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The authors have declared that no competing interests exist.

Conceived and designed the experiments: MH. Performed the experiments: MH JB. Analyzed the data: MH JB WB. Contributed reagents/materials/analysis tools: MH JB WB. Wrote the paper: MH JB.

Indirect estimation methodologies of the total fertility rate (TFR) have a long history within demography and have provided important techniques applied demographers can use when data is sparse or lacking. However new methodologies for approximating the total fertility rate have not been proposed in nearly 30 years. This study presents a novel method for indirectly approximating the total fertility rate using an algebraic rearrangement of the general fertility rate (GFR) through the known relationship between GFR and TFR. It then compares the proposed method to the well-known Bogue-Palmore method. These methods are compared in 196 countries and include overall errors as well as characteristics of the countries that contribute to fertility behavior. Additionally, these methods were compared geographically to find any geographical patterns. We find this novel method is not only simpler than the Bogue-Palmore method, requiring fewer data inputs, but also has reduced algebraic and absolute errors when compared with the Bogue-Palmore method and specifically outperforms the Bogue-Palmore method in developing countries. We find that our novel method may be useful estimation procedure for demographers.

Estimates of population fertility characteristics are of critical importance for understanding short-term shifts in population age-structure and related growth dynamics

Regression-based methods, such as the Bogue-Palmore procedure, rely upon the powerful least-squares criteria to predict TFR in light of symptomatic indicators

In spite of known relationships between child/woman ratios and gross reproductive rate

The method described here relies upon known relationships between TFR and the General Fertility Rate

Where

_{n}F_{x} is equal to the ratio of births to the population at risk for giving birth:

The relationship of TFR to the General Fertility Rate (GFR) is well known. The GFR is calculated as the ratio of births observed among women of childbearing ages:

Where B_{t} is the total number of births and _{40}W_{10} is the number of women aged 10 to 50. We can rewrite the GFR equation to be in the same notation as _{n}F_{x},

Which restates the GFR in a form where one may insert the GFR function into the TFR function and just multiply by the width of the interval to approximate the Total Fertility Rate. Summing across age intervals, as is the case with TFR, is unnecessary since the age intervals are simplified into one single age interval. Mathematically, the sum of the ratios (TFR) is different from the ratio of the sums (GFR), but empirically the results should be and are very similar

Age-specific Fertility Rates are commonly calculated in one or five year intervals, denoted as

_{n}F_{x} for a five-year interval is related to the sum of its individual age groups and thus the age-specific fertility rate can be approximated as both the sum of the ratios and as the ratio of the sums.

Age | Population | Births | ASFR |

50,505 | 1,601 | 0.032 | |

51,325 | 2,073 | 0.04 | |

51,796 | 2,480 | 0.048 | |

51,803 | 2,916 | 0.056 | |

52,582 | 3,391 | 0.064 | |

258,010 | 12,461 | 0.241 | |

0.241 |

It should be noted that Bogue and Palmore ^{2} = 0.992 ^{2} = 0.943 for 1970

With certain strong assumptions including no infant mortality and no migration in the previous five-years, the child/woman ratio may be substituted into the term representing the General Fertility in

Where

This derivation we have dubbed the implied Total Fertility Rate (iTFR) due to it reflecting the implied total fertility rate present in the age structure of the population. It suggests that, subject to the aforementioned assumptions, the child/woman ratio may be used directly to make an approximation of TFR. The iTFR is nothing more than an extension of the algebra already used to construct grouped rates extrapolated out to include the entire fertility interval. Instead of 40 1-year ASFRs, or 8 5-year ASFRs, or 4 10-year ASFRs, we are simply calculating 1 40-year ASFR. And since it accommodates the entire fertility interval (menarche to menopause), 1 40-year ASFR essentially is the TFR, simply constructed as a single interval instead of multiple intervals with the assumption that ASFRs are constant over time, keeping in the tradition of Brass’s P/F ratio

The Hamilton-Perry method

The Hamilton-Perry method essentially projects child populations through the use of a fertility wide (menarche to menopause) ratio of children to women.

Given the strong observed relationship between GFR and TFR

Data on populations at the country level, including population counts used to estimate the child/woman ratio, levels of infant mortality, the proportion of women married, and estimates of TFR were taken from the United Nations Population Division for the year 2000

The method provides estimates of B_{0}, B_{1}, etc. that were fitted using standard least-squares criteria

In evaluating the accuracy of population estimates, applied demographers typically apply

The darker the country, the larger the absolute percent error. Countries in gray could not be calculated due to data limitations.

