Demography is a science of rates and if paleodemography is to become a science, its practioners must approximate demographic rates for prehistoric populations. In demography, mortality rates are "simply" ratios of the number of deaths at certain ages and the population at risk at those ages over a defined period of time. As is well known, paleodemography lacks information on the population at risk as well as the period of time in which skeletons were deposited, and consequently lacks true mortality rates. To circumvent these problems, conventional paleodemography assumes that skeletal age profiles are equivalent to the deaths column in a life table (d_x)--that is, the observed population was stationary, the excavated skeletal data, representative, and estimated ages, valid. For decades following the birth of the discipline, paleodemographers satiated their curiousity about one of the fundamental facts of a population, "the mean length of life," by simply computing the mean of imputed ages in a skeletal collection--notwithstanding cautions from the field's founders (Angel 1969; see Buikstra 1997). Despite the seductive simplicity of this mean, the measure is flawed, because the assumptions of stationarity, representativeness and validity--"whopper assumptions" ("false assumptions" according to Angel, 1969)--are rarely met. The fortuitous happenstances required for these assumptions to be attained are daunting indeed (Hammel 1996). All the more astonishing is how infrequently paleodemographers check, compare or calibrate their data against models or other empirical data. Hooton's assessment of the problem of the age-representativeness of skeletal deposits, published in 1930, remains a salutary warning to would-be paleodemographers (1930:15):

Louis Henry, the "father" of historical demography and a by-stander at the birth of paleodemography, observed in 1954 that French prehistoric skeletal data compiled by Noguier (1954) were not likely to be representative of the age and sex structure of deaths in a human population because age distributions were too heaped, there were too few older adults and females were clearly under-represented. A few years later, Carrier (1958), recognizing the difficulty of estimating mortality solely from an age profile of deaths, developed a method for calibrating age-ratios from empirical populations against those from model stable populations with various demographic growth rates. The idea of using model stable populations, rather than the more constricted special case of a stationary population, did not flourish among paleodemographers until the 1970s (Weiss 1973; Asch 1976). The ensuing boom persisted for scarcely a decade before Bocquet-Appel and Masset (1982) attacked the raw materials of the discipline, the skeletal age profiles, arguing that "the information conveyed by the age indicators is so poor that the age distributions thus available can hardly reflect anything but random fluctuations and errors of method" (329). Recently, Wood, Milner, Harpending and Weiss (1992) broadened the assault to a theoretical front, arguing that attempts to defend paleodemography by means of paleopathology were doomed by osteological paradoxes--demographic nonstationarity, selective mortality, and individual heterogeneity in the risks of disease and death. These criticisms and others provoked a flurry of publications in defense (for a summary of recent publications, see Paine 1997a).

The reasoning of fertility-centered paleodemography is more biological than demographic. Imagine the age distribution of a population as a pyramid. In human populations, mortality strikes the pyramid most forcefully at its base and peak, killing the young and the elderly in greater proportions than at other ages. In contrast, the fertility effect strikes solely at the base of the pyramid, at the instant of birth. Since death may occur at any age and birth occurs only at a single instant, the force of fertility is concentrated at a single point whereas the force of mortality is dissipated because it may strike at any age. Thus, if the paleodemographer is to compare empirical age distributions for skeletal or model populations, fertility offers a larger target than mortality. This fact, puzzling to the untrained, is illustrated in Figure 1 , panels "A"-"D".^{*(a note on graphics)}

The powerful influence of fertility on the age structure of deaths is easily visualized. High fertility dampens the proportion of deaths at middle and older ages, while low fertility inflates proportions at older ages, as is seen in the stable populations depicted in Figure 1. When fertility is held constant and mortality is allowed to vary (left panels, designated "A" and "C"), the impact on the age structure of deaths is trivial and is scarcely perceptible at any age between fifty years and five, the ages at which the paleodemographic evidence is strongest. The left panels in Figure 1 show that the force of mortality is weak, with little discrimination between age distributions, notwithstanding the fact that in these graphics life expectancies at birth (e0) are allowed to vary from 20 to 50 years and fertility is held constant in the left panels--in the top at GRR=3 and in the bottom, at GRR=4 (average completed family sizes of six and eight children, respectively). Much greater variation is present in the right panels of Figure 1, which display fertility effects ranging from GRR=2 to GRR=6, with mortality held constant. In the top right panel mortality is held at e0=20 years while in the bottom, e0 is placed at 50 years.

