An adaptive approach to the biometric analysis of infant mortality
Tommy Bengtsson and Göran Broström
2026-10-22
Abstract
The first year of life is not only the period when mortality is highest, but also the period when categories: endogenous and exogenous. While endogenous deaths are due to inherited factors is finally investigated. Bourgeois-Pichat model, endogenous and exogenous mortality, competing risks, seasonal mortality, probabilistic approach, Sweden.1 Introduction
Jean Bourgeois-Pichat argued that infant mortality should be divided into two categories: endogenous and exogenous deaths. Endogenous deaths are, by definition, due to inherited factors or acquired during gestation and delivery. They typically occur in the initial period of life, though sometimes later, and include congenital debility, prematurity, and malformation. Exogenous infant mortality, which Bourgeois-Pichat regards as accidental, is deaths for which society must hold itself responsible. To reduce endogenous mortality, medical intervention is essential (Bourgeois-Pichat, 1952).
It is difficult to distinguish between endogenous and exogenous mortality, either because they are difficult to diagnose or because information on causes of death is lacking. Bourgeois-Pichat therefore proposed a solution to the problem based on an idea of a universal law governing the distribution of deaths in the first year of life (Bourgeois-Pichat 1951a, 1951b; Bourgeouis-Pichat 1952). He assumed that all deaths taking place in the last eleven months of the first year were exogenous and argued that, although there are also some endogenous mortalities during this period, they do not affect the age-pattern of mortality much.
Supported by examples from mid-twentieth-century Western and Southern Europe, the US and Canada, New Zealand, and other countries, Bourgeois-Pichat showed that the cumulative infant deaths after the first month followed a linear development given a log-cube transformation of age. He argues that our disease resistance follows our gain in birth weight, which follows this distribution (Bourgeois-Pichat 1951b). As an example, we show the development of birth weight on this scale using male data from WHO in Figure 1.1.
Figure 1.1: Development of boys’ weight after birth (WHO) and the log-cube transformation shifted 3.41 units upwards.
Mortality in the first month of life is then divided into an endogenous and an exogenous component, assuming that the same linearity also exists in this period. It leads to the conclusion that exogenous deaths in the first 28 days of life represent 22.8 percent of the deaths from the 29th to the 365th day, or 18.6 percent of the total number of exogenous infant deaths.
2 Background
Today, 2.8 percent of all newborns die before reaching one year of age (UNICEF 2025). This is the result of a process that began in the twentieth century, except in the Western world, where it started much earlier. In Sweden, for example, it started around 1800 when 20 percent of all newborns died before age one, to decline to 0.2 percent today (Folkhälsomyndigheten 2026; SCB 1955). Mortality has, however, not declined evenly for children of all ages. Instead, it has occurred primarily after the first month of life. As a result, in England and Wales, post-neonatal mortality fell below neonatal mortality in the 1930s, just like in Sweden (Galley 2023; SCB 2020). This process has continued and, amazingly enough, today half of all deaths below age five take place in the first month of life (UNICEF 2025).
Since mortality at other ages has not been falling as rapidly, the decline in infant mortality has been a key driver of life expectancy, not only in Western countries but even more so in Asia and South America. During the second half of the 20th century, they experienced an increase in period life expectancy of one year per calendar year, compared with about three months per calendar year in Western countries (Oeppen and Vaupel 2002).
The decline in infant mortality is the result of a complex process. Epidemics like smallpox and whooping cough became less common even before vaccines were introduced, and the same held for later declines in measles, chickenpox, and tuberculosis (Bengtsson 2015). Vaccinations then helped push these diseases down further and keep them from coming back. When knowledge about the spread of diseases developed in the second part of the nineteenth century and investments in public health were made, infant mortality further declined, and so did the education of midwives.
The first year of life is not only the period when mortality is highest, but also the period when improvements should be possible. It is important to know the causes so that the right measures can be taken both at birth and during the first year of life. Unfortunately, access to causes of death is often poor, if available at all. Usually, only easily diagnosed infectious diseases are recorded. In other cases, “unknown childhood disease” is often listed, even in a country like Sweden, which has cause-specific mortality data back to 1749.
