The influence of time averaging and space averaging on the application of foraging theory in zooarchaeology

1. Introduction
For more than 20 years, Americanist zooarchaeologists
have called on foraging theory [58] to provide
models of human-resource use [3,68]. Even biologists
have occasionally applied such models to zooarchaeological
materials [35,55]. During the 1990s, use of these
models in western North America resulted in the development
of a set of methods for monitoring the history of
human resource use as evidenced by the zooarchaeological
record [8–11,15,19,32–34,50,51,63]. Some of the
more interesting implications of these models concern
the fact that pre-industrial humans had significant influences
on prehistoric faunas the world over; these implications
have now been abundantly confirmed by the
zooarchaeological record (summarized in Ref. [29]). In
particular, as a result of their selective exploitation of
prey taxa that provided either or both low costs and
high returns, humans with primitive technologies often
caused changes in faunal diversity (taxonomic richness
and evenness), independent of changes in climate. Therefore,
humans sometimes had to alter what they were
exploiting as a response to a change in the availability of
animal prey that they themselves had caused.
Many significant new insights into the past have been
provided by applying foraging theory to zooarchaeological
problems. Yet such application is not always
straightforward and thus the voice of caution has
occasionally been heard. For example, many studies
conclude that changes in the list of exploited prey to
include more small, low-value prey taxa were the result
of human exploitation depressing the availability of
large, high-value prey taxa. The voice of caution
requires that this explanation be substantiated by disconfirmation
of alternative causes of depressed prey
availability, such as changes in technology, changes
in how technology was used, and environmentally or
climatically driven changes in taxonomic abundances
[30]. Further, it has been pointed out from an
Old-World context that zooarchaeological abundances
of Linnaean taxa do not always reveal changes in prey
availability with respect to particular spatio-temporal
coordinates [59]. Finally, it has been noted that the
traditional means of quantifying prey abundances—the
number of identified (faunal) specimens (NISP) per
taxon—entails various epistemological and interpretive
difficulties [66]. Importantly, analytical techniques have been and are being developed for addressing a number
of these issues [18,31,60,61,67].
One potential difficulty for which there presently is
no obvious analytical solution is that, as Grayson and
Delpech [31, p. 1119] put it, if one is to apply explanatory
models derived from foraging theory to the zooarchaeological
record, “concepts that are meant to apply
in ecological time must be translated to archaeological
time.” In particular, they underscore the fact that with
respect to prey-choice or diet-breadth models, where
either is measured as the number of taxa exploited or
taxonomic richness (NTAXA), the time period represented
by a zooarchaeological assemblage results in
NTAXA comprising the maximum number of taxa
exploited during the total duration of time represented
by that assemblage. Paraphrasing their example, if five
taxa are exploited over 99 years, but those five taxa plus
five additional taxa are procured in the 100th year, then
diet breadth rendered as NTAXA for the assemblage
resulting from that 100 years will be 10—the maximum
number of taxa exploited (see also Ref. [12]). The
significant point of this example is that variation
between the first 99 years and the 100th year will be
masked by the fact that multiple annual assemblages
have been lumped by the formational history of the
zooarchaeological assemblage. It is the effect of such
lumping, whether along the temporal dimension, the
spatial dimension, or both, that is one concern here.
A second concern is that there is another potential
difficulty not previously mentioned in the pertinent
literature. It resides in the variable typically referred to
as sample size. The influences of sample size rendered as
NISP are often acknowledged, and most zooarchaeologists
who apply models derived from foraging theory
to their data take steps to account for sample size
when taxonomic richness or taxonomic abundance is
measured with NISP (see Ref. [20] for a particularly
important method in this respect). No zooarchaeologist
I am aware of, however, has considered the potential
influence of sample size rendered as the number of
analyzed spatio-temporally distinct assemblages—what
I will term NASM—of faunal remains, which arguably
comprises a measure of sample size at a different scale
than NISP.
The third and final concern resides in the fact that
zooarchaeological assemblages comprise samples of
different portions of the spatial and temporal continua.
