Abstract
The previous models are of poor quality for the early ages because of the wide distribution of observed values. In this chapter, life tables are constructed using the logit method, based on the same sample of observed tables and on the mean of the probabilities of dying by age. This model appears to be one of the best ways of studying infant and child mortality.
This work owes much to the initial approaches of Magali Belaigues-Rossard in 1999–2000. We are also grateful to Brahim Ahmedou for writing a macro for automatically generating new tables and to Arnaud Bringé for the statistical validation.
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Notes
- 1.
After Frank W. Notestein’s pioneering work in 1944 (cited by Josianne Duchêne 1999, p. 155), the first set of model tables was that proposed by the UN (1955, 1956), which started from the statistical link between two successive probabilities and estimated the entire mortality curve from one indicator (in this case, the infant probability of dying).
- 2.
The various x q 0 are calculated as follows:
5 q 0 is obtained from 1 q 0 and 4 q 1, such that 5 q 0 = 1 q 0 + 4 q 1 − (1 q 0 × 4 q 1)
10 q 0 is obtained from 5 q 0 and 5 q 5 such that 10 q 0 = 5 q 0 + 5 q 5 − (5 q 0 × 5 q 5)
15 q 0 is obtained from 10 q 0 and 5 q 10 such that 15 q 0 = 10 q 0 + 5 q 10 − (10 q 0 × 5 q 10) etc.
- 3.
Brass introduces a fixed constant into the usual transformation:
$$ {\mathrm{logit}}_x{q}_0= \ln \left(\frac{{}_xq_0}{1-{}_xq_0}\right)= \ln \left(\frac{1-{l}_x}{l_x}\right) $$This fixed constant is intended to counterbalance the weight of the logarithmic transformation of the l x (probabilities of surviving from birth to age x). The two terms in the equation are both very small when x = 0 and l x = 1. On the logit scale this amounts to modifying parameter α (which must be multiplied by 0.5) and leaving parameter β unchanged.
- 4.
The equation they propose is
logit x q 0 = α + β * T(κ, λ), with T such that
if x q 0 ≥ 0.5, T(κ, λ) = \( \frac{{\left(\frac{{}_x{q}^{\prime}_0}{1-{}_x{q}^{\prime}_0}\right)}^k-1}{2\lambda } \), and if x q 0 < 0.5, T(κ, λ) = \( \frac{{\left(\frac{{}_x{q}^{\prime}_0}{1-{}_x{q}^{\prime}_0}\right)}^{-k}-1}{-2\lambda } \).
- 5.
Parameter κ measures mortality at ages 0–5 relative to mortality at around age 35. If κ > 0, the number of survivors falls more sharply in the youngest age groups in the observed table than in the reference table; and vice versa, if κ < 0. In the youngest age groups, therefore, the survival curve is determined by β and κ. Parameter λ measures the mortality of the 70–74 age group compared with the 60–64 age group. If λ > 0, the number of survivors in the oldest age groups declines more gradually in the observed table; and vice versa, if λ < 0.
- 6.
As defined by Abdel Omran (1971), the epidemiological transition comprises three successive sequences: the first, which he calls the “age of pestilence and famine”, is characterized by high mortality and a life expectancy at birth of 20–40 years. The “age of receding pandemics”, is marked by a regular decline in mortality and a gain in length of life (life expectancy at birth close to 50 years) enabling the population to enter a growth phase. The current third phase of “degenerative and man-made diseases” sees the emergence of new diseases and of pathologies related to population ageing. As child mortality falls and length of life increases, the shape of the mortality curve changes from one phase to the next.
- 7.
0.5 ≤ β ≤ 1.5 and −1.5 ≤ α ≤ 1.
- 8.
The minimum and maximum values are sometimes outliers, particularly for 5 q 0, for which we preferred to use the thresholds of 0.175 and 0.574.
- 9.
It will be recalled that the palaeodemographic indicators constructed on the juvenile segment can only be used with the “both sexes” models, because of the extreme difficulty of determining a child’s sex with current techniques (see Chap. 4). “Male” and “female” models may, however, be of use with the input “mean adult age at death”. These will be provided later.
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Séguy, I., Buchet, L. (2013). Definition and Exploration of a Pre-industrial Standard. In: Handbook of Palaeodemography. INED Population Studies, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-01553-8_9
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