In statistics, **cluster sampling** is a sampling plan used when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. It is often used in marketing research. In this sampling plan, the total population is divided into these groups (known as clusters) and a simple random sample of the groups is selected. The elements in each cluster are then sampled. If all elements in each sampled cluster are sampled, then this is referred to as a "one-stage" cluster sampling plan. If a simple random subsample of elements is selected within each of these groups, this is referred to as a "two-stage" cluster sampling plan. A common motivation for cluster sampling is to reduce the total number of interviews and costs given the desired accuracy. For a fixed sample size, the expected random error is smaller when most of the variation in the population is present internally within the groups, and not between the groups.

The population within a cluster should ideally be as heterogeneous as possible, but there should be homogeneity between clusters. Each cluster should be a small-scale representation of the total population. The clusters should be mutually exclusive and collectively exhaustive. A random sampling technique is then used on any relevant clusters to choose which clusters to include in the study. In single-stage cluster sampling, all the elements from each of the selected clusters are sampled. In two-stage cluster sampling, a random sampling technique is applied to the elements from each of the selected clusters.

The main difference between cluster sampling and stratified sampling is that in cluster sampling the cluster is treated as the sampling unit so sampling is done on a population of clusters (at least in the first stage). In stratified sampling, the sampling is done on elements within each stratum. In stratified sampling, a random sample is drawn from each of the strata, whereas in cluster sampling only the selected clusters are sampled. A common motivation of cluster sampling is to reduce costs by increasing sampling efficiency. This contrasts with stratified sampling where the motivation is to increase precision.

There is also multistage cluster sampling, where at least two stages are taken in selecting elements from clusters.

Without modifying the estimated parameter, cluster sampling is unbiased when the clusters are approximately the same size. In this case, the parameter is computed by combining all the selected clusters. When the clusters are of different sizes there are several options:

One method is to sample clusters and then survey all elements in that cluster. Another method is a two-stage method of sampling a fixed proportion of units (be it 5% or 50%, or another number, depending on cost considerations) from within each of the selected clusters. Relying on the sample drawn from these options will yield an unbiased estimator. However, the sample size is no longer fixed upfront. This leads to a more complicated formula for the standard error of the estimator, as well as issues with the optics of the study plan (since the power analysis and the cost estimations often relate to a specific sample size).

A third possible solution is to use probability proportionate to size sampling. In this sampling plan, the probability of selecting a cluster is proportional to its size, so that a large cluster has a greater probability of selection than a small cluster. The advantage here is that when clusters are selected with probability proportionate to size, the same number of interviews should be carried out in each sampled cluster so that each unit sampled has the same probability of selection.

An example of cluster sampling is area sampling or geographical cluster sampling. Each cluster is a geographical area. Because a geographically dispersed population can be expensive to survey, greater economy than simple random sampling can be achieved by grouping several respondents within a local area into a cluster. It is usually necessary to increase the total sample size to achieve equivalent precision in the estimators, but cost savings may make such an increase in sample size feasible.

Cluster sampling is used to estimate high mortalities in cases such as wars, famines and natural disasters.^{ [1] }

- Can be cheaper than other sampling plans – e.g. fewer travel expenses, administration costs.
- Feasibility: This sampling plan takes large populations into account. Since these groups are so large, deploying any other sampling plan would be very costly.
- Economy: The regular two major concerns of expenditure, i.e., traveling and listing, are greatly reduced in this method. For example: Compiling research information about every household in a city would be very costly, whereas compiling information about various blocks of the city will be more economical. Here, traveling as well as listing efforts will be greatly reduced.
- Reduced variability: in the rare case of a negative intraclass correlation between subjects within a cluster, the estimators produced by cluster sampling will yield more accurate estimates than data obtained from a simple random sample (i.e. the design effect will be smaller than 1). This is not a common place scenario.

Major use: when the sampling frame of all elements is not available we can resort only to the cluster sampling.

- Higher sampling error, which can be expressed by the design effect: the ratio between the variance of an estimator made from the samples of the cluster study and the variance of an estimator obtained from a sample of subjects in an equally reliable, randomly sampled unclustered study.
^{ [2] }The larger the intraclass correlation is between subjects within a cluster the worse the design effect becomes (i.e. the larger it gets from 1. Indicating a larger expected increase in the variance of the estimator). In other words, the more there is heterogeneity between clusters and more homogeneity between subjects within a cluster, the less accurate are our estimators become. This is because in such cases we are better off sampling as many clusters as we can and making do with a small sample of subjects from within each cluster (i.e. two stage cluster sampling). - Complexity. Cluster sampling are more sophisticated and requires more attention with how to plan and on how to analyze (i.e.: to take into account the weights of subjects during the estimation of parameters, confidence intervals, etc.)

