Introduction
For population studies it is essential to estimate population growth or decline. For anthropological data sets this is rarely attempted, probably because the data quality is considered too limited. Nevertheless, the calculating these measures seems worthwhile, at least it should indicate whether the resulting values are unrealistically high or low.
Indices of reproduction
Mortality rate
The mortality rate m is calculated following C. Masset
and J.-P. Bocquet-Appel (1977) as a
percentage of the population, on the basis of the indices
and
.
A value of 1.0 means that 1 out of 100 individuals died per year. In
archaeological cases m is usually the same as the natality
rate n because the population is assumed to be
stationary.
schleswig <- life.table(schleswig_ma[c("a", "Dx")])
lt.reproduction(schleswig)[1,]## method value description
## 1 m 4.39 MortalityBirth and death rates according to Buikstra et al. 1986
Buikstra et al. (1986) proposed an alternative formula. They observed that the relation of those individuals aged 30 years and above to those aged 5 years and above is strongly correlated with the birth rate and, to a lesser extent, with the death rate. Despite the high correlations, Buikstra et al. decided against calculating direct birth and death rates but rather used the D30+/D5+-index to compare different archaeological populations.
Here, apart from the -ratio, the birth and death rates are calculated as well from the regression formulas supplied by Buikstra et al., knowing that their merit is questionable. The values are intended to estimate the number of births and deaths, respectively, per year per 100 individuals.
schleswig <- life.table(schleswig_ma[c("a", "Dx")])
lt.reproduction(schleswig)[c(1,8:10),]## method value description
## 1 m 4.39 Mortality
## 8 D30_D5 0.57 ratio D30+/D5+
## 9 BR 6.20 Birth rate
## 10 DR 3.18 Death rateFor the Schleswig data, the ratio of and is 0.57, which means that the number of individuals aged 5 and above nearly doubles that of those aged 30 and above. This indicates a moderate to high birth rate. Applying the formulas by Buikstra et al., the birth rate estimated is 6.2, nearly doubling the death rate of 3.2. The latter value is lower than the value of 4.4 computed by the formula of Masset and Bocquet-Appel.
odagsen <- life.table(list("corpus mandibulae" = odagsen_cm[c("a", "Dx")],"margo orbitalis" = odagsen_mo[c("a", "Dx")]))
lt.reproduction(odagsen)## $`corpus mandibulae`
## method value description
## 1 m 3.37 Mortality
## 2 dep 62.32 Dependency ratio
## 3 TFR 6.46 Total fertility rate
## 4 GRR 3.15 Gross reproduction rate
## 5 NRR 1.82 Net reproduction rate
## 6 r 2.98 Rate of natural increase
## 7 Dt 23.24 Doubling time in years
## 8 D30_D5 0.69 ratio D30+/D5+
## 9 BR 4.74 Birth rate
## 10 DR 2.51 Death rate
##
## $`margo orbitalis`
## method value description
## 1 m 2.42 Mortality
## 2 dep 34.05 Dependency ratio
## 3 TFR 5.83 Total fertility rate
## 4 GRR 2.84 Gross reproduction rate
## 5 NRR 1.45 Net reproduction rate
## 6 r 1.87 Rate of natural increase
## 7 Dt 37.09 Doubling time in years
## 8 D30_D5 0.69 ratio D30+/D5+
## 9 BR 4.77 Birth rate
## 10 DR 2.52 Death rateThe case of Odagsen is particularly informative, as age and sex were estimated using two different skeletal elements ( and ). The other reproduction indices differ between the two data sets, but the -ratios are nearly identical at 0.69. This may indicate the robustness of this index. The value itself indicates a higher percentage of older individuals and thus a lower birth rate than we have seen for Schleswig. The birth rate is estimated at 4.7% and the death rate at 2.5%.
Reproduction rates
There are several approaches to calculating reproduction rates (e. ´g. Henneberg 1976). We largely follow the methodology by F. A. Hassan (1981). Typically, a Total fertility rate (TFR) of 6–8 is assumed for prehistoric populations (Acsadi and Nemeskeri 1970, 177; Henneberg 1976; Hassan 1981, 130). That means that a woman living at least to her climacteric period is expected to have given birth to an average of 6 to 8 children.
