Methods for Constructing Life Tables for Canada, Provinces and Territories

Release date: July 12, 2017 Correction date: (if required)

by
Laurent Martel (Demography Division)
Martin Provost (Household Survey Methods Division)
André Lebel (Demography Division)
Simon Coulombe (Demography Division)
Adam Sherk (Household Survey Methods Division)

Introduction

Life tables are used to compute life expectancy at birth and at different ages, but they are also used to compute many other indicators: death probabilities, probabilities of survival between two ages, years of life lived and the number of survivors at different ages. The construction of a life table makes it possible to summarize mortality within a population at a given time (period life table) or within a cohort (cohort life table). Accordingly, life tables meet many statistical needs, particularly in the fields of health, epidemiology, and actuarial science, and facilitate comparisons between regions or cohorts.

Statistics Canada has disseminated life tables, both complete (by single years of age) and abridged (by five-year age groups) for Canada, the provinces and territories, since 1939 and at irregular intervals, most often centered on census years. The methodology on which these period life tables are based has also been revised in some occasions, following advances in the field of mortality studies. An important revision was made for the construction of the life tables for the period 1995 to 1997 released in August 2002, with the introduction of a model for estimating old-age mortality.

A major revision was undertaken in 2013 of the methodology described in this document to take into account current methods for constructing life tables. This revision was also an opportunity to take into account the work conducted in the context of the development of the Human Mortality Database (HMD),Note 1 a database aiming to facilitate comparisons of mortality data between a large number of countries and regions, including Canada.Note 2 While differences remain between the two methods, the increased consistency between the methodology used by Statistics Canada and that of the HMD offers many advantages, especially for demographic projections and analysis.

Two methodology changes were then introduced in the elaboration of the life tables. The first change relates to the method of estimating old-age mortality. A logistic model based on the work of Kannisto (1992) replaced the quadratic model of Coale-Kisker (1990) first used in the construction of the tables for the period 1995 to 1997. This change was made because studies showed that in countries with high-quality data, the increase, from one age to the next, in mortality rates tended to follow a logistic curve (Kannisto et al. 1994; Thatcher et al. 1998). This logistic model is also used in the construction of the HMD life tables.

The second change applies to the method of smoothing age-specific death probabilities. The previous method, used since the 1970s and relying, among other factors, on pivotal ages, was replaced by a method based on splines. This method is more flexible and does not rely on arbitrary pivotal ages.

Statistics Canada has implemented another change related to the dissemination of the life tables, making them an annual product to better meet the needs of its many clients. As soon as new data on deaths from Vital Statistics is disseminated by Statistics Canada, the life tables are produced using the methodology presented in this document and made available in the publication Life Tables, Canada, Provinces and Territories in CANSIM and Excel formats. These tables cover three reference years, starting with the period 1980 to 1982, then 1981 to 1983, and so on.

This methodology document is divided into three parts. The first part presents data sources and steps for the construction of complete life tables. These tables are calculated for Canada and all of the provinces except Prince Edward Island. Complete life tables have the advantage of being more detailed than abridged tables, especially for old ages, where most of the decline in mortality has been concentrated for some years now.

The second part presents data sources and steps for the construction of the abridged life tables used for Prince Edward Island, Yukon, the Northwest Territories and Nunavut. Abridged life tables must be used when population sizes and death counts are too small to calculate complete life tables accurately.

Finally, the last part presents a short analysis of the impact of the changes in the methodology on the comparability of life expectancy data over time.

Methodology for complete life tables

Using the revised methodology of Statistics Canada’s life tables, the construction of such tables is a seven-step process:

Step 1: Calculation of observed mortality rates for ages 5 to 109 and for the open age group of 110 years and over;
Step 2: Modeling of observed mortality rates for ages 95 to 109 and for the open age group of 110 years and over;
Step 3: Calculation of death probabilities for ages 5 to the open age group of 110 years and over;
Step 4: Calculation of death probabilities for ages 0 to 4;
Step 5: Smoothing of death probabilities for ages 1 to 94;
Step 6: Calculation of the elements of the life table;
Step 7: Calculation of the margins of error for death probabilities and life expectancy.

Input data

Two data sources are used to construct complete life tables: Statistics Canada’s Vital Statistics and Population Estimates Program.

More specifically, for a given sex and region, the following four data sets are required to calculate a complete life table for a period extending from calendar year α-1 to year α+1:

In general, the input data on deaths by age and sex from Canadian Vital Statistics are considered to be of very good quality (Bourbeau and Lebel 2000), even between 80 and 100 years of age (Beaudry-Godin 2010). Each year, there are very few deaths with unknown age or sex; any that exist are redistributed according to the known structure of observed deaths by age and sex. There are also few late registrations of deaths in Canada.

Similarly, Statistics Canada’s population estimates are considered to be of very good quality. These estimates, used in the context of the Federal-Provincial Fiscal Arrangements Act, are based on the last available census, are adjusted for census net undercoverage and take into account demographic events since the last available census, by age and sex. Most often, postcensal population estimates are used to compute the life tables in a timely fashion, and these life tables are revised once the new population estimates are updated.

Estimating mortality at 100 years and over presents a challenge, since population sizes and observed death counts are lower and records are more subject to reporting errors. However, the use of a logistic model makes it possible to obtain a consistent series of mortality rates in old age, since this series is modeled.

Step 1: Calculation of observed mortality rates for ages 5 to 109 and for the open age group of 110 years and over

For each year of age x included between 5 and the open age group of 110 years and over, the observed mortality rate is computed by taking the ratio of the sum of deaths during the three-years period to the sum of the populations in the middle of each of the three years of the three-year of the same period. By doing so, it allows differentials that can be observed in the population growth rate of the three calendar years used in the calculation of the mortality rates to be taken into account.

