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Is NRR Time-Sensitive in Measuring Population Replacement Level by Arni S.R. Srinivasa Rao

By Emily Merchant posted 01-26-2021 18:56

By Arni S.R. Srinivasa Rao, Professor and Director, Laboratory for Theory and Mathematical Modeling, Division of Infectious Diseases, Medical College of Georgia.

This is a non-technical summary - download the full article here.

The NRR is a frequently computed reproductive measure in demography that is often used for evaluating replacement-level fertility in a population. Alfred Lotka and Robert Kuczynski popularized it in the United States while understanding population growths and replacement levels [1,2].

This technical piece introduces a concise debate on the implications of NRR (net reproduction rate), given that NRR only represents what we think it represents in a stable population, and no human population in practice is stable. Click on the supplementary file for fuller technical details. The quantity NRR = 1 computed for any given year should not be treated as representing the attainment of replacement fertility in that year, because NRR = 1 represents replacement only in a stable population. A new measure is proposed through this opinion piece. Finally, the comment concludes that we either need a different measure for what we want or a different way of interpreting NRR. The standard formula for NRR can be found in several textbooks, for example, see [3]. When NRR = 1 for a year, we often say that the population has attained a replacement level of fertility in that year. However, such a statement is only true when the population is stable. 

In a population that is not stable, NRR provides a futuristic population replacement value. NRR is therefore not sensitive enough to evaluate the population replacement levels in the year in which the data on the number of total children born to mothers of various ages is collected. Through technical details, this article provides arguments on how the futuristic value described through standard NRR formulae creates uncertainty on the time to replace a woman with a girl child, and certainly, the replacement of the population is not happening in the year in which children born data is collected.  

We, therefore, need a new measure. This article provides a formula (called Q0) to understand retrospective replacement fertility. The structure of Q0 expressed in the article is ideologically different from Lotka’s or Kuczynski’s principles of population growth using life tables. The quantity Q0 can be computed only retrospectively and its value does not indicate the average number of women replaced by girl child in the year in which the above said data was collected. A hypothetical numerical example was demonstrated to compute Q0 in Table 1. 

The article ends by providing two more formulae for computing replacement fertility, called Q`0 and Q``0. The quantity Q`0 uses the data on future births of women in the reproductive ages and the quantity Q``0 uses the data on women who have completed the childbearing in the reference year. There are several ways to address the uncertainty explained in this note. One could also obtain certain metrics to study differences between various proposed measures under different stability conditions as in [4]. All such discussions are out of the scope of the current technical opinion piece.


  1. Dublin L.I. and Lotka A.J. (1925). On the True Rate of Natural Increase. Journal of the American Statistical Association, Sep., 1925, Vol. 20, No. 151, pp. 305-339
  2. Kuczynski R.R. (1932). Fertility and Reproduction: Methods of Measuring the Balance of Births and Deaths. Falcon Press, New York.
  3. Preston S, Heuveline P, Guillot M (2000). Demography: Measuring and Modeling Population Processes 1st Edition Wiley-Blackwell.
  4. Rao, A.S.R.S. (2014). Population stability and momentum. Notices of American Mathematical Society. 61 (2014), no. 9, 1062–1065.