Technical Guide on Demographic Estimates at Statistics Canada
Chapter 3
Base population

Censuses play an essential role in calculating population estimates. Every five years, population estimates are linked to the results of the last census to calculate a new base population. The Demographic Estimates Program’s (DEP) base population corresponds to the population used as a reference point or a starting point for the estimation process.

For postcensal estimates, the base population is the population enumerated at the last census, adjusted for census net undercoverage (CNU), incompletely enumerated reserves and settlements (IERS), and demographic adjustments at certain ages. Formal reviews of population counts are also included at the subprovincial and subterritorial levels. The base population is also referred to as the censal estimate. The reference date for the censal estimate is the same as the date of the corresponding census.

Postcensal estimates as of July 1^st of a census year are calculated using the component method based on the base population and take into account demographic events that occurred between Census Day and June 30. Intercensal estimates are derived from postcensal estimates corrected for error of closure. Error of closure is the difference between the base population and the postcensal estimate on the same census date. Chapter 13 of this guide presents the quality measures of population estimates.

This chapter presents the methodology for calculating each component of the DEP base population. It also includes the methods used to calculate the base population by age and gender, as well as for subprovincial and subterritorial estimates.

Base population by province and territory

At the provincial and territorial level, the base population is calculated as follows:

$Pop=Censuscounts+CNU+IERS$

The DEP base population makes the following corrections and additions to the census data:

• Adjustment of the population to account for CNU.
• Addition of independent estimates of the population of IERS in 1991, 1996, 2001, 2006, 2011, 2016 and 2021.
• Addition of demographic adjustments to certain ages. These adjustments do not affect the provincial/territorial total of the base population.
• Adjustment for early enumeration in parts of northern Quebec, Newfoundland and Labrador, Yukon and the Northwest Territories in 1991 and 1996.
• Addition of estimates of non-permanent residents in 1971, 1976, 1981 and 1986. Since 1991, non-permanent residents have been included in the census universe.

The table below illustrates the various components that make up the base population, by province and territory from 2001 to 2021.

Census

Census counts are the foundation of population estimates. Information on censuses and their universes and concepts can be found in the introduction in this guide.

Census net undercoverage

Census net undercoverage (CNU) measures census coverage errors, which are defined as errors caused by miscounting the population on Census Day.

There are two types of coverage errors. First, population undercoverage occurs when people who should have been counted were not, while population overcoverage counts those who should not have been counted or counts them more than once. Undercoverage is more common than overcoverage. The net impact of undercoverage and overcoverage on the size of the population is referred to as census net undercoverage (CNU).

Census coverage studies

After each census, Statistics Canada conducts coverage studies to measure coverage errors. Three studies examined coverage error in the 2021 Census. First, the 2021 Dwelling Classification Survey (DCS) examined coverage errors resulting from the misclassification of dwelling occupancy. Census data were corrected to account for this type of coverage error. Second, the Census Undercoverage Study (CUS), previously called the Reverse Record Check (RRC), measured population undercoverage. Third, the 2021 Census Overcoverage Study (COS) measured population overcoverage caused by some people being counted more than once by the census.

Census data are not adjusted for the population coverage errors measured by the CUS and COS. Instead, CNU estimates are used to produce Statistics Canada’s population estimates.

More information on the methodology of the 2021 Census coverage studies is available in chapters 6 to 8 of the Coverage Technical Report, Census of Population, 2021, Statistics Canada Catalogue no. 98-303-X.

Coverage studies provide undercoverage estimates in the 1991 to 2021 censuses at the provincial and territorial levels, and in the 1971 to 1986 censuses only at the provincial level. Estimates of overcoverage for the provinces and territories are only available as of the 1991 Census. To estimate overcoverage in censuses before 1991, it was assumed that the ratio of overcoverage to undercoverage for each census between 1971 and 1986 was the same as in 1991. To estimate the CNU for Yukon and the Northwest Territories prior to 1991, it was assumed that the ratio of the CNU for each territory to the 10 provinces for each census between 1971 and 1986 was the same as in 1991.

For consistency, the 1991 Census undercoverage and overcoverage estimates were revised in 1998 to reflect methodological improvements made to the 1996 Census coverage studies. This update changed the CNU for the 1971 to 1986 censuses. Similarly, the 1996 Census undercoverage and overcoverage estimates were revised in 2003.

The DEP requires an estimate of the CNU for various estimation domains. The CNU estimates from coverage studies have a good level of reliability for large domains (e.g., provinces and territories or broad age groups), but they are less accurate for smaller domains due to the small sample sizes in these domains. This is one reason why a demographic adjustment is required at certain ages; this will be described in another section of this chapter.

