An Essay on Inverse Projection

©Robert McCaa (posted February 2001)

IntroductionInverse projection is a logical inversion of conventional population projection techniques. The method is used to infer refined demographic statistics--mortality, fertility, and population age structures--from crude data--annual totals of births and deaths and an estimate of initial population size. Instead of deriving counts from age-specific rates as with conventional cohort projection, inverse projection estimates age-specific rates from counts. The technique is particularly suited for studying populations of the past where age details are scarce.

Surprisingly accurate demographic statistics can be computed with the inverse projection method. As with most demographic techniques, the better the data, the better the estimates. What is startling about inverse projection is how little data is required to simulate the demographic history of places, large and small, over long periods of time within a range of a few percentage points (Lee 1985, McCaa 1989a, McCaa and Vaupel 1992). Population pyramids, life expectancy, total fertility, net reproduction ratios, and many other refined demographic statistics may be estimated, even where authentic data are sparse. Recent innovations (see Table 1) fine-tune the method to offer several different algorithms, including a stochastic microsimulation process which estimates not only a wide range of age-dependent demographic rates and measures but also their standard errors (Bertino and Sonnino, 1995).

Inverse projection is a method for estimating basic demographic statistics where the vital or parish registration system is relatively efficient, but where population censuses or surveys are lacking, infrequent or unreliable. The technique requires a count of births and deaths year-by-year and the size of the initial population. In the absence of reliable empirical age data, model age structures of mortality, fertility, migration, and, most importantly, the initial population may be relied upon.

In contrast to the family reconstitution method, which also relies on vital events data, inverse projection requires only aggregate counts of births and deaths, tallied by year. Thus, with inverse methods the demographic history of large areas is easily reconstructed. Where family reconstitution is labor intensive, inverse projection demands little effort to tally and computerize annual counts of births and deaths. Finally, family reconstitution's stringent methodological rules exclude from consideration a sizeable fraction of births and deaths. The reconstitutable minority is unlikely to be representative of the majority of the population. Inverse projection applies powerful non-linear models to data for the entire population. It is superior to stable population methods because no assumption of stability is required.

The inverse method has been used to construct population histories of states (England 1541-1871, Norway 1735-1974, Sweden 1750-1875, Denmark, Philippines, Valencia 1610-1899, Italy 1750-1911, Chile 1855-1964, Bulgaria, Costa Rica, and Cuba 1900-1959), regions (Northern Italy 1650-1881, the Canary Islands 1680-1850, Sardinia 1862-1921, and Scania 1650-1760), cities, and parishes or missions (Colyton 1545-1834, Pays de Caux 1530-1700, Lucerne 1700-1930, Berne 1720-1920, Amsterdam 1680-1921, Velletri 1595-1740, and California). This list, although incomplete, includes studies which test a wide range of demographic hypotheses, from models of population change in industrializing Europe to demographic responses to epidemics.

History and ScopeSince Lee first created inverse projection in 1971, seven additional implementations of the method have been developed (Table 1). In each case macrodemographic data for a parish, city, region, nation or other geographical or administrative unit are required. They differ in the use of models to supplement missing or unknown parameters. In recent decades, as computing power increased so has the sophistication and computational demands of the various algorithms.

Lee's classic forward inverse algorithm (1974) remains the simplest and most straightforward. Total population is required to initiate the projection. Counts of births for each period (quinquennia in Lee's first program) are added and deaths subtracted to obtain the population at the end of each interval. Where a count of the total population is known for a subsequent period, the projected total is subtracted from the observed to obtain an estimate of net migration, which is apportioned equally among the intervening periods.

The initial age structure of the population may be obtained from either a census or a model. Lacking an empirically derived age structure, the demographically naive might stop at this point, but Lee (1974, 1985) and others have shown that even an arbitrarily chosen model age structure yields remarkably robust estimates (Brunborg 1976, Wachter 1986, McCaa 1989a, McCaa and Vaupel 1992, McCaa 1993). Moreover the bias of a poor choice rapidly disappears as the stream of vital rates molds the projected age structure period-by-period with births added to the base of the pyramid at age zero and deaths subtracted from the slope, age-by-age.

"Vital rates, not initial states" is the inverse projectionist's mantra, but this is not a matter of faith. The theoretical foundations of inverse projection lie in the 1920s with the work of H. T. J. Norton on the weak ergodicity theorem and in the 1960s with its rediscovery by Alvaro Lopez. The method rests on what Kenneth Wachter has called "the cornerstone of formal demography", the notion that, in the long run, vital rates, not initial states, determine what a population will look like (Wachter 1986). Lee has shown that population age structures are quickly forgotten, even when punctuated by demographically critical events such as epidemics, wars, baby booms or busts (Lee 1985). The weak ergodicity theorem teaches that, when it comes to age, populations have no memory. As populations are projected through time, the initial age structure is displaced by birth, death and migration rates. For all practical purposes, populations subject to the same initial population size and identical streams of births, deaths and migration will converge, even though they may begin from radically different age structures. A century into a projection erases even the most peculiar initial age structure. On the other hand population size remains a powerful constraint. If mis-specified and the population growth rate is negative, distortions will mushroom. Where the population is growing, the bias will diminish over time (McCaa and Vaupel 1992).

