Editing Sampling (statistics) (section)

==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

=== <span id="Random sampling"> Simple random sampling </span> ===
{{Main|Simple random sampling}}
[[File:Simple random sampling.PNG|thumb|300px|A visual representation of selecting a simple random sample]]
In a simple random sample (SRS) of a given size, all subsets of a sampling frame have an equal probability of being selected. Each element of the frame thus has an equal probability of selection: the frame is not subdivided or 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.

Simple random sampling can be vulnerable to sampling error because the randomness of the selection may result in a sample that does not 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 over represent 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.

Also, simple random sampling can be cumbersome and tedious when sampling from a 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. Simple random sampling cannot accommodate the needs of researchers in this situation, because it does not provide subsamples of the population, and other sampling strategies, such as stratified sampling, can be used instead.

===Systematic sampling===
{{main|Systematic sampling}}
[[File:Systematic sampling.PNG|thumb|350px|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 ''k''th 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 ''k''th 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 [[randomization|randomized]], systematic sampling is a type of [[probability sampling]]. It is easy to implement and the [[Stratified sampling|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. (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 ''un''representative 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===
{{main|Stratified sampling }}
[[File:Stratified sampling.PNG|thumb|300px|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.<ref name="Robert M. Groves, et al"/> The ratio of the size of this random selection (or sample) to the size of the population is called a [[sampling fraction]].<ref name=sampling-minimax/> There are several potential benefits to stratified sampling.<ref name=sampling-minimax/>

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:
# Variability within strata are minimized
# Variability between strata are maximized
# The variables upon which the population is stratified are strongly correlated with the desired dependent variable.

; Advantages over other sampling methods
# Focuses on important subpopulations and ignores irrelevant ones.
# Allows use of different sampling techniques for different subpopulations.
# Improves the accuracy/efficiency of estimation.
# Permits greater balancing of statistical power of tests of differences between strata by sampling equal numbers from strata varying widely in size.

; Disadvantages
# Requires selection of relevant stratification variables which can be difficult.
# Is not useful when there are no homogeneous subgroups.
# Can be expensive to implement.

; Poststratification
Stratification is sometimes introduced after the sampling phase in a process called "poststratification".<ref name="Robert M. Groves, et al"/> 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.<ref name="Robert M. Groves, et al"/>

; Oversampling
Choice-based sampling or oversampling is one of the stratified sampling strategies. In choice-based sampling,<ref>{{cite journal|last1=Scott|first1=A.J.|last2=Wild|first2=C.J.|year=1986|title=Fitting logistic models under case-control or choice-based sampling|journal=[[Journal of the Royal Statistical Society, Series B]]|volume=48|issue=2|pages=170–182|doi=10.1111/j.2517-6161.1986.tb01400.x |jstor=2345712}}</ref> the data are stratified on the target and a sample is taken from each stratum so that rarer target classes will be more represented in the sample. The model is then built on this [[Sampling bias|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===

{{main|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.

<blockquote>
''Example: Suppose we have six schools with populations of 150, 180, 200, 220, 260, and&nbsp;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&nbsp;to&nbsp;150, the second school 151 to 330&nbsp;(=&nbsp;150&nbsp;+&nbsp;180), the third school 331 to 530, and so on to the last school (1011 to&nbsp;1500). We then generate a random start between 1 and 500 (equal to&nbsp;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&nbsp;1137, i.e. the first, fourth, and sixth schools.''
</blockquote>

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&nbsp;– 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.<ref name="MySwedeLohr">
* {{cite book|author=Lohr, Sharon L.|title=Sampling: Design and Analysis}}
* {{cite book|author=Särndal, Carl-Erik |author2=Swensson, Bengt |author3=Wretman, Jan|title=Model Assisted Survey Sampling}}</ref>

===Cluster sampling===
[[File:Cluster sampling.PNG|thumb|300px|A visual representation of selecting a random sample using the cluster sampling technique]]
{{main|Cluster sampling }}

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.<ref name="MySwedeLohr"/> However, each sample may not be a full representative of the whole population.

===Quota sampling===
{{main|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|minimax sampling]]. The minimax sampling has its origin in [[Theodore Wilbur Anderson|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 [[Linear Discriminant Analysis|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. In this case, the sampling ratio of classes is selected so that the worst case classifier error over all the possible population statistics for class prior probabilities, would be the best.<ref name=sampling-minimax/>

===Accidental sampling===
{{Main|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 they 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:

# 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?
# 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?
# 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===
{{further|Self-selection bias}}

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.<ref name=":1">{{Cite web|url = https://heal-info.blogspot.com/2017/07/voluntary-sampling-method.html|title = Voluntary Sampling Method combined with Social Media advertising|access-date = 18 December 2018 |website = heal-info.blogspot.com |series= Health Informatics |last = Ariyaratne | first = Buddhika | date= 30 July 2017}}{{unreliable source?|date=December 2018}}</ref> 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===
{{Main|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 campaign]]s.<ref>Lazarsfeld, P., & Fiske, M. (1938). The" panel" as a new tool for measuring opinion. The Public Opinion Quarterly, 2(4), 596–612.</ref> This [[longitudinal study|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.<ref name="SM" >
Groves, et alia. ''Survey Methodology''
</ref> There have been several proposed methods of analyzing [[panel data]], including [[MANOVA]], [[growth curve (statistics)|growth curves]], and [[structural equation modeling]] with lagged effects.

===Snowball sampling===
{{Main|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 ===
{{expand section|date=July 2015}}
{{Main|Theoretical sampling}}

Theoretical sampling<ref name=":0">{{Cite web|url = http://www.fao.org/ag/humannutrition/32428-0613f516cb07eade922c8c19b4d0452c0.pdf|title = Examples of sampling methods}}</ref> 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. An initial, general sample is first collected with the goal of investigating general trends, where further sampling may consist of extreme or very specific cases might be selected in order to maximize the likelihood a phenomenon will actually be observable.

=== Active sampling ===
In [[active sampling]], the samples which are used for training a machine learning algorithm are actively selected, also compare [[active learning (machine learning)]].

=== Judgmental selection ===
{{main|Judgment sample}}Judgement sampling, also known as expert or purposive sampling, is a type non-random sampling where samples are selected based on the opinion of an expert, who can select participants based on how valuable the information they provide is.

=== Haphazard sampling ===
{{expand section|date=July 2024}}
Haphazard sampling refers to the idea of using human judgement to simulate randomness. Despite samples being hand-picked, the goal is to ensure that no conscious bias exists within the choice of samples, but often fails due to [[selection bias]].<ref>{{Cite web |date=7 January 2024 |title=Haphazard sampling definition |url=https://www.accountingtools.com/articles/haphazard-sampling |website=AccountingTools}}</ref> Haphazard sampling is generally opted for due to its convenience, when the tools or capacity to perform other sampling methods may not exist.

The major weakness of such samples is that they often do not represent the characteristics of the entire population, but just a segment of the population. Because of this unbalanced representation, results from haphazard sampling are often biased.<ref>{{Cite book |title=IRS Statistical Sampling Handbook |publisher=USA: Department of the Treasury, Internal Revenue Service |year=1988 |pages=8}}</ref>