---
title: "Vignette >Reproduction<"
author: "Nils Müller-Scheeßel"
date: "December 2020"
output: rmarkdown::html_vignette
bibliography: ../inst/REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{Vignette >Reproduction<}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

## Load libraries

```{r, message=FALSE}
library(mortAAR)
library(magrittr)
```

## Introduction

For population studies it is of vital importance to estimate growth or decline of a population. For anthropological datasets this is rarely attempted, probably because the data quality seems to scanty. Nevertheless, the calculation of such measures seems worth the try, at least it should give an impression if the resulting values are unrealistic high or low.


## Indices of reproduction

### Mortality rate

The mortality rate `m` is calculated after C. Masset and J.-P. Bocquet-Appel [-@masset_bocquet_1977] in per cent of a given population, on the basis of the indices $D_{5-14}$ and $D_{20+}$.

$m = 12.7 * \frac{D_{5-14}}{D_{20+}}$

A value of 1.0, for example, means that 1 among 100 individuals died per year. In archaeological cases `m` is usually the same as the natality rate `n` because it has to be assumed that the population is stationary.

```{r}
schleswig <- life.table(schleswig_ma[c("a", "Dx")])
lt.reproduction(schleswig)[1,]
```


### Birth and death rates according to Buikstra et al. 1986

Buikstra et al. [-@buikstra_et_al_1986] have offered a different formula. They made the interesting observation that the relation of those individuals aged 30 years and above to those aged 5 years and above is very closely related to the birth rate and also closely (but less significantly) related to 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 $\frac{D_{30+}}{D_{5+}}$-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 supposed to specify the number of births and deaths, respectively, per year and per 100 individuals.
$BR = -11.493 * \frac{D_{30+}}{D_{5+}} + 12.712$
$DR = -5.287 * \frac{D_{30+}}{D_{5+}} + 6.179$

```{r}
schleswig <- life.table(schleswig_ma[c("a", "Dx")])
lt.reproduction(schleswig)[c(1,8:10),]
```

For the Schleswig data, the ratio of $D_{5+}$ and $D_{30+}$ is 0.57, which means that the number of individuals aged 5 and above nearly doubles that of those aged 30 and above. This speaks for a moderate to high birth rate. Applying the formulas by Buikstra et al., the birth rate would be 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.

```{r}
odagsen <- life.table(list("corpus mandibulae" = odagsen_cm[c("a", "Dx")],"margo orbitalis" = odagsen_mo[c("a", "Dx")]))
lt.reproduction(odagsen)
```

Interesting is the case of Odagsen, where age and sex were determined on two different parts of the skeleton (margo orbitalis and corpus mandibulae). The other reproduction indices differ between the two data-sets, but the $\frac{D_{30+}}{D_{5+}}$-ratios are nearly identical at 0.69. This might be seen as speaking for 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 computed to have been 4.7%. and the death rate 2.5%.


### Reproduction rates

There are different approaches to calculate reproduction rates [e. g. @henneberg_1976]. We largely follow the methodology by F. A. Hassan [-@hassan_1981]. Typically, a Total fertility rate (TFR) of 6--8 is assumed for prehistoric populations [@acsadi_history_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 on average 6 to 8 children.

Recently, C. McFadden and M. F. Oxenham [-@mcfadden_oxenham_2018a] have published a formula to estimate the Total fertility rate from archaeological data, provided that infants are represented fully in the archaeological record. Unfortunately, this will not be the case for most archaeological datasets. Therefore, we used the data published by McFadden and Oxenham to apply it to the $P_{5-19}$-index after J.-P. Bocquet-Appel [-@bocquet_appel_2002]. We approximated the ratio by three different methods of fitting (linear, logistic, power) and recommend logistic fitting, but the others are available as well.

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 arrived at by summing the product of the GRR, the age specific fertility rate calculated from the data given by Hassan [-@hassan_1981, 137 tab. 8.7] and the age specific survival taken from the life table and dividing the result by 10000.

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 [-@hassan_1981, 140]. An alternative way to calculate the intrinsic growth rate has recently described by C. McFadden and M. F. Oxenham [-@mcfadden_oxenham_2018b]. They present a regression calculation based on the index $D_{0-15}/D$ also used for fertility calculations (see above) in connections with modern data. It must not astonish 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 highly diverge, 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:

$Dt = \frac{100 \;* \;ln(2)}{r}$

In the following, the dataset of the medieval cemetery of Schleswig is used again to demonstrate the different outcome of the varying options.

```{r}
lt.reproduction(schleswig, fertility_rate = "McFO", growth_rate = "fertility")[c(-1,-2),]
```

