Settlement surveys that yield counts of the number of houses or rooms dated to a chronological period can be used to indicate population and to estimate rates of population growth or decline. Population estimates and population growth rates are, of course, interpretively critical for describing and explaining many aspects of social dynamics. For this sort of survey data, population growth rates can be estimated by applying an ordinary, compound interest formula to room counts, standardized by period length, for successive time periods. This article offers a more nuanced approach that simulates a continuous process of room construction and abandonment, yielding a total number of rooms occupied during a period. The modeled growth rate is the value at which the model total most closely matches the value observed in the settlement data, given the set of model parameters. The model results are sensitive to both structure use-life and the founding room count. This sensitivity does not reflect a defect of the model. Instead it points to a major problem of equifinality—a great variety of different processes could account for the observed data. Exploration of three cases shows that reviewing the model results for a range of combinations of reasonable parameter values can provide demographic insights that are more informative—and can be markedly different—than those provided by the standard formula.

Accurate demographic reconstructions are critical for addressing issues of considerable importance. To name a few, these include assessing prehistoric migration and population circulation, arguments that involve resource availability or carrying capacity, and for identifying and explaining major settlement pattern transformations such as episodes of population aggregation or nucleation. Demographic reconstructions from archaeological data are often based on settlement pattern data conveyed by some aggregate population-related indicator, e.g., house count, or area covered by structures, separated into archaeologically identifiable periods that are often longer than we would like. Thus, our task is to extract the best possible demographic information from the untidy archaeological record (

At a minimum, we’d like to be confident in our understanding of the relative trends, that growth in one period is faster or slower than another. In terms of the absolute values of growth rates, it is important to know, in particular, whether the rate of growth is too high to be accounted for by in-place growth, a level that Cowgill (

In this article we consider cases in which systematic survey produces estimates of house counts assigned to a series of reasonably well-dated chronological periods but with no temporal resolution within periods. In different areas and time periods, houses could include separate residential structures, e.g., as indicated by house mounds, pit structure depressions, tipi rings, etc., or they could be portions of larger residential buildings (see

With rare exceptions, these conditions obtain over much of the northern Southwest USA from about AD 800 to 1400. Our main examples come from the Cíbola or Zuni region in the Southwest. There, after AD 800, the dominant architectural form was the “pueblo,” with contiguous, rectangular masonry- or adobe-walled surface rooms (Figure

Plan of a typical pueblo room block from the Cíbola region (Techado Springs Pueblo, Roomblock A). Note the consistent layout with living rooms with hearths in the front (south) and storage and other activity rooms in the back (after

Ethnographic and experimental evidence indicate limited use lives for these wood, mat, and dirt-roofed structures (see

As our initial example, we use our 1990–1991 full coverage survey of 10.4 km^{2} surrounding the prehistoric Zuni settlement of Heshotauthla on the Zuni Indian Reservation in west-central New Mexico. The area was densely occupied, with 305 prehistoric and historic sites, not all of which were occupation sites, recorded in this reasonably small area. A ceramic seriation (

Heshotauthla survey, room counts by period.

Period | Dates | Sites | Rooms | Rooms/25 Years |
---|---|---|---|---|

A | 850–900 | 2 | 12 | 6.00 |

B | 900–950 | 9 | 38 | 19.00 |

C | 950–1050 | 19 | 105 | 26.25 |

D | 1050–1125 | 8 | 97 | 32.33 |

E | 1125–1175 | 11 | 129 | 64.50 |

F | 1175–1225 | 18 | 323 | 161.50 |

G | 1225–1275 | 22 | 468 | 234.00 |

To estimate population growth rates in situations satisfying the assumptions detailed above, we must address the problem of what Ammerman (

Standardization can be done by dividing the number of rooms by the period length or, more intuitively, by the number of use-lives, or by the number of 25- or 50-year intervals in a given period. In our first example, shown in Table

The compound interest approach uses a simple model of population growth, calculated using a formula applied to the standardized room counts from two, temporally adjacent periods (see also

which can be rewritten as

where _{1}_{2}_{DE}

In effect,

Estimated growth rates for the Heshotauthla survey derived from the formula and the simulation.

Period-to-period growth rate using compound interest formula.