That the age pattern of deaths should be determined by fertility rather than mortality is counter-intuitive. Common sense tells us that the age structure of deaths is determined by mortality, but common sense is wrong. As noted above, the force of mortality works over the entire range of ages, whereas fertility effects are concentrated at a single moment, birth. Although paleodemographers have known this fundamental demographic truth at least since the mid-1980s (Sattenspiel and Harpending 1983; Johansson and Horowitz 1986; Horowitz, Armelagos and Wachter 1988), the mortality mindset persists (Mendsforth 1990; Van Gerven, Guise, Sheridan and Adams 1995:471-472; Saunders, Herring, Sawchuck and Boyce 1995:104; Bocquet-Appel and Bacro 1997; discussions of earlier versions of this paper at the History of Health and Nutrition in the Western Hemisphere conference, Columbus OH, March 7, 1996 and the American Association of Physical Anthropology Annual Meeting, Salt Lake City UT, April 2, 1998).

A methodological extension of the importance of fertility in determining the age distribution of deaths is the appeal to non-skeletal evidence for insight on likely population growth rates. With fertility and growth rates known and a stable population assumed, mortality levels may be readily deduced (Johansson and Horowitz 1986; Horowitz, Armelagos, and Wachter 1988). Since the data on which this paper is based do not include indicators of demographic change, no attempt is made here to estimate growth rates for individual populations. However, once fertility is estimated, then various mortality levels may be hypothesized by simply considering alternative rates of population change, including negative or no change at all.

Table 2 suggests a way of re-thinking paleodemographic statistics in fertility terms from which mortality may be surmised at given rates of population change--if the uniformitarian hypothesis is accepted. For example, if an age distribution of skeletal material points to a gross reproduction ratio in the 3.8-4.2 range, and if other archaeological data suggests that the rate of population change was zero, then we may deduce that the crude birth rate was 60. Since this is a stationary population we know that life expectancy at birth is the reciprocal of the crude birth rate, in this case 1/.060, or 17 years. In cases where the inferred growth rate is not zero, the lower panel in Table 2 offers a chart for inferring life expectancy at birth (crude death rates are derived by subtracting the rate of natural increase from the birth rate). Thus, if the estimated GRR is ~4.0 and the presumed annual rate of natural increase 1%, then the table reveals that the corresponding crude birth rate would be 50, the death rate 40, and life expectancy at birth 26 years. The table provides a wide range of combinations for facilitating the derivation of coherent estimates. An even-wider range of combinations may be obtained from model life tables. Here, I follow the lead of Johansson and Horowitz (1986:235, 238 note 1) in using Coale and Demeny's female-based models. Alternatives to the Coale and Demeny models exist (see Paine and Harpending 1996 or Bocquet-Appel and Bacro 1997), but differences between the various systems are relatively slight. Whatever system of stable population parameters used, the uniformitarian hypothesis is assumed (see also Hammel 1996).

In this paper "faux" hazard rates are computed for ages 0, 1, 5, 15, and for each ten-year interval up to, but not including, the last interval for which ages of death are reported for the specific dataset, usually 45-54. Since we do not know the age structures of skeletal populations, if paleodemographers are to exploit hazard analysis they must first accept what I have dubbed the "whopper" assumptions: that empirical age profiles of skeletal populations are representative, valid, and comparable to model population death profiles (d_x). The whopper assumptions provide the basis for computing hazard rates. In fact, these are "faux," rather than real, hazard rates because they are computed solely from the age distribution of deaths without reference to the population at risk of dying. By accepting the whopper assumptions, paleodemographers suppose that the skeletal population studied is in fact stationary. If the comparison between observed data and model is to be fair, this assumption also must be made for the model population. If my reading of the recent paleodemographic hazard rate literature is correct, researchers compare observed data transformed by the whopper assumption against untransformed mortality quotients from model populations. Graphing empirical over model rates maximizes the reader's ability to visually compare patterns in ways that are not possible with conventional paleodemographic displays of skeletal age profiles. Graphing successive hazard rates minimizes the arbitrariness of age groupings. It must be noted that criticisms directed against the use of the mortality quotient (q_x) in paleodemography are equally applicable to the central death rate (m_x), or hazard rates.

