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 for 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 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,Advantage
* Can be cheaper than other sampling plans – e.g. fewer travel expenses, and 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 aDisadvantage
* 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. 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 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 is more sophisticated and requires more attention with how to plan and 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 ofInference when the number of clusters is small
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. 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.Cameron, C. and D. L. Miller (2015): A Practitioner's Guide to Cluster-Robust Inference. Journal of Human Resources 50(2), pp. 317–372. Small numbers of clusters are 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.Angrist, J.D. and J.-S. Pischke (2009): Mostly Harmless Econometrics. An empiricist's companion. Princeton University Press, New Jersey. Several studies have highlighted the consequences of serial correlation and highlighted the small-cluster problem. 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 observations 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 indicates 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 the correct coverage. 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. 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.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.See also
*References
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