More specifically:

M n x = t=a1 a+1 D x,t t=a1 a+1 P x,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaab2eadaWgaaWcbaGaaeiEaaqabaGc cqGH9aqpdaWcaaqaamaaqahabaGaaeiramaaBaaaleaacaqG4bGaae ilaiaabshaaeqaaaqaaiaabshacqGH9aqpcaWGHbGaeyOeI0IaaGym aaqaaiaadggacqGHRaWkcaaIXaaaniabggHiLdaakeaadaaeWbqaai aabcfadaWgaaWcbaGaaeiEaiaabYcacaqG0baabeaaaeaacaqG0bGa eyypa0JaamyyaiabgkHiTiaaigdaaeaacaWGHbGaey4kaSIaaGymaa qdcqGHris5aaaaaaa@5600@

where M n x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaab2eadaWgaaWcbaGaaeiEaaqabaaa aa@3B18@ is the observed mortality rate between ages x and x+n (in the case of complete tables, n = 1); t=a1 a+1 D x,t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaqahabaGaaeiramaaBaaaleaacaqG4bGaaeilaiaabshaaeqaaaqa aiaabshacqGH9aqpcaWGHbGaeyOeI0IaaGymaaqaaiaadggacqGHRa WkcaaIXaaaniabggHiLdaaaa@44D3@ is the sum of deaths between ages x and x+n for calendar years a1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggacqGHsislcaaIXaaaaa@3A88@ , a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggaaaa@38E0@ and a+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggacqGHRaWkcaaIXaaaaa@3A7D@ of the reference period, and; t=a1 a+1 P x,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaqahabaGaaeiuamaaBaaaleaacaqG4bGaaeilaiaabshaaeqaaaqa aiaabshacqGH9aqpcaWGHbGaeyOeI0IaaGymaaqaaiaadggacqGHRa WkcaaIXaaaniabggHiLdaaaa@44DE@ is the sum of the population counts estimated on July 1 between ages x and x+n for calendar years a1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggacqGHsislcaaIXaaaaa@3A88@ , a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggaaaa@38E0@ and a+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggacqGHRaWkcaaIXaaaaa@3A7D@ of the reference period.

of the reference period.

In the case where, for a given age, no deaths are observed during the reference period, a mortality rate is imputed based on a geographic approach according to Table 1.

Table 1
Regions associated to the different provinces, for imputation

Table 1
Regions associated to the different provinces, for imputation
Table summary
This table displays the results of table 1 regions associated to the different provinces. The information is grouped by province (appearing as row headers), region for imputation (appearing as column headers).
Province Region associated for imputationNote 1
Newfoundland and Labrador Atlantic provincesNote 2
Nova Scotia
New Brunswick
Quebec Canada
Ontario Canada
Manitoba Prairie provinces
Saskatchewan
Alberta
British Columbia Canada

Step 2: Modeling of observed mortality rates for ages 95 to 109 and for the open age group of 110 years and over

From 95 years up to the open age group of 110 years and over, the mortality rates calculated in Step 1 can exhibit important variations due to the small number of deaths and persons submitted to the risk of dying. It can even happen that at some very old ages, often beyond age 105, it is impossible to calculate rates because of lack of deaths or at-risk population, a situation that arises frequently where there are small populations.

In these circumstances, it is better to use a model for estimating old-age mortality rates, which can yield both a more accurate representation of the mortality pattern, and a complete series of rates up to the open age group of 110 years and over. Therefore, a simplified logistic model based on the work of Kannisto (1992) was used. This model was adjusted by the maximum likelihood method, and it features an upper asymptote equal to 1.

The model takes the following form:

μ x = αe βx 1+ αe βx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabY7adaWgaaWcbaGaaeiEaaqabaGccqGH9aqpdaWcaaqaaiaabg7a caqGLbWaaWbaaSqabeaacaqGYoGaaeiEaaaaaOqaaiaaigdacqGHRa WkcaqGXoGaaeyzamaaCaaaleqabaGaaeOSdiaabIhaaaaaaaaa@4627@

where μ x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabY7adaWgaaWcbaGaaeiEaaqabaaaaa@3A63@ is the mortality force (mortality hazard in continuous time) at age x, and; α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai abeg7aHbaa@3999@ and β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabk7aaaa@3932@ are the parameters to be estimated.

The parameter α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai abeg7aHbaa@3999@ corresponds roughly to the baseline mortality at age 0 and the parameter β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabk7aaaa@3932@ corresponds to the rate of increase (logistic) of mortality from one age to the next. Parameters α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai abeg7aHbaa@3999@ and β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabk7aaaa@3932@ are assumed to be positive, meaning that this condition is enforced as the model is estimated. The model is estimated by SAS software’s NLIN procedure,Note 4 and for optimization, the Newton method is used (SAS Institute Inc. 2008A). The modeled mortality rate between age x and x+n corresponds to:

M ^ n x = μ ^ (x+n)/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiqad2eagaWeamaaBaaaleaacaqG4baa beaakiabg2da9iqbeY7aTzaataWaaSbaaSqaaiaabIhacaqGRaGaae OBaiaab+cacaqGYaaabeaaaaa@423D@

To ensure that the logistic model properly adjusts the observed old-age data, a minimum of 15 observed mortality rates between ages 80 up to the open age group of 110 years and over, calculated in Step 1, must be computable (and not imputed) for estimating the model. If this threshold were not reached, the model would not be estimated and an abridged life table, rather than a complete life table, would be calculated.

Step 3: Calculation of death probabilities for ages 5 to the open age group of 110 years and over

Steps 1 and 2 yield a series of mortality rates for ages 5 to 110. These rates are then converted to death probabilities by the so-called actuarial method:

q n x = 2xnx M n x * 2+(nx M n x * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEaaqabaGc cqGH9aqpdaWcaaqaaiaaikdacqGHflY1caqGUbGaeyyXIC9aieeGBe aaleacbbOaieeGb6gaaeqcbbiakiacbbyGnbGaaGjcVlaayIW7dGaG aA0=aaahaaWcbKaGaA0=aaqaiaiGg9paaiacGayGQaaaaOWaaSbaaS qaaiacacO=+daabIhaaeqaaaGcbaWaaeWaaeaacaaIYaGaey4kaSIa aeOBaiabgwSixpaaBeaaleaacaqGUbaabeaakiaab2eacaaMi8+aia fGBaaaleacqbOaiafGbIhaaeqcqbiakmacuaihaaWcbKafagacuaOa iqbGcQcaaaaakiaawIcacaGLPaaaaaaaaa@6A1D@

where q n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEaaqabaaa aa@3B3D@ is the death probability between ages x and x+n, that is the probability that an individual aged x die before reaching age x+n ; n is the age interval (in the case of complete life tables, n = 1), and; M n x * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaab2eadaqhaaWcbaGaaeiEaaqaaiaa bQcaaaaaaa@3BC7@ is either the observed mortality rate ( M n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaab2eadaWgaaWcbaGaaeiEaaqabaaa aa@3B19@ ) between ages x and x+n for x comprised between 5 and 94 years or the modeled mortality rate ( M ^ n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakmaaHaaabaGaaeytaaGaayPadaWaaSba aSqaaiaabIhaaeqaaaaa@3BDB@ ) for x comprised between 95 and 110 years.