Non-stationary populations. Third, fitting the age distribution of skeletons to model life tables ranging over all likely combinations of fertility (and mortality), may aid the interpretation of demographic conditions for that population. Paleodemographers advocate combining graphical analysis of the shape function of deaths with statistical methods, such as maximum likelihood estimates or proportional hazard analysis (Buikstra and Konigsberg 1985; Paine 1989; Konigsberg and Frankenberg 1992).

While paleodemographers endorse the idea of comparing the shape of the mortality curve from an empirical age array with model populations, researchers often do not actually graph empirical data over a comprehensive range of stable populations.
Figure 2 provides a framework for making comparisons. With mortality held constant (upper panels), age patterns of deaths from various model populations do not overlap. However when various combinations of both fertility and mortality are considered (bottom panel, Figure 2), the difficulty of identifying the best fitting model becomes readily apparent. Comparing the pattern of hazard rates computed from a stable population with a gross reproduction ratio of two (yielding a completed family size of four children) with others ranging upwards to a GRR of six, as in the upper-left panel, reveals striking differences in the pattern of hazard rates by age (here, the level of mortality is held constant at e0=50 years). When the mortality level is shifted (to say, e0=20, as in the upper-right panel), the general pattern remains unchanged, although the two graphs are clearly different. When the panels are laid on top of each other (lower left), then the picture becomes more complex, even confusing, with many over-lapping combinations. Adding intermediate mortality levels to the picture produces a mass of over-lapping lines (not shown). Patterns unique to a particular fertility regime emerge only from age 30 and beyond. Figure 2 is a warning for paleodemographic modelers who rely solely on goodness-of-fits statistics instead of examining empirical data against fitted models and their residuals.

Bocquet-Appel and Masset (1982) proposed a slightly different ratio, d5-14/d20+, to minimize the effects of aging errors for young adults. The authors carefully designed their ratio to be sensitive to the effects of fertility on the age distribution of deaths in stable populations. Indeed, the Buikstra ratio is much less sensitive to different levels of fertility and mortality than for d5-14/d20+, but unfortunately confidence intervals are not as readily computed for the latter.

I turn the challenge to uniformitarianism on its head. The problem with uniformitarianism, it seems to me is not in the inappropriateness of model populations, but in biases associated with the deposition, recovery and aging of skeleton material. Figure 3 illustrates the argument. Using model data in Figure 2 as a starting point, we place the skeletal population of bronze-age Lerna over a wide range of stable populations. The upper left panel ("A") in Figure 3 shows that by age thirty the difference between observed and model data forms a large open jaw--just as the challengers to the uniformitarian orthodoxy predict. Deviations from the model data are statistically significant, as demonstrated by confidence intervals in panel "B". From age thirty, empirical hazard rates and the confidence intervals for ancient Lerna range much higher than any model stable populations. The same jaw-shaped pattern is observed for a twentieth-century Afro-American skeletal population from Dallas Texas (lower left, panel "C"). Overlaying the two skeletal populations (lower right, panel "D") reveals almost identical match for these skeletal populations, even though separated by four millennia and an ocean.

