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    6. Concluding remarks

    Phillip S. Kott and Dan Liao

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    In Section 4, we noted two reasons to prefer calibration weighting in two steps: to make implicitly fitting a logistic response model easier and to incorporate nearly quasi-optimal calibration. A side benefit of two-step calibration is more efficient estimation of the response model in step one since there is no sampling error to confound the estimation. This is useful when one wants to analyze the causes of unit nonresponse for its own sake.

    We must concede, however, that the reduction in mean squared error using two steps was modest in our simulation experiments in Section 5. Moreover, the practical appeal of the simplicity of calibrating in a single step cannot be denied.

    When calibration-weighting is used to adjust for nonresponse that is not missing at random as described in Chang and Kott (2008) and Kott and Chang (2010), the efficiency gains from a second step involving only calibration variables and functions of calibration variables model variables is likely to be sizeable.

    When the finite population correction factors can be ignored, replication offers a much simpler approach to variance estimation than equation (3.7) even though the second summation on the right-hand side can be dropped in this situation. A different attractive alternative is the “collapsed” version of equation (4.2) that ignores the impact of the first calibration step:

    v ˜ ( t y ) = ∑ k , j ∈ S ( 1 − π k π j π k j ) [ w k e 2 k ] [ w j e 2 j ] + ∑ k ∈ R d k ( h k 2 α k 2 − h k α k ) e 2 k 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWG2bGbaG aadaqadeqaaiaadshadaWgaaWcbaGaamyEaaqabaaakiaawIcacaGL PaaacqGH9aqpdaaeqbqaamaabmaabaGaaGymaiabgkHiTmaalaaaba GaeqiWda3aaSbaaSqaaiaadUgaaeqaaOGaeqiWda3aaSbaaSqaaiaa dQgaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadUgacaWGQbaabeaaaa aakiaawIcacaGLPaaadaWadaqaaiaadEhadaWgaaWcbaGaam4Aaaqa baGccaWGLbWaaSbaaSqaaiaaikdacaWGRbaabeaaaOGaay5waiaaw2 faaaWcbaGaam4AaiaacYcacaWGQbGaeyicI4Saam4uaaqab0Gaeyye IuoakmaadmaabaGaam4DamaaBaaaleaacaWGQbaabeaakiaadwgada WgaaWcbaGaaGOmaiaadQgaaeqaaaGccaGLBbGaayzxaaGaey4kaSYa aabuaeaacaWGKbWaaSbaaSqaaiaadUgaaeqaaOWaaeWaaeaacaWGOb Waa0baaSqaaiaadUgaaeaacaaIYaaaaOGaeqySde2aa0baaSqaaiaa dUgaaeaacaaIYaaaaOGaeyOeI0IaamiAamaaBaaaleaacaWGRbaabe aakiabeg7aHnaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaiaa dwgadaqhaaWcbaGaaGOmaiaadUgaaeaacaaIYaaaaOGaaiOlaaWcba Gaam4AaiabgIGiolaadkfaaeqaniabggHiLdaaaa@7A13@

    This estimator clearly estimates the prediction-model variance if that model holds. A version of it − with the second summation removed − fared well in our simulation experiments (not shown). Some caution is needed before one draws too strong a conclusion from that result since the linear model was never too far from holding in our investigations.

    Finally, a number of assumptions were made to simplify the exposition. The interested reader can extend the results to unbounded d k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGKbWaaS baaSqaaiaadUgaaeqaaOGaaiilaaaa@3B24@  more general and not-necessarily-bounded weight-adjustment functions, or to allow the prediction-model errors to be correlated within primary sampling units. When N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobaaaa@3938@  grows faster than n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaai ilaaaa@3A08@  the assumption that σ k 2 = z k T η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaam4AaaqaaiaaikdaaaGccqGH9aqpcaWH6bWaa0baaSqa aiaadUgaaeaacaWGubaaaOGaaC4Tdaaa@4157@  can sometimes be dropped. See, for example, Kott (2009, page 69).

    Acknowledgements

    This paper was prepared for the Symposium on the Analysis of Survey Data and Small Area Estimation, in honour of the 75th Birthday of Professor J.N.K. Rao sponsored by the Fields Institute for Research in Mathematical Sciences. The authors would like to thank the organizers of the conference for the invitation to present this paper and the Institute for its generous funding of the conference without which this paper would never have been written. They would also like to thank a number of editors and referees for their helpful comments.

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