For discussion purposes I refer to these portions
as contexts, irrespective of size or duration. Each
assemblage of faunal remains occupies a unique context
in terms of its spatial and temporal coordinates. The
influence of the number of unique spatio-temporal contexts
included in an analysis on interpretations of
changes in prehistoric foraging is only now beginning to
be examined (see subsequently). I refer to the frequency
of these contexts generally as NCTX, the number of
spatial contexts as NCTXspace, and the number of
temporal contexts as NCTXtime. Note that sometimes
NASM = NCTX, but the more typical relation will be
NASM > NCTX as a result of analytical lumping. In
this article, I explore some of the effects that NASM and
NCTX have on analytical results produced by application
of methods derived from foraging-theory models
and used by zooarchaeologists working in western
North America.
2. The usual method
I will not review the reasoning behind the application
of foraging-theory models to zooarchaeological data
here, as there are numerous such discussions now available
[8,9,30]. It suffices to note that as used in western
North America, the typical basic model—the preychoice
model—is that human foragers will preferentially
exploit the largest prey first because these taxa are the
most valuable. The model further holds that if valuable
prey decrease in availability and thus the frequency at
which they are encountered decreases, then foragers
will turn to progressively more kinds of less valuable,
generally smaller, prey—each individual considered less
valuable than a single larger individual—to maintain a
constant level of nutrition. Thus, the ratio of large to
small + large prey will fluctuate over time, decreasing
(becoming progressively smaller than 1.0) as large prey
become less available relative to small prey and increasing
(becoming progressively closer to 1.0) as large prey
become more available relative to small prey. Typically,
prey abundances measured as NISP are rendered as an
index value between 0.0 and 1.0 that expresses the ratio
of large to small + large prey. Ugan and Bright [66,
p. 1309] refer to these index values as “relative abundance
indices (AIs),” and I use their acronym in the
remainder of my discussion. Plotting each AI against
the temporal midpoint of the time span during which the
faunal remains on which it is based were deposited—
whether stratigraphically, radiometrically, or culturally
determined—in a bivariate scatterplot, visually reveals
what are interpreted to be changes in prey return rates
that are in turn interpreted to reflect changes in prey
availability or abundance. Interpretation of the scatterplot
is sometimes aided by calculating a simple best fit
regression line through the point scatter in order to
reveal temporal trends [8,9,33].
Many applications of foraging-theory models to
zooarchaeological data concern temporally distinct
(usually stratigraphically delimited) faunal assemblages
recovered from a single site, or faunal assemblages from
several geographically propinquitous sites distributed
across several temporal periods. In a bivariate scatterplot
of data points, with the x-axis usually representing
time and the y-axis usually representing the AIs, there
are few data points. This makes both the visual and the
596 R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610
statistical detection of trends in changing faunal abundances
relatively easy and the interpretation of the
covariation between the two variables rather straightforward.
The critical issue to recognize is that if NASM
is small, then few points comprising AIs will be plotted.
For example, Butler [16] plotted AIs for eight temporally
distinct assemblages from a single site against
their ages; Broughton [9] plotted AIs for nine assemblages
from nine sites against their ages; Potter [50]
plotted AIs for six assemblages from two sites against
four time periods; Janetski [33] plotted AIs for 20
assemblages from 19 sites against their ages; Broughton
[8] plotted AIs for 18 assemblages from 14 sites against
their ages; and Broughton [11] plotted 14–16 AIs against
10 temporally distinct strata in a single site. In each of
these applications, NASM%20.
Not only does NASM tend to be small, but in each of
the examples just listed, spatial and/or temporal variation
was variously muted over multiple assemblages.