Within the open age group of 110 years and over, the death probability takes the value of 1.

Step 4: Calculation of death probabilities for ages 0 to 4

Between ages 0 and 4, death probabilities are estimated directly, since mortality at these ages has its own particular pattern. For example, between 0 and 1 year of age, deaths are not distributed uniformly over the year, but are instead concentrated in the first days of life. The method used is the same as the one used in the most recent editions of the complete life tables produced by Statistics Canada (Statistics Canada 2006).

Schematically, and for a given year α, the calculation of the death probability at age 0 (1q0), also called the infant mortality rate, is based on a complement to 1 of a product of two ratios, the first representing the probability that a person of exact age x will survive to the end of the calendar year in which that person reaches age x, and the second representing the probability that a person living at the end of the calendar year in which he or she reaches age x will survive until the exact age of x+1.

Thus, according to the Lexis diagram (Figure 1):

Figure 1
Lexis diagram

Figure 1 Lexis diagram

Description for Figure 1

This Lexis diagram shows age on the vertical axis and calendar years on the horizontal axis. Deaths of year x are distinguished according to year of birth, so deaths occurring during year x for persons born during year x are distinguished from deaths occurring during year x, for persons born during year x-1. We also show population on January 1 of year x and x+1. Finally, we show the population of exact age 0 and 1 year.

q 1 0 =1( P 0,a+1 * E 0,a x E 1,a P 0,a * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaaIXaaabeaakiaabghadaWgaaWcbaGaaGimaaqabaGc cqGH9aqpcaaIXaGaeyOeI0YaaeWaaeaadaWcaaqaaiaabcfadaqhaa WcbaGaaGimaiaacYcacaGGHbGaey4kaSIaaGymaaqaaiaacQcaaaaa keaacaqGfbWaaSbaaSqaaiaaicdacaGGSaGaamyyaaqabaaaaOGaey yXIC9aaSaaaeaacaqGfbWaaSbaaSqaaiaaigdacaGGSaGaamyyaaqa baaakeaacaqGqbWaa0baaSqaaiaaicdacaGGSaGaamyyaaqaaiaacQ caaaaaaaGccaGLOaGaayzkaaaaaa@51B9@

The value E 0,a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabweadaWgaaWcbaGaaGimaiaacYcacaWGHbaabeaaaaa@3B3E@ is obtained by adding to the population aged 0 on January 1 of year a+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggacqGHRaWkcaaIXaaaaa@3A7D@ (value P 0,a+1 * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabcfadaqhaaWcbaGaaGimaiaacYcacaWGHbGaey4kaSIaaGymaaqa aiaacQcaaaaaaa@3D95@ in Figure 1) the number of deaths at age 0 during year a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggaaaa@38E0@ , among children born during that same year a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggaaaa@38E0@ ( D 0,a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaG qaaiqa=reagaqbamaaBaaaleaacaaIWaGaaiilaiaadggaaeqaaaaa @3B4F@ ). The value E 1,a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabweadaWgaaWcbaGaaGymaiaacYcacaWGHbaabeaaaaa@3B3F@ is obtained by subtracting from the population aged 0 on January 1 of year a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggaaaa@38E0@ ( P 0,a * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabcfadaqhaaWcbaGaaGimaiaacYcacaWGHbaabaGaaiOkaaaaaaa@3BF8@ ) the number of deaths at age 0 during year a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggaaaa@38E0@ , among children born during the previous year ( D 0,a ′′ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai qabseagaqbgaqbamaaBaaaleaacaaIWaGaaiilaiaadggaaeqaaaaa @3B54@ ). Description for formula 12 step 4

By transposition and using three years for the calculations:

q 1 0 =1( t=a1 a+1 P 0,t+1 t=a1 a+1 P 0,t+1 + t=a1 a+1 D 0,t x t=a1 a+1 P 0,t t=a1 a+1 D 0,t ′′ t=a1 a+1 P 0,t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaaIXaaabeaakiaabghadaWgaaWcbaGaaGimaaqabaGc cqGH9aqpcaaIXaGaeyOeI0YaaeWaaeaadaWcaaqaamaaqahabaGaae iuamaaBaaaleaacaaIWaGaaiilaiaabshacqGHRaWkcaaIXaaabeaa aeaacaqG0bGaeyypa0JaamyyaiabgkHiTiaaigdaaeaacaWGHbGaey 4kaSIaaGymaaqdcqGHris5aaGcbaWaaeWaaeaadaaeWbqaaiaabcfa daWgaaWcbaGaaGimaiaacYcacaqG0bGaey4kaSIaaGymaaqabaaaba GaaeiDaiabg2da9iaadggacqGHsislcaaIXaaabaGaamyyaiabgUca Riaaigdaa0GaeyyeIuoakiabgUcaRmaaqahabaGabeirayaafaWaaS baaSqaaiaaicdacaGGSaGaaeiDaaqabaaabaGaaeiDaiabg2da9iaa dggacqGHsislcaaIXaaabaGaamyyaiabgUcaRiaaigdaa0GaeyyeIu oaaOGaayjkaiaawMcaaaaacqGHflY1daWcaaqaamaabmaabaWaaabC aeaacaqGqbWaaSbaaSqaaiaaicdacaGGSaGaaeiDaaqabaaabaGaae iDaiabg2da9iaadggacqGHsislcaaIXaaabaGaamyyaiabgUcaRiaa igdaa0GaeyyeIuoakiabgkHiTmaaqahabaGabeirayaafyaafaWaaS baaSqaaiaaicdacaGGSaGaaeiDaaqabaaabaGaaeiDaiabg2da9iaa dggacqGHsislcaaIXaaabaGaamyyaiabgUcaRiaaigdaa0GaeyyeIu oaaOGaayjkaiaawMcaaaqaamaaqahabaGaaeiuamaaBaaaleaacaaI WaGaaiilaiaabshaaeqaaaqaaiaabshacqGH9aqpcaWGHbGaeyOeI0 IaaGymaaqaaiaadggacqGHRaWkcaaIXaaaniabggHiLdaaaaGccaGL OaGaayzkaaaaaa@95B0@

