Using balanced sampling in creel surveys
Section 5. Discussion

In the context of creel surveys, balanced sampling techniques such as the cube method or the rejective algorithm are used to ensure a predetermined sample size in small domains of the survey population. The cube method is very effective at doing so especially in complex survey designs with several stages of sampling. It does not change the selection probabilities and it yields domain sample sizes that are very close their target values. The rejective method, on the other hand, changes the selection probabilities slightly and produce domain sample sizes that are more variable. With a large number of constraints, Fuller’s rejective sampling scheme is not really applicable as it requires the evaluation and the inversion of a large covariance matrix in (2.3); alternative acceptation criteria for a sample need to be investigated.

Acknowledgements

We thank the Associate Editor and the referees for their constructive comments on the first version of this manuscript. The assistance and the suggestions of Valérie Bujold, Michel Legault and of Hélène Crépeau who participated to the initial phase of this project is gratefully acknowledged. This project benefitted from the financial assistance of the Canada Research Chair in Statistical Sampling and Data Analysis and form a discovery grant (5244/2012) from the Natural Sciences and Engineering Research Council of Canada.

Appendix

Calculation of the joint selection probabilities when N = 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8urps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGobGaaGypaiaaiodaaaa@3415@

Consider a population of size 3 and let π 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaaGymaaqaba GccaGGSaaaaa@34E3@ π 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaaGOmaaqaba GccaGGSaaaaa@34E4@ and π 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaaG4maaqaba aaaa@342B@ be the marginal selection probabilities when drawing a sample of size n = 2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbGaaGypaiaaikdacaGGUaaaaa@34AD@ The joint selection probabilities π i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaaiilaaaa@3605@ i j = 1,2,3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyiyIKRaamOAaiaai2daca aIXaGaaGilaiaaikdacaaISaGaaG4maaaa@3990@ satisfy

( 1 1 0 1 0 1 0 1 1 ) ( π 12 π 13 π 23 ) = ( π 1 π 2 π 3 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaauaabaqadmaaaeaacaaIXa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaWaae WaaeaafaqaaeWabaaabaGaeqiWda3aaSbaaSqaaiaaigdacaaIYaaa beaaaOqaaiabec8aWnaaBaaaleaacaaIXaGaaG4maaqabaaakeaacq aHapaCdaWgaaWcbaGaaGOmaiaaiodaaeqaaaaaaOGaayjkaiaawMca aiaai2dadaqadaqaauaabaqadeaaaeaacqaHapaCdaWgaaWcbaGaaG ymaaqabaaakeaacqaHapaCdaWgaaWcbaGaaGOmaaqabaaakeaacqaH apaCdaWgaaWcbaGaaG4maaqabaaaaaGccaGLOaGaayzkaaGaaGOlaa aa@50AE@

Thus

( π 12 π 13 π 23 ) = ( 1 1 0 1 0 1 0 1 1 ) 1 ( π 1 π 2 π 3 ) = 1 2 ( 1 1 1 1 1 1 1 1 1 ) ( π 1 π 2 π 3 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaauaabaqadeaaaeaacqaHap aCdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeqiWda3aaSbaaSqa aiaaigdacaaIZaaabeaaaOqaaiabec8aWnaaBaaaleaacaaIYaGaaG 4maaqabaaaaaGccaGLOaGaayzkaaGaaGypamaabmaabaqbaeaabmWa aaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaigdaaeaacaaIWa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaaaaiaawIca caGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaauaaba qadeaaaeaacqaHapaCdaWgaaWcbaGaaGymaaqabaaakeaacqaHapaC daWgaaWcbaGaaGOmaaqabaaakeaacqaHapaCdaWgaaWcbaGaaG4maa qabaaaaaGccaGLOaGaayzkaaGaaGypamaalaaabaGaaGymaaqaaiaa ikdaaaWaaeWaaeaafaqabeWadaaabaGaaGymaaqaaiaaigdaaeaacq GHsislcaaIXaaabaGaaGymaaqaaiabgkHiTiaaigdaaeaacaaIXaaa baGaeyOeI0IaaGymaaqaaiaaigdaaeaacaaIXaaaaaGaayjkaiaawM caamaabmaabaqbaeaabmqaaaqaaiabec8aWnaaBaaaleaacaaIXaaa beaaaOqaaiabec8aWnaaBaaaleaacaaIYaaabeaaaOqaaiabec8aWn aaBaaaleaacaaIZaaabeaaaaaakiaawIcacaGLPaaacaaIUaaaaa@6979@

Using these equations, the entries of the covariance matrix (4.1) can be evaluated using the stage 1 and the stage 2 selection probabilities.

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