# Cluster sampling

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.

Cluster elements

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.

When clusters are of different sizes

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.

Applications of cluster sampling

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).
• 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.)

More on cluster sampling

Two-stage cluster sampling

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.

{\displaystyle {\frac {V_{c}({\hat {\beta }})}{V({\hat {\beta }})}}=1+(n-1)\rho }

References

1. ^David Brown, Study Claims Iraq’s ‘Excess’ Death Toll Has Reached 655,000, Washington Post, Wednesday, October 11, 2006. Retrieved September 14, 2010.
2. ^Kerry and Bland (1998). Statistics notes: The intracluster correlation coefficient in cluster randomization. British Medical Journal, 316, 1455–1460.
3. ^Ahmed, Saifuddin (2009). Methods in Sample Surveys(PDF). The Johns Hopkins University and Saifuddin Ahmed.
4. ^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.
5. ^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.
6. ^Cameron A. C. and P. K. Trivedi (2005): Microeconometrics: Methods and Applications. Cambridge University Press, New York.
7. ^ Jump up to:ab Cameron, C. and D. L. Miller (2015): A Practitioner’s Guide to Cluster-Robust Inference. Journal of Human Resources 50(2), pp. 317–372.
8. ^ Jump up to:ab Angrist, J.D. and J.-S. Pischke (2009): Mostly Harmless Econometrics. An empiricist’s companion. Princeton University Press, New Jersey.
9. ^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.
10. ^Kezdi, G. (2004): Robust Standard Error Estimation in Fixed-Effect Panel Models. Hungarian Statistical Review 9, pp. 95–116.
11. ^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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