where q 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaaIXaaabeaakiaabghadaWgaaWcbaGaaGimaaqabaaa aa@3AC6@ is the death probability at age 0; t=a1 a+1 P 0,t+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaqahabaGaaeiuamaaBaaaleaacaaIWaGaaiilaiaabshacqGHRaWk caaIXaaabeaaaeaacaqG0bGaeyypa0JaamyyaiabgkHiTiaaigdaae aacaWGHbGaey4kaSIaaGymaaqdcqGHris5aaaa@463B@ is the sum of the population counts of age 0 estimated on January 1 of years a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggaaaa@38E0@ , a+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggacqGHRaWkcaaIXaaaaa@3A7D@ and a+2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aadggacqGHRaWkcaaIYaaaaa@3A7E@ , that is the last two years of the reference period and the year that follows; t=a1 a+1 D 0,t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaqahabaGabeirayaafaWaaSbaaSqaaiaaicdacaGGSaGaaeiDaaqa baaabaGaaeiDaiabg2da9iaadggacqGHsislcaaIXaaabaGaamyyai abgUcaRiaaigdaa0GaeyyeIuoaaaa@449F@ is the sum of deaths at age 0 during the reference period, for children born in the same year as the year of their death; t=a1 a+1 P 0,t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaqahabaGaaeiuamaaBaaaleaacaaIWaGaaiilaiaabshaaeqaaaqa aiaabshacqGH9aqpcaWGHbGaeyOeI0IaaGymaaqaaiaadggacqGHRa WkcaaIXaaaniabggHiLdaaaa@449F@ is the sum of the population counts of age 0 estimated on January 1 during the reference period, and; t=a1 a+1 D 0,t ′′ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaqahabaGabeirayaafyaafaWaaSbaaSqaaiaaicdacaGGSaGaaeiD aaqabaaabaGaaeiDaiabg2da9iaadggacqGHsislcaaIXaaabaGaam yyaiabgUcaRiaaigdaa0GaeyyeIuoaaaa@44AA@ is the sum of deaths at age 0 during the reference period, for children born during the year preceding the year of their death.

From ages 1 to 4 years, an equation following the same principle was used:

q n x =1( t=a1 a+1 P x,t+1 t=a1 a+1 P x,t+1 + t=a1 a+1 D x,t x t=a1 a+1 P x,t t=a1 a+1 D x,t ′′ t=a1 a+1 P x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEaaqabaGc cqGH9aqpcaaIXaGaeyOeI0YaaeWaaeaadaWcaaqaamaaqahabaGaae iuamaaBaaaleaacaqG4bGaaeilaiaabshacaqGRaGaaeymaaqabaaa baGaaeiDaiabg2da9iaadggacqGHsislcaaIXaaabaGaamyyaiabgU caRiaaigdaa0GaeyyeIuoaaOqaamaabmaabaWaaabCaeaacaqGqbWa aSbaaSqaaiaabIhacaqGSaGaaeiDaiabgUcaRiaaigdaaeqaaaqaai aabshacqGH9aqpcaWGHbGaeyOeI0IaaGymaaqaaiaadggacqGHRaWk caaIXaaaniabggHiLdGccqGHRaWkdaaeWbqaaiqabseagaqbamaaBa aaleaacaqG4bGaaeilaiaabshaaeqaaaqaaiaabshacqGH9aqpcaWG HbGaeyOeI0IaaGymaaqaaiaadggacqGHRaWkcaaIXaaaniabggHiLd aakiaawIcacaGLPaaaaaGaeyyXIC9aaSaaaeaadaqadaqaamaaqaha baGaaeiuamaaBaaaleaacaqG4bGaaeilaiaabshaaeqaaaqaaiaabs hacqGH9aqpcaWGHbGaeyOeI0IaaGymaaqaaiaadggacqGHRaWkcaaI XaaaniabggHiLdGccqGHsisldaaeWbqaaiqabseagaqbgaqbamaaBa aaleaacaqG4bGaaeilaiaabshaaeqaaaqaaiaabshacqGH9aqpcaWG HbGaeyOeI0IaaGymaaqaaiaadggacqGHRaWkcaaIXaaaniabggHiLd aakiaawIcacaGLPaaaaeaadaaeWbqaaiaabcfadaWgaaWcbaGaaeiE aiaabYcacaqG0baabeaaaeaacaqG0bGaeyypa0JaamyyaiabgkHiTi aaigdaaeaacaWGHbGaey4kaSIaaGymaaqdcqGHris5aaaaaOGaayjk aiaawMcaaaaa@976C@

In the case where, for each age between 1 and 4 years, no death is observed during the reference period, a death probability will be interpolated for this age in Step 5, where death probabilities are smoothed.

Step 5: Smoothing of death probabilities for ages 1 to 94

In steps 3 and 4, a complete series of death probabilities was computed for ages 0 to 110 years. However, between ages 1 and 94, these probabilities can exhibit an irregular pattern over time, especially for regions with small populations. To ensure that death probabilities evolve consistently from one age to another and to estimate, if needed, missing probabilities between ages 1 and 4, smoothing was applied using B-splines. This smoothing is applied to death probabilities from 0 to 109 years of age, to ensure a harmonious link between smoothing by means of B-splines and the model for estimating old-age mortality (Step 2).Note 5

The B-splines smoothing technique has the advantage of being flexible, that is, to provide users many options to fit the data in the best way possible. Since B-splines, just like any splines, are constructed using piecewise polynomial functions joined together, it is appropriate to choose various positions on the horizontal axis–or knots–where these junctions occur. The greater the number of knots, the better the smoothed curve fits the original curve of death probabilities by age; conversely, a small number of knots provide more power to the smoothing. As a result, the fluctuations between ages are eliminated and a curve with a more regular appearance is obtained.

There are algorithms that can be used to find both the optimal number of knots to use and their position on the age distribution in the context of constructing life tables (Kaishev et al. 2009). However, these algorithms are complex to use. For these complete tables, the number and the position of the knots were instead determined empirically, a series of tests were performed to evaluate both the neutrality and the adjustment of the smoothing method chosen.Note 6 The smoothing method used must have the smallest effect possible on the age-specific life expectancy. Also, each series of smoothed probabilities is compared to the observed non smoother series of probabilities, so as to ensure goodness of fit.