Interval | Years y | p_{1} |
p_{2} |
Growth Rate r (%) | |
---|---|---|---|---|---|

A–B | 875–925 | 50 | 6 | 19 | 2.3 |

B–C | 925–1000 | 75 | 19 | 26.25 | 0.4 |

C–D | 1000–1087.5 | 87.5 | 26.25 | 32.33 | 0.2 |

D–E | 1087.5–1150 | 62.5 | 32.33 | 64.5 | 1.1 |

E–F | 1150–1200 | 50 | 64.5 | 161.5 | 1.9 |

F–G | 1200–1250 | 50 | 161.5 | 234 | 0.7 |

It is important to recognize that the calculation does not depend on standardizing using a 25 year basis. To see this, note that the relevant term in the equation is _{2}p_{1}

One should be skeptical especially of the calculated A–B growth rate. In the survey area there is no indication of any occupation prior to Period A. This implies that there was an initial migration at some time during period A, but we have no evidence to suggest when that occurred. The calculated rate (2.3%) assumes that the initial migration was at the beginning of the period. If it occurred late in period A, for example at AD 890, then y is diminished to 30 years and the calculated rate increases dramatically (to 3.9%). For this reason, we do not use the period A estimates below.

Contrast the contemporary example with the typical archaeological situation. We almost never have indicators that allow us to estimate momentary populations, instead we have cumulative counts of structures (our population indicators) accumulated over the course of period. While these counts are standardized, as described above, it is not at all obvious that the standardized value is an appropriate stand-in for the momentary population at the midpoint of a period, as is assumed in this approach.

By making a few more assumptions and simulating the process of room construction and abandonment, we can estimate growth rates and, in addition, get estimates of the momentary population or rooms at any point in time. This approach yields estimates of population growth or decline

As in the compound interest approach, we need to know each period’s length and the total room count dated to each period. We assume that the growth rate we are attempting to estimate is constant, though unknown,

Starting with the number of rooms occupied at the beginning of a given period, each year it simulates the abandonment of rooms that have reached their use-life and the construction of replacement rooms and, in addition, the construction of new rooms or abandonment of occupied rooms as dictated by the application of a constant growth rate. Thus, for a given growth rate, the simulation produces a cumulative number of rooms that have a majority of their occupation dated to that period. The objective, for each period, is to find the growth rate for which the

Ignoring, for now, some of the messy details, running the simulation for the Heshotauthla survey produces the results presented in Table

Modeled growth rate by period for the Heshotauthla survey, 25 years structure use-life.

Period | Dates | Observed Rooms | Period Start Rooms | Period End Rooms | Cumulative Rooms | Growth Rate (%) |
---|---|---|---|---|---|---|

A | 850–900 | 12 | — | — | — | — |

B | 900–950 | 38 | 13 | 18 | 38 | 0.7 |

C | 950–1050 | 105 | 18 | 36 | 105 | 0.7 |

D | 1050–1125 | 97 | 36 | 29 | 97 | -0.3 |

E | 1125–1175 | 129 | 29 | 122 | 129 | 2.9 |

F | 1175–1225 | 323 | 122 | 208 | 323 | 1.1 |

G | 1225–1275 | 468 | 208 | 169 | 468 | –0.4 |

The initial period begins with a user-stipulated number of rooms; all other periods begin with an inventory of rooms that had not reached their use-lives by the end of the previous period. Each period also starts with an arbitrary, provisional, growth rate. Stepping through each year of the period, a room that has reached its use-life in that year is abandoned and a replacement room is built. The provisional growth rate is then applied to that year to determine if one or more additional rooms need to be constructed (positive growth rate) or abandoned (negative growth rate). Once the final year of a period is reached, the model tabulates the beginning and ending number of rooms for the period, and most importantly, counts the simulated rooms — abandoned or still in use — that are dated to the period. A room whose occupation spans a period boundary is dated to the period with a majority of its occupation. The number of simulated rooms dated to the period is compared to the number of rooms archaeologically observed to date to that period. If they match, then the provisional growth rate is reported as fitting the growth rate for that period. If the observed number of rooms is greater than the modeled number, then the provisional growth rate is increased. If the observed number is lower, then the tentative rate is decreased, and the model is run again, until the room counts match.

More concretely, referring to Table

In the foregoing discussion we set aside the issue of what the starting number of rooms should be for the initial period, since that cannot be generated by the preceding period. It turns out that the model is, in fact, quite sensitive to the value chosen. Relatively high and relative low values for the initial number of rooms often produce estimated growth rates that oscillate dramatically in successive periods. Choice of a number that is small relative to the total number of rooms dated to the period, results in a high growth rate for the period that in turns results in the construction of a large number of rooms near to the end of the period that are dated to the subsequent period. This, in turn, can result in negative growth rate in that next period. Choice of a number that is high relative to the number of dated rooms has the opposite effect.

In the absence of relevant data about the correct value, we suggest basing the selection on the difference between modeled growth rate

Sensitivity to the number of initial rooms, Heshotauthla survey.

While sensitivity to the initial number of rooms may appear to be a defect of the model, what it really demonstrates is the problem of equifinality of the demographic processes. That is, numerous, quite different possible scenarios could have produced the periodized archaeological record that we face. The compound interest approach does not eliminate this problem, it simply obscures it.

