Alternative estimates of fertility control by using parity distributions: a comment on David et al.

David, Mroz, Sanderson, Wachter, and Weir (1988) proposed a procedure for inferring the existence of fertility control by using parity distributions.' There is a need for such procedures to complement those that use age-specific marital fertility rates (Coale & Trussell 1974). The reasoning and techniques developed by David et al. constitute a major and promising step in this direction. Their method consists of two components. The first is the calculation of a "natural" parity distribution, specific for a given population, which serves as a reference against which the observed parity distribution is compared. This discussion will not deal with the calculation of the natural distribution, despite its critical importance. My focus is on the second component, which compares the observed and natural distributions to generate an interval estimate of the proportion at each observed parity who have practiced control. David et al. produce "a pair of easily implemented formulas for upper- and lowerbound estimates of the expected proportion of the population ever controlling and the distribution of controllers by parity" (p. 163). They imply that their upper- and lower-bound estimates, "derived from the polar cases of pure stopping and pure spacing" (p. 179), embrace the full range of the proportion of controllers under any possible pattern of control. The authors stated that "they yield upper and lower bounds . . . even when the actual behavior of effective controllers in the target [observed] cohort does not conform to either of the polarcase specifications" (p. 172). The main example in David et al.'s article will be reanalyzed in this comment. It consists of urban women from the 1911 Census of Ireland who married at the ages of 2024 and at the time of the census had been married for 5-9 years. David et al. calculated that a maximum of 30.68% and a minimum of 20.57% of these women had effectively practiced some form of control. It is noteworthy that the minimum is so high in a cohort that was only married for 5-9 years and preceded modern contraception, and that the range between the lower and upper estimates is relatively small. Findings such as these have great significance for inferring the adoption of contraception in historical populations. I will show that in this example, and more generally, the range of controllers may be much wider than implied by David et al. Of particular importance, the minimum number may be much smaller. I shall suggest that their method is more sensitive to its assumptions than potential users might at first recognize. For notation, t( j) and n( j) are the observed and natural parity distributions, and T( j) and N( j) are the corresponding cumulative distributions, cumulated from the high to the low parities. Upper and lower estimates of the proportion of women at parity j who may be regarded as effective controllers are labelled cu( j) and CL( I); their sums across j are the overall limits Cu and CL. If one wished to estimate the maximum proportion of controllers without David et al.'s methodology, the simplest approach would be to assume that each controller had prevented the minimum number of births, that is, exactly one birth. Let C be the proportion of