Two series of knots are used, depending on the size of the population for which a complete mortality table is produced. The first series includes 11 knots, set at the following ages: 0, 1, 9, 15, 18, 24, 30, 35, 40, 50 and 90.Note 7 instead of 9 knots in the previous release of life tables (2016) to take into account recent evolution of mortality between ages 30 and 50, with two new knots, at ages 35 and 40. Between ages 50 and 94, the number and position of the knots is of lesser importance than for younger ages, as the mortality pattern is mostly linear. Before age 50, chosen knots correspond often to inflexion points on the classic death probabilities curve. By choosing these knots, it was possible to enforce a similar pattern in death probabilities from one region to the next, while allowing enough flexibility to allow period and regional variations in mortality to be taken into account. This 11-knot series is always used for Canada, Quebec, Ontario, Alberta and British Columbia. It is also used most of the time for the other provinces except Newfoundland and Labrador, although an analysis is performed each time to select the most appropriate series of knots.

The second series consists of 7 knots, set at the following ages: 0, 9, 18, 24, 30, 50 and 90. Using this series, smoothing is “stronger” than for the other series and is used for provinces with smaller populations. This 7-knots series is used for Newfoundland and Labrador and, at times, Nova Scotia, New Brunswick, Manitoba and Saskatchewan.

The B-splines smoothing of the death probabilities between 1 and 94 years in the current life tables has been fitted using the TRANSREGNote 8 procedure in the SAS statistical software (SAS Institute Inc., 2008B).

Step 6: Calculation of the elements of the life table

By creating a series of smoothed death probabilities between ages 0 and 110, it is possible to calculate all the life table elements based on a fictitious cohort of 100,000 newborns, according to the following equations:

Number of survivors at exact age 0, also called the root of the life table ( l 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabYgadaWgaaWcbaGaaGimaaqabaaaaa@39CF@ ): 

l 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabYgadaWgaaWcbaGaaGimaaqabaaaaa@39CF@ = 100,000 newborns

Number of deaths between ages x and x+n ( d n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabsgadaWgaaWcbaGaaeiEaaqabaaa aa@3B30@ ):

d n x = l x x q n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabsgadaWgaaWcbaGaaeiEaaqabaGc caqG9aGaaeiiaiaabYgadaWgaaWcbaGaaeiEaaqabaGccqGHflY1da WgbaWcbaGaaeOBaaqabaGccaqGXbWaaSbaaSqaaiaabIhaaeqaaaaa @444A@

Number of survivors at exact age x ( l x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabYgadaWgaaWcbaGaaeiEaaqabaaaaa@3A10@ ):

l x = l x-n - d n x-n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabYgadaWgaaWcbaGaaeiEaaqabaGccaqG9aGaaeiiaiaabYgadaWg aaWcbaGaaeiEaiaab2cacaqGUbaabeaakiaab2cadaWgbaWcbaGaae OBaaqabaGccaqGKbWaaSbaaSqaaiaabIhacaqGTaGaaeOBaaqabaaa aa@44C5@

Probability of survival between ages x and x+n ( p n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabchadaWgaaWcbaGaaeiEaaqabaaa aa@3B3C@ ):

p n x = 1- q n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabchadaWgaaWcbaGaaeiEaaqabaGc caqG9aGaaeiiaiaabgdacaqGTaWaaSraaSqaaiaab6gaaeqaaOGaae yCamaaBaaaleaacaqG4baabeaaaaa@4150@

Number of years lived between ages x and x+n (stationary population) ( L n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabYeadaWgaaWcbaGaaeiEaaqabaaa aa@3B18@ ):

L n x =nx l x+n + d n x x f n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabYeadaWgaaWcbaGaaeiEaaqabaGc cqGH9aqpcaqGUbGaeyyXICTaaeiBamaaBaaaleaacaqG4bGaae4kai aab6gaaeqaaOGaey4kaSYaaeWaaeaadaWgbaWcbaGaaeOBaaqabaGc caqGKbWaaSbaaSqaaiaabIhaaeqaaOGaaeylamaaBeaaleaacaqGUb aabeaakiaabAgadaWgaaWcbaGaaeiEaaqabaaakiaawIcacaGLPaaa aaa@4CBF@ for x comprised between 0 to 109 years

where, for complete life tables,

f 1 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaaIXaaabeaakiaabAgadaWgaaWcbaGaaeiEaaqabaaa aa@3AFC@ (separation factor) =

{ 1( t=a1 a+1 D x,t t=a1 a+1 ( D x,t + D x,t ′′ ) ) and 0.5 for x5 years for x = 0 to 4 years if the numerator and denominator are greater than 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaceaaeaqabeaacaaIXaGaeyOeI0YaaeWaaeaadaWcaaqaamaaqaha baGabeirayaafaWaaSbaaSqaaiaabIhacaqGSaGaaeiDaaqabaaaba GaaeiDaiabg2da9iaadggacqGHsislcaaIXaaabaGaamyyaiabgUca Riaaigdaa0GaeyyeIuoaaOqaamaaqahabaWaaeWaaeaaceWGebGbau aadaWgaaWcbaGaamiEaiaacYcacaWG0baabeaakiabgUcaRiqadsea gaqbgaqbamaaBaaaleaacaWG4bGaaiilaiaadshaaeqaaaGccaGLOa GaayzkaaaaleaacaqG0bGaeyypa0JaamyyaiabgkHiTiaaigdaaeaa caWGHbGaey4kaSIaaGymaaqdcqGHris5aaaaaOGaayjkaiaawMcaaa qaaiaabggacaqGUbGaaeizaiaabccacaqGWaGaaeOlaiaabwdacaqG GaGaaeOzaiaab+gacaqGYbGaaeiiaiaabIhacqGHLjYScaqG1aGaae iiaiaabMhacaqGLbGaaeyyaiaabkhacaqGZbaaaiaawUhaaiaabAga caqGVbGaaeOCaiaabccacaqG4bGaaeiiaiaab2dacaqGGaGaaeimai aabccacaqG0bGaae4BaiaabccacaqG0aGaaeiiaiaabMhacaqGLbGa aeyyaiaabkhacaqGZbGaaeiiaiaabMgacaqGMbGaaeiiaiaabshaca qGObGaaeyzaiaabccacaqGUbGaaeyDaiaab2gacaqGLbGaaeOCaiaa bggacaqG0bGaae4BaiaabkhacaqGGaGaaeyyaiaab6gacaqGKbGaae iiaiaabsgacaqGLbGaaeOBaiaab+gacaqGTbGaaeyAaiaab6gacaqG HbGaaeiDaiaab+gacaqGYbGaaeiiaiaabggacaqGYbGaaeyzaiaabc cacaqGNbGaaeOCaiaabwgacaqGHbGaaeiDaiaabwgacaqGYbGaaeii aiaabshacaqGObGaaeyyaiaab6gacaqGGaGaaeimaiaab6caaaa@AD6A@

If either the numerator or the denominator equals to zero in the calculation of f 1 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGXaaabeaakiaabAgadaWgaaWcbaGaaeiEaaqabaaa aa@3AF5@ for x comprised between 0 to 4 years, a value of f 1 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGXaaabeaakiaabAgadaWgaaWcbaGaaeiEaaqabaaa aa@3AF5@ is imputed based on a geographic approach like the one used in Step 1.