There are two additional startup considerations with the model. First, contrary to what the model assumes, the occupation of the initial period may not start when that period is generally dated to begin. If the occupation were to have started well into that initial period, then the predicted growth rates will be too low and the inventory of rooms beginning the next period will reflect that fact, affecting the growth rates in that second period. In this circumstance there may be relevant external data. For example, if there are deposits that are chronometrically dated early in the period, or if there is evidence of occupation in an earlier period not under consideration, it may be reasonable to assume that there was occupation throughout the period. Indeed, we have omitted period A in our example because we don’t know when, within that period, occupation began, and that can have a substantial impact on subsequent periods.

The second has to do with the age distribution of the rooms that are ‘constructed’ at the beginning the first period under consideration. A cohort of brand-new rooms built at one time may make sense in the case of a major migration into the area, but if there is evidence of occupation prior to the initial period under consideration, as is the case here, it makes sense to randomly age the rooms between 0 years and their use life. This choice is given the user, but the randomized ages are used in the analyses presented here.

In Table

As long as we select numbers of initial period rooms as described above, Figure

Sensitivity to room use-life, Heshotauthla survey.

The model, as now implemented, not only allows the user to stipulate a fixed structure use-life, but also allows the user to assign a normal (Gaussian) distribution for the use-life with a user-specified mean use-life and standard deviation, with modeled use-life truncated to have a minimum of one year and a maximum of twice the mean use-life. Figure

Modeled growth rates using normally distributed structure use lives with mean of 25 years and different standard deviations (standard deviation of 0 is a fixed structure use life). The red, S.D. = 0 line is very close to the dark blue, std = 5 line.

Thus far, we have focused mainly on the mechanics of the modeled and compound interest approaches to estimating population growth. Referring back to Figure

Number of occupied rooms through time for the Heshotauthla survey, computed annually for the simulation and using the compound interest formula.

The survey of the Ojo Bonito area on the Hinkson Ranch immediately south of the Zuni Indian Reservation serves as our second example. This full coverage survey of 57.6 km^{2}, conducted between 1984 and 1994, recorded 560 archaeological sites, not all of which were prehistoric habitation sites. Similar to the Heshotauthla survey, sites were assigned to chronological periods based on the ceramic complexes represented in their surface assemblages. Table

Room counts by period, Ojo Bonito survey.

Period | Dates | Rooms | Rooms/25 Years |
---|---|---|---|

C | 1000–1050 | 82 | 41.0 |

D | 1050–1125 | 154 | 51.3 |

E | 1125–1175 | 77 | 38.5 |

F | 1175–1225 | 244 | 122.0 |

G | 1225–1275 | 574 | 287.0 |

The Ojo Bonito and Heshotauthla growth rate and population curves have similar shapes (Figures

Estimated growth rates for the Ojo Bonito survey derived from the formula (red line) and the simulation (blue line).

Number of occupied rooms through time for the Ojo Bonito survey, computed annually by the simulation and using the compound interest formula.

Blake, LeBlanc and Minnis’ (1986) influential publication of the Mimbres Foundation’s survey of the Mimbres Valley approaches many of the questions attacked here. The article is based on extensive data and does an admirable job of not only presenting their source data and results but also explaining their assumptions and methods. It is useful to revisit their work in light of the discussion in the paper, to see what additional or different conclusions might be drawn.

The original paper can be consulted for more detail, but we note that the Mimbres Foundation systematically surveyed a stratified sample of about 11% (100 km^{2}) of the 903 km^{2} Mimbres Valley and reported room counts where possible and room area, in five periods from AD 200 to about 1400. Based on the by-stratum sampling fraction, they extrapolate the room counts and room area to the entire valley as presented in Table

They assume a constant growth rate though out the entire sequence, which they calculate using the compound interest formula, using data from the two best understood periods. For their sample survey of the Mimbres area, Blake, LeBlanc and Minnis (^{2}) through the Classic Period (AD 1000–1150; 80,405 m^{2}). They use the Early Pithouse period (AD 200–550; 12,904 m^{2}) room area as the initial area for the Late Pithouse Period and the Classic Period room area as the terminal area calculate the rate over the entire 600 years of the Late Pithouse and Classic Periods, yielding a rate of (80405/12904)^{1/600} = 0.31%. Had they used their standardized room area^{2}; AD 375) and Classic Period (40,203 m^{2}; AD 1075) they would have gotten a similar rate of (40,203/2718)1/700 = 0.39%. Assuming the 0.3% growth rate throughout the sequence, they then explored the implications of different structure use life on population, with little consideration of the possibility that the growth rates were not constant.