The CNU counts must be whole numbers (positive or negative). They must be produced for the estimation domains corresponding to the cross-tabulation of variables: province and territory, year of age, and gender. In addition, the modelled CNU figures, when summed across the areas of interest, must match the published CNU totals for the broad coverage study domains. In addition, the CNU must be a smooth function of the year of age. In other words, the evolution of the CNU from one year of age to the next must be regular and, ideally, should not fluctuate too rapidly and abruptly. The methodology for estimating the CNU by age and gender is presented later in this chapter.

Incompletely enumerated reserves and settlements

Enumeration is not authorized or cannot be completed on some reserves and settlements. These areas, which numbered 63 in the 2021 Census, are referred to as incompletely enumerated reserves and settlements (IERS). Health and safety restrictions put in place to slow the spread of COVID-19 as well as natural disasters that occurred during collection (including wildfires that led to evacuations) contributed to the partial enumeration of many reserves and settlements. As a result, 2021 Census population and dwelling counts are not available for the 63 IERS and are not included in the 2021 Census tabulations.

Neither the 2021 Census nor the CUS are able to produce estimates of the population living in IERS. A model-based methodology is used to produce official population estimates for these geographic areas.

An adjustment is then applied to the model-based estimates to account for CNU. This adjustment is based on national CNU estimates for enumerated reserves.

For more information on the methodology used to estimate IERS, please consult the following:

• Section 12.2 of the Coverage Technical Report, Census of Population, 2021, Statistics Canada Catalogue no. 98-303-X;
• Section 4.2 of the Indigenous Peoples Technical Report, Census of Population, 2021, Statistics Canada Catalogue no. 98-307-X.

Base population by age and gender

The components of the base population by age and gender and of the total population are the same, except for the addition of demographic adjustments at specific ages. These adjustments are necessary because the CNU can be less accurate at certain ages for which other reliable administrative data are available.

The base population by age and gender for each province and territory is calculated as follows:

$Po{p}_{a,g}=Censuscount{s}_{a,g}+CN{U}_{a,g}+IER{S}_{a,g}+demographicadjustment{s}_{a,g}$

Where, for each province and territory:

The table below shows the effect of the various adjustments to the census to calculate the base population by age group.

Census net undercoverage by age and gender

CNU estimates from coverage studies, which are calculated using samples, have a good level of reliability for large domains (such as provinces and territories or broad age groups). However, they are less accurate for smaller domains due to the small sample sizes in these domains. Therefore, a model better adapted to small domains is used to produce CNU estimates by year of age and gender for the provinces and territories: the Fay-Herriot model^Note or the empirical Bayes model. This model produces more accurate estimates in small domains, despite their potential bias. As a result, estimation using this model involves a trade-off between bias and variance of the CNU estimates.

A number of elements are needed to produce estimates using the Fay-Herriot model. Direct estimates and associated estimated variances are required for all domains of interest. Independent variables related to the CNU must also be available. The direct estimates used are of the log of the net undercoverage adjustment factors (ratio of the population count adjusted for coverage to the census population count) and are derived from the coverage studies estimates. These estimates are by province or territory, age group and gender. Age groups are used since direct estimates by year of age are considered too volatile to be successfully modelled. In the provinces, the age groups are as follows: 0 to 4 years; 5 to 9 years; 10 to 17 years; 18 to 24 years; 25 to 34 years; 35 to 44 years; 45 to 54 years; 55 to 64 years; 65 to 74 years; 75 to 84 years; and 85 years and older. In the territories, the age groups are the same up to age 64, followed by a single group for 65 years and older.

The independent variables come from the census questionnaire and include level of education, mobility, type of dwelling, tenure type, and mother tongue among others^Note . The variable selection process is performed using Mallows' Cp criterion as well as other standard statistical selection processes and selection criteria, such as backward selection, forward selection, Akaike information criterion (AIC), and p-values. Mallows’ Cp, which minimizes model error while applying a penalty to models with more variables, is used to obtain a relatively sparse model (one that has as few terms as possible but remains highly predictive).

Once the appropriate independent variables are selected, the parameters of the Fay-Herriot model are estimated using the restricted maximum likelihood method^Note The parameters estimated include the model variance and the regression coefficients. The model produces estimates of the log of the net undercoverage adjustment factor by province or territory, age group and gender.

Next, an assumption is required to produce the modelled estimates for the required domains, i.e., census net undercoverage by year of age and gender for each province and territory. This assumption is that the adjustment factor is constant for all years of age in a domain. For example, for a given province and gender, the correction factor for the CNU at age 1 is assumed to be the same as the one that will be applied at age 3, since both ages are in the 0-to-4 age group.