L 110+ =l 110 x e 110 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabYeadaWgaaWcbaGaaeymaiaabgdacaqGWaGaae4kaaqabaGccaqG 9aGaaeiBamaaBaaaleaacaqGXaGaaeymaiaabcdaaeqaaOGaeyyXIC TaaeyzamaaBaaaleaacaqGXaGaaeymaiaabcdaaeqaaaaa@4541@ for the open age group 110 years and over where e 110 = 1 M ^ 110 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabwgadaWgaaWcbaGaaeymaiaabgdacaqGWaaabeaakiaab2dadaWc aaqaaiaabgdaaeaaceqGnbGbambadaWgaaWcbaGaaeymaiaabgdaca qGWaaabeaaaaaaaa@3FE7@

Total number of cumulative years of life lived starting at age x (cumulative stationary population) ( T x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabsfadaWgaaWcbaGaaeiEaaqabaaaaa@39F8@ ):

T x = i=x 110 L n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabsfadaWgaaWcbaGaaeiEaaqabaGccaqG9aWaaabCaeaadaWgbaWc baGaaeOBaaqabaGccaqGmbWaaSbaaSqaaiaabMgaaeqaaaqaaiaabM gacaqG9aGaaeiEaaqaaiaabgdacaqGXaGaaeimaaqdcqGHris5aaaa @44CA@

Life expectancy between ages 0 and 109 ( e x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabwgadaWgaaWcbaGaaeiEaaqabaaaaa@3A09@ ):

e x = T x l x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabwgadaWgaaWcbaGaaeiEaaqabaGccaqG9aWaaSaaaeaacaqGubWa aSbaaSqaaiaabIhaaeqaaaGcbaGaaeiBamaaBaaaleaacaqG4baabe aaaaaaaa@3F00@

Step 7: Calculation of the margins of error for death probabilities and life expectancy

Since publishing the complete life tables for the period 1995 to 1997, Statistics Canada has disseminated the coefficients of variation associated with death probabilities and life expectancies from life tables. This quality indicator gives the reader an idea of the variability of the estimate, which largely depends on the number of deaths on which the estimate is based.

The quality indicator used this time is the margin of error, which is used to directly calculate 95% confidence interval of an estimate. The margin of error (m.e.) of death probabilities at age x is calculated as follows:

m.e.( q n x ) = 1.96x s .e.( q n x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aab2gacaqGUaGaaeyzaiaab6cacaqGOaWaaSraaSqaaiaab6gaaeqa aOGaaeyCamaaBaaaleaacaqG4baabeaakiaabMcacaqGGaGaaeypai aabccacaqGXaGaaeOlaiaabMdacaqG2aGaeyyXICTaaeyzaiaab6ca caqG0bGaaeOlamaabmaabaWaaSraaSqaaiaab6gaaeqaaOGaaeyCam aaBaaaleaacaqG4baabeaaaOGaayjkaiaawMcaaaaa@4F19@

where the standard error (s.e.) of q n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEaaqabaaa aa@3B3D@ is given by:

s.e.( q n x )= V( q n x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabohacaqGUaGaaeyzaiaab6cadaqadaqaamaaBeaaleaacaqGUbaa beaakiaabghadaWgaaWcbaGaaeiEaaqabaaakiaawIcacaGLPaaaca qG9aWaaOaaaeaacaqGwbWaaeWaaeaadaWgbaWcbaGaaeOBaaqabaGc caqGXbWaaSbaaSqaaiaabIhaaeqaaaGccaGLOaGaayzkaaaaleqaaa aa@4699@

and the variance of q n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEaaqabaaa aa@3B3D@ is given by the Chiang formula (1984):

V( q n x )= q n x 2 ( 1 q n x ) D x * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabAfadaqadaqaamaaBeaaleaacaqGUbaabeaakiaabghadaWgaaWc baGaaeiEaaqabaaakiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaaBe aaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEamaaCaaameqa baGaaeOmaaaaaSqabaGcdaqadaqaaiaaigdacqGHsisldaWgbaWcba GaaeOBaaqabaGccaqGXbWaaSbaaSqaaiaabIhaaeqaaaGccaGLOaGa ayzkaaaabaGaaeiramaaDaaaleaacaqG4baabaGaaeOkaaaaaaaaaa@4C13@

where D x * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabseadaqhaaWcbaGaaeiEaaqaaiaabQcaaaaaaa@3A96@ are estimated deaths between the age x and x+n in the population based on smoothed mortality rates, the latter being computed from smoothed death probabilities.

The margin of error and the standard deviation of life expectancies at age x are calculated using the same equations, with the exception of the equation used for the variance, which according to Chiang (1984), is:

V( e x )= i=x 110 l i 2 x [ ( 1 f n i )+ e i+n ] 2 xV( q n i ) l x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aabAfadaqadaqaaiaabwgadaWgaaWcbaGaaeiEaaqabaaakiaawIca caGLPaaacqGH9aqpdaWcaaqaamaaqahabaGaaeiBamaaBaaaleaaca qGPbWaaWbaaWqabeaacaqGYaaaaaWcbeaaaeaacaqGPbGaaeypaiaa bIhaaeaacaaIXaGaaGymaiaaicdaa0GaeyyeIuoakiabgwSixpaadm aabaWaaeWaaeaacaaIXaGaeyOeI0YaaSraaSqaaiaab6gaaeqaaOGa aeOzamaaBaaaleaacaqGPbaabeaaaOGaayjkaiaawMcaaiabgUcaRi aabwgadaWgaaWcbaGaaeyAaiaabUcacaqGUbaabeaaaOGaay5waiaa w2faamaaCaaaleqabaGaaGOmaaaakiabgwSixlaabAfadaqadaqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeyAaaqabaaa kiaawIcacaGLPaaaaeaacaqGSbWaaSbaaSqaaiaabIhadaahaaadbe qaaiaabkdaaaaaleqaaaaaaaa@62A3@

For example, a margin of error of 0.00020 for a death probability at age 0 of 0.00556 enables to build a 95% confidence interval with lower and upper limits of 0.00536 and 0.00576. In other words, the death probability is precise within a range of 0.00020, 19 times out of 20. In rare cases, subtracting the margin of error to the associated probability of dying may yield a negative result. If so, the lower limit can be considered to be exactly zero.