Recently, C. McFadden and M. F. Oxenham (2018b) have published a formula to estimate the Total fertility rate from archaeological data, provided that infants are fully represented in the archaeological record. Unfortunately, this will not be the case for most archaeological data sets. Therefore, we used data published by McFadden and Oxenham to apply it to the index after J.-P. Bocquet-Appel (2002). We approximated the ratio by three different methods of fitting (linear, logistic, power) and recommend logistic fitting as the default, but the others are available as well.
More recently, B. Taylor, M. Oxenham and C. McFadden (2023) have presented a formula for the Total fertility rate for cases where pre-adult individuals are known to be underrepresented.
The Gross reproduction rate (GRR) is calculated by multiplying the TFR with the ratio of female newborns, assumed to be a constant of 48.8% of all children (Hassan 1981, 136). The Net reproduction rate is calculated by summing the product of the GRR, the age specific fertility rate calculated from the data given by Hassan (1981, 137 tab. 8.7) and the age specific survival taken from the life table and dividing the result by 10,000.
The Rate of natural increase or Intrinsic growth rate r
(growth in per cent per year) can be computed from the fertility
calculations following Hassan (1981, 140).
An alternative way to calculate the intrinsic growth rate has recently
been described by C. McFadden and M. F. Oxenham (2018a). They present a regression calculation
based on the index
also used for fertility calculations (see above) in connection with
modern data. It is not surprising that even with the
McFadden/Oxenham-index used for the fertility rate, the actual numbers
for the computed Intrinsic rate of growth and the Rate of Natural
Increase can differ substantially, as with the former formula, further
life table data is taken into account.
Depending on the values chosen for the Total fertility rate and the
Intrinsic growth rate, the Doubling time Dt will also vary.
This can be found in any textbook as follows:
In the following, the data set of the medieval cemetery of Schleswig is used again to demonstrate the different outcome of the varying options.
lt.reproduction(schleswig, fertility_rate = "McFO", growth_rate = "fertility")[c(-1,-2),]## method value description
## 3 TFR 4.85 Total fertility rate
## 4 GRR 2.37 Gross reproduction rate
## 5 NRR 1.17 Net reproduction rate
## 6 r 0.79 Rate of natural increase
## 7 Dt 87.77 Doubling time in years
## 8 D30_D5 0.57 ratio D30+/D5+
## 9 BR 6.20 Birth rate
## 10 DR 3.18 Death rateThe McFadden/Oxenham-index gives a Total fertility rate of 4.85 which corresponds to approximately 5 children per woman. Of these children 48% will be female which leads to a Gross reproduction rate of 2.37. Taking into account the years lived by women in Schleswig during their reproductive period, the Net reproduction rate would be 1.17, meaning that on average every woman – regardless of their time of death – would have had 1.17 daughters who themselves reproduced. This translates to a rate of natural increase of 0.79%: Every year the Schleswig population would have increased by roughly 0.8% and under stable growth conditions, would have doubled every 88 years.
lt.reproduction(schleswig, fertility_rate = "TOMc", growth_rate = "fertility")[c(-1,-2),]## method value description
## 3 TFR 6.92 Total fertility rate
## 4 GRR 3.38 Gross reproduction rate
## 5 NRR 1.67 Net reproduction rate
## 6 r 2.56 Rate of natural increase
## 7 Dt 27.04 Doubling time in years
## 8 D30_D5 0.57 ratio D30+/D5+
## 9 BR 6.20 Birth rate
## 10 DR 3.18 Death rateThe index of Taylor et al., which concentrates on the adult population only (age 15 and above), arrives at a much higher estimate for the Total fertility rate. This is computed with 6.92, and therefore also the Gross reproduction rate, the Net reproduction rate and the Rate of natural increase are higher than with the McFadden/Oxenham-index. Conversely, the doubling time is much lower, at 27.04 years.
lt.reproduction(lt.correction(schleswig)$life_table_corr, fertility_rate = "McFO", growth_rate = "fertility")[c(-1,-2),]## method value description
## 3 TFR 6.53 Total fertility rate
## 4 GRR 3.19 Gross reproduction rate
## 5 NRR 1.06 Net reproduction rate
## 6 r 0.28 Rate of natural increase
## 7 Dt 247.36 Doubling time in years
## 8 D30_D5 0.38 ratio D30+/D5+
## 9 BR 8.34 Birth rate
## 10 DR 4.17 Death rateAs stated above, the McFadden/Oxenham-index assumes – somehow
unrealistically – that the youngest age-groups are fully represented in
the skeletal population. If we apply the life table corrections as
advised by C. Masset and J.-P. Bocquet-Appel via the function
lt.correction (see vignette “Life table corrections”), the
values change dramatically. The Total fertility and Gross reproduction
rates increase, but because of the much higher mortality among the
youngest, the Net reproduction rate and thus the Rate of natural
increase shrink. The doubling time is extended to nearly 250 years.
