἐμπειρία (empeiría).

After Immanuel Kant, in philosophy, it is common to call the knowledge gained a posteriori knowledge (in contrast to a priori knowledge).

Healthy user bias

The healthy user bias is a bias that can damage the validity of epidemiologic studies testing the efficacy of particular therapies or interventions. Specifically, it is a sampling bias: the kind of subjects that voluntarily enroll in a clinical trial and actually follow the experimental regimen are not representative of the general population. They can be expected, on average, to be healthier as they are concerned for their health and are predisposed to follow medical advice, both factors that would aid one's health. In a sense, being healthy or active about one's health is a precondition for becoming a subject of the study, an effect that can appear under other conditions such as studying particular groups of workers (i.e. someone in ill health is unlikely to have a job as manual laborer).

Infrastructure bias

In economics and social policy, infrastructure bias is the influence of the location and availability of pre-existing infrastructure, such as roads and telecommunications facilities, on social and economic development.

In science, infrastructure bias is the influence of existing social or scientific infrastructure on scientific observations.

In astronomy and particle physics, where the availability of particular kinds of telescopes or particle accelerators acts as a constraint on the types of experiments that can be done, the data that can be retrieved is biased towards that which can be obtained by the equipment.

Procedural bias, related to infrastructure bias, is shown by a case of irregular genetic sampling of Bolivian wild potatoes. A 2000 report of previous studies' sampling found that 60% of samples had been taken near towns or roads, where 22% would be the average, had the samples been taken at random (or from equidistant points, or at specifically varying distances from towns, representative of the average terrain density).

Inherent bias

The phrase "inherent bias" refers to the effect of underlying factors or assumptions that skew viewpoints of a subject under discussion. There are multiple formal definitions of "inherent bias" which depend on the particular field of study.

In statistics, the phrase is used in relation to an inability to measure accurately and directly what one would wish to measure, meaning that indirect measurements are used which might be subject to unknown distortions.

Margin of error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the "true" figures; that is, the figures for the whole population. Margin of error is positive whenever a population is incompletely sampled and the outcome measure has positive variance (that is, it varies).

The term "margin of error" is often used in non-survey contexts to indicate observational error in reporting measured quantities.

Microdata (statistics)

In the study of survey and census data, microdata is information at the level of individual respondents. For instance, a national census might collect age, home address, educational level, employment status, and many other variables, recorded separately for every person who responds; this is microdata.

Recall bias

In epidemiological research, recall bias is a systematic error caused by differences in the accuracy or completeness of the recollections retrieved ("recalled") by study participants regarding events or experiences from the past. Sometimes also referred to as response bias, responder bias or reporting bias, this type of measurement bias can be a methodological issue in research involving interviews or questionnaires, in which case it could lead to misclassification of various types of exposure. Recall bias is of particular concern in retrospective studies that use a case-control design to investigate the etiology of a disease or psychiatric condition. For example, in studies of risk factors for breast cancer, women who have had the disease may search their memories more thoroughly than members of the unaffected control group; those who had the disease may recall a greater variety of risk factors they had been exposed to, including those falsely attributed to the disease in the media, such as use of oral contraceptives. To minimize recall bias, some clinical trials have adopted a "wash out period", i.e., a substantial time period that must elapse between the subject's first observation and their subsequent observation of the same event.

Replication (statistics)

In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "the repetition of the set of all the treatment combinations to be compared in an experiment. Each of the repetitions is called a replicate."

Replication is not the same as repeated measurements of the same item: they are dealt with differently in statistical experimental design and data analysis.

For proper sampling, a process or batch of products should be in reasonable statistical control; inherent random variation is present but variation due to assignable (special) causes is not. Evaluation or testing of a single item does not allow for item-to-item variation and may not represent the batch or process. Replication is needed to account for this variation among items and treatments.

Sample (statistics)

In statistics and quantitative research methodology, a data sample is a set of data collected and the world selected from a statistical population by a defined procedure. The elements of a sample are known as sample points, sampling units or observations.

Typically, the population is very large, making a census or a complete enumeration of all the values in the population either impractical or impossible. the sample usually represents a subset of manageable size. Samples are collected and statistics are calculated from the samples, so that one can make inferences or extrapolations from the sample to the population.

The data sample may be drawn from a population without replacement (i.e. no element can be selected more than once in the same sample), in which case it is a subset of a population; or with replacement (i.e. an element may appear multiple times in the one sample), in which case it is a multisubset.