Similarly, a margin of error of 0.2 on a life expectancy at birth of 81.9 years enables to build a 95% confidence interval with lower and upper limits of 81.7 and 82.1 years.

Methodology for abridged life tables

An abridged life table is used when the size of a population of a given region is too small to accurately compute a complete life table (by single years of age) since, often, no deaths are recorded for a number of ages, a common situation between the ages of 1 and 15 years. An abridged life table is produced for Prince Edward Island and for Yukon, the Northwest Territories and Nunavut, separately.

These abridged life tables are constructed using the methodology described in this section. It is mostly based on the methodology used in the complete life tables, to ensure maximum consistency between the two series of tables. In some cases, as in the calculation of the death probability at age 0, the two methods are identical. However, it is not necessary to use the model to estimate old-age mortality, since the abridged life table ends at the open age group of 90 years and over. Also, no method of smoothing death probabilities is used for the abridged life tables, as there are less random fluctuations. The methodology used is moreover very close to the one used in previous editions of the abridged life tables produced by Statistics Canada (2006), with the exception of the imputation of mortality rates from an aggregated region in the case where population sizes or death counts are too low.

Constructing abridged life tables is a five-step process:

Step 1: Calculation of observed mortality rates for age group 1 to 4 to the open age group 90 years and over;
Step 2: Calculation of death probabilities for age group 1 to 4 to the open age group 90 years and over;
Step 3: Calculation of death probabilities at age 0;
Step 4: Calculation of the elements of the life table;
Step 5: Calculation of the margins of error for death probabilities and life expectancy.

Input data

Calculating abridged tables requires the same data series as for complete tables.

Age groups

Abridged tables are produced using 20 age groups, for which the notation is of the form x to x+(n-1) as indicated in Table 2.

Table 2
Age interval by age group

Table 2
Age interval by age group
Table summary
This table displays the results of table 2: age interval by age group. The information is grouped by age group (appearing as row headers), n (age interval) (appearing as column headers)
Age group n (age interval)
0 year 1
1 to 4 years 4
5 to 9 years to 85 to 89 years 5
90 years and over --

Step 1: Calculation of observed mortality rates for age group 1 to 4 to the open age group 90 years and over

For the age group 1 to 4, for each 5-year age group x to x+(n-1) from 5 to 89 years and the open age group 90 years and over, the observed mortality rate is computed by taking the ratio of the sum of deaths during the three-year (calendar year) period to the sum of the populations on July 1 for the same age group and during the same period. The formula used is the same as that of the complete life tables, adjusted to take into account age groups (see Step 1 of the section on complete life tables).

In a case where, for a given age group and sex, no deaths are observed during the reference period, an observed mortality rate is imputed on the basis of a geographic approach. First, the region is considered; for Prince Edward Island, the observed mortality rate imputed will be that of the region comprised of all of the Atlantic provinces. For Yukon, the Northwest Territories and Nunavut, the reference region consists of the three territories pooled together. This imputation procedure is also used for mortality rates above age 49 when, for a given age group, population counts are lower than 50 or death counts are lower than 10.

If no deaths were observed in these two major regions, a very rare situation, the observed mortality rate for Canada as a whole is used.

Step 2: Calculation of death probabilities for age group 1 to 4 to the open age group 90 years and over

The observed mortality rates obtained in Step 1 are converted into death probabilities by the Greville method (1943). This method yields very similar results to the actuarial method used for complete tables (Ng and Gentleman 1995), while ensuring that the probabilities obtained are never greater than 1.

According to Greville,

q n x = m n x 1 n + m n x x [ 0.5+ n 12 x ( m n x lnC ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEaaqabaGc cqGH9aqpdaWcaaqaamaaBeaaleaacaqGUbaabeaakiaab2gadaWgaa WcbaGaaeiEaaqabaaakeaadaWcaaqaaiaaigdaaeaacaqGUbaaaiab gUcaRmaaBeaaleaacaqGUbaabeaakiaab2gadaWgaaWcbaGaaeiEaa qabaGcdaWadaqaaiaaicdacaGGUaGaaGynaiabgUcaRmaalaaabaGa aeOBaaqaaiaaigdacaaIYaaaamaabmaabaWaaSraaSqaaiaab6gaae qaaOGaaeyBamaaBaaaleaacaqG4baabeaakiaab2cacaqGGaGaciiB aiaac6gacaqGdbaacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaaaa@55D2@

where q n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaabghadaWgaaWcbaGaaeiEaaqabaaa aa@3B3D@ is the death probability between ages x and x+n; m n x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aaBeaaleaacaqGUbaabeaakiaab2eadaWgaaWcbaGaaeiEaaqabaaa aa@3B19@ is the observed mortality rate between ages x and x+n; n is the size of the age group interval, which is 4 years in the case of age group 1 to 4 and 5 years after except for the last age group, and; C is equal to:

C=( 1 45 )ln( m 5 85 m 5 40 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaai aaboeacqGH9aqpdaqadaqaamaalaaabaGaaGymaaqaaiaaisdacaaI 1aaaaaGaayjkaiaawMcaaiGacYgacaGGUbWaaeWaaeaadaWcaaqaam aaBeaaleaacaaI1aaabeaakiaab2gadaWgaaWcbaGaaGioaiaaiwda aeqaaaGcbaWaaSraaSqaaiaaiwdaaeqaaOGaaeyBamaaBaaaleaaca aI0aGaaGimaaqabaaaaaGccaGLOaGaayzkaaaaaa@4844@

Within the open age group of 90 years and over, the death probability takes the value of 1.

Step 3: Calculation of death probabilities at age 0

The method of calculating death probabilities at age 0 is identical to the method used for complete tables.