lt.reproduction(schleswig, fertility_rate = "BA_log", growth_rate = "fertility")[c(-1,-2),]## method value description
## 3 TFR 7.01 Total fertility rate
## 4 GRR 3.42 Gross reproduction rate
## 5 NRR 1.69 Net reproduction rate
## 6 r 2.63 Rate of natural increase
## 7 Dt 26.39 Doubling time in years
## 8 D30_D5 0.57 ratio D30+/D5+
## 9 BR 6.20 Birth rate
## 10 DR 3.18 Death rateIn contrast, the log-regression derived from the -index by Bocquet-Appel arrives at a Total fertility rate of around 7, also in line with the expectation (see above). This higher value, not surprisingly, leads to much higher numbers in all the other indices, including a much higher Rate of natural increase and a much shorter doubling time of only 26 years.
lt.reproduction(lt.correction(schleswig)$life_table_corr, fertility_rate = "BA_log", growth_rate = "fertility")[c(-1,-2),]## method value description
## 3 TFR 7.01 Total fertility rate
## 4 GRR 3.42 Gross reproduction rate
## 5 NRR 1.13 Net reproduction rate
## 6 r 0.63 Rate of natural increase
## 7 Dt 109.76 Doubling time in years
## 8 D30_D5 0.38 ratio D30+/D5+
## 9 BR 8.34 Birth rate
## 10 DR 4.17 Death rateBecause the -index does not depend on the age-group 0–4 years, the Total fertility and Gross reproduction rates do not change if the life table correction is applied. Still, the other values are affected, and the Net reproduction rate decreases to 1.13. The Rate of natural increase is then 0.63 and the doubling time 110 years.
lt.reproduction(schleswig, growth_rate = "MBA")[c(6,7),]## method value description
## 6 r 0.00 Rate of natural increase
## 7 Dt 26163.39 Doubling time in yearsIf we skip the fertility computation and consider the growth rate devised by Masset and Bocquet-Appel, the picture changes again: Now the Rate of natural increase is virtually zero, the population is stationary and the doubling time is more than 25000 years.
lt.reproduction(schleswig, growth_rate = "McFO")[c(6,7),]## method value description
## 6 r 1.81 Rate of natural increase
## 7 Dt 38.27 Doubling time in yearsThe McFadden/Oxenham-index for the Rate of natural increase lies between these extremes. It gives a value of 1.81 and thus a doubling time of 38 years.
In sum, the differing outcomes underline the importance of fertility and mortality in younger age groups for the growth direction of a given population. Following Hassan (1981, 140), one would credit the McFO-fertility index with lifetable correction after Masset and Bocquet-Appel to produce the most plausible result as Hassan deems a growth rate of 0.5% per year as the maximum for prehistoric populations. On the other hand, also growth rates of 2.7% per year are known from ethnographic cases (Hassan 1981, 142).
Rate of dependent individuals
The ratio of dependent individuals (Hassan 1981, 147) is usually – but probably erroneously for archaic societies Grupe et al. (2015), 423 – assumed to apply to those aged below 15 or 60 and above, and compared to individuals aged 15–-59:
schleswig <- life.table(schleswig_ma[c("a", "Dx")])
lt.reproduction(schleswig)[2,]## method value description
## 2 dep 105.83 Dependency ratioThe example data set of the Schleswig yields a dependency ratio of
around 106. That would mean that in Schleswig there would have been
roughly as many dependent individuals as those available to support
them. This number even doubles if we apply the life table corrections
devised by C. Masset and J.-P. Bocquet-Appel via the function
lt.correction (see vignette “Life table corrections”).
lt.reproduction(lt.correction(schleswig)$life_table_corr)[2,]## method value description
## 2 dep 206.75 Dependency ratioRelation between female and male individuals
The ratio of adult males to females (= Masculinity index) is calculated for individuals aged 15 and older:
This index is informative for several reasons (Herrmann et al. 1990, 310):
It can point to basic problems in the data sets in that, say, one sex is grossly over- or underrepresented.
