The estimation of fertility from incomplete cohort data by means of the transformed gompertz model

Among the several models of fertility that have been developed, the Gompertz function has been shown to perform quite well. The fit to cumulative fertility by age of women is better in the middle age range than in the tails of the distribution, especially for high fertility populations. An empirical transformation of the age scale is developed to produce a better fit to the entire reproductive period. The substitution of age by this transformed scale results in the transformed Gompertz model of fertility: F(x)=FPQYs(x) 0 < P, Q < 1 0 < F where F(x) is cumulative fertility by age of women, x, F is completed fertility, P and Q are parameters and Ys(x) is the transformed age scale. The model can also be written as: Y(x)= a+ BYs(x) where Y(x)= -1n(-1n F(x)/F), a=-1n(-1nP) and B=-1nQ. Ys(x) is obtained by averaging over a selection of transformed patterns of high fertility schedules generated by the empirically based Coale-Trussell model of fertility. The model is tested using several sets of good quality fertility rates for birth cohorts of women, including a series of simulated rates developed especially for this purpose. Both the goodness of fit of the model and its projection capabilities from incomplete cohort data are shown to be good. Fitting is by the least squares method with equal weights. Several sets of poorer quality, high fertility, cohort data obtained from maternity histories collected in Bangladesh, Sri Lanka and West New Guinea are used to illustrate the application of the model. Fitting for these data is by least squares with an infinite weight on reported parity at the time of the survey. The estimates of completed fertility are plausible, and the fitted curves provide evidence of reporting errors in the data. There is also evidence of trends over time in the level and pattern of fertility.