The scarcity of older adults in the large number of skeletal samples analyzed here, whether ancient or modern, confirms the increasingly widespread perception of age bias (Henry 1954; Angel 1969; Bocquet-Appel and Masset 1982; Buikstra and Konigsberg 1985; Lovejoy, Owen, Meindl, Mensforth and Barton 1985; Johansson and Horowitz 1986; Walker, Johnson and Lambert 1988; Konigsberg and Frankenberg 1992; Paine 1997b). Skeletal datasets of American, European and African ancestries ranging from as long ago as four millennia to as recently as four decades reveal the uniform scarcity of old adults. The strikingly uniformitarian patterns of death observed in the empirical populations depicted in these graphs suggest biases in the skeletal data themselves as the most likely explanation rather than a structure of fertility peculiar to skeletal populations (Figure 4). ¹

Visual examination of faux hazard rates for these data reveals systematic departures from model life tables--unusually low mortality rates at young adult ages (15-34 years) and unusually high rates for middle aged adults (35-54 years). This unique pattern could be considered real, if it did not appear in almost all skeletal datasets--ancient Native American populations (from Libben, the Dickson Mounds, and Tlatilco--Figure 5 top left panel--to paleolithic Brazil), nineteenth century African Americans (Dallas TX--Figure 5 top right panel--Philadelphia PA, and Cedar Grove AK), Euro-Americans (Figure 5 bottom left panel--Quito, Ecuador, Bellville, Canada, and the Highland Park NY poor-house), or European populations (from Early Neolithic Çatal Hüyük to Classic Greece--see below)--but not in data from historical burial books for American Indians (Figure 4 bottom right panel) or in Coale and Demeny's model life tables (Figure 5).

Faux hazard rates computed from parish records for a Native American population at La Purísima Mission, near Lompoc, California, from 1813 to 1849, do not show exaggerated mortality rates for middle-aged adults (Figure 4, lower right panel; Walker, Johnson and Lambert 1988). On the contrary, burial records suggest that too many deaths were assigned to older ages, from about 45 years, or that the population was not stable. The large number of burials (n=1140 at age 5 and older) yields gratifyingly narrow confidence intervals, ranging +/-1 female child (GRR), much smaller than those for skeletal datasets, where the range is typically +/-2 female children or more.

The strongest proof of age-bias in skeletal data comes from a 19th century Canadian site, St. Thomas' Anglican Church Cemetery in Bellville, Ontario, where skeletal data, burial registers, and a census list are available for comparison (Saunders, Herring, Sawchuk and Boyce, 1995). The exceptionally large skeletal collection offers every advantage to the paleodemographer. The skeletal material lay in well-drained sandy soil and "is extremely well-preserved and in excellent condition" (Saunders et al., 1995). Ages were determined for 97% of the sample of 577 individuals. The parish records are also well-preserved and reasonably accurate. Yet, the hazard rates for this historical skeletal population are similar to those for paleopopulations. The Bellville skeletal population is unlike any model population and is strikingly different from the pattern drawn from burial registers (Figure 6 --note that my analysis is based on data eye-balled from Saunders et al., 1995, Figure 2, p. 106). The skeletal data yield hazard "rates" for older adults slightly beyond the pale of likely models (left panel), while the age distribution of deaths from the burial registry fits comfortably within the range of stable populations (right panel).

The authors of this extraordinary test case construct life tables for each dataset using conventional paleodemographic assumptions and privileging mortality effects over fertility. They conclude that the skeletal and parish book data are significantly different, with life expectancies at birth of 19.4 years for the former and 26.5 years for the latter. Then Saunders and her colleagues take their data an important step further, confronting a skeletal lifetable with a true lifetable constructured from empirical death rates for the same population. By coupling the population age structure for the town obtained from a census list with registered burials, they are able to compute true death rates and a third life table. These data yield real rates and probably the most accurate estimate of life expectancy at birth for this population: 36.5 years.

Differences among the three estimates are attributed to composition bias, specifically the effects of migration on a frontier community undergoing rapid settlement (Saunders et al, 1995:104-110). Bias in the aging of skeletal material is not discussed; nor is the whopper assumption of stationarity. Yet, the skeletal data (Figure 6 left panel) display an upward shift of hazard rates for older adults, beyond the bounds of even the highest fertility and mortality rates for a model stable population. This deviance is consistent with patterns found in other skeletal datasets, regardless of period, place or population.