The genius of Lee's method is how, in the absence of data, deaths are apportioned by age (Lee 1974). As a first approximation, an arbitrary set of age specific death rates from a single parameter family of life tables is applied for the quinquennium. The product of these rates and the age structure at the beginning of the interval yields an estimate of the number of deaths for each age group. Summing over all groups and dividing by the total number of observed deaths yields an adjustment factor, the normalized death ratio, or k. This factor is a measure of the discrepancy between the observed number of deaths and that projected given the age structure and level of mortality from the model life table rates. As a second and final approximation, k is used to compute the exact number of deaths at each age which will sum to the total for the entire interval. This is accomplished directly by, first, subtracting age specific rates between the model life table used in step one and an adjacent table from the same family. The result is a domain of mortality variability at each age. This domain is used to interpolate an adjusted age-specific death rate. The calculation is accomplished by simply multiplying k and the mortality domain at each age and adding the result to the first approximation. This yields a final estimate of deaths for each age group. Summed they equal the total deaths for the period.

Net migration by age is apportioned crudely as a function of age-specific rates and a level parameter for each period. Where migration is a minor factor relative to mortality, this solution--required due to a lack of data--produces acceptable results.

Fertility statistics are by-products of inverse projection and are wholly independent of age estimation and projection. The number of births for each interval is determined by the stream of vital events. What is lacking is a summary fertility statistic which takes into account the age structure of the population. A solution common to all inverse projection algorithms is the estimation of the gross reproduction ratio from the number of births, age structure of the population and normalized age patterns of fertility.

ExpansionsFour important expansions of Lee's inverse method have emerged in recent decades: a full range of age groups, annual projections, crisis mortality, and two-sex models. Extending the number of age groups beyond age 55 and over so that the final, open-ended age accounts for only a small fraction of total deaths was the first, most significant and least controversial innovation in inverse projection methods (Brunborg 1976). To reduce computation costs Lee's 1971 algorithm crudely lumped the oldest ages into a "55 and over" group. Brunborg quickly discovered that this shortcut introduces a substantial bias, particularly where a large fraction of total deaths occur in the open-ended group. As long as life expectancy at birth is less than thirty years, the effect of lumping is small, but where life expectancy is moderately good, highly variable, or improving, error becomes substantial. Brunborg's 1976 algorithm and all subsequent ones allow the user to set a more reasonable upper age limit, such as 75, 90, or even 100 years and older.

A second important enhancement of the inverse method was the development of single year projections, in place of five or ten year periods. Anticipated by Lee and Brunborg, annual projection cycles were first implemented by Bonneuil's trend method (1990). Initially, there were objections to annual projections as spurious precision until Bonneuil demonstrated that aggregation into five or ten year periods introduces serious distortions. Whenever a mortality crisis occurs, aggregation flattens mortality variation and thereby flattens the projected age structure. With quinquennial or decennial projections, mortality domain interpolation dampens variation throughout the population age structure. Annual mortality spikes are transformed, at best, into waves which continue to undulate through the age pyramid over subsequent cycles until the cohort vanishes into the open-ended age group. Now that computation costs are no longer a constraint, annual projections are becoming common, although some inverse algorithms still permit quinquennial or even decennial projections as an option.

Crisis mortality, due to epidemics, famine or other demographic catastrophe, is poorly approximated by model life tables. For the inverse projectionist the alternative is to either apply a special set of empirical or model age-specific death rates for the crisis years or fix a crude death rate ceiling or threshold to mimic the impact of crises on the population age structure (Bonneuil 1990). In crisis years, mortality domain interpolation will assign too many deaths to age groups where variability is greatest, principally for infants, children and the oldest or open-ended age group. The optimal solution is to use empirically derived age-specific death rates peculiar to the type of crisis (Bertino and Sonnino, 1995; Rosini, 1996). Less desirable, although widely used due to the scarcity of age-specific data for these situations, is to set a mortality threshold, say a crude death rate of 40 per thousand population (Bonneuil, 1990; McCaa and Vaupel 1992; McCaa 1993). Deaths below the threshold are apportioned according to standard inverse methodology. Any excess is distributed at a flat rate for each age group. Although an arbitrary fix, the effect is to produce more robust estimates than that provided by the standard allocation algorithm.

Single sex projections were the norm with the first inverse projection programs, while computer time was still scarce and expensive. Now that this is no longer a constraint, two sex projections are becoming frequent. Indeed, it is recommended where annual totals by sex of births, deaths, and total population are readily available, and significant sex-differentials in mortality and migration are suspected. Precision is an obvious benefit as well as the ability to model gendered differences in mortality and migration.

