92 SECTION 6 CLUSTER SAMPLING AND SYSTEMATIC LIST SAMPLING 6.1 What is Cluster Sampling? 6.1.1 Definition Cluster Sampling --- Probability sampling in which sampling units at some point in the selection process are collections, or clusters, of population elements 6.1.2 Selecting a Simple One-Stage Cluster Sample A. Specify appropriate clusters. In most instances clusters would consist of population elements which are physically close together and therefore relatively similar to each other (i.e., with high intra-cluster homogeneity). Clusters almost always have varying numbers of population members.93 B. Select a simple random sample of clusters from a complete list of clusters. C. Collect survey data from all population elements falling in the selected clusters. Note: We have considered here a special type of cluster sampling in which the probability sampling method used to select clusters is simple random sampling. However, any form of probability sampling as well as stratification can be used to choose the clusters. 6.1.3 Some Examples A. In a household survey for a small city, a probability sample of blocks is selected. Each block in this case represents a cluster of households. Households may or may not be subsampled within selected blocks.94 B. In a national sample of inpatient hospital visits for individuals with multiple sclerosis during some calendar year, a probability sample of hospitals is chosen. Each hospital in this instance represents a cluster of visits by patients with multiple sclerosis who were discharged during that year. C. In a survey of first graders in the schools of a state, a probability sample of schools is selected. All first graders in a school would represent a cluster in this design. 6.1.4 Some Implications of Using Cluster Sampling A. If we collect survey data on all members of each selected cluster (i.e., we have a "one-stage" sample of members), the probability of choosing each element in this cluster is the same as the probability of choosing the cluster regardless of the number of elements in the cluster. For example, if we have 100 clusters of unequal size and we randomly choose 10 of them, the probability of selection for each element in the 10 selected clusters is 0.1 regardless of the sizes of the selected clusters.95 B. Cluster sampling generally yields estimates with relatively larger variances (i.e., lower precision) than samples of the same size which are chosen by element (i.e., non cluster) sampling. The amount of the increase in variance is directly related to the average sample cluster size. C. Because members of clusters are often close in geographic proximity, the average cost per sample element can be reduced substantially over element sampling if cluster sampling is used. The amount of the reduction in costs is directly related to the average size of the clusters that are used. D. Since elements in clusters are usually similar (i.e., clusters are internally homogeneous), the amount of information gathered by the survey may not be increased substantially as new measurements are taken within clusters. This tells us that sample cluster sizes should not be too large. As a general rule, the number of clusters in the population should be large which means that the average size of clusters should be kept as small as possible.96 E. NOTE RE-WORDING: The survey statistician frequently has some choice in the size of clusters that are used in a survey. In making this choice, the cost advantages of large (sample) clusters must be properly weighed against the statistical advantages of smaller (sample) clusters. F. Analysis of data from cluster samples is frequently somewhat more complex than analysis of data from element samples. G. Cluster sampling eliminates the need for a sampling frame consisting of a list of all elements in the population. Since clusters are the units being sampled, a listing of all clusters in the population constitutes an appropriate frame. H. V(n)>0 is possible if clusters vary in size. (Skip to Section 6.5 on p. 111) 6.2 Estimating a Population Mean from a Simple Cluster Sample 6.2.1 Setting A. A population of elements is divided into clusters so that each element is a member of only one cluster.97 B. We wish to estimate the mean per element for some characteristic in the population (denoted by the symbol, Y). C. The sampling method we decide to use in selecting our sample of clusters is simple random sampling. Other more complex sampling methods involving stratification might have also been used. D. Data are collected from all elements in the sample of clusters. 6.2.2 Some Definitions N = Number of elements in the population A = Number of clusters in the population BNA= = Average number of elements per cluster in the population a = Number of clusters chosen in the sample nj = Number of population elements which fall in the j-th cluster98 n = njja=1 = Total number of elements in the clusters chosen in the sample bna= = Average number of elements per cluster in the sample yj = Total of all observations in the j-th cluster 6.2.3 Estimator of Y rynynjjajjajja= ====111 (6.1) Note: The symbol "r" is used since the estimator in formula (6.1) is usually called a ratio mean.99 6.2.4 Estimated Variance of r A. If the average cluster size in the population B is known; 2c2A av(r) sAaB− =   (6.2) where sc2 is a measure of variability among clusters calculated by the formula sy rnay r n r y nacj jjajjajjaj jja2212 2121 1121=−−=+ −−= = = =   ( ) (6.3) B. If the average cluster size in the population B is not known: 2c2(A a)av(r) sAn− =   (6.4) where sc2 is defined in formula (6.3).100 6.2.5 Standard Error of r se(r) v(r)= (6.5) where the value of v(r) would depend on whether formula (6.2) or (6.4) was used. 6.2.6 Confidence Intervals for Y (n>50) Lower Boundary: r t se r−( ) (6.6) Upper Boundary: r t se r+( ) (6.7) The value of t depends on the confidence level we wish for the confidence interval (see Section 2.2.7). Interpretation: We are 95 percent sure that Y is covered by the interval whose (t=1.96) boundaries are computed by formulas (6.6) and (6.7). A. Formulas in Section 6.2 can be used to estimate a