Theory and applications of backward probabilities and prevalences in cross-longitudinal surveys

Résumé 0

In this chapter we introduce the backward probability and the backward prevalence. Both measures are of minor interest compared to forward probability and forward prevalence; however, they can bring a better understanding of the past population dynamic. More- over, we show that through the calculation of backward probabilities, one can reconstruct prevalences at any age for older generations. Here, the demographic interest lies in the comparison of the three prevalences: cross-sectional, forward, and backward. In order to accomplish our task, we first review theories of Markov chains with (i) an age- independent transition matrix and, (ii) when transitions vary with age. The second theory leads to an interesting property called “weak ergodicity” that allows us to predict future prevalences for younger generations. It is important to mention here that backward probability was rapidly defined in 1980 (Brouard, 1980) using longitudinal information of French women’s participation in economic activity between 1977 and 1978, and most results presented in the publication of 1980 are reviewed for this study. This chapter also shows that in a stationary multistate population, cross-sectional, forward, and backward prevalences are identical at each age. If they are not, as in the case of the economic activity of French women which changed after the 1968s revolution (women’s liberation, contraception, and abortion laws), our approach enables a clearer, faster, and synthetic analysis of these changes without the need to wait 20 or 30 years until these women meave the labor market. Then, we review demographic tools, now widely used in mortality analysis that compare “cross-sectional prevalence of survival” and period mortality table. We extend them to multistates methods, particularly to methods developed in the mid-1990s to estimate disability-free life expectancies.