3 Construction of the B-P plot
(Bourgeois-Pichat 1951a) described a graphical method to both separate endogenous from exogenous mortality and at the same time get estimates of both. Central in the Bourgeouis-Pichat (B-P) construction is the time transformation \(g\):
\[\begin{equation}\label{eq:g} g(t) = C log^3(1 + t), \quad 0 \le t \le 365, \end{equation}\]
where \(t\) is age measured in days. Its graph is shown in Figure 3.1.
The constant \(C\) is chosen so that \(g(365) = 365\). Note that \(C\) is not part of the original definition of the log-cube transform but is provided here only to make graphical comparisons easier to interpret. It makes no difference otherwise.
Figure 3.1: The log-cube transform \(g\).
Three postulates govern Bourgeois-Pichat’s procedure.
The exogenous mortality follows a uniform distribution on the log-cube transformed time scale.
The post-neonatal mortality is strictly exogenous.
The exogenous and endogenous death processes act independently of each other.
A simple example shows how the B-P graphical construction works. Assume , of which infancy, and ten death ages Day:
\(Day = c(2, 6, 10, 15, 25, 41, 90, 120, 180, 309)\).
Calculate \(Rate\): \(Rate = c(\frac{1}{100}, \frac{2}{100}, \frac{3}{100}, \ldots, \frac{10}{100}).\)
In a graph with time scale transformed by the log-cube transform, plot Rate against Day, see Figure 3.2.
Figure 3.2: The two steps in the construction of the B-P plot (log-cube time scale).
Mark two points corresponding to (i) the smallest and (ii) the largest death age in the post-neonatal period (left panel). Next, draw a straight line from \((309, 0.1)\) through (41, 0.06) to the \(y\) axis, that is, \(Day = 0\), and a horizontal line from \((309, 0.1)\) to \(Day = 0\) (right panel).
Now, in the right panel of Figure 3.2 we can read total mortality as \(0.1\) and endogenous mortality as \(0.045\), and by subtraction we get the exogenous mortality equal to \(0.1 - 0.045 = 0.055\). Roughly one fifth of the total exogenous mortality will occur during the neonatal period by design. So during the neonatal period, it is expected to have roughly four times more endogenous than exogenous deaths.
The exact proportion of exogenous deaths in the neonatal period is \(g(30) / 365 \approx\) 20 per cent. So, given that the total number of neonatal deaths is \(D_1\) and the total number of postneonatal deaths is \(D_2\), the estimated number \(R\) of endogenous deaths is given by
\[\begin{equation} R = D_1 - \frac{D_2}{4} \end{equation}\]
Very simple, no fancy plots needed!
Note that this procedure requires data with no left-truncation and no internal right censoring. How to handle these exceptions is showed next.
3.1 The B-P plot in a probabilistic environment
The first two postulates imply together that a left-truncated sample of postneonatal deaths follows a uniform distribution on the log-cube time scale, and the straight line drawn in the diagram is a simple attemt to estimate the underlying cumulative density function (CDF), and not the cumulative hazard function (CHF), mistakenly suggested by Lynch et al. (1998).
However, that slip in interpretation is understandable, because it is natural to speak of the CHF rather than the CDF in this context. However, let us first describe the maximum likelihood approach to find the best linear fit in the B-P model.
Let \(D\) be the total number of observed postneonatal deaths, and let \((N - D)\) be the total number of infants surviving one year of age. On the log-cube scale with the neonatal period starting after age 30 days, the slope \(\alpha\) of the line is estimated by
\[\begin{equation} \hat{\alpha} = \Biggl\{365 + \frac{(365 - g(30)) (N - D)}{D}\Biggr\}^{-1} \end{equation}\]
Let us try it on our toy example, see Figure 3.3.
Figure 3.3: The B-P plot with ML estimate of the exogenous mortality.
This result does not differ much from the original (Figure 3.2, right hand panel).
But, as stated above, no fancy plots are needed, just use Equation (2)!
