This part details the data and methods used to derive the subnational abridged life tables and standardised death rates presented in chapter 4.
Data
The data used to construct the 1995–97 (revised), 2000–02, and 2005–07 subnational abridged life tables and standardised death rates were:
 deaths registered in New Zealand of people resident in each area in the December years 1995–97, 2000–02, and 2005–07, by age group (0, 1–4, 5–9, 10–15, ... , 80–84, 85 and over) and sex
 live births registered in New Zealand to mothers resident in each area in the December years 1995–97, 2000–02, and 2005–07, by sex
 the estimated resident population of each area at 30 June 1996, 30 June 2001 and 30 June 2006, by age group (1–4, 5–9, 10–15, ... , 80–84, 85 and over) and sex.
Deaths numerator
The life tables were compiled from deaths registered, rather than deaths occurring, in each respective threeyear period. Most death statistics refer to registrations rather than occurrences for a given time period, because of the time lag between when the death occurred and when it is registered. Hence, for a given period, the number of death registrations can be confirmed before the number of death occurrences. For periods of a year or more, the difference between registrations and occurrences is generally small, so death statistics referring to registrations are suitable for most purposes.
An adjustment for address nonresponse among deaths was made. There was no response to the address question for 0.0 percent of deaths in 1995–97, 0.2 percent of deaths in 2000–02, and 0.5 percent of deaths in 2005–07.
Population denominator (exposedtorisk population)
The estimated resident population of each area at 30 June (the midpoint) for each period was used as the denominator to calculate death rates. The estimated resident populations at 30 June 1996, 30 June 2001, and 30 June 2006 were based on the census usually resident population counts at 5 March 1996, 6 March 2001, and 7 March 2006, respectively, with adjustments for:
 net census undercount
 residents temporarily overseas on census night
 births, deaths, and net migration between census night and 30 June of the census year
 reconciliation with demographic estimates at ages 0–9 years.
For more information about the estimated resident population, refer to “Information about the population estimates” on the Statistics NZ website (www.stats.govt.nz).
Derived rates
The life tables were based on deaths averaged over three years. This is designed to reduce the impact of yeartoyear statistical variations, particularly at younger ages where there may be a small number of deaths and at very old ages where the population at risk may be small. In some cases, the subnational data does not enable death rates to be reliably estimated at all ages.
The construction of each abridged life table involved three stages. First, central death rates (mx) were calculated for each age interval, except the first year of life. Second, the Brass logit system was used to smooth agespecific death rates for all areas. Third, the smoothed rates were used to calculate a set of agespecific probabilities of death (qx), which were then used to derive other life table functions. The derivation of the mortality rate in the first year of life differed from all other ages and required special formulae, as detailed below in ‘Age 0 years’.
Life table notation
x 
Exact age (eg exact age 5 corresponds to 5 years and 0 days). 
l_{x} 
Number of people alive at exact age x from the original group of 100,000 (l0). 
L_{x} 
Average number of people alive in the age interval x to x + 1. 
d_{x} 
Number of deaths in the age interval x to x + 1. 
q_{x} 
Probability that a person at exact age x dies within a year. 
_{5}q_{x} 
Probability that a person at exact age x dies within 5 years. 
p_{x} 
Probability that a person at exact age x lives another year. 
_{5}p_{x} 
Probability that a person at exact age x lives another 5 years. 
_{5}m_{x} 
Central death rate for population in the age group x to x + 5. 
_{5}s_{x} 
Proportion of population in the age group x to x + 5 surviving another 5 years. 
e_{x} 
Expected number of years of life remaining at exact age x. 
Age 0 years
The probability of dying in the first year of life (q_{0}) required special treatment because infant deaths are skewed towards the early part of this age. The following example shows the formula for calculating q0 for 2005–07, where the denominator approximates the exposedtorisk population:
The value for q_{0} is then used to derive the following life table functions:

the radix of a life table 



where 0.85 and 0.15 approximate the proportion of infant deaths occurring in the first 6 months of life and second 6 months of life, respectively 


Age 1–4 years
The central death rates (m_{x}) for this age group were calculated by dividing the average annual deaths of residents of each area by the estimated resident population of each area at the midpoint of the period. For 2005–07:
Age 5–84 years
The central death rates (mx) were calculated for each fiveyear age group by dividing the average annual deaths of residents of each area by the estimated resident population at the midpoint of the period. For 2005–07:

for x = 5, 10, 15, ..., 80 

for x = 5, 10, 15, ..., 80 

for x = 5, 10, 15, ..., 80 

for x = 5, 10, 15, ..., 80 

for x = 5, 10, 15, ..., 80 

for x = 5, 10, 15, ..., 80 

for x = 5, 10, 15, ..., 80 
Age 85 years and over
Data for those aged 85 years and over were combined into one age group. Because it is an openended interval, some unique formulae were required:
For all ages:

for x = 0, 1, 5, 10, 15, ..., 85 where x + h = 85. 
Brass logit system
The Brass logit technique enables the calculation of smooth abridged life tables for areas that have unreliable and/or zero agespecific death rates, by adjusting the observed rates with reference to a standard life table. The technique does not alter the overall level of mortality, but the agespecific functions of the life table are smoothed. Essentially, the technique compares mortality between the area and a standard life table across ages, then a line of best fit is calculated to describe that relationship by age. The line of best fit is then used in conjunction with the standard life table to determine death rates for the small area life table. An example of observed and smoothed death rates is given in figure 5.01. For a more detailed description of the Brass logit system refer to Brass (1975).
Figure 5.01
The subnational abridged life tables for 1995–97 to 2005–07 presented in this report use the Brass logit system and the complete life tables for New Zealand for 1995–97 to 2005–07, respectively, as the standard.
Abridged life tables compared with complete period life tables
There are small differences in life table measures derived from abridged and complete period life tables (table 5.01). Abridged life tables use grouped age data and an openended upper age group. By comparison, complete period life tables use singleyear of age data. The abridged and complete life tables presented here also use different methods for smoothing death rates. The abridged life tables use the Brass logit system and the complete life tables for each respective period as the standard life tables.
Table 5.01
Standardised death rates
Standardised death rates (SDRs) provide a summary measure of the mortality experience of an area, while allowing for the different agesex composition of each area. Using the direct method of standardisation, SDRs indicate the overall death rate (deaths per 1,000 population) if the observed agesex specific death rates were applied to a standard population. The SDRs presented in this report use the age and sex distribution of the estimated resident population of New Zealand at 30 June 1996 as the standard:
where
m_{a} is the agesex specific death rate of the ethnic group
P_{a} is the standard population at each age and sex
P is the total standard population
_{a} are age groups 0, 1–4, 5–9, 10–15, ..., 80–84 and 85+ years