It was therefore considered appropriate to only adjust males in this age group. The adjustment of 7, was 1. Age Group: 10—14 years For the 10—14 year age group, the sex ratio of the new population base was too high compared with data coherence sources. An adjustment was introduced to reduce the level of males by 3,, reducing the sex ratio from Like the 5—9 year age group, the level of males was higher than data coherence sources while female levels were closer to supporting data.
The adjustment of 3, was 0. Age Group: 40—44 years For the 40—44 year age group, the sex ratio of the new population base was too low compared with data coherence sources. An adjustment was introduced to raise the male level by 6, and reduce female level by 11,, for a net adjustment of —5, This raised the sex ratio from The ratio achieved was slightly lower than comparative data sources for two reasons: 1 any greater ratio adjustment would have required a level adjustment that was too far outside Census undercount standard errors, and 2 in all data coherence sources, the sex ratio for this age group was lower than the age group on either side and a ratio of The female level differed from data coherence sources more than the male level and this was reflected in the adjustments made.
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The increase of 6, for males was 1. Age Group: 45—49 years For the 45—49 year age group, the sex ratio of the new population base was too high compared with data coherence sources. An adjustment was introduced to raise the level of females by 8,, reducing the sex ratio from The female level was below coherence data sources while the male level compared well with the sources so it was considered appropriate to adjust females only in this age group.
The adjustment of 8, was 1. Age Group: 55—59 years For the 55—59 year age group, the sex ratio of the new population base was too low compared with data coherence sources. An adjustment was introduced to raise the level of males by 5,, increasing the sex ratio from For higher age groups, the Census undercount estimation method is less optimised for sex ratio figures. This results in an adjustment that tends towards a census-based sex ratio that become more accurate for higher age groups.
Labor Force Statistics from the Current Population Survey
Data coherence sources suggested that the level decline for males aged 55—59 years in compared to this cohort in the Census should not be as large as the new population base indicated. The level of females matched the data coherence sources more closely, so it was considered appropriate to adjust males only in this age group. The increase of 5, was 1. Age Group: 60—64 years Back to top For the 60—64 year age group, the sex ratio of the new population base was too high compared with data coherence sources.
An adjustment was introduced to raise the level of females by 10,, reducing the sex ratio from In the data coherence sources, the peak sex ratio for older age groups occurs in the 65—69 years age group. This required the sex ratio for the 60—64 years age group in to be lower than some of the data coherence sources, but closer to the sex ratio based on unrebased ERP as at 30 June The female level was comparatively low, whereas the male level which was comparable with data coherence sources. Congressional apportionment, for example, requires only population totals by state, whereas the revenue sharing formula uses population and per capita income for each incorporated place.
Whether adjustment of population totals by state—resulting in more accurate congressional apportionment—will also result in more accurate distribution of revenue sharing monies may depend on how the adjustment is distributed within the state and what, if any, adjustments are made to the per capita income estimates. How the different loss functions can or should be reconciled in order to preserve consistency between the uses of census data for different applications is an issue for which the panel has in the abstract little advice to offer.
Even for any single given use of census data, the benefit of adjustment may vary from place to place. Suppose, for example, that midwestern central cities were grouped into a domain and the census results for each city adjusted by the same formula. Adjustment might improve accuracy in these cities as a group, but not in all cities equally, nor in every city.
Nor would they benefit equally, and some might be adversely affected lose federal funds or representation. It must be accepted that no adjustment procedure can be expected to simultaneously reduce the error of all information for every location in the United States. Rather, adjustment should be undertaken when there is reasonable certainty that appreciable reduction in the general differential coverage error will be achieved.
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A relatively trivial reduction would not be worthwhile, since adjustment will surely cost time and resources to implement, and doubt about whether the adjustment did or did not reduce differential coverage error would impair public confidence in census figures. Furthermore, knowledge of a subsequent adjustment might reduce public cooperation, thus lowering the completeness of the census count. For an effective adjustment procedure to be widely accepted, given that not all localities will benefit, it is important that there be as widespread understanding and agreement as possible within the professional community of statisticians that a general reduction in differential coverage error is sufficiently desirable to accept adverse impacts on some individual localities.
More important but difficult to obtain is this understanding throughout all levels of government see Keyfitz, In other words, localities need to recognize two important points regarding adjustment. First, the standard of comparison should not be the raw census count. That is, an adjustment that lowers the population count for an area may have reduced the error in the estimate for that area as much as an adjustment that raises the count for another area.
Second, although adjustment may increase error for some localities, the country as a whole may still benefit if adjustment has reduced overall differential error. The panel believes that it is substantially more important to reduce the general error per person than the general error per place. Hence, the panel does not recommend the use of loss functions for measuring the total error that weight each political jurisdiction equally, for example, that determine the proportion of the 39, revenue sharing jurisdictions that gained or lost through adjustment, regardless of the number of people in each jurisdiction.