Step 4: Calculation of the elements of the life table

The various elements of the life table are computed in the same way as for the complete tables, adjusting for age groups (see Step 6 of the section on complete life tables).

Step 5: Calculation of the margins of error for death probabilities and life expectancy

The same equations are used to calculate margins of error as for the complete tables, adjusting for age groups (see Step 7 of the section on complete life tables).

Impact of methodology changes

Numerous authors have shown that life expectancy at birth calculated using a life table is a robust statistical indicator, relatively insensitive to changes in methodology (e.g., Ng and Gentleman 1995). However, it is appropriate to examine the impact of the changes made to the construction of complete and abridged life tables in comparison with Statistics Canada’s previous edition (2006).

Complete life tables

Previously, two major changes have been made to the construction of the complete life tables: The Coale-Kishker model for estimating mortality at advanced ages has been replaced by a logistic model, and the method for smoothing the mortality rates has been replaced by a smoothing technique based on B-splines applied to the death probabilities.

These changes have very little impact on the comparability of life expectancy data through time. A comparison made between the life expectancy at birth drawn from the complete life table already published by Statistics Canada for the period 2000 to 2002 and that based on the life tables reproduced for the same period using the present methodology show a difference of at most 0.1 year for Canada as a whole.

For the nine provinces for which a complete life table is computed (males and females separately), this difference in life expectancy at birth between the two series is lower than 0.1 year in 13 of the 18 cases. For the 5 additional cases, this difference in life expectancy at birth do not exceed 0.4 year. In these 5 cases, the difference can be attributed to an improvement in the junction between the series of smoothed death probabilities and modeled death probabilities for old-ages, junction which occurred between 87 and 88 years in the life tables for the period 2000 to 2002 already released (Statistics Canada 2006) and between 94 and 95 years with the revised methodology. Figure 2 provides an overview of the differences in age-specific death probabilities computed using the previous and revised methodology for the life table of females in Saskatchewan for the period 2000 to 2002.

Figure 2
Age-specific death probabilities for females in Saskatchewan for the period 2000 to 2002 according to the previous and revised methodology

figure 2 Age-specific death probabilities for females in Saskatchewan for the period 2000 to 2002 according to the previous and revised methodology

Description for Figure 2
Data table for Figure 2
Age Previous methodology (Statistics Canada 2006 Revised methodology
0 year 0.00494 0.00499
1 year 0.00039 0.00036
2 years 0.00024 0.00026
3 years 0.00019 0.00020
4 years 0.00016 0.00016
5 years 0.00011 0.00013
6 years Note F: too unreliable to be published 0.00012
7 years Note F: too unreliable to be published 0.00012
8 years Note F: too unreliable to be published 0.00012
9 years 0.00012 0.00013
10 years 0.00015 0.00014
11 years 0.00016 0.00017
12 years 0.00024 0.00020
13 years 0.00030 0.00025
14 years 0.00038 0.00033
15 years 0.00046 0.00044
16 years 0.00053 0.00056
17 years 0.00057 0.00063
18 years 0.00057 0.00060
19 years 0.00055 0.00055
20 years 0.00051 0.00050
21 years 0.00048 0.00047
22 years 0.00045 0.00045
23 years 0.00044 0.00043
24 years 0.00043 0.00043
25 years 0.00043 0.00042
26 years 0.00043 0.00043
27 years 0.00044 0.00044
28 years 0.00046 0.00046
29 years 0.00048 0.00048
30 years 0.00051 0.00051
31 years 0.00055 0.00055
32 years 0.00060 0.00059
33 years 0.00067 0.00064
34 years 0.00076 0.00069
35 years 0.00085 0.00074
36 years 0.00094 0.00080
37 years 0.00101 0.00087
38 years 0.00106 0.00094
39 years 0.00109 0.00101
40 years 0.00112 0.00109
41 years 0.00115 0.00118
42 years 0.00121 0.00128
43 years 0.00127 0.00139
44 years 0.00134 0.00150
45 years 0.00142 0.00163
46 years 0.00153 0.00177
47 years 0.00169 0.00192
48 years 0.00188 0.00209
49 years 0.00210 0.00227
50 years 0.00236 0.00247
51 years 0.00267 0.00269
52 years 0.00303 0.00293
53 years 0.00348 0.00319
54 years 0.00403 0.00348
55 years 0.00460 0.00379
56 years 0.00516 0.00414
57 years 0.00564 0.00453
58 years 0.00599 0.00495
59 years 0.00624 0.00542
60 years 0.00649 0.00593
61 years 0.00682 0.00650
62 years 0.00732 0.00713
63 years 0.00803 0.00782
64 years 0.00889 0.00859
65 years 0.00985 0.00944
66 years 0.01086 0.01038
67 years 0.01188 0.01142
68 years 0.01281 0.01257
69 years 0.01370 0.01385
70 years 0.01466 0.01527
71 years 0.01583 0.01685
72 years 0.01733 0.01861
73 years 0.01913 0.02056
74 years 0.02115 0.02273
75 years 0.02344 0.02515
76 years 0.02603 0.02784
77 years 0.02898 0.03085
78 years 0.03211 0.03420
79 years 0.03538 0.03794
80 years 0.03907 0.04212
81 years 0.04346 0.04679
82 years 0.04881 0.05202
83 years 0.05495 0.05787
84 years 0.06170 0.06443
85 years 0.06932 0.07177
86 years 0.07809 0.08001
87 years 0.08829 0.08925
88 years 0.08878 0.09964
89 years 0.09716 0.11130
90 years 0.10633 0.12442
91 years 0.11635 0.13893
92 years 0.12727 0.15470
93 years 0.13919 0.17178
94 years 0.15218 0.19020
95 years 0.16631 0.21000
96 years 0.18168 0.23121
97 years 0.19838 0.25384
98 years 0.21648 0.27790
99 years 0.23607 0.30338
100 years 0.25726 0.33027
101 years 0.28012 0.35852
102 years 0.30473 0.38809
103 years 0.33117 0.41892
104 years 0.35950 0.45091
105 years 0.38980 0.48398

Abridged life tables

Concerning the abridged life tables, only very minor changes were made to the methodology used in previous life tables produced by Statistics Canada. Like the complete tables, the death probability at age 0 was calculated directly, to maintain consistency. Where possible, the separation factor (fx) used in the equation on years of life lived (stationary population) is calculated for the province or territory of interest. These changes have very little impact on the computed life expectancies.

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