3.2 Historical comments on Bourgeois-Pichat’s method
A study based on data from 1889 to 1891 for three rural districts with low infant mortality and three cities in England with high infant mortality, where causes of death are available, broadly supports Bourgeois-Pichat’s thesis (Galley and Woods 1999). The deaths from ‘Convulsions,’ ‘Atrophy,’ and ‘All other causes,’ which were likely endogenous, occur primarily during the first month of life. Another important finding of this study is that changes in the distribution or virulence of individual diseases, such as smallpox and scarlet fever, have the potential to alter the linear pattern of cumulative infant mortality on the log-cube scale.
Bourgeois-Pichat found deviations from the linear pattern as well, for example, for Quebec 1944–1947, where the curve bends downward quite strongly after the sixth month (Bourgeouis-Pichat 1952). Downward bends have also been found using Swedish 18th- and 19th-century parish data (Bengtsson 1999; Bengtsson and Lundh 1999; Lynch et al. 1998; Sundin and Tedebrand 1981). National data for Sweden 1910–-1946 show, however, no such downward bend but a straight line on the log-cube scale (Bourgeois-Pichat 1951b; Bourgeouis-Pichat 1952).
In other places, the curve bends upward instead. For example, the curve for Sardinia, in 1948, bends strongly upward already after four months (Bourgeouis-Pichat 1952). Knodel and Kintner (1977) show ample examples of upward-bending curves and argue that this form is due to increasing mortality after stopping breastfeeding. Taken together, even though cumulative infant mortality is commonly found to be linear on the log cube age scale, there are many examples of deviations, both upward- and downward-bending patterns, not only back in time but also for the twentieth century.
A study of Italian regions during the period 1872 to 1879 focuses on the distribution of endogenous and exogenous mortality during the first month of life (Manfredini 2004). The colder northern regions, characterized by higher numbers of exogenous deaths during the first month of life, were estimated to have very high levels of endogenous mortality using Bourgeois-Pichat’s method, considerably higher than the warmer southern regions. Therefore, using the linear relationship for exogenous mortality after the first month to predict the proportion of neonatal exogenous mortality could be misleading due to the influence of local factors, such as climate, which is why it must be used with care.
Bourgeois-Pichat’s biometric method has also been used to identify problems with data recording (Wrigley 1977; Wrigley and Schofield 1981). While a high level of endogenous mortality cannot be used as a criterion for high data quality, the opposite holds true, they argue. A very low level of endogenous mortality indeed indicates data problems, just like large deviations from the normal sex ratio at birth do. These two criteria are, in fact, often used in historical studies to evaluate data quality (Bengtsson and Lundh 1995). These two criteria become even more precise if they are applied to different social strata, since under-recording of early deaths often has a social gradient, possibly due to the costs involved in a burial (Bengtsson 1999; Bengtsson and Lundh 1995).
To this, Lynch et al. (1998) remarked: “From a historical standpoint, we are not persuaded by Bengtsson and Lund’s explanation of what they believe to be the cause of widespread underregistration of the earliest infant deaths.” And they continue “burial fees were quite modest in the nineteenth century and were often waived for the poor (Kangas 1985). Fees paid in kind were often accepted.”
To conclude, these historical remarks are somewhat irrelevant in the light of the findings in this paper: Deviations from the postulates given by Bourgeois-Pichat for a particular data set rarely indicates anything else than that at least one of the postulates are invalid, probably the one about a uniform distribution of the exogenous mortality, for which there is no data available from the neonatal period.
3.3 A slight modification
A slightly better approach to the method of Bourgeois-Pichat is to formulate the estimation problem in terms of hazards rather than densities. The first step on this road is to utilize an assumption of constant hazard function on the post-neonatal age period rather than a constant density function. The practical difference is negligible when infant mortality is no too high, and it motivates the B-P procedure as a good approximation of the true state of affairs, useful in the absence of suitable software for survival analysis, as was the case for Jean Bourgeois-Pichat in the nineteenfifties.
So with infant mortality data on the log-cube time scale, right censored and eventually left truncated, estimation of the exogeneous and endogeneous mortality is straightforward:
Left-truncate your data at age \(g(30)\) time units (30 days), and fit an exponential distribution. The estimated cumulative distribution function of the exogenous mortality is \[\begin{equation} \hat{F}_{\text{exo}}(t) = 1 - \exp(\hat{\alpha} t), \quad 0 \le t \le 365, \end{equation}\] where \(\hat{\alpha}\) is equal to the total number of deaths divided by total exposure time in the truncated sample. Note that we are extrapolating back in age through the first month of life.