Excess skeletons for middle-aged and older adults is also found in European populations ranging over five millennia (Figure 7 ; Angel 1969). In three of the most ancient populations, the excess is apparent from age 25. Figures 6 and 7 confirm Bocquet-Appel and Masset's argument that the age distribution of deaths for skeletal populations is strikingly distinct from historical ones (1982:320) and that the problem of aging skeletal material is grave (1982:320, 329; see also Buikstra and Konigsberg 1985:316, 323; and Konigsberg and Frankenberg 1992:235). Defenders of current or revised aging techniques (Van Gerven and Armelagos 1983; Lovejoy, Owen, Meindl, Mensforth and Barton 1985) rely on a Kolmogorov-Smirnov two-sample test for cumulative frequencies to check for differences in age structure of samples, but this non-parametric test is relatively weak (STATA 1995:2:492). Parametric tests, such as proportional hazard models, are much stronger and provide greater precision in weighing differences between two populations. Unfortunately, since the authors do not publish the data on which their analyses are based--a frustratingly common practice in the pages of the American Journal of Physical Anthropology--no further statistical tests could be made. Likewise, one might wish to construct proportional hazard models for modern day, traditional human societies, such as the !Kung or Yanomamo, but the data are not republished by paleodemographers (Milner, Humpf and Harpending 1989).

Figures 3-8 reveal the classic problem of excess mortality from middle adulthood and suggest that probably the most reliably aged skeletons are those of older children and younger adults, that is, from age 5 to 24, or perhaps, 34 years. Yet, statistical modelers typically assign all age groups equal weight, proportional to the number of deaths in each. The results may point to a single best-fitting model, but there may be other good-fitting models in the vicinity that satisfy one's curiosity. For example, Paine's analysis of the raw Oneota data finds that a GRR of 3.1 yields the best fitting model. His graph of the log likelihood as a function of GRR reveals a broad band of pretty-good fits for GRRs ranging from two to five (Paine 1989, his figure 1 lower panel). Hazard rates in Figure 8 suggest that any match between model data and the Oneota faux hazard rates is rather vague and ill-defined.

Some paleodemographers and all anti-uniformitarians object to the use of region "West" model life tables on the grounds that they are derived from the historical experience of Western Europe and therefore do not apply to paleopopulations. However, as these graphs make clear the gap is between kind--skeletal data versus burial registers--not time. Model populations also "fail to fit" modern skeletal data. Moreover West model tables offer the best case scenario, with greater proportions of deaths at mature adult ages than found in East, North or South tables.

There are five reasons why poor fits stand out in figures 3-8. First, the graphs show hazard rates instead of simple percentages of deaths. Second, rates are drawn to a logarithmic scale to emphasize proportional over absolute differences. A conventional graph highlights variations in age groups with the greatest frequencies (usually the first and last, particularly if they are excessively broad, say beginning from the thirties or forties), where most deaths are assigned, but also where paleodemographers are most inclined to dismiss deviations as due to error. Third, here confidence intervals are favored over point estimates and are drawn for each estimate (if only to take into account sample size--not imprecision in skeletal aging; see Konigsberg and Frankenberg 1992:251). Fourth, the empirical data are displayed over a wide range of model hazard rates extending over all likely stable demographic conditions. Finally, graphs permit the reader to view and evaluate an entire array of empirical and model data to interpret whole patterns without the obfuscation which may occur with statistical model-fitting or mathematical solutions (Gage 1988).

Paleodemographers have long noted the paucity of skeletal data for the older ages and the rapid increase in estimated mortality "rates" from mid-adulthood (say, from age 30 or 40). The possibility that this is due to "consistent bias in the estimates of adult ages at death" is accepted by some researchers (Buikstra and Konigsberg 1985:323, 326; Konigsberg and Frankenberg 1992:235) while others consider it as one of several hypotheses worth testing (Gage 1988:433; Walker, Johnson and Lambert 1988; Paine and Harpending 1997; Storey and Hirth 1997).

