The Denominator

A method for detailed small area projections


The U.S. has experienced several notable demographic changes in recent decades, including rapid aging and diversifying of the general population. These changes can be especially impactful at the local level, such as within a county, city, or even neighborhood, impacting policy and planning decisions regarding where to locate new housing, healthcare resources, childcare facilities, and more. While demographic forecasting methods, such as the cohort component method, give us an analytical tool to prepare for future population changes, data limitations often prohibit these methods from being applied at the local level where local government policies are directed. Furthermore, issues with small numbers of populations in local areas often can lead to traditional estimates producing nonsensible forecasts with extreme population growth or decline.  In this post we outline a method for making localized, henceforth referred to as small area, projections of populations by detailed groups, such as Age, Sex, Race and Ethnicity, in the absence of traditional data requirements necessary for projections. We then apply this method to the population of King County, Washington to make projections to 2045 by Age, Sex, Race and Ethnicity for all census tacts in the county.


The most well known method used for demographic projections for populations that have adequate vital registration systems is the cohort component method (CCM).  The CCM requires age and sex specific data about fertility, mortality, and net migration for the population of interest. Using this data, age specific fertility, mortality, and migration rates are estimated and applied to the most recently observed population to project what the future population may look like. This method has been used by many agencies in the US to build baseline forecasts of what future populations may look like (Swanson et al 2010), While often used to make projections at larger areas, this method is insufficient at smaller areas for two reasons.

First, data on fertility, mortality, and migration in the United states are often not available at small area designations. Most national vital reporting agencies, such as the Center for Disease Controls Wide-ranging Online Data for Epidemiologic Research, only report vital statistics data on births and deaths for counties at the most granular level. Migration data is even more elusive with the Census reporting limited detailed information on county to county flows. While some local governments produce more detailed estimates of this demographic data, they are rare.

Second, traditional methods of estimating underlying mortality, fertility and migration rates, are susceptible to large amounts of error when populations are small, which is more common for small area estimates. For example, a single year with a few more deaths than average can result in wildly different annual mortality rates across time which may lead to population projections that are insensibly small if we expect that year to be an outlier. The same possibility is true for migration and fertility rates and can become more common when populations are broken down even further such as by race and/or ethnicity.

To overcome both of these issues we utilize the Hamilton Perry (HP) method that is only reliant on population data from two time points, rather than data on fertility, mortality, and migration. The HP method is a variant of the cohort component method (Swanson et al. 2010). It makes several simplifying assumptions about the components of population change, namely, that observed changes in population size capture the effects of fertility, mortality, and net migration. The HP method estimates population change using only data on population counts of different age- and sex-specific groups from two time points for the geographic area of interest. Because data on fertility, mortality, and migration rates are often unavailable for sub-county areas such as Census tracts, the HP method is a valuable way to measure and project population change at the Census tract level.

The Hamilton-Perry method projects population change using two primary measures: the Cohort Change Ratio (CCR) and the Child Woman Ratio (CWR).

Cohort Change Ratio (CCR)

The Cohort Change Ratio (CCR) captures the change in the size of a given population over the course of the time period being examined, and thereby proxies for age-specific population change arising from survival and in-migration, net of mortality and out-migration. The CCR can be calculated for various geographic levels and can be calculated separately by sex, race and ethnicity, and other demographic characteristics.

5CCRx = 5Px+5, T1 / 5Px, T0

5CCR20 = 5P25,2020 / 5P20,2015

A CCR value less than 1 indicates that the population in question has shrunk across the time period through mortality and/or out-migration, while a value greater than 1 indicates that it has grown through in-migration.

