Data from the usage survey, specifically the number of in-use vessels and vessel uselife, allow prediction of the number of vessels that will be discarded annually using Equation 5. The variation in annual discard with household size is plotted for various ceramic vessel morphologies in Figure 10. (Tables 6 and 8 give these data as total annual discard, dividing by the number of families yields the data of Figure 10.) There is a strong correlation between the predicted annual discard of ceramic ollas and household size (r = .86, N = 9, p = .003), with a slope of .06 ollas/person. This is significant since ollas also

Household Size

Household Size

Figure 11. Predicted sherd accumulation, quantified as surface area (cm2) per (a) family-year and (b) person-year, as a function of household size. Sherd discard increases with household size, giving a positive slope when plotted per family-year (a), but incrementally fewer sherds are discarded per person, giving a negative slope when plotted per person-year (b). These curves suggest that bulk sherd accumulation may indicate the numbers of families present, but not total population without some means for estimating household size.

Figure 10. The mean annual number of ceramic vessels discarded for (a) ollas, (b) large ollas, (c) chatas, and (d) tostaderas per household are plotted as a function of household size. Dashed lines give the best fit linear regression.

account for the highest discard level, with a mean of 1.15 ollas/family-year. For large ceramic ollas, we calculate the annual discard by adjusting the uselife to be 14 years (see discard survey discussion above). However, even with this shortened uselife, the annual discard level for large ollas is only .09 ollas/family-year, and there is little correlation between discard and household size (r = . 18, N = 9, p = .65). For mod-erate-size chatas (small and medium), the mean annual discard level is .48 chata/family-year. The discard also is uncorrelated with household size (r = .13, N = 9, p = .75), primarily because of the nearly zero slope in the best fit line (dash) for small families and the scattered values for large families. For tostaderas, a uselife of 1.32 years yields a mean annual discard of .75 tostaderas/family-year, and there is some correlation between annual discard and household size (r = .45, N = 9, p = .22), with a slope of .02 tostaderas/person.

Similar estimates can be made for the annual discard of metal ollas, yielding .20 metal ollas/family-

Figure 11. Predicted sherd accumulation, quantified as surface area (cm2) per (a) family-year and (b) person-year, as a function of household size. Sherd discard increases with household size, giving a positive slope when plotted per family-year (a), but incrementally fewer sherds are discarded per person, giving a negative slope when plotted per person-year (b). These curves suggest that bulk sherd accumulation may indicate the numbers of families present, but not total population without some means for estimating household size.

year and .05 large metal ollas/family-year (Table 6). Since metal ollas have only been available in historic times, we would like to estimate what the ceramic olla discard rate would be if only ceramic vessels were used. One approach is to assume that in the absence of metal, a larger number of ceramic vessels would be used, in particular, that a one-for-one vessel replacement of ceramic by metal has occurred. This approach is not without problems since metal and ceramic vessels may not be functionally equivalent. However, under this assumption, the sum of ceramic (546) and metal (336) ollas in use (Table 5), divided by the ceramic olla uselife (2.37 years) and the number of families (199), yields an estimated discard rate of 1.87 ollas/family-year in the absence of metal. Likewise the estimated discard rate for large ollas, in the absence of metal, would be. 14 ollas/family-year.

The above mean discard rates allow prediction for the number of households and the years of discard represented by the discard survey data (Table 2). A total of 210 discarded ceramic ollas were recorded (Table 2); dividing by 1.15 ollas/family-year yields 183 family-years. For chatas, 83 vessels divided by .48 chatas/family-year gives 173 family-years. For tostaderas, 68 vessels divided by .75 tostaderas/fam-ily-year yields 91 family-years. These values (183, 173, and 91 family-years) compare reasonably well with the figure of 223 families and suggest that the actual reported discard was for somewhat less than one year.

