Using the American Community Survey Benefits and Challenges (2007) / Chapter Skim
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Appendix B Controlling the American Community Survey to Postcensal Population Estimates
Appendix B Controlling the American Community Survey to Postcensal Population Estimates
Pages 269-289
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From page 269... ...
Census Bureau has proposed the use of postcensal population estimates as population controls for the American Community Survey at a fine level of geographic and demographic stratification. These population estimates are known to be imperfect.
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From page 270... ...
0 USING THE ACS: BENEFITS AND CHALLENGES 270 USING THE ACS: BENEFITS AND CHALLENGES levels of demographic and/or geographic aggregation. The panel's charge for me was to evaluate these options from a theoretical, not an empirical, perspective.
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From page 271... ...
, we have that Tcell = Dk l , so that the sample is controlled to the postcensal population estimates in every cell. The PSE with cell controls therefore is similar to the Census Bureau's plan to control at a fine level of demographic stratification within estimation areas.
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From page 272... ...
USING THE ACS: BENEFITS AND CHALLENGES 272 USING THE ACS: BENEFITS AND CHALLENGES The second estimator is the PSE with control on one margin. Without loss of generality, take this as the row margin: �� � i ∈U g h Fi Si Ri yi /πi h ˆ Dg• � � Tmarg = (B.2)
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From page 273... ...
APPENDIXB: CONTROLLING TO POSTCENSAL POPULATION ESTIMATES APPENDIX B 273 B–1.3 Assumptions Assume that the postcensal population estimation errors {D g h − C g h } are independent random variables, independent of the design variables {Fi , Si , Ri } and of the data {yi }. These independence assumptions might be questionable in a real application.
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From page 274... ...
USING THE ACS: BENEFITS AND CHALLENGES 274 USING THE ACS: BENEFITS AND CHALLENGES Note that either of these sets of assumptions imply that the cell-level stratification is sufficient (fine enough) to account for differences in coverage and response, and so it is an appropriate level of aggregation for poststratification adjustments.
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From page 275... ...
APPENDIXB: CONTROLLING TO POSTCENSAL POPULATION ESTIMATES APPENDIX B 275 the PSE with cell-level controls will be biased. It is useful to express this bias as � � � � � � � δg h δg h ˆ , µ g h cvU E m Tcell − T (C•• − 1)
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From page 276... ...
USING THE ACS: BENEFITS AND CHALLENGES 276 USING THE ACS: BENEFITS AND CHALLENGES � � cvU δ g h /C g h �= 0, and the correlation is negative (overestimate, low income) , so a negative term is contributed to the bias.
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From page 277... ...
APPENDIXB: CONTROLLING TO POSTCENSAL POPULATION ESTIMATES APPENDIX B 277 B–2.2 Design Properties of PSE with Cell-Level Controls The PSE is nonlinear in the design variables, so its expectation and variance under the design are approximated from the usual Taylor series linearization, �� � � � � ˆ Cg h + δg h yg h + yg h Dg h − Cg h − δg h Tcell ≃ ¯ ¯ g ,h g ,h � C g h + δ g h � Fi Si R i � � yi − y g h .
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From page 278... ...
USING THE ACS: BENEFITS AND CHALLENGES 278 USING THE ACS: BENEFITS AND CHALLENGES B–2.3 Model Properties of PSE with Margin-Level Controls � Define Ag h = i ∈U g h Fi Si Ri /(πi C g h ) , the estimated observation probability in cell g , h.
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From page 279... ...
APPENDIXB: CONTROLLING TO POSTCENSAL POPULATION ESTIMATES APPENDIX B 279 The first term is bias attributable solely to population estimation error. It would appear even in the absence of any bias due to undercoverage or nonresponse.
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From page 280... ...
0 USING THE ACS: BENEFITS AND CHALLENGES 280 USING THE ACS: BENEFITS AND CHALLENGES The relative design bias is then given approximately by � � ˆ RelBias Tmarg � � � C g • aveU g (¯g h )
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From page 281... ...
APPENDIXB: CONTROLLING TO POSTCENSAL POPULATION ESTIMATES APPENDIX B 281 ling on the overall count is analogous to controlling on the row margin when there is a single row. Thus the overall-control estimator has relative model bias given approximately by � � ˆ RelBias m Toverall � � δg h ≃ aveU Cg h � � + corr U Ag h , µ g h cvU (Ag h )
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From page 282... ...
USING THE ACS: BENEFITS AND CHALLENGES 282 USING THE ACS: BENEFITS AND CHALLENGES The interpretation of this relative design bias directly parallels the relative model bias described above. The first term is bias attributable solely to estimation error.
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From page 283... ...