Using the model proposed here, we can use their data to ask what the within-period growth rates might have been. The model provides very similar results for growth rates for use-lives of 15, 20, 25, and 75 years (75 is the number used by Blake and his colleagues, page 453–454). Recalling that we set the initial number of rooms for the first period to minimize the difference between the growth rate for the first two periods and the compound interest growth rate between those periods (Table

Mimbres Valley data from Blake, LeBlanc and Minnis (

Period | Dates | Rooms | Rooms/75 Years |
---|---|---|---|

Early Pithouse | 200–550 | 646 | 138.4 |

Late Pithouse | 550–1000 | 1215 | 202.5 |

Classic | 1000–1150 | 4848 | 2424.0 |

Black Mountain | 1150–1275 | 560 | 336.0 |

Cliff | 1275–1350 | 286 | 286.0 |

Period-to-period growth rate using compound interest formula.

Interval | Growth Rate r (%) | |
---|---|---|

EPH – LPH | 375–775 | 0.10 |

LPH – Classic | 775–1075 | 0.83 |

Classic – Black Mountain | 1075–1213 | –1.43 |

Black Mtn – Cliff | 1213–1313 | –0.16 |

If we rerun the model for the Early Pithouse Period through the Classic Period (this is similar to the approach taken by Blake and his colleagues), the model dates all the rooms in the terminal period to that period, rather than assuming that late-constructed rooms will have assemblages dating to the subsequent period (Table

Simulated growth rates for the Mimbres Valley.

Period | Use Life | ||||
---|---|---|---|---|---|

Dates | 15 years | 25 years | 30 years | 75 years | |

Early Pithouse | 200–550 | 0.09 | 0.11 | 0.12 | 0.12 |

Late Pithouse | 550–1000 | 0.10 | 0.10 | 0.11 | 0.11 |

Classic | 1000–1150 | 2.25 | 2.16 | 2.12 | 1.80 |

We suggest that a low growth rate on the order of 0.1% during the Early and Late Pithouse periods, rapid growth of about 2% in the Classic, followed by a discontinuity and depopulation in the Black Mountain and Cliff phase provides a more refined, and more useful and accurate characterization of what happened in the Mimbres than a steady growth rate of 0.3% to 0.4% throughout the Early Pithouse through Classic periods.

Given the interpretive value of accurate assessments of population dynamics, what steps might be taken to improve those estimates for survey data? For either approach discussed here, the most useful improvement would be more refined dating that allow temporal assignments to shorter time periods and better dating of the initial occupation of an area. Seriations that position site occupations within periods would allow demographic reconstructions that do not assume constant growth rates over substantial temporal intervals would be even more helpful (e.g.,

Reconstructing the dynamics of past populations is a task central to understanding numerous social processes from migrations, to population aggregation, to changes in social complexity. Systematic settlement surveys provide some of the best archaeological data for that purpose (

In archaeology, one approach to this problem has been to use a compound interest formula and standardized room counts by period to calculate growth rates

As we have shown, the approaches can yield substantially different results with quite different implications for the underlying social processes. The simulation makes a number of explicit, simplifying assumptions, any of which may be unrealistic in a given circumstance. However, application of the formula to the same data

The question is not which approach provides the correct answer. It is almost unimaginable that the assumptions of either approach are realistic, notably that of a constant growth rate within [simulation] or between [formula] lengthy archaeological periods. Instead, we need to ask to what extent they can inform us about the broad outlines of past population dynamics.

The first, and we believe unarguable, conclusion is that there is an enormous problem of equifinality in the reconstruction of population dynamics. There are countless plausible and quite different scenarios that can result in exactly the same set of observed data—the room counts by period. Unfortunately, additional evidence that would allow us to discriminate among the alternatives will often be lacking.

Certainly, the equifinality problem indicates that we should not take any result as gospel. Instead, we need to evaluate the assumptions that underlie the results and the sensitivity of the results to changes in assumptions. While we recommend maintaining a healthy skepticism, we believe that in many cases the simulation approach relies on assumptions that are more plausible and will yield results that are superior to those produced by the formula. Based on our knowledge of the archaeology of these areas, this certainly appears to be so for the cases presented here.

All data (room counts by period) needed to reproduce these results are provided in tables in this article, some of which are derived from data presented by Blake, LeBlanc and Minnis (

Hassan (_{12} = (1/_{2}/p_{1}

In this example, obtaining a match required between 8 and 15 trials. Because room counts are integers a limited range of growth rates will produce the same results.

Blake, LeBlanc and Minnis (

We use the adjusted number of rooms for the Classic period from their Table

An earlier version of the approach presented here was used in Kintigh, Glowacki and Huntley (

The authors have no competing interests to declare.