It may hint towards cultural reasons like sex-specific mobility.
Note that with a higher mortality rate of adult females, an MI < 1 does not necessarily indicate imbalance in the living population.
As example, we choose the Early Neolithic cemetery of Nitra in Slovakia.
nitra_prep <- prep.life.table(nitra, group="sex", agebeg = "age_start", ageend = "age_end")
nitra_life <- life.table(nitra_prep)
lt.sexrelation(nitra_life$f, nitra_life$m)[1,]## method value description
## 1 MI 0.67 Masculinity indexThe first row of the function lt.sexrelation reports the
Masculinity index. With 0.67 it is quite low and points to a surplus of
female individuals. However, as pointed out above, this must be
interpreted with caution as it could also be due to a higher mortality
of female adults.
Maternal mortality
Maternal mortality rate (MMR) is a key indicator of the health system of a given population. Maternal mortality is defined as dying during pregnancy or within the first 42 days after birth due to complications. Recently, C. McFadden and M. F. Oxenham (2019) have provided a formula to calculate it from archaeological data. They show that with modern data a very high correlation is achieved by only comparing the absolute numbers of the age group 20 to 24. This has the additional advantage that for this age group anthropological aging methods are reasonably exact. Recently, McFadden et al. (2020) have presented a stabilized version of this formula to account for highly skewed ratios in terms of the relation between male and female individuals.
lt.sexrelation(nitra_life$f, nitra_life$m)[c(-1),]## method value description
## 2 Ratio_F_M 2.36 Ratio of females to males aged 20--24
## 3 MMR1 449.00 Maternal mortality per 100,000 births
## 4 MMR2 4.49 Maternal mortality per 1,000 birthsIn Nitra, more than twice the number of females of the age 20–24 have been found than males of the same age. This gives a unstabilized Maternal mortality rate of 712 (not shown). However, because the ratio of female and male individuals is so unbalanced (Masculinity index = 0.67, see above), the stabilized version used here calculates an actual rate of 449 per 100,000 births or 4.49 per 1,000 births. In percentage, this number corresponds to 0.45% maternal deaths per birth.
McFadden and Oxenham based their formula on modern data which, of course, allow the most precise calculations. However, for early modern census data, much higher Maternal mortality rates are estimated (Schofield 1986), easily topping 1,000 and more. For these numbers, the formula by McFadden and Oxenham does not work, as their sample data sets do not exceed 700. This is a problem which still needs more in-depth research.
Therefore, Maternal mortality rates computed for prehistoric populations should be compared to modern data with great caution. Regardless if the actual regression calculation is in need of adjustment or not, the McFadden/Oxenham-formula still seems capable to allow comparisons between prehistoric populations.
Population size
The estimation of the population size for a given cemetery is only possible if a stationary population is assumed. In this case, the number of deaths is simply multiplied with the life expectancy at birth and divided by the time span in years the cemetery was in use. Additionally, it is assumed that an unknown number of individuals is not represented in the cemetery and, therefore, the result is multiplied by an adjustment factor k (Herrmann et al. 1990). By default, this is assumed to be 1.1. Still, in many cases, the actual numbers calculated seem too small to yield viable population sizes.
The simple formula is as follows:
Here, we use all individuals from Nitra to calculate the size of the population burying their dead here.
The median of the span of a phase-model (OxCal 4.4.) of the available 14C-dates (Griffiths 2013) indicates that the cemetery was used for 147 years. However, the actual 95.4%-span is 0–260 years, and a strong wiggle in-between probably distorts the picture. Therefore, 80 years could be a more realistic value.
lt.population_size(nitra_life$All,t = 147)## method value description
## 1 D 75.0 Number of deaths
## 2 e0 28.5 Life expectancy
## 3 k 1.1 Correction factor
## 4 t 147.0 Time span
## 5 P 16.0 Population size
lt.population_size(nitra_life$All,t = 80)## method value description
## 1 D 75.0 Number of deaths
## 2 e0 28.5 Life expectancy
## 3 k 1.1 Correction factor
## 4 t 80.0 Time span
## 5 P 29.4 Population sizeIn the first case, the Nitra population is estimated at only 16 individuals. In the second, the population burying in this cemetery would have comprised about 29 individuals at any given time during its existence.