Two-stage cluster sampling, a simple case of multistage sampling, is obtained by selecting cluster samples in the first stage and then selecting a sample of elements from every sampled cluster. Consider a population of *N* clusters in total. In the first stage, *n* clusters are selected using ordinary cluster sampling method. In the second stage, simple random sampling is usually used.^{ [3] } It is used separately in every cluster and the numbers of elements selected from different clusters are not necessarily equal. The total number of clusters *N*, number of clusters selected *n*, and numbers of elements from selected clusters need to be pre-determined by the survey designer. Two-stage cluster sampling aims at minimizing survey costs and at the same time controlling the uncertainty related to estimates of interest.^{ [4] } This method can be used in health and social sciences. For instance, researchers used two-stage cluster sampling to generate a representative sample of the Iraqi population to conduct mortality surveys.^{ [5] } Sampling in this method can be quicker and more reliable than other methods, which is why this method is now used frequently.

Cluster sampling methods can lead to significant bias when working with a small number of clusters. For instance, it can be necessary to cluster at the state or city level, units that may be small and fixed in number. Microeconometrics methods for panel data often use short panels, which is analogous to having few observations per clusters and many clusters. The small cluster problem can be viewed as an incidental parameter problem.^{ [6] } While the point estimates can be reasonably precisely estimated, if the number of observations per cluster is sufficiently high, we need the number of clusters for the asymptotics to kick in. If the number of clusters is low the estimated covariance matrix can be downward biased.^{ [7] }

Small numbers of clusters is a risk when there is serial correlation or when there is intraclass correlation as in the Moulton context. When having few clusters, we tend to underestimate serial correlation across observations when a random shock occurs, or the intraclass correlation in a Moulton setting.^{ [8] } Several studies have highlighted the consequences of serial correlation and highlighted the small-cluster problem.^{ [9] }^{ [10] }

In the framework of the Moulton factor, an intuitive explanation of the small cluster problem can be derived from the formula for the Moulton factor. Assume for simplicity that the number of observation per cluster is fixed at *n*. Below, stands for the covariance matrix adjusted for clustering, stands for the covariance matrix not adjusted for clustering, and ρ stands for the intraclass correlation:

The ratio on the left-hand side provides an indication of how much the unadjusted scenario overestimates the precision. Therefore, a high number means a strong downward bias of the estimated covariance matrix. A small cluster problem can be interpreted as a large n: when the data is fixed and the number of clusters is low, the number of data within a cluster can be high. It follows that inference when the number of clusters is small will not have correct coverage.^{ [8] }

Several solutions for the small cluster problem have been proposed. One can use a bias-corrected cluster-robust variance matrix, make T-distribution adjustments, or use bootstrap methods with asymptotic refinements, such as the percentile-t or wild bootstrap, that can lead to improved finite sample inference.^{ [7] } Cameron, Gelbach and Miller (2008) provide microsimulations for different methods and find that the wild bootstrap performs well in the face of a small number of clusters.^{ [11] }

In statistics and probability theory, the **median** is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result.

**Statistics** is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.

The **weighted arithmetic mean** is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

In statistics, multistage sampling is the taking of samples in stages using smaller and smaller sampling units at each stage.

In statistics, quality assurance, and survey methodology, **sampling** is the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population in question. Sampling has lower costs and faster data collection than measuring the entire population and can provide insights in cases where it is infeasible to sample an entire population.

In statistics, the **Pearson correlation coefficient** ― also known as **Pearson's r**, the

In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. **Estimation of covariance matrices** then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in **R**^{p×p}; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has a normal distribution, the sample covariance matrix has a Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data require deeper considerations. Another issue is the robustness to outliers, to which sample covariance matrices are highly sensitive.

In probability theory and statistics, the **discrete uniform distribution** is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of *n* values has equal probability 1/*n*. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen".

This **glossary of statistics and probability** is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics.

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- Estimating the precision of sample statistics by using subsets of available data (
**jackknifing**) or drawing randomly with replacement from a set of data points (**bootstrapping**) - Permutation tests are exact tests: Exchanging labels on data points when performing significance tests
- Validating models by using random subsets

**Bootstrapping** is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

In statistics, the **intraclass correlation**, or the **intraclass correlation coefficient** (**ICC**), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other. While it is viewed as a type of correlation, unlike most other correlation measures it operates on data structured as groups, rather than data structured as paired observations.

**Forest inventory** is the systematic collection of data and forest information for assessment or analysis. An estimate of the value and possible uses of timber is an important part of the broader information required to sustain ecosystems. When taking forest inventory the following are important things to measure and note: species, diameter at breast height (DBH), height, site quality, age, and defects. From the data collected one can calculate the number of trees per acre, the basal area, the volume of trees in an area, and the value of the timber. Inventories can be done for other reasons than just calculating the value. A forest can be cruised to visually assess timber and determine potential fire hazards and the risk of fire. The results of this type of inventory can be used in preventive actions and also awareness. Wildlife surveys can be undertaken in conjunction with timber inventory to determine the number and type of wildlife within a forest. The aim of the statistical forest inventory is to provide comprehensive information about the state and dynamics of forests for strategic and management planning. Merely looking at the forest for assessment is called taxation.

