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The transition probabilities are calculated as follows:
where q x, y, j, i+1 is the probability that a woman aged x in year y of parity j with her j th birth i years ago progresses to birth j +1 in ageing from x years in year y to x +1 years in year y +1, and Px, y, j, i is the number of women aged x in year y of parity j with the j th birth i years ago. These probabilities are calculated as [1 - (probability that a woman does not progress to another birth)].
An additional calculation is required for transition probabilities where the interval between births is zero years. These cases will appear in the Census data as two (or more) own children of the same age, and will usually occur for multiple births. For example, consider the case of a woman in 2006 aged 26 years with three own children, aged five, zero and zero years. In 2005 she will appear in the data as 25 years old, parity one, four years since last birth. In 2006 she will have shifted to 26 years old, parity three, 0 years since last birth. Her transition to second birth is captured by equation , but her transition to third birth is not. The latter transition is captured by considering the difference between the number of 'expected' and 'observed' women in each age/parity group in each year.
The expected number of women, age x +1, year y +1, parity j +1, is given by:
that is, the number of women age x in year y of parity j +1, plus the number of women age x in year y of parity j who have another birth in transitioning to age x +1 in year y +1, less the number of women age x in year y of parity j +1 who have another birth in transitioning to age x +1 in year y +1. This equation simplifies to:
The observed number of women, age x +1, year y +1, parity j +1, is given by:
For women age x in year y with parity zero who transition to parity two in ageing to x +1, y +1, the number of their second births is given by  - , where j = 0:
2nd births with interval 0
For women who transition from parity j, j ≥ 1, to parity j + 2 in the course of one year, the number of their j + 2 births is given by:
[j + 2] births with interval 0
The probability that a woman aged x in year y of parity j who transitions to parity j +1 in ageing to x +1, y +1, also transitions to parity j +2 in the same year is given by:
In the discussion above, the number of own children in the household has been equated with parity. However, the two are not necessarily equal, since children may have died or be resident outside the maternal home.
The transition probabilities calculated above were adjusted so that they refer to true parity rather than to own children. This was done by comparing the own-children transition probabilities for each year 1991—2000 to the parity transition probabilities for 1991—2000 calculated previously by Kippen (2003, 2004). We find that the ratio of transition probabilities for each age of woman and own children/parity is relatively constant across the ten years. We therefore use ratios averaged across the decade to inflate the own-children transition probabilities to parity transition probabilities.
The 1981, 1986, 1996 and 2006 Censuses asked the number of children each woman had ever had. This information on number of children ever born(3) (parity) was used to restrict the distributions Px, j, j, i to women whose number of own children in the household was equal to their parity in the relevant Censuses for comparison.
We found that there was no need to adjust birth intervals to take account of children missing from the household. A comparison of interval distributions for all women, and women for whom number of children is equal to parity, showed that they are virtually identical. This suggests that if mothers have children missing from their household, it is likely that all their children are missing, or that the oldest or youngest are missing.
1 - There are limitations to using the relationship in household variable to identify natural mother-child relationships in a household. The identification of a 'child' to a female reference person does not imply that the child is born of that mother. A parent-child relationship is defined in the Census as "a relationship between two persons usually resident in the same household. The nominal child is attached to the nominal parent via a natural, adoptive, step, foster or child dependency relationship." While in later Censuses it is possible to separately identify the relationships of 'step', 'foster' or 'other dependent' in the parent-child relationship, it has never been possible to distinguish between 'natural' and 'adoptive' parent-child relationships across the Censuses using the relationship in household variable. <back
2 - It should be noted that there are known data quality issues related to the reporting of 'age' and these may impact the data and analysis. For further information please read the data quality statements included in 2006 Census of Population and Housing - Reference and Information. <back
3 - It should be noted that there are known data quality issues related to the reporting of 'number of children ever born' and these may impact on the data and analysis. For further information please read the data quality statements included in 2006 Census of Population and Housing - Reference and Information. <back
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