Calculate \(p_1 = \hat{F} (g(30))\) and \(p_2 = \hat{F}(365) - \hat{F}(g(30))\) and let \(D_2\) be the observed number of postneonatal deaths.
The number of exogeneous neonatal deaths \(D_1\) is then estimated by:
\[\begin{equation}\label{eq:cumhaz} \hat{D}_1 = \frac{p_1}{p_2} D_2 \end{equation}\]
- If the total number of neonatal deaths is \(D\), then the estimated number of endogenous deaths is \(D - D_1\).
With our toy example, we get \(\hat{\alpha} = 0.000185\), \(p_1 = 0.0132\) and \(p_2 = 0.0520\). With \(D_2 = 5\) we finally get
\[\begin{equation} \hat{D_1} \approx 3.73 \end{equation}\]
3.4 Tabular data
Often the Bourgeois-Pichat method is applied to tabular data. The most extreme case is that the only available data are number of births \(B\), number of neonatal deaths \(D_1\), and total number of deaths \(D\). The Bourgeois-Pichat line fitting is very simple to do in this case, but how about the hazards approach?
It turns out that it is easy to find given the data at hand followíng the procedure given in Section 4.3: The probability of dying in the post-neonatal period, given survival of the neonatal period is estimated by \(\hat{p} = (D-D_1) / (B - D_1)\), and the length of the post-neonatal time span is \(T = g(365) - g(28) \approx 297\) “g-units”. The estimate of the rate \(\alpha\) in the post-neonatal period is then given by
\[\begin{equation} \hat{\alpha} = -\frac{log(1 - \hat{p})}{T}. \end{equation}\]
And so on.
4 The assumptions about exogenous mortality
Exogenous causes of death are due to social, cultural, economic, and environmental factors, and also accidents. An argument for the constant hazards and the uniform approaches to exogenous infant mortality is that, after compensating for age via the log-cube transformation, these factors are random and evenly spread in time, like white noise. However, this disregards seasonal variations in weather, and following that the spread of infectious diseases.
To illustrate this, data will be split in two ways, first by “birth month” (cohort data) and second by “calendar month” (period data).
4.1 Cohorts by birth month
In Figure 4.1 we see that the post-neonatal part of the total cumulative hazards are more or less convex for those born in spring (red) or summer (green), while the curves are concave for births in the winter (blue). Bad fits to “constant hazard” and “uniform distribution”. However, as the solid black curve shows, the 12-month mixture has an excellent fit to a straight line.
Note the two outliers, February and October. Born in February means a tough first month of life, but surviving that, the future prospects are great. The opposite holds for those born in October, no problems during the neonatal phase, but then comes the tough winter months.
Figure 4.1: Cumulative hazards by birth month, cohorts, Västerbotten 1861–1950, log-cube time scale.
Figure 4.2: Postneonatal cumulative hazards by birth season (cohorts, log-cube transform).
In Figures 4.3 and 4.4, the B-P plots for all cases and born during fall, respectively, are shown split up in two periods, the neonatal and the postneonatal. The left hand panels are the interesting ones, showing the split of total neonatal mortality into its components endogenous and exogenous. The right hand panels show how well the uniform assumption fits to data, quite well with the full data set, not so well with data only for births during fall. This shows that the validity of the “straight line assumption” is highly dependent on the composition of the data at hand: Is it well balanced with respect to birth season, geography, time period, and possibly other factors?
Figure 4.3: Exponential procedure, Västerbotten, all cases.
Figure 4.4 also suggests that the postneonatal data should be used in order to get a better estimate of the exogenous mortality in the postneonatal period. It remains to figure out how that will affect the result in the left hand panel, and that is suggested later, but one possibility is shown in Figure 4.5. It is constructed by assuming a Gompertz distribution for the exogenous mortality over the wholw first year of life, and estimate its parameters from available data in the postneonatal period.
Figure 4.4: Exponential procedure, Västerbotten, born during fall.
Figure 4.5: Gompertz procedure, Västerbotten, born during fall.