In general the results displayed in these graphs are disappointing to anyone who expects much precision from such a large collection of data for such diverse populations. Invariably, samples are too small. Patterns deviate greatly from anything in the non-skeletal record. One might conclude that the uniqueness of mortality regimes of paleopopulations are the explanation, if the HNWH project's historical skeletal datasets did not show deviations similar to those for paleopopulations (and if one forgets that fertility rather than mortality determines age structure). Instead, the constancy of the patterns revealed here points to bias as the likely answer--that the collections are not representative of any once-living population, stable or otherwise, and that skeletal aging procedures distort true age structures.

The middle panel of the table shows that ninety-five percent confidence intervals for ratios in the HNWH database average ten points and more, whenever the number of cases is less than 100. The ratio, then, is sensitive only to very large differences in fertility, and sizeable datasets are required if a narrow confidence interval is desired, corresponding to a GRR of, say, +/-0.5. The table also proves that for most sites variations among datasets are not statistically significant. For example, the Arriaza data for the Pre-Ceramic era show 60% of those reaching age five surviving to age 30 (n=87), pointing to a GRR of 4.0, compared with only 52% for the second set (n=96), suggesting a GRR of 5; yet, confidence intervals for the first range from 50 to 70% and 42-62% for the second (or GRRs of 3-5 and 4-6, respectively). The overlap suggests that not much should be made of differences in point estimates between sets with fewer than one hundred cases.

Likewise, the Lallo datasets for Dickson Mounds, widely discussed in the literature as evidence for fertility or mortality change from foraging to agrictural modes of production (Lallo et al., 1980; Johansson, and Horowitz (1986)), reveal intriguing variations, but the number of cases is so small that it is useful to generate hypotheses, but not to test them. In contrast Larsen's large, chronologically ordered collections for the Southeastern United States reveal only minor variations in two of three cases, suggesting that fertility varied little over almost two millennia (1900-250 BP), notwithstanding enormous cultural, epidemiological, political and material change. While two sites (Irene Mound and Santa Catalina) point to implausibly high fertility, with gross reproduction ratios well above 6, data for Amelia Island suggest a more plausible fertility level (GRR 2-3.5). Indeed if our range of plausibility for the Bauikstra ratio is 42-80 (GRR 6-2, e0=20-50), nearly half of the twenty-six sets in the HNWH Database range beyond one--at times both--of these extremes. Of the published skeletal datasets considered here fully one-half fall beyond the limits of the uniformitarian calibrations. In contrast, all the historical datasets in Table 3 , including skeletal as well as textbased sets, fall within the bounds of the calibration standards computed from model life tables.

The Bocquet-Appel and Masset Index (d5-14/d20+) offers unique advantages for analyzing skeletal data. Even the most unrelenting critic of skeletal aging would concede that for the two age classes required by this index (5-14 and 20+), mis-classifications should be relatively rare. The index is also powerful from a demographic viewpoint (Table 4 --upper panel). In low fertility populations, the ratio falls below 10, but in high fertility populations, exceeds 35. The Bocquet-Appel and Masset index has extremely interesting properties that place it above all other indicators, although the lack of confidence intervals is a serious drawback. The correlation between the Buikstra and Bocquet-Appel indices exceeds -.98 for the model populations used in calibrating the skeletal data.

Table 4 reports the Bocquet-Appel and Masset index for each site and the corresponding gross reproduction for three levels of mortality according to model life table calibrations. Sites falling beyond the calibrated limits (8-45) are fewer than in the case of the Buikstra ratio, but this is largely because only point estimates, as opposed to confidence intervals, are considered. Yet, six sites in the HNWH Database, three of the published skeletal datasets and two of the historical sets exceed plausible limits. In most cases, the problem is at the lower limit, signalling a paucity of deaths for children, but in the case of Pearson and Nea Nikomedeia there seem to be too few for adults.