Child Woman Ratio (CWR)

The Child Woman Ratio (CWR) proxies for fertility by measuring the ratio of the child population to the population of women of childbearing age (between 15-44). The CWR is calculated separately for male and female children to account for the empirical sex ratio at birth and differential infant mortality rates (Parazzini et al. 1998). The two ratios are then multiplied by the size of the population of women of childbearing age to obtain an estimated number of children aged 0-4. The general form of the CWR equations are below. The first two equations represent the calculation of the CWR used to proxy for fertility rates for female and male infants respectively, and the second two equations demonstrate how the CWR is then used to create projections of the female and male infant and child population (FP and MP).

CWRFP =  5FP0, T1 / 30FP15, T1

CWRMP =  5MP0, T1 / 30FP15, T1

5FP0,T1 = CWRFP * 30FP15,T1

5MP0,T1 = CWRMP * 30FP15,T1


The HP method deals with data limitation concerns but not issues caused by small population numbers. To mitigate issues of unreasonable growth in geographies with small, rapidly growing populations, we implement a multi-stage smoothing process to produce our CCRs and CWRs. This smoothing process is a variant of a smoothing method originally applied to small area forecasts in Japanese prefectures (Inoue 2017). To implement the smoothing process, we produce our tract-level population forecasts incrementally: first we compute county-level projections, then we create county-level projections by race and ethnicity smoothed to the overall projected county population size. Then, we create our tract-level projections and smooth them to the pre-smoothed race- and ethnicity-specific projected estimates of the county population size. Performing this multi-stage smoothing method is necessary in constructing tract-level projections as the population sizes of different census tracts can vary greatly. Moreover, the age, sex, and race and ethnicity distribution can be drastically different in one tract compared to another. These differences in distribution can be further compounded by racial and ethnic subpopulations residing in clusters in certain parts of the county. Our multi-step smoothing process takes into account these differential distributions, helping create more accurate small-area estimates. For more mathematical details on this smoothing approach see here.


To evaluate this method we apply it to the population of King county where multiple agencies have made population projections which we may compare it against. Using Age, Sex, Race, and Ethncity data for each tract we compute smoothed estimnates of CCR and CWR using 2010 census data and 2015 intercensal estimates from the Washington state Office of Finance and Mangement (OFM). We compare these forecasts to other forecasts made by OFM as well as the Puget Sound Regional Council (PSRC) to show that the estimation process makes reasonable forecasts as seen in Figure 1. 

Change in population is shown using a stacked ribbon chart. While in 2000 the white population was nearly 75% we estimate that by 2045 it will be just below 50%.
Similar to national projections, our method estimates that the population of King County Washington will become a minority majority population by 2045.

In addition we are able to make more detailed examinations of where populations might experience growth. For example, similar to OFM’s state projections we estimate that one of the fastest growing racial and ethnic groups in King county state will be the Hispanic population. With our estimates we can show where this growth is estimated to occur by looking at growth by census tract for these two groups. In Figure 2, we show that within King county, growth is greatest in the areas outside of the city center for these two groups despite the high density of the county population being towards the city center.

A map of King County is shown will fill colors for the rate of growth for the Hispanic population. We show that growth is greatest in the peripheries of the county rather than the center.
While the King County Hispanic population is projected to grow across almost all census tracts in the county, this growth will be particularly rapid for locations in the periphery of the county rather than the city center.

This method any be applied to any population as long as age and sex specific data is available and there are a number of ways which smoothing may be applied. To make it easier to use this method, our team has created an R library to apply this method for projections.


Inoue, T. (2017). A new method for estimating small area demographics and its application to long-term population projection. In D. Swanson (Ed.) The frontiers of applied demography, pp. 473-489). New York, NY: Springer. Swanson, D.A., Schlottmann, A., & Schmidt, B. (2010). Forecasting the population of census tracts by age and sex: An example of the Hamilton-Perry method in action. Population Research and Policy Review, 29(1), 47-63. Parazzini, F., La Vecchia, C., Levi, F., Franceschi, S. (1998). Trends in male∶female ratio among newborn infants in 29 countries from five continents. Human Reproduction13(5), 1394–1396

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