Comparing Figures 4.4 and 4.5 shows, as expected, a better fit with the Gompertz assumption (right hand panels), but also that the estimates of how the neonatal mortality is split between endogenous and exogenous mortality change.
4.2 Periods by month
Figure 4.6: By month, period data, Västerbotten 1861–1950.
Figure 4.7: Period data, Västerbotten 1861–1950, fall.
The periods are of course more homogeneous regarding weather and external threats, and it shows off in that for all months the suggested models fit reasonably well, although on different levels, here shown by the “constant hazards model” for exogenous mortality.
4.3 Conclusion
The assumptions about constant hazard and uniform distribution of exogenous mortality are only applicable on the population level, it can never(?) hold for an individual. It can be thought of as an average over birth cohorts, where it is important to have birth dates spread evenly within calendar years.
Looking at Figure 4.2 gives a feeling that something is wrong with the postulates: The exogenous mortality doesn’t change, while the endogenous postneonatal mortality varies when it should be zero on all occasions. A better approach must be to estimate the exogenous mortality for the postneonatal period and then make an assumption about its connection to the exogenous mortality in the neonatal period. Let’s call it the adaptive approach.
5 The adaptive approach
The adaptive approach to the problem rests on the observation that the first postulate of Bourgeois-Pichat is not necessary, and it is in fact misleading: The second postulate implies that we in fact have full information in the sense of data of the exogenous mortality in the postneonatal period, since there total mortality is equal to exogenous mortality, and the latter can be estimated by the estimator of the first.
What remains is the extrapolation from age 30 days back to birth, where we have no information of the division of deaths between endogenous and exogenous. Bourgeois-Pichat solved this problem simply by continuing the straight line from age 30 back to birth, and we can do something similar.
We replace the first postulate by
- The exogenous mortality in the neonatal period is an “extension” of the post-neonatal mortality model.
This can be done by assuming a suitable parametric model for the exogenous mortality in the full first-year age span, and then estimate its parameters from the left-truncated data in the postneonatal period. We exemplify by assuming a Gompertz distribution with two unknown parameters.
The Gompertz distribution has hazard function given by
\[\begin{equation} h_G(t) = \lambda \exp(r t), \quad t > 0, \end{equation}\]
where \(r\) (rate) and \(\lambda\) (level) are parameters to be estimated. If \(r = 0\) we have a constant hazard, that is, an exponential distribution, and in practice the Bourgouis-Pichat model. In our example, \(\hat{r} \approx -3.5\), significanty diffent from zero, which also Figure 5.1 seems to imply.
Figure 5.1: Exogenous mortality, births during fall, Västerbotten 1861–1950, on the log-cube time scale.
It should be noted that the problem we consider here is in fact a competing risks problem: Two causes of death, exogenous and endogenous, and the estimation of subdistributions is not straightforward (Beyersman et al. 2011). Our historical data contain information of cause of death to some extent, more in later time periods. See Section 11 for more information.
6 In practice
With the Västerbotten data, we investigate the development over time by splitting the time interval 1861–1950 into nine intervals, each of length ten years. For the neonatal results, see Figure 6.1. Besides the general decrease in neonatal mortality over time, it is obvious that it gets more and more dominated by endogenous causes with time.
Figure 6.1: Neonatal mortality by time period, Västerbotten 1861–1950.
And in Figure 6.2 the development of postneonatal (exogenous) mortality is shown.
Figure 6.2: Neonatal mortality by time period, Västerbotten 1861–1950.
And it is obvious that the fit to the Gompertz distribution is excellent, and probably also to the Exponential one (straigth lines). Moreover, the postneonatal mortality starts its dramatic decrease in the early nineteenth century.
7 Biometric analysis on the natural time scale
Now when we are free of the rigid “straight line” assumption it is time to get rid of the unnatural “log-cube time transformation”. The question is only if the Gompertz distribution will will survive this change. If not, an option is to transform the Gompertz distribution similarly. If we do that, the result is
Figure 7.1: Gompertz procedure, Västerbotten, born during fall, natural time scale.
which is the same as reported in Figure 4.5, but maybe esier to interpret.
