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Handout3-W09 - STAT 454/854 Winter 2009 Handout#3...

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STAT 454/854 Winter 2009 Handout #3 Single-stage Cluster Sampling (1) Single-stage cluster sampling with clusters of equal sizes We have M i = M , i = 1 , · · · , N and K = NM . The population mean is μ y = N - 1 N i =1 μ i , where μ i = M - 1 M j =1 y ij . Under single-stage cluster sampling with clus- ters selected by SRSWOR, an unbiased estimator of μ y is given by ¯ y = n - 1 n i =1 μ i , with theoretical variance V y ) = 1 - n N σ 2 M n , where σ 2 M = ( N - 1) - 1 N i =1 ( μ i - μ y ) 2 . An unbiased variance estimator is given by v y ) = 1 - n N s 2 M n , where s 2 M = ( n - 1) - 1 i s ( μ i - ¯ y ) 2 . (2) Intra-cluster correlation coefficient Suppose that we randomly select a cluster from the population, and then randomly select two elements without replacement in the selected cluster, and let z 1 and z 2 be the respective y -value of the two selected elements. Define the intra-cluster correlation coefficient as ρ = Cov ( z 1 , z 2 ) / { V ( z 1 ) V ( z 2 ) } 1 / 2 , where Cov ( z 1 , z 2 ) = 1 NM ( M - 1) N X i =1 M X j =1 M X k 6 = j ( y ij - μ y )( y ik - μ y ) and V ( z 1 ) = V ( z 2 ) = { ( NM -
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