Related methods. As a means of dealing with the closure problem, back projection was developed by Wrigley and Scofield for their massive reconstruction of the population of England, 1541-1871 (Wrigley and Scofield, 1981; Oeppen 1993b). Most populations, over the long run, are not closed, and England is no exception. While forward inverse projection takes into account authentic migration data, Lee's method cannot produce migration flows (Lee 1993a). Back projection seeks to generate migration estimates from a terminal age structure by backcasting the population against the flow of births and deaths. While it might seem commonsensical to begin with the best, i.e. most recent, data and project backward, Lee argues that this ignores the weak ergodic theorem. In Lee's words (1993a), "the problem originates with the attempt to resurrect people who have died into the oldest age group, an attempt that is, in practice, hypersensitive to error." The Cambridge team used iteration procedures to settle upon a single series of best estimates.

Generalized inverse projection responds to Lee's criticisms of back projection and broadens the method into an analytical system which exploits whatever data are available as well as a broad range of assumptions or constraints, including components derived from back projection (Oeppen 1993a, b). Generalized inverse projection uses a standard method of demographic accounting and a standard non-linear optimization algorithm to overcome a range of empirical and theoretical problems. Among these is the temptation to estimate migration endogenously by means of smoothing parameters.

Other flavors of inverse projection were developed to extend projections back to the earliest years of parish registration (Bonneuil 1990, 2000) and to maximize the use of data available in specific countries, such as population censuses or status animarum containing age details (Rosini 1996). The most innovative, recent algorithm is stochastic inverse projection, a microsimulation technique which seeks to assess the variability in a population's evolution (Bertino and Sonnino 1995). Instead of a single deterministic solution as with forward inverse projection, the stochastic method offers many solutions by simulating individual mortality events as a non-homogeneous Poisson process. Three additional assumptions are required: that deaths are uniformly distributed over the year, and that deaths are independent of each other, and that every person has the same mortality function as contemporaries at each point in time.

Calibration and limitations.Lee first calibrated the method against family reconstitution results for the parish of Colyton, 1545-1834 (Lee, 1974). In addition to confirming the general findings of an excellent family reconstitution study for this parish, Lee was able to gain insight into periodic shifts in fertility and mortality that are not readily discernible with the family reconstitution method. For example, he estimated life expectancy at birth for the plague years 1645-49 to have been less than nine years. A second important calibration was Brunborg's reconstruction of the population history of Norway, which he compared with statistics derived from highly accurate age-specific data (1976). The results were remarkably consistent, tracking the official life expectancy at birth figures from a low of 43.7 years in the 1830s to a high of 74 years in the late 1960s. Average error was roughly two years. What most surprised Brunborg was how little difference the various mortality suppositions made on the estimates.

The lesson from Brunborg's experiments was that the quality of data is the critical factor for successful inverse projections. Later, his findings were confirmed by a series of wide-ranging tests by a skeptical demographer who challenged an enthusiast of inverse projection to a double-blind experiment, indeed a dozen such experiments (Vaupel and McCaa 1992). Vaupel used conventional cohort component projection techniques to simulate a wide range of demographic histories, and challenged McCaa to reproduce the results for each from the bare minimum required for inverse projection: the birth and death series and initial population size. The results were spectacularly successful, so much so that they exposed a flaw, or "bug" in the conventional projection program.

This experiment demonstrated that the most robust age data from models could be selected directly by minimizing the normalized mortality ratio (k). This statistic can be used as a measure of goodness of fit where model age data are required, such as for the initial population and the age structure of mortality (McCaa 1991). The ratio is an indicator of the age-standardized intensity of mortality relative to the age structure of the initial population and to the pair of age-specific mortality schedules used to project the population to the next period. The sum of squares of this ratio (K2) is a useful measure of goodness of fit, when the birth and death series, and initial population size are held constant. In other words, where two inverse projections differ only in terms of model age structures of mortality or initial population, K2 can be used to identify the better fitting suppositions. Best fitting models have the lowest K2s. When comparing sets of projections the stream of counts and initial population size must remain the same for the K2 statistic to be a reliable guide.

The experiment confirmed that data quality remains the most important constraint for successfully applying the method. Careful attention should be given to spikes in mortality or surges in migration which may confound the normality assumption of how these events should be distributed over the age structure.

ProspectsThe inverse method requires good vital registration data, which for historical studies limits its usefulness primarily to the European cultural area. Nevertheless, even here, only a small number of potential applications have been attempted. Lee's challenge that the method be extended to the modeling of marriage and legitimate fertility has gone unanswered. The inverse method could be extended to estimate other demographic measures where authentic scale parameters are available and age patterns can be modeled, as in the case of parity progression ratios (Feeney 1985). The most important contribution of the method remains the light that it sheds on demographic processes in the past. In the case of England, for example, we now know, thanks to inverse projection, that the great surge of population in the late eighteenth century was due to a rise in fertility rather than a fall in mortality as has long been supposed (Wrigley and Scofield 1981).

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