Using balanced sampling in creel surveys
Section 2. Balanced sampling

Suppose that U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbaaaa@325F@ is a finite population of size N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGobaaaa@3258@ that is sampled with a design having selection probabilities given by { π i : i = 1, , N } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaGadaqaaiabec8aWnaaBaaaleaaca WGPbaabeaakiaayIW7caGG6aGaaGjbVlaadMgacaaI9aGaaGymaiaa iYcacqWIMaYscaWGobaacaGL7bGaayzFaaGaaGjcVlaac6caaaa@41D1@ If x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4baaaa@3282@ is an auxiliary variable known for all population units, then the sample is balanced on x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4baaaa@3282@ if the Horvitz-Thompson estimator for the total of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4baaaa@3282@ is equal to the known total of x . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bGaaiOlaaaa@3334@ In other words, for any balanced sample s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbGaaiilaaaa@332D@ the following equation has to be satisfied,

i s x i π i = i = 1 N x i . ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaaeqbqaamaalaaabaGaamiEamaaBa aaleaacaWGPbaabeaaaOqaaiabec8aWnaaBaaaleaacaWGPbaabeaa aaaabaGaamyAaiabgIGiolaadohaaeqaniabggHiLdGccaaI9aWaaa bCaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacaaI9aGa aGymaaqaaiaad6eaa0GaeyyeIuoakiaai6cacaaMf8UaaGzbVlaayw W7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaigdacaGGPaaaaa@507A@

For the surveys considered here, we balance on indicator variables I i ( ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaSbaaSqaaiaadMgaaeqaaO WaaeWaaeaacqaHjpWDaiaawIcacaGLPaaaaaa@36CD@ equal to 1 if unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@3273@ is of type ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHjpWDaaa@3352@ and 0 otherwise. If all the units i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@3273@ for which I i ( ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaSbaaSqaaiaadMgaaeqaaO WaaeWaaeaacqaHjpWDaiaawIcacaGLPaaaaaa@36CD@ is equal to 1 have the same selection probability π ω , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaeqyYdChabe aakiaacYcaaaa@35F5@ then equation (2.1) reduces to i s I i ( ω ) / π ω = i = 1 N I i ( ω ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaWcgaqaamaaqababaGaamysamaaBa aaleaacaWGPbaabeaakmaabmaabaGaeqyYdChacaGLOaGaayzkaaaa leaacaWGPbGaeyicI4Saam4Caaqab0GaeyyeIuoaaOqaaiabec8aWn aaBaaaleaacqaHjpWDaeqaaaaakiaai2dadaaeWaqaaiaadMeadaWg aaWcbaGaamyAaaqabaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaa WcbaGaamyAaiaai2dacaaIXaaabaGaamOtaaqdcqGHris5aOGaaiOl aaaa@4C0A@ In this context the balancing equation simply requests that the number of sampled units of type ω , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHjpWDcaGGSaaaaa@3402@ n ω = i s I i ( ω ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiabeM8a3bqaba GccaaI9aWaaabeaeaacaWGjbWaaSbaaSqaaiaadMgaaeqaaOWaaeWa aeaacqaHjpWDaiaawIcacaGLPaaacaGGSaaaleaacaWGPbGaeyicI4 Saam4Caaqab0GaeyyeIuoaaaa@4087@ is equal to its expectation,

n ω = i = 1 N I i ( ω ) π ω . ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiabeM8a3bqaba GccaaI9aWaaabCaeaacaWGjbWaaSbaaSqaaiaadMgaaeqaaOWaaeWa aeaacqaHjpWDaiaawIcacaGLPaaacqaHapaCdaWgaaWcbaGaeqyYdC habeaaaeaacaWGPbGaaGypaiaaigdaaeaacaWGobaaniabggHiLdGc caaIUaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmai aac6cacaaIYaGaaiykaaaa@4FC4@

To implement balanced sampling we use the cube method of Deville and Tillé (2004), and the extension of Hasler and Tillé (2014) to cope with highly stratified populations. In Section 4 this method is compared with the implementation of the rejection method proposed by Fuller (2009). In the context of this study, we are balancing on T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGubaaaa@325E@ types of units; we want the sampled numbers of units for the T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGubaaaa@325E@ types, n ˜ = ( n 1 , , n T ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGUbGbaGaacaaI9aWaaeWaaeaaca WGUbWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiablAciljaaiYcacaWG UbWaaSbaaSqaaiaadsfaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabe aatCvAUfeBSn0BKvguHDwzZbqegeKCPfgBGuLBPn2BKvginnfaiqaa caWFsedaaOGaaGzaVlaacYcaaaa@49F8@ to be equal to their expectations, E ( n ˜ ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGfbWaaeWaaeaaceWGUbGbaGaaai aawIcacaGLPaaacaGGSaaaaa@358A@ under the sampling design. Under rejective sampling, the sample is said to be balanced if

Q T , n = ( n ˜ E ( n ˜ ) ) [ Var ( n ˜ ) ] 1 ( n ˜ E ( n ˜ ) ) < γ 2 ( 2.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGrbWaaSbaaSqaaiaadsfacaaMb8 UaaGilaiaaykW7caWGUbaabeaakiaai2dadaqadaqaaiqad6gagaac aiabgkHiTiaadweadaqadaqaaiqad6gagaacaaGaayjkaiaawMcaaa GaayjkaiaawMcaamaaCaaaleqabaWexLMBbXgBd9gzLbvyNv2CaeHb bjxAHXgiv5wAJ9gzLbsttbaceaGaa8NeXaaakmaadmaabaGaaeOvai aabggacaqGYbWaaeWaaeaaceWGUbGbaGaaaiaawIcacaGLPaaaaiaa wUfacaGLDbaadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaai qad6gagaacaiabgkHiTiaadweadaqadaqaaiqad6gagaacaaGaayjk aiaawMcaaaGaayjkaiaawMcaaiaaiYdacqaHZoWzdaahaaWcbeqaai aaikdaaaGccaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI YaGaaiOlaiaaiodacaGGPaaaaa@6AB3@

where Var ( n ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGwbGaaeyyaiaabkhadaqadaqaai qad6gagaacaaGaayjkaiaawMcaaaaa@36C2@ represents the design based covariance matrix of n ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGUbGbaGaaaaa@3287@ and γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHZoWzdaahaaWcbeqaaiaaikdaaa aaaa@3415@ is a tolerance value that determines the balancing condition. Samples that do not meet the balancing equation Q T , n < γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGrbWaaSbaaSqaaiaadsfacaaMb8 UaaGilaiaaykW7caWGUbaabeaakiaaiYdacqaHZoWzdaahaaWcbeqa aiaaikdaaaaaaa@3B7E@ are simply rejected.


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