**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 that which 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 we need 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, our '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 we seek 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 we actually want information.
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, we might study rats in order to get a better understanding of human
health, or we 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 sentencing of
a batch of material from production (acceptance sampling by lots), it is
possible 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 possible. 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
actually 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.

^{}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.**PROBABILITY AND NONPROBABILITY SAMPLING**

A probability
sampling scheme is one 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:

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

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 [[accidental 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 frame 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

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 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 minimises 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.
However,
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, discussed below, 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, which is discussed below,
addresses this weakness of SRS.

Simple
random sampling is always an EPS design (equal probability of selection), but
not all EPS designs are simple random sampling

Systematic sampling relies on arranging the target
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 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.*Example: Suppose we wish to sample people from a long street that starts in a poor district (house #1) and ends in an expensive district (house #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

*un*representative of the overall population, making the scheme less accurate than simple random sampling.*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.*

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.

Main
article: Stratified sampling

Where
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.

^{[1]}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

- 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".

^{[1]}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^{}
Oversampling

Choice-based
sampling is one of the stratified sampling strategies. In choice-based
sampling,

^{[2]}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. To address this
problem, PPS may be combined with a systematic approach.

*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

^{}
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 generally increases the variability of sample estimates above that of
simple random sampling, depending on how the clusters differ between
themselves, as compared with 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

^{}
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 many years

Convenience sampling (sometimes known as grab
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:

- 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.

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 the same
information again several times over a period of time. Therefore, each
participant is given the same survey or interview at two or more time points;
each period of data collection is called a "wave". 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.

^{}There have been several proposed methods of analyzing panel data, including MANOVA, growth curves, and structural equation modeling with lagged effects.
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