Given that the exogenous mortality follows an Exponential distribution on the log-cube scale, it follows that the cumulative hazards function on the natural time scale is
\[\begin{equation*} H(t; \lambda) = \lambda \log^3(t + 1), \; t > 0. \end{equation*}\]
Note the duality: \(\text{log-cube transform} \Longleftrightarrow \text{cumulative hazards function}\)!
It follows that the hazard function \(h\) is given by
\[\begin{equation*} h(t; \lambda) = \lambda \frac{3 \log^2(t + 1)}{t + 1}, \; t > 0, \end{equation*}\]
see Figure 7.2.
Figure 7.2: Exogeneos hazard function on the normal time scale.
We may be relatively confident that the shape is OK in the post-neonatal period, where data are available, but the shape during the neonatal period is a pure guess. Examining data with known causes of death might be enlightening.
We repeat the hazards method for Västerbotten 1961–1890 on both the log-cube and the natural time scales for comparison, see Figure 7.3.
Figure 7.3: Cumulative hazards, Västerbotten 1861–1890, log-cube and natural time scales.
The picture on the natural time scale is of course what we prefer, so why did Bourgeois-Pichat use the log-cube time transform instead of the log-cube cumulative hazards function? The answer is of course that he had no access to a computer back in the fifties, and his way of compensating for that was in fact very clever. Similar pencil-and-ruler techniques have earlier been used, for instance in Weibull probability plotting (Weibull 1951).
8 Estimation
8.0.1 Data from Scania, infants born 1812–1850
A case with higher total infant mortality. First, see Figure 8.1
Figure 8.1: Scania 1812–1850, Gompertz assumption, normal time scale.
Then a comparison between boys and girls, Exponential assumption, see Figure 8.2.
Figure 8.2: Estimated cumulative hazards, exponential assumption for the exogenous part, Scania 1812–1850, normal time scale.
9 Experiences from data with some information on cause of death
Our data from Västerbotten contain information on cause of death for \(7\,186\) cases, while the total number of infant deaths is \(17\,675\). That is, to 41 percent of all infant deaths we have information of cause of death. The quality of this information varies over time and space, naturally. However, we will look into the deaths with an associated cause in an experiment with the methods for estimating the relative frequency of endogenous deaths in the neonatal period.
9.1 Frequencies of endogenous causes of death by death age
We assume that Chapters I, N, P, and Q refer to endogeneoous deaths, the rest, except cases with no information of cause of death, are exogenous. With that in mind, the Västerbotten data neonatal deaths consist of 2747 cases with known cause of death, of which 1842 are endogenous and 905 exogenous. Thus, 67 percent of the neonatal deaths are endogenous.
We look at all cases of infant death with a cause noted and show the relative frequency of endogenous deaths by time of death in Figure 9.1.
Figure 9.1: Relative frequency of endogenous deaths by death week, Västerbotten 1860–1950.
Obviously, there is no sharp limit between neonatal and postneonatal periods regarding relative frequency of endogenous causes of death.
9.2 A numerical experiment
We created a data set from the Västerbotten data by removing all infants that survived one year of age. This will of course not affect the fraction endogenous deaths in the neonatal period, the number that this is all about. We compare methods of estimating this fraction, which is known to us (67%).
9.2.1 IMR of 100 percent
With only deaths (all surviving individuals removed), we get with the Gompertz method, see Figure 9.2,
Figure 9.2: No survivors, the adaptive method.
with an endogenous rate of 80 percent.
The exponential model breaks down in this case, because the slope of the exogenous curve will be too high (the cumulative hazard rate tends to infinity as age gets close to one). See Figure 9.3,
Figure 9.3: No survivors, exponential method.
The uniform method, on the other hand, works well, see Figure 9.4.
Figure 9.4: No survivors, B-P method.
with an endogenous rate of 59 percent.
9.2.2 IMR of 25 percent
We can achieve this by adding a suitable number of survivors to our “all-deaths” data set, Figure 9.5,
Figure 9.5: 25 percent IMR, the adaptive method.
with an endogenous rate of 66 percent.
The B-P method, See Figure 9.6,
Figure 9.6: 25 percent IMR, B-P method.
with an endogenous rate of 59 percent. So obviously the adaptive method is superior.
9.2.3 IMR of 10 percent
We can achieve this by adding a suitable number of survivors to our “all-deaths” data set, Figure 9.7,
Figure 9.7: 10 percent IMR, the adaptive method.
with an endogenous rate of 67 percent.