The darker the country, the larger the absolute percent error. Countries in gray could not be calculated due to data limitations.

Child Woman Ratio Method | Bogue-Palmore Method | ||||||

n | MAPE | MALPE | RMSE | Percentage PointImprovement (Average) | MAPE | MALPE | RMSE |

196 | 5.71% | 1.89% | 7.35% | 3.89 | 9.60% | 0.92% | 27.11% |

Number of Countries With Highest Performance | Average of DHS Estimates | |||||

n | Population Characteristic | CW | BP | CW | BP | p-value of t-test |

150 | Total Fertility Rate | 95 | 55 | 2.87 | 2.41 | |

43 | House has Electricity | 29 | 14 | 40.59% | 72.86% | |

39 | Percent Literate | 26 | 13 | 57.35% | 75.82% | 0.081 |

48 | Percent Births in Health Facility (3 years prior) | 28 | 20 | 56.33% | 63.39% | 0.358 |

35 | Percent of Women Using Contraceptives | 23 | 12 | 49.50% | 63.29% | |

41 | Percent of Women Never Married | 26 | 15 | 27.37% | 29.31% | 0.199 |

47 | Infant Mortality Rate Previous 5 Years | 28 | 19 | 35/10,000 | 20/10,000 | |

145 | Infant Mortality Rate (CIA Factbook) | 95 | 55 | 28/10,000 | 29/10,000 | 0.3069 |

The performance of each method appears to depend on a number of the proximate determinants of fertility (

The results of this study suggest that when compared to the Bogue-Palmore method, dramatic improvements in the accuracy of estimates of TFR may be made by employing a simple method based only on the child/woman ratio. The method assumes no migration among women of child-bearing age and their dependent young aged 0 to 4 years, and insignificant infant mortality. These results are surprising given the strong assumptions present in the method and may speak to the value of locally-tailored demographic estimates instead of estimates that minimize error over large, diverse, settings that may have little in common with one another. A lack of consideration of locally-specific error measures in standard least-squares estimation algorithms

In spite of these encouraging results, it is worth noting that the use of child/woman ratio for approximating TFR is subject to some important limitations that demographers should consider when implementing the method. It is obvious that successful use of the method is dependent upon accurate census enumeration data. Unfortunately, it is known that enumerations of children aged 0–4 years are biased by age-heaping

Furthermore, the evaluation made in this paper depended on two forms of uncertain or omitted data in its comparisons. First, symptomatic indicator data were not available for all countries, limiting the ability to compute Bogue-Palmore estimates in all nations for which TFR estimates were available. This would pose (at least conceptual) limitations on the results presented here in two ways. First, the estimation of average errors associated with the Bogue-Palmore estimates could be directionally-biased by this lack of availability if this availability and the potential performance of the method were systematically related. In this sense, the performance of the Bogue-Palmore estimates could be either better or worse if data were available to make these estimates. In other words, the performance observed here is a function of both the appropriateness of the original regression model for the countries analyzed as well as the limitations of the sample of countries for which the method could be used. The greater availability of the simple data used in the Child/Woman ratio method speaks to its practical utility and certain advantages over the Bogue-Palmore method; however, the omission of n = 46 countries from the analysis is not to be underappreciated. While the average errors associated with the Child/Woman ratio method is likely robust, the errors associated with the Bogue-Palmore method cannot be certainly characterized as such.

In addition to these limitations, it should also be noted that both methods were used to predict a TFR that was itself estimated by the United Nations. It is largely agreed within the demographic community that the UN estimates are likely the best available estimates, but it should be remembered that these estimates themselves are subject to remediations for incomplete data

In spite of these limitations, the successful application of a very simple method of approximating TFR using the child/woman ratio in conjunction with simple algebraic rearrangement is highly promising. If one may recreate the UN-based estimates using such simple procedures it suggests that applied demographers working with limited resources may make very successful use of this simple technique in creating quality estimates of an important demographic indicator that is used to summarize fertility patterns in a meaningful way, similar to the way that life-expectancy provides summarized information on mortality. The method may also be locally-tailored to specific circumstances where vital-records data are not collected at all and such local-tailoring may even suggest the successful extension of the method to sub-national levels where the Bogue-Palmore approach has been problematic

^{nd}edition. New York: Springer.

^{th}edition. New York: McGraw-Hill.