In an archaeological setting, the number of discarded vessels is more difficult to estimate than the total accumulation of sherds, although techniques or estimating vessel numbers and morphologies are under development (e.g., Hagstrum and Hildebrand 1990). Larger vessels, more vessels in use, and shorter vessel uselives all increase the rate of sherd accumulation in the archaeological record. Data from the usage survey demonstrate how household size may be related to vessel volume, numbers, and uselife. A summation of these factors allows household size to be related to the rate of sherd accumulation, quantified as sherd surface area discarded per family-year (Figure 11 a) or as surface area discarded per person-year (Figure lib). Sherd discard is calculated from the annual rate of vessel discard (Figure 10), multiplied by average surface area per vessel (Table 3) for ollas (2,100 cm2), large ollas, (4,200 cm2), chatas ( 1,645 cm2), and tostaderas (1,150 cm2). The average sherd discard per family-year is 4,789 cm2 (consistent with the sum of discard from all vessel types in Figure 10). Larger families have somewhat higher rates of sherd accumulation; sherd discard rate is correlated with household size (r = .84, N = 9, p = .005) with a positive slope of 168 cm2 per person. To compensate for ceramic replacement by metal, we assume a one-for-one replacement (as above). In this case, the average sherd discard per family-year is 6,710 cm2, and the increase with household size (r = .84, N = 9, p = .004) is 251 cm2 per person (Figure 1 la).

Sherd accumulation, however, is not constant when calculated per person (Figure lib), making household size an important parameter for population estimation from ceramic data.

Average sherd discard per person-year for ceramics is 859 cm2; accounting for metal replacement increases discard to 1,206 cm2. When plotting sherd discard rate per person-year against household size, there is a negative slope of-150 cm2 per person for ceramics and -202 cm2 per person when accounting for metal replacement. This suggests that grossly different rates of sherd accumulation will result from the same population distributed into smaller or larger families. For example, a population distributed into small families (3 persons per family) would discard sherds at 2.8 times the rate of the same population distributed into large families (9 persons per family). Conversely, if the number of families is held fixed between two different populations, one with small households and another with large households, three times the population produces only a 25 percent increase in sherd discard (Figure 11a). These data suggest that when using bulk sherd accumulation (counts, weights, or surface area), it may be possible to estimate accurately the number of families, but not the total number of persons. Additional knowledge of household size is required to estimate the total population from ceramic data.

By knowing the relation between discard and household size for several vessel morphologies, as in Figure 10, it is possible to turn the problem around so that household size can be calculated from the relative proportion of discarded vessels, a direct analog to what might be done with archaeological data. To test this approach, we use data from the vessel discard survey to predict the mean household size and, simultaneously, the numbers-of-households and years-of-discard product. The basis for this approach is the varying relationship between household size and the discard rate for different vessel types. Olla and tostadera discard increases with household size, whereas chata discard is relatively constant or uncorrected with household size (Figure 10). We do not include large ollas because of their low discard rate.

For a given ceramic morphology, the annual discard per household, D, and the household size, H, are assumed to be linearly related (dashed lines in Figure 10), where m is the slope and b is the intercept of a linear regression:

The total discard for each vessel morphology, Tp is a product of the annual discard ZX, and the num-ber-of-households time product NH *t. Substituting into the equation above yields a relation between household size and discard as:

For archaeological data, the unknowns in the above equation are the household size, H, and the number-of-households time product NH Knowledge of at least two vessel morphologies allows solution for these two unknowns. We use three vessel morphologies (ollas, chatas, and tostaderas), giving an overly determined least-squares problem. In matrix form, the equations become:

NH*T

T1T2T3

Using the discard survey values for T (Table 2) and the usage survey values for m and b (Figure 10) yields:

NH*T

- 063 .04.021 210 83 68
- 799 .470 .625] (13)

Applying a matrix inverse to both sides (e.g., Moore-Penrose pseudoinverse) allows solution for the household size, 5.0, and the number-of-households time product, 171 family-years. (If only ollas and chatas are used, the result is a household size of 7.4 and a number-of-households time product of 165 family-years.) These values are comparable to the actual household size of 5.7 and the 223 households reporting discard for somewhat less than one year.

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