Formally, this is described as follows:

Equation 3.1:            ${\widehat{M}}_{ijka}=C_{ijka}\text{X}(e\widehat{{}^{\log (Fijk)}}-1)$

where:

After modelling, the CNU estimates are no longer consistent with the coverage studies estimates. In other words, the total of the modelled estimates will not equal the CNU produced by the coverage studies in certain broad domains such as the provinces and territories. The modelled estimates must therefore be adjusted to restore coherence with the coverage studies. This adjustment is made by applying reconciliation methods^Note to the modelled CNU estimates, which ensures that the totals of the coverage studies are complied with while minimizing the changes made to the modelled estimates as much as possible. Reconciliation is also referred to as raking or balancing. Specifically, the modelled estimates must correspond simultaneously to two margins: the provincial and territorial totals and the smoothed national total by year of age.

The totals of the first margin are taken directly from the coverage studies and are used without being modified. The totals of the second margin are obtained by smoothing the direct estimates by year of age at the national level using the penalized B-spline technique.^Note Smoothing is done because it is preferable for the CNU to evolve gradually from one year of age to the next and to not fluctuate too quickly and abruptly. The second margin is then calibrated^Note to ensure they are consistent with the national estimates from the coverage studies for each of seven age groups (0 to 17 years; 18 to 24 years; 25 to 34 years; 35 to 44 years; 45 to 54 years; 55 to 64 years; and 65 years and older). Calibration is a technique that imposes the totals from coverage studies while preserving the trend of the smoothed margin as much as possible.

A consequence of the reconciliation process is that the modelled estimates generally exhibit a smoother trend than the equivalent direct estimates produced by the coverage studies. However, some jumps may still appear at the edge years of the age groups. To reduce the effect of these jumps, a smoothing process using splines is applied again to the modelled provincial and territorial estimates. Since this smoothing disrupts the coherence with the provincial and territorial totals from the coverage studies, it is followed by a prorating adjustment to restore coherence.

Finally, the CNU estimates by age and gender are rounded in a controlled fashion such that each estimate is rounded to a whole number while preserving the provincial and territorial totals. Most of the time, this controlled rounding is equivalent to rounding to the nearest whole number.

Incompletely enumerated reserves and settlements by age and gender

The method described above provides a provincial or territorial estimate of the population of incompletely enumerated reserves and settlements. Estimates by age and gender are obtained through distributions for each province and territory, based on enumerated reserves and settlements adjusted for CNU. The resulting distributions are applied to incompletely enumerated reserves and settlements. The resulting age and gender estimates are then rounded and adjusted to ensure coherence with the totals by province and territory.

Demographic adjustments

Adjustments are applied at age 0 and older ages because of greater variability in CNU estimates from coverage studies and the availability of alternative data sources at these ages.

Adjustment at age 0

The postcensal estimate of the population at age 0 is derived almost entirely from the estimated number of births (from vital statistics) the previous year. The estimate is therefore very reliable and can be used as a benchmark to adjust the census estimate. Census estimates for the population aged 0 are directly obtained by taking postcensal estimates of the same age to minimize inconsistencies with the vital statistics information. The methods for calculating births in the Demographic Estimates Program are described in the chapters on births and deaths in this guide.

The differences between census counts and the results of the demographic adjustment are redistributed among the population aged 5 to 74, by their relative weight per province or territory and gender. This way, the total population by province or territory remains unchanged.

Adjustment at advanced ages

An analysis of the age and gender structure of census counts and postcensal population estimates shows that the population at older ages, particularly those aged 95 and older, can be affected by overestimation or underestimation. Coverage studies fail to correct these differences due to the sample sizes at these ages. The type and scope of these errors in older populations can vary from one census to another, ranging from reporting errors (intentional or unintentional) to data entry and/or process errors.

The demographic adjustment at older ages is carried out using data from the Office of the Chief Actuary of Canada (OCA), the T1 Family File (T1FF) and the most recent mortality tables published by Statistics Canada.

Adjustments were made for the population aged 95 and older for the 2021 base population. Adjustments to the base population of previous years were made for the 90-and-older group.

Two methods were used to calculate this adjustment. The extinct cohort method was used for cohorts deemed to have no survivors. For these cohorts, population estimates were calculated based on the number of deaths from vital statistics. All deaths in each cohort are added to reconstruct its population based on the assumption that these cohorts will not experience any other demographic event.

For non-extinct cohorts (cohorts deemed to still have survivors), the adjustment for the 2021 base population used two data sources: administrative data from the OCA and the extinct cohorts (EC) estimates. For extinct cohorts from previous years, the relationship between the EC estimates and Old Age Security (OAS) data from the OCA is relatively stable. The demographic adjustment was calculated in two steps, using this relationship. First, average ratios of the EC data to the population estimates based on OAS data were calculated by province or territory, age and gender. Next, these ratios were applied to 2021 OAS population estimates for non-extinct cohorts.