Sample size determination

Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is determined based on the expense of data collection, and the need to have sufficient statistical power. In complicated studies there may be several different sample sizes involved in the study: for example, in a stratified survey there would be different sample sizes for each stratum. In a census, data are collected on the entire population, hence the sample size is equal to the population size. In experimental design, where a study may be divided into different treatment groups, this may be different sample sizes for each group.

Sample sizes may be chosen in several different ways:

experience – A choice of small sample sizes, though sometimes necessary, can result in wide confidence intervals or risks of errors in statistical hypothesis testing.

using a target variance for an estimate to be derived from the sample eventually obtained, i.e. if a high precision is required (narrow confidence interval) this translates to a low target variance of the estimator.

using a target for the power of a statistical test to be applied once the sample is collected.

using a confidence level, i.e. the larger the required confidence level, the larger the sample size (given a constant precision requirement).

Sampling bias

In statistics, sampling bias is a bias in which a sample is collected in such a way that some members of the intended population are less likely to be included than others. It results in a biased sample, a non-random sample of a population (or non-human factors) in which all individuals, or instances, were not equally likely to have been selected. If this is not accounted for, results can be erroneously attributed to the phenomenon under study rather than to the method of sampling.

Medical sources sometimes refer to sampling bias as ascertainment bias. Ascertainment bias has basically the same definition, but is still sometimes classified as a separate type of bias.

Sampling distribution

In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were separately used in order to compute one value of a statistic (such as, for example, the sample mean or sample variance) for each sample, then the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample is observed, but the sampling distribution can be found theoretically.

Sampling distributions are important in statistics because they provide a major simplification en route to statistical inference. More specifically, they allow analytical considerations to be based on the probability distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.

Sampling error

In statistics, sampling error is incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics on the sample, such as means and quantiles, generally differ from the characteristics of the entire population, which are known as parameters. For example, if one measures the height of a thousand individuals from a country of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country. Since sampling is typically done to determine the characteristics of a whole population, the difference between the sample and population values is considered an error. Exact measurement of sampling error is generally not feasible since the true population values are unknown.

Selection bias

Selection bias is the bias introduced by the selection of individuals, groups or data for analysis in such a way that proper randomization is not achieved, thereby ensuring that the sample obtained is not representative of the population intended to be analyzed. It is sometimes referred to as the selection effect. The phrase "selection bias" most often refers to the distortion of a statistical analysis, resulting from the method of collecting samples. If the selection bias is not taken into account, then some conclusions of the study may not be accurate.

Self-selection bias

In statistics, self-selection bias arises in any situation in which individuals select themselves into a group, causing a biased sample with nonprobability sampling. It is commonly used to describe situations where the characteristics of the people which cause them to select themselves in the group create abnormal or undesirable conditions in the group. It is closely related to the non-response bias, describing when the group of people responding has different responses than the group of people not responding.

Self-selection bias is a major problem in research in sociology, psychology, economics and many other social sciences. In such fields, a poll suffering from such bias is termed a self-selected listener opinion poll or "SLOP".

The term is also used in criminology to describe the process by which specific predispositions may lead an offender to choose a criminal career and lifestyle.

While the effects of self-selection bias are closely related to those of selection bias, the problem arises for rather different reasons; thus there may be a purposeful intent on the part of respondents leading to self-selection bias whereas other types of selection bias may arise more inadvertently, possibly as the result of mistakes by those designing any given study.

Statistical inference

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.

Stratified sampling

In statistics, stratified sampling is a method of sampling from a population which can be partitioned into subpopulations.

In statistical surveys, when subpopulations within an overall population vary, it could be advantageous to sample each subpopulation (stratum) independently. Stratification is the process of dividing members of the population into homogeneous subgroups before sampling. The strata should define a partition of the population. That is, it should be collectively exhaustive and mutually exclusive: every element in the population must be assigned to one and only one stratum. Then simple random sampling or systematic sampling is applied within each stratum. The objective is to improve the precision of the sample by reducing sampling error. It can produce a weighted mean that has less variability than the arithmetic mean of a simple random sample of the population.

In computational statistics, stratified sampling is a method of variance reduction when Monte Carlo methods are used to estimate population statistics from a known population.Assume that we need to estimate average number of votes for each candidate in an election. Assume that country has 3 towns: Town A has 1 million factory workers, Town B has 2 million office workers and Town C has 3 million retirees. We can choose to get a random sample of size 60 over the entire population but there is some chance that the random sample turns out to be not well balanced across these towns and hence is biased causing a significant error in estimation. Instead if we choose to take a random sample of 10, 20 and 30 from Town A, B and C respectively then we can produce a smaller error in estimation for the same total size of sample.

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