Small-Area Estimates And Projections Research Paper

1. Smallness

Small area is usually administratively defined. It can be a county, province, city, or municipality within a state or country, for example. In some cases one is interested in a subset of the large population defined by criteria other than geographic affiliation, such as age, sex, income, educational level, health status, etc. The latter subpopulations are referred to as domains.

The ‘smallness’ of a subpopulation can be under-stood either in terms of its size or in terms of the data available for it. In the case of municipalities or city blocks, populations can be small in the sense that they may vary considerably over time. Domains of a particular educational or health status may be especially variable. This poses challenges for the evaluation of educational policies or for environmental epidemiologic studies, for example. On the other hand, large cities or counties may have populations in the millions, so the populations of interest can be stable over time.

Frequently direct population estimates are available for the large area only, and survey data or administrative records must be used to estimate the population totals for the small areas. When survey data are used, the sample size available for each small area is often small and sampling variability is high. Independently of the size of the target population, estimation is then uncertain. If administrative records or other symptomatic data from the small area are used, then the biases of such data and of the statistical models used, may also be major sources of inaccuracy.

From an administrative point of view, all such estimates may have a similar status. This is the case in the allocation of funds, or apportionment of political representation, based on population size, for example.

2. Data For Small Area Estimation

Direct estimates of small area populations are continuously available for countries with high-quality population registers. In principle, direct estimates of population by educational level and subarea may similarly be obtained from computerized registers in real time. Primary problems in the statistical analysis of small area data are then caused by changes in administrative classifications, the timeliness of information updates, and quality of registration. Moreover, there are typically domains that cannot be distinguished in the register data. An example is the distribution of population by health status. It is also possible that the register based information refers to a legally resident (de jure) population, whereas the information needs involve the population that is de facto present in the area.

In countries that rely on censuses for population estimation, demographic bookkeeping, administrative records, or sampling techniques are used for small area estimation during intercensal periods. Data on births, deaths, and migration can be used for population estimation. Migration may be estimated from symptomatic data, such as school enrollment, for example.

Population estimation can also be based on housing units and estimates of persons per unit. The latter may be further decomposed into a product of an occupancy rate and persons per occupied housing unit. Estimates of the number of housing units can be difficult to obtain and update if there are many unauthorized additions, demolitions, or conversions. Natural dis-asters may also wipe out housing units. In addition to housing units, it is possible to use the number of registered passenger cars, income tax returns, etc. to estimate the population. Statistical regression analysis can be applied to census data to estimate the relationship between population or population change, and such symptomatic variables.

If the completeness of a population register (or a census) is in doubt, then the so-called capture– recapture (or dual systems) methods can be used to evaluate the level of undercount for small areas. It is possible to recount the population for a sample of the small areas and deduce the number missed by the registration system (or the census) in demographic, ethnic, or social domains. If not corrected, differential undercount may create biases if population data are used to allocate funds to small areas, for example.

3. Methods For Survey Based Estimation

Small area estimation may involve many other types of totals besides population. It can be income, un-employment, crop and livestock production, volume of wood in forests, etc. Such estimates may have similar administrative uses as population figures. Data on income or unemployment can be used in the allocation of funds, for example. On the other hand, for crop and forest statistics there may be additional data sources such as images taken by satellites.

In the so-called synthetic estimation method it is assumed that small areas have the same relevant characteristics as the large area. For example, suppose the average income is accurately known by age and sex for the large area from a sample survey, but the sample is too small for reliable estimation for small areas. Assume that the population size is known by age and sex for a small area. Then, the synthetic estimate of the total income in the small area assumes that the area has the same average income by age and sex as the large area. The estimator has a low variance, but it can be severely biased if income varies between areas. Sample average from the small area is an unbiased estimate of income, but it has a high variance. The trade-off between bias and variance is the basic problem of small sample estimation.

Composite or combined methods try to alleviate the biases inherent in synthetic methods by using weighted averages of estimators derived according to different principles. A natural approach is to try to find weights that would minimize the mean squared error of the estimator. One problem in the use of such weights for administrative purposes is whether or not the same weights can be used for all small areas. Weights that have been tailored for each area separately may be statistically unstable and their use may be difficult to defend in a political process involving monetary allocations, for example.

Better accuracy is available if realistic statistical models can be developed to represent the relationship between the variable of interest (such as income) and possible auxiliary variables (such as age and sex). Possible spatial correlations across small areas may also be accounted for. Regression models that include random effects are typically used.

In the so-called Empirical Bayes method the bias of the regression model is represented by an area-specific random effect. An optimal weighted average of the regression estimator and a direct sample based estimator is, in principle, obtained by weights deter-mined by the variance of the random effect and the sampling variance of the direct estimator. Both must be estimated from the data so exact optimality is difficult to achieve. The estimation of the variance of the random effect is particularly challenging. Different approaches have been proposed that use both analytical approximations and simulation techniques, such as bootstrap. The computationally intensive Hierarchical Bayes methods solve the problem in a unified probabilistic framework that offers conceptual advantages.

Depending on the spatial setting, complex inter-relationships between the small areas are possible. Several layers of random effects may be warranted, and the effects need not be hierarchical. Such models are also called variance component models or multilevel models.

4. Forecasts Of Small Area Populations

Migration typically has a bigger effect on forecasts of small areas than on forecasts of national populations. International migration is frequently concentrated on certain small areas so its relative impact may be large. Internal migration related to labor markets can be highly volatile. In countries without a population register, direct data on both types of migration are often lacking. Estimates must then be based on symptomatic techniques of the type discussed above. In consequence, the uncertainty of small area forecasts is typically higher. Probability theory provides a way to describe the uncertainty to forecast users in a realistic manner.

Should one forecast the populations of the small areas separately, and then aggregate to the national level, or should one forecast the national population first and then disaggregate to small areas? If the former method is applied independently for each small area by a local authority, say, then the sum of forecasts is often too high, because administrative units are reluctant to forecast declines in population for fear that it might become a self-fulfilling prophecy. On the other hand, if the forecasts are prepared by a central authority, there are many technical difficulties. The simplest approach is synthetic: one assumes that changes in the populations for small areas are a fixed proportion of a forecasted national change. At the other end of the spectrum is a multistate cohort-component model that describes migration in terms of full origin–destination migration flows by age and sex. The difficulty with the latter models is that the number of migration flows can be large. In a system of n areas it is n(n–1). A compromise is to forecast out-migration for each area separately, and to pool the out-migrants. In-migrants are then forecasted synthetically as fractions of the pool for each receiving region. As in other applications, the synthetic method may lead to biases.

5. From Administrative Units To User Defined Areas

Much of the interest in small area estimation and forecasting comes from administrative uses such as allocation of funds. However, small area data are also used in social and medical research to measure the effects of interventions, for example. Administrative boundaries are frequently not well suited to such analyses. Domains of interest may intersect several areas, or they may be too small for their effect to be discernible in small area data. A new development in small area estimation and forecasting is the introduction of the geographic information systems (GIS). When all housing units are given geographic co-ordinates, the data user may define new aggregates based on the coordinates, irrespective of any existing administrative boundaries.

Estimation methods from spatial statistics will then become an essential part of small area estimation. Special care is needed when user defined regions are used to locate disease clusters, for example. Both the random variation inherent in small populations and the selection effects must be properly handled. Existing techniques of forecasting must similarly be modified because information about migration may be more difficult to describe. On the other hand, user defined regions provide opportunities for more meaningful estimates of the population at risk.

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