APPENDIXB: CONTROLLING TO POSTCENSAL POPULATION ESTIMATES APPENDIX B 283 For given levels of postcensal population estimation biases {δ g h }, the exact biases and relative biases for the various estimators, under either the model or design assumptions, can be computed directly from these values for cell counts and totals, margin (row or column) counts and totals, and the overall counts and totals.
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From page 284... ...
B–4 DISCUSSION The derivations and numerical examples described in earlier sections illustrate some important points that may be relevant to the use of population estimates as controls in weighting for the American Community Survey. First, the poststratification estimator that is controlled at the cell level is unbiased only if the postcensal population estimation biases (in correctly specified cells)
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From page 285... ...
TABLE B.1 Exact Relative Biases for Case of Unbiased Postcensal Population Estimates Column 1 Column 2 Row Sums ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Trow Trow Trow Tcell Tcolumn Toverall Tcell Tcolumn Toverall Tcell Tcolumn Toverall Row 1 0 Count 0 0 0.067 0 −0.067 −0.13 −0.18 −0.11 −0.059 −0.12 −0.12 Total 0 0 0.067 0 −0.067 −0.13 −0.18 −0.11 −0.059 −0.022 −0.12 −0.14 Row 2 0.13 0.059 0.11 0.18 0 0.12 0.12 Count 0 0 0.053 0 −0.053 0.13 0.059 0.11 0.18 0.13 0.078 Total 0 0 0.053 0 −0.053 −0.035 Column Sums 0 0 0.059 0 0 0 Count 0 0 0.060 0 −0.060 −0.059 0.083 0.020 0.037 0.098 0.074 0.035 Total 0 0 0.057 0 −0.055 −0.033 NOTE: Shown are exact relative biases for estimators controlled to postcensal population estimates at cell or margin (row or column) , or controlled only to overall postcensal population estimate ("overall")
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From page 286... ...
286 TABLE B.2 Exact Relative Biases for Case of Equally Biased Postcensal Population Estimates Column 1 Column 2 Row Sums ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Trow Trow Trow Tcell Tcolumn Toverall Tcell Tcolumn Toverall Tcell Tcolumn Toverall Row 1 0.1 0.027 0.1 0.17 0.035 0.1 0.1 Count −0.038 −0.094 −0.022 −0.030 −0.029 0.1 0.027 0.1 0.17 0.035 0.1 0.076 Total −0.038 −0.094 −0.022 −0.032 −0.051 Row 2 0.1 0.042 0.24 0.16 0.1 0.16 0.22 0.29 0.1 0.1 0.23 0.23 Count 0.1 0.042 0.24 0.16 0.1 0.16 0.22 0.29 0.1 0.061 0.24 0.19 Total Column Sums 0.1 0.034 0.1 0.035 0.1 0.17 0.1 0.16 0.1 0.1 0.1 0.1 Count 0.1 0.040 0.19 0.12 0.1 0.16 0.14 0.21 0.1 0.064 0.18 0.14 Total NOTE: Shown are exact relative biases for estimators controlled to postcensal population estimates at cell or margin (row or column) , or controlled only to overall postcensal population estimate ("overall")
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From page 287... ...
TABLE B.3 Exact Relative Biases for Case of Unequally Biased Postcensal Population Estimates Column 1 Column 2 Row Sums ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Trow Trow Trow Tcell Tcolumn Toverall Tcell Tcolumn Toverall Tcell Tcolumn Toverall Row 1 0 0.013 0.022 0.035 Count −0.1 −0.11 −0.081 −0.094 −0.05 −0.05 −0.03 −0.029 0 0.013 0.022 0.035 Total −0.1 −0.11 −0.081 −0.094 −0.067 −0.071 −0.047 −0.051 Row 2 0.2 0.18 0.18 0.16 0.3 0.32 0.28 0.29 0.25 0.25 0.23 0.23 Count 0.2 0.18 0.18 0.16 0.3 0.32 0.28 0.29 0.22 0.21 0.20 0.19 Total Column Sums 0.05 0.035 0.05 0.035 0.15 0.16 0.15 0.16 0.1 0.1 0.1 0.1 Count 0.15 0.13 0.14 0.12 0.2 0.22 0.19 0.21 0.16 0.15 0.15 0.14 Total NOTE: Shown are exact relative biases for estimators controlled to postcensal population estimates at cell or margin (row or column) , or controlled only to overall postcensal population estimate ("overall")
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From page 288... ...
Control on both margins through some kind of raking procedure was not treated here but is worthy of further consideration. The results in this paper indicate that the Census Bureau's plan for control at a fine level of demographic stratification within estimation areas may be problematic.
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From page 289... ...
APPENDIXB: CONTROLLING TO POSTCENSAL POPULATION ESTIMATES APPENDIX B 289 Särndal, C.-E., Swensson, B., and Wretman, J
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