The **Heckman correction** is a statistical technique to correct bias from non-randomly selected samples or otherwise incidentally truncated dependent variables, a pervasive issue in quantitative social sciences when using observational data. Conceptually, this is achieved by explicitly modelling the individual sampling probability of each observation together with the conditional expectation of the dependent variable. The resulting likelihood function is mathematically similar to the tobit model for censored dependent variables, a connection first drawn by James Heckman in 1974. Heckman also developed a two-step control function approach to estimate this model, which avoids the computational burden of having to estimate both equations jointly, albeit at the cost of inefficiency. Heckman received the Nobel Memorial Prize in Economic Sciences in 2000 for his work in this field.

In various science/engineering applications, such as independent component analysis, image analysis, genetic analysis, speech recognition, manifold learning, and time delay estimation it is useful to **estimate the differential entropy** of a system or process, given some observations.

In survey methodology, the **design effect** is the ratio between the variances of two estimators to some parameter of interest. Specifically the ratio of an actual variance of an estimator that is based on a sample from some sampling design, to the variance of an alternative estimator that would be calculated (hypothetically) using a sample from a simple random sample (SRS) of the same number of elements. It measures the expected effect of the design structure on the variance of some estimator of interest. The design effect is a positive real number that can indicate an inflation, or deflation in the variance of an estimator for some parameter, that is due to the study not using SRS.

In statistics, the **Horvitz–Thompson estimator**, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the total and mean of a pseudo-population in a stratified sample. Inverse probability weighting is applied to account for different proportions of observations within strata in a target population. The Horvitz–Thompson estimator is frequently applied in survey analyses and can be used to account for missing data, as well as many sources of unequal selection probabilities.

The **ratio estimator** is a statistical parameter and is defined to be the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical and symmetrical tests such as the t test should not be used to generate confidence intervals.

In statistics, **stratified randomization** is a method of sampling which first stratifies the whole study population into subgroups with same attributes or characteristics, known as strata, then followed by simple random sampling from the stratified groups, where each element within the same subgroup are selected unbiasedly during any stage of the sampling process, randomly and entirely by chance. Stratified randomization is considered a subdivision of stratified sampling, and should be adopted when shared attributes exist partially and vary widely between subgroups of the investigated population, so that they require special considerations or clear distinctions during sampling. This sampling method should be distinguished from cluster sampling, where a simple random sample of several entire clusters is selected to represent the whole population, or stratified systematic sampling, where a systematic sampling is carried out after the stratification process. Stratified random sampling is sometimes also known as "**quota random sampling**".

- ↑ David Brown, Study Claims Iraq's 'Excess' Death Toll Has Reached 655,000,
*Washington Post*, Wednesday, October 11, 2006. Retrieved September 14, 2010. - ↑ Kerry and Bland (1998). Statistics notes: The intracluster correlation coefficient in cluster randomization.
*British Medical Journal*, 316, 1455–1460. - ↑ Ahmed, Saifuddin (2009).
*Methods in Sample Surveys*(PDF). The Johns Hopkins University and Saifuddin Ahmed. - ↑ Daniel Pfeffermann; C. Radhakrishna Rao (2009).
*Handbook of Statistics Vol.29A Sample Surveys: Theory, Methods and Infernece*. Elsevier B.V. ISBN 978-0-444-53124-7. - ↑ LP Galway; Nathaniel Bell; Al S SAE; Amy Hagopian; Gilbert Burnham; Abraham Flaxman; Wiliam M Weiss; Julie Rajaratnam; Tim K Takaro (27 April 2012). "A two-stage cluster sampling method using gridded population data, a GIS, and Google EarthTM imagery in a population-based mortality survey in Iraq".
*International Journal of Health Geographics*.**11**: 12. doi:10.1186/1476-072X-11-12. PMC 3490933 . PMID 22540266. - ↑ Cameron A. C. and P. K. Trivedi (2005): Microeconometrics: Methods and Applications. Cambridge University Press, New York.
- 1 2 Cameron, C. and D. L. Miller (2015): A Practitioner's Guide to Cluster-Robust Inference. Journal of Human Resources 50(2), pp. 317–372.
- 1 2 Angrist, J.D. and J.-S. Pischke (2009): Mostly Harmless Econometrics. An empiricist's companion. Princeton University Press, New Jersey.
- ↑ Bertrand, M., E. Duflo and S. Mullainathan (2004): How Much Should We Trust Differences-in-Differences Estimates? Quarterly Journal of Economics 119(1), pp. 249–275.
- ↑ Kezdi, G. (2004): Robust Standard Error Estimation in Fixed-Effect Panel Models. Hungarian Statistical Review 9, pp. 95–116.
- ↑ Cameron, C., J. Gelbach and D. L. Miller (2008): Bootstrap-Based Improvements for Inference with Clustered Errors. The Review of Economics and Statistics 90, pp. 414–427.

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