The B-P method, See Figure 9.8,
Figure 9.8: 10 percent IMR, B-P method.
with an endogenous rate of 59 percent. So obviously the adaptive method is superior.
9.2.4 IMR of 5 percent
We can achieve this by adding a suitable number of survivors to our “all-deaths” data set, Figure 9.9,
Figure 9.9: 5 percent IMR, the adaptive method.
with an endogenous rate of 67 percent.
The B-P method, See Figure 9.10,
Figure 9.10: 5 percent IMR, B-P method.
with an endogenous rate of 59 percent. So obviously the adaptive method is superior.
10 Competing risks
See Figure 10.1 for an analysis with the R package survival.
Figure 10.1: Cumulative incidence curves, Skellefteå and Umeå 1861–1951 (log-cube time scale).
Note that the curves represent cumulative probabilities, not hazards, the probability of being in one particular state at a given time. It looks as if there are three states (all deaths), but there is a fourth, implicitly given, state, still being alive.
We note that by calendar time, the cause-less deaths gets fewer.
11 Conclusion
The purpose of the “biometric” analysis needs to be specified: Obviously, the prime (and only reasonable) goal is to find an estimation of the proportion of endogenous deaths among all the neonatal deaths with a data set on infant mortality that lacks information of cause of death.
We want to let go the assumption, made by Bourgeouis-Pichat, that exogenous postneonatal mortality follow a uniform distribution on the log-cube time scale. On the other hand, we want to keep his assumption that all postneonatal deaths are exogenous. So the goal must be to estimate the postneonatal mortality from data at hand, on the natural time scale. And then find a way to find a model for exogenous infant mortality from the analysis of the postneonatal mortality.
So, the efforts must start with the postneonatal period, and find a good model for infant mortality that can be estimated with data only from the postneonatal period. Earlier in this paper we found that a Gompertz model worked nicely with the log-cube time transform, but will it do on the natural time scale? If not, what candidates are there? Of course, we can calculate the distribution on the normal time scale, given a Gompertz distribution on the log-cube scale, but the result will not be nice to work with (but it is absolutely possible).
And, finally, we want the time unit be year!
11.1 Tests of postneonatal distributions with the Västerbotten data
Two natural candidates are the Gompertz and Weibull distributions.We check their respective fits on the full Västerbotten data set in Figure 11.1.
Figure 11.1: Fit comparison, Weibull and Gompertz. Västerbotten data.
The Weibull distribution is our choice. Now to business, the steps are
Left-truncate your data file at age 30 days (but in time unit year, 30 / 365).
Fit a Weibull model to the left-truncated data set.
Plot the estimated distribution on the full first year, see Figure 11.2.
Figure 11.2: CDF of the exogenous infant mortality.
From Figure 11.2 we see that the probability of dying in the neonatal period from an exogenous cause is 0.027, and dying in infancy of an exogenous cause is 0.072, so we can expect that 100 (0.027 / 0.072) = 37.5 percent of the exogenous deaths will occur in the neonatal period, and therefore 62.5 per cent will occur in the postneonatal period.
But we know the number of deaths in our data in the postneonatal period, it is \(9\,975\).
- Conclusion: We may expect 0.375 * 9975 / 0.625 = 5985 exogenous deaths in the neonatal period. But we know the total number of deaths in this period, it is 7700. So the expected fraction of exogenous deaths in the neonatal period is 100 * 5985 / 7700 = 77.7 percent.
Important: The numbers in this example must be checked!
12 Summary
In this article, we propose a variation of the Bourgeois-Pichat biometric analysis of infant mortality. Instead of assuming that cumulative mortality during the last eleven months of infancy follows a uniform distribution, given a log-cubic transformation of age, we suggest to estimate this distribution, and do so without the log-cube time transformation. This is more satisfactory, since we have data available to do so.
While the Bourgeois-Pichat model fits well with many countries in the late nineteenth century and in the first half of the twentieth century, there are many exceptions (see Knodel and Kintner, 1977). It is surprising that, despite this fact, the assumption of a uniform distribution on the log-cube time scale has survived so many decades.