Rather, the panel believes that the contribution to total loss attributable to an area should reflect the size of its population. In measuring the total loss associated with an adjustment procedure, we recommend that the contribution to this loss attributable to a geographic region should reflect its population size. Thus, we recommend against loss functions based solely on the number of political entities losing or gaining through adjustment. The next section discusses the properties of several kinds of loss functions and considers specifically how they take into account population size.
The classical yardstick used by sample survey researchers to assess the accuracy of a single number, chosen principally for its convenient mathematical properties, is the square of the deviation between the number and its true value. Whatever loss function we use to assess the accuracy of a single number, we still must determine a rule for amalgamating the losses associated with each number into an overall loss function for the entire set of numbers produced.
The usual tack taken is to sum the individual loss functions. Using this rule for squared error applied to population gives disproportionate weight to large localities. Consider the following example. Suppose there are two areas, one with true population of 10, The larger area with twice the population of the smaller area and the same percentage error counts for four times as much in the overall loss function.
Using this rule for squared error applied to relative or percentage error i. In this case, the larger area counts for no more than the smaller area in the overall loss function. The following argument gives an in-between notion that may be about right, although we make no absolutist claim for either the argument or the resulting loss functions.
In this case, the larger area with twice the population of the smaller area and the same percentage error also counts for twice as much in the overall loss function. Tukey argues for the use of relative squared error on the grounds of its invariance properties. That is, relative squared error has the property that the contribution of the error for one area to the overall loss function is proportional to its size, assuming that the percentage error for all subareas is the same.
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Another loss function that has this invariance property and is more tractable computationally see Kadane, is squared error divided by the estimated value. Again, the area with twice the population size contributed twice as much to the overall loss function. Both of these loss functions, squared error divided by the true value and by the estimated value, are commonly used in the analysis of contingency tables. All are weighted versions of one another.
Thus, relative squared error and squared relative error relative to the true value are simple squared error weighted by the reciprocal of the true value and the reciprocal squared, respectively; relative squared error is squared relative error weighted by the true value; and so on. The foregoing discussion pertains to the construction of a loss function associated with census-produced numbers wherein it is the absolute accuracy of each number that is to be assessed.
Concern with minimizing a differential coverage error indicates the need for a loss function that reflects not the error in each number eo ipso but rather that error in relation to the errors in the other numbers within a set of census-produced numbers. For example, we do not so much want to gauge the accuracy of population counts for each county in a state as to gauge whether the inaccuracies are relatively evenly distributed across the counties. Relative error is defined as the error divided by the true number or, equivalently, as the ratio of the census-produced number to the unknown true number minus 1.
If the aggregate is a state total and each component number is a county total, then this measure would be the squared difference between the relative error of the county number and that of the state number or, equivalently, between the ratio of the census to the true number for the county and the ratio for the state. Suppose, for example, that our two areas of population size 10, and 5,, each of which had a relative error of 10 percent, were part of a larger area of population 50, that had a relative error of 8 percent.
Then the measure of misproportionality for each of the two component areas would be 0.
cpanel.openpress.alaska.edu/abc-of-medical-law-abc-series.php If the area of 50, population instead had a 10 percent relative error, then the loss function for each of the components would be zero. To aggregate this loss function across counties, say, Tukey suggests a weighted sum of the component i. In our example, the loss for the area of 10, population would count twice as heavily as the loss for the smaller area in the overall loss function.
One characteristic of the loss functions given above is that they are general in nature and not specific to census data uses, except in distinguishing absolute and differential inaccuracy. Some consideration has been given to use-specific loss functions, in particular in the work by Kadane on congressional seat allocation and the work by Spencer b on alloca-. Kadane demonstrates the close relationship between loss functions proposed by Tukey and the loss function underlying the method currently used in seat allocation.
See Appendix 7. In modeling the revenue-sharing loss function, Spencer suggests that the components are not merely the units receiving revenue sharing dollars but also the one source of funding. For each component, he postulates as loss function a constant multiple of the magnitude of the overallocation if one exists or a possibly different constant multiple of the underallocation if any.
The overall loss function is an unweighted sum of component loss functions. The Kadane paper exhibits a loss function for allocating congressional seats among states whose minimization results in the allocation procedure actually used in Congress. In the case of revenue sharing and other major uses of census data, the task of ascertaining an appropriate loss function is more complex.
The loss function studied by Spencer was merely a convenient construct, a springboard from which he could proceed to investigate the central issue, the implications of data inaccuracy for revenue sharing. Research on the effect of the choice of loss function on the effectiveness of adjustment procedures has been limited. Spencer a provides some evidence that the degree of improvement resulting from adjustment is not very sensitive to choice of loss function.
Schirm and Preston studied the effect of a very simple synthetic adjustment see the discussion of synthetic estimation in a subsequent section , using only two demographic groups, on population proportions across geographic areas using a number of different loss functions:. They found, in a limited simulation study, that the decision to adjust is sensitive to selection of the loss function in that the first two loss functions offer consistent recommendations about whether or not to adjust in only about 60 percent of the simulated cases.
Seeking the source of the 40 percent inconsistency can bring us closer to an understanding of the impact of choice of loss function on the adjust-.
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