The McFadden/Oxenham-index gives a Total fertility rate of 4.85 which equals a total of around 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 lived years of the Schleswig woman during their reproductive years, 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 in turn would have had children. This translates to a Rate of natural increase of 0.79%: Every year the Schleswig population would have increased by roughly 0.8% and with stable growth would have doubled every 88 years.

```{r}
lt.reproduction(lt.correction(schleswig)$life_table_corr, fertility_rate = "McFO", growth_rate = "fertility")[c(-1,-2),]
```

As stated above, the 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.

```{r}
lt.reproduction(schleswig, fertility_rate = "BA_log", growth_rate = "fertility")[c(-1,-2),]
```

On contrast, the log-regression derived from the $P_{5-19}$-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.

```{r}
lt.reproduction(lt.correction(schleswig)$life_table_corr, fertility_rate = "BA_log", growth_rate = "fertility")[c(-1,-2),]
```

Because the $P_{5-19}$-index is not depended 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.

```{r}
lt.reproduction(schleswig, growth_rate = "MBA")[c(6,7),]
```

If 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 stationary and the doubling time more than 25000 years.

```{r}
lt.reproduction(schleswig, growth_rate = "McFO")[c(6,7),]
```

The McFadden/Oxenham-index for the Rate of natural increase lies somewhere in-between. It gives a value of 1.81 and thus a doubling time of 38 years.

In sum, the differing outcomes underline the importance of the fertility and the mortality among the youngest for the growth direction of a given population. Following Hassan [-@hassan_1981, 140], one would credit the corrected McFO-fertility rate to produce the most probable 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 this is put into relation of the individuals aged in-between:

$DR = \frac{\sum{D_{0-14}} + \sum{D_{60+}}}{\sum{D_{15--59}}}$

```{r}
schleswig <- life.table(schleswig_ma[c("a", "Dx")])
lt.reproduction(schleswig)[2,]
```

The example dataset of the Schleswig produces a Dependency ratio of around 106. That would mean that in Schleswig there would have been roughly as much 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").

```{r}
lt.reproduction(lt.correction(schleswig)$life_table_corr)[2,]
```


### Relation between female and male individuals

The proportional relation between adult males and females (= Masculinity index) is defined for juvenile and older individuals:

$MI = \frac{D_{\geq15 _{male}}}{D_{\geq15 _{female}}}$

It is interesting for a number of reasons [@herrmann_prahistorische_1990, 310]:

1. It can point to basic problems in the datasets in that, say, one sex is grossly over- or underrepresented.

2. 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 speak for an unbalanced MI in life.

As example, we choose the Early Neolithic cemetery of Nitra in Slovakia.

```{r}
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,]
```

The 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 has to be weighted carefully as it could also be due to a higher mortality of female adults.


## Maternal mortality

Maternal mortality rate (MMR) is a basic indicator for 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 [-@mcfadden_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 reasonable exact. Recently, McFadden et al. [-@mcfadden_et_al_2020] have presented a stabilized version of this formula to cater for highly biased ratios in terms of the relation between male and female individuals.

$Stabilized\; MMR = 333.33 * \frac{D_{20-24_{female}}}{D_{20-24_{male}}} * \frac{D_{15+_{male}}}{D_{15+_{female}}} - 76.07$

```{r}
lt.sexrelation(nitra_life$f, nitra_life$m)[c(-1),]
```

In 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 translates 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 only very carefully compared to modern data. 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 be 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 resulting number is multiplied by an arbitrary value k [@herrmann_prahistorische_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:

$P = \frac{D\; *\; e0\; *\; k}{t}$

This time, we use all individuals from Nitra to calculate the size of the population burying their dead here.

```{r message=FALSE, warning=FALSE, include=FALSE}
knitr::knit_engines$set(oxcal = function(options) {
 # no changes
})
```

```{oxcal}
 Sequence()
 {
  Boundary("A");
  Phase("Nitra-Gräber")
  {
   R_Date("OxA-24574", 6222, 37);
   R_Date("OxA-24575", 6226, 36);
   R_Date("OxA-24095", 6298, 33);
   R_Date("OxA-24576", 6196, 36);
   R_Date("OxA-24577", 6216, 36);
   R_Date("OxA-23793", 6221, 35);
   R_Date("OxA-24578", 6138, 34);
   R_Date("OxA-24579", 6317, 34);
   R_Date("OxA-24580", 6227, 35);
   R_Date("OxA-23794", 6219, 35);
   R_Date("OxA-24582", 6328, 36);
   First("Nitra-Beginn");
   Last("Nitra-Ende");
   Span("Nitra-Dauer");
  };
  Boundary("B");
 };
```

The median of the span of a phase-model (OxCal 4.4.) of the available <sup>14</sup>C-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.

```{r}
lt.population_size(nitra_life$All,t = 147)
lt.population_size(nitra_life$All,t = 80)
```

In the first case, the Nitra population would have had a size of 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.

## References