For the base populations associated with previous censuses, the non-extinct cohorts were estimated using the survival ratio method. This method assumes that deaths from non-extinct cohorts are distributed by age like those from extinct cohorts. The population at a given age is estimated using the survival ratios of non-extinct cohorts based on the ratios of the last five extinct cohorts.

As for the demographic adjustment at age 0, the differences between census counts and the results of this demographic adjustment are redistributed among the population aged 5 to 74, by their relative weight by province or territory and by gender.

Base population for subprovincial geographies

The production of population estimates for subprovincial geographies requires the creation of a base population that is coherent with the provincial and territorial base population. The base population by age and gender is first calculated at the census subdivision (CSD) level, then the base populations for higher-level subprovincial geographies are obtained by aggregating CSD base populations.

The base populations of subprovincial geographies and the provinces and territories share many conceptual similarities, such as the same population universe. However, some differences must be considered to ensure that the subprovincial base population is representative of the adjustments that directly affect the CSD population.

The base population for subprovincial areas, derived at the CSD level, first considers census count modifications following a formal review process. Next, census net undercoverage and demographic adjustments are applied and the result is then calibrated to its equivalent at the provincial and territorial level to ensure coherence between all levels of geography. Finally, modelled population estimates for incompletely enumerated reserves and settlements are added to complete the subprovincial base population. These steps are described in more detail below.

Integrating modifications into census population counts

When Statistics Canada publishes population and dwelling counts from the census, data users sometimes question the validity of the figures for a particular geographic area, such as a municipality (census subdivision) or a submunicipal area (designated place). As a result, Statistics Canada implemented the Policy on response to formal review requests of 2021 Census population and dwelling counts. If a correction is deemed necessary, which generally only affects a few CSDs each census, it is reviewed at the census subdivision (CSD) level.

Those count modifications are sent separately to the Demographic Estimates Program as they are not reflected in raw census counts for the CSDs in question. Since counts modified following a formal review are only provided as a revised total population, they are distributed so as to maintain the same distribution by age and gender as the original, uncorrected, census count for each CSD in question.

Census net undercoverage and demographic adjustments

To calculate the base populations for subprovincial areas, the corresponding provincial and territorial CNU rates, available by age and gender, are applied to the published census counts after accounting for population count modifications following formal reviews. In this synthetic estimate, it is assumed that, in a given province or territory and for a given age and gender, the CNU rate is constant across all levels of geography. For example, in British Columbia, the CNU rate for 20-year-old men is considered to be the same across the province.

By design, census coverage studies aim to provide accurate net undercoverage by age and gender for the provinces and territories. It would be impractical and costly to produce accurate results for subprovincial areas across the country since it would require a much larger sample. Those limitations are the reason why the net census undercoverage rate at the provincial and territorial level must be reused consistently across subprovincial areas when more specific data is not available.

For subprovincial areas, census net undercoverage and demographic adjustments by age and gender (described above in the section on demographic adjustments)—both of which are available only for the provinces and territories—are applied simultaneously to adjust CSD counts.

Calibration to the provincial and territorial population, excluding incompletely enumerated reserves and settlements

To maintain coherence with the provincial and territorial base population, the CSD adjusted population derived up to this point must be calibrated. This is done before including incompletely enumerated reserves and settlements (IERS), as these are the result of modelling that is not subject to the adjustments applied to other CSDs.

As a result, CSD adjusted populations by age and gender (excluding IERS) are calibrated with the base population of their respective province or territory, from which the IERS population has been subtracted. The two-way raking calibration process used is described in the chapter on Population estimation methods – subprovincial areas.

To limit the impact the calibration process could have on the population distribution for subprovincial areas, the CSD adjusted populations by age and gender (excluding IERS) are first aggregated to obtain intermediate adjusted census division (CD) populations (excluding IERS) by age and gender. Those CD populations are first calibrated to their provincial and territorial counterparts from which IERS have been subtracted. The adjusted CSD populations by age and gender (excluding IERS) are then calibrated to these calibrated intermediate CD populations (excluding IERS).

Integration of incompletely enumerated reserves and settlements

The final step in preparing the subprovincial CSD base population is to add the estimated population of IERS that were removed from the base population up to this point. Since IERS populations are the result of an estimation model, they are not subject to the adjustments described above.

The CSD population estimates that correspond to IERS on Census Day are derived from a two-step estimation model.^Note The population on Census Day for each individual IERS is obtained by applying a linear growth from their population at the time of the previous census. The growth rate is derived from all reserves and settlements that were completely enumerated over the same period in their respective province or territory.

The model then applies a correction to account for a net undercoverage rate of IERS by calculating the national net undercoverage rate from all completely enumerated reserves and settlements. The resulting populations by age and gender for CSDs that correspond to IERS are incorporated into the other CSDs to form the complete base population for all CSDs. This base population can then be aggregated to obtain base populations for other geographical levels, such as census divisions or census metropolitan areas.