There are several ways in which spatio-temporal variation
can be analytically muted. These can be illustrated
by considering two fictional assemblages (NASM = 2),
each from a different site. Assume that the sites are
geographically close, but vary in microhabitat (NCTXspace
= 2), that the assemblages were deposited over
similar durations of time (say, 400 years), that the
assemblages are temporally sequential in age
(NCTXtime = 2), and that the assemblages belong to the
same archaeological culture based on their associated
artifacts. What are the analytical alternatives? Because
the assemblages differ in spatial and temporal coordinates,
each AI might be plotted against its age in a
separate graph or plotted in the same graph with a
different symbol denoting a different geographic position,
thereby maintaining the distinctive geographic
(and temporal) coordinates of the two. Or, the two AIs
might be plotted in the same graph using the same
symbol, thereby muting their geographic distinctiveness
(NCTXspace = 1), but maintaining their temporal distinctiveness
(NCTXtime = 2). Alternatively, the AIs
might be plotted with different symbols in the same
graph to maintain their geographic contexts, but according
to the cultural period to which they belong, thereby
muting their temporal distinctiveness (NCTXtime = 1).
Finally, the NISP per taxon of the two assemblages
might be summed, an AI calculated, and that value
plotted against either the absolute (average) age of the
two assemblages or the cultural period represented by
the two. This last possibility would mute the spatial and
temporal distinctiveness of the two assemblages (NCTX
= 1) as well as mute their distinctive taxonomic richnesses,
evennesses, and heterogeneities in favor of what
we can call a general or universal (with respect to the
two assemblages) spatio-temporal-taxonomic measure
(NASM = 1). In the first three alternatives, variation
in geographic space and/or time between the two
assemblages is muted; NCTX is reduced analytically. In
the fourth alternative, not only are both spatial and
temporal variations muted, but so are variables
of taxonomic structure and composition; NASM is
reduced.
Examples of the potential ways that one might
analytically lump contexts or assemblages are found in
the extant literature. For example, Janetski [33] plotted
(NASM = 16) AIs from a single physiographic area and
representing a single cultural period against the midpoint
of their absolute ages. The 16 assemblages were
derived from an area approximately 400150 km, but
NCTXspace was analytically reduced from 16 to 1 by
lumping. Szuter and Bayham [64] summed NISP data
from 70 sites (NASM = 70, NCTX = 70) into 11
temporally distinct sets (NASM = 11, NCTXtime = 11)
in a single area (NCTXspace = 1) and plotted the AIs
against the midpoint of the temporal period represented,
thereby muting both temporal and spatial variations.
Butler [15] summed NISP data from eight sites (NASM
= 9, NCTXspace = 8, NCTXtime<9) by cultural period
(NCTXtime = 4) for an area (NCTXspace = 1) and
plotted AIs against the appropriate temporal midpoint
of a cultural period, thereby potentially muting both
temporal variation and spatial variation.
With no explicit warrant for analytically reducing
NCTX or NASM by lumping temporally and/or
spatially propinquitous assemblages, the question of
what effect that lumping might have on results is begged.
The source of such a question resides in a fundamental
dictum of archaeology that the subject of study comprises
three dimensions of variation [43]. Those three
dimensions are form, space, and time. Form in this case
is the value taken by the AI; space comprises geographic
variation in the locations of the assemblages from which
AIs are derived; and time comprises variation in the
temporal positions of the assemblages from which AIs
are derived.
In many cases, the lumping of otherwise spatially
and/or temporally distinct assemblages is not explicitly
justified in the literature. Some possible reasons for such
lumping can, however, be suggested. It could be argued
that lumping removes the effects of seasonal, geographic,
and individual variation in resource exploitation
in the interest of determining an annual, regional,
or overall pattern; such a warrant has been characterized
as the “‘noise-filtering’ of time averaging” [47, p. 226]. In
zooarchaeology, such noise filtering would provide what
Uerpmann [65] referred to 30 years ago as the “average
diet” of prehistoric peoples across a geographic area
(rather than a single site locality) and/or across a greater
or lesser span of time (rather than a single meal or day’s
worth of meals). Paleobiologists have long been concerned
about the influence of what they refer to as “time
averaging” on ecological interpretations of faunal
assemblages [4,48,52], and they have developed various
R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610 597
methods for analytically detecting and contending with
time averaging (Refs. [14,37], and references therein).
Paleobiologists have also long been aware that some
accumulations of faunal remains represent multiple
biotic communities [53,54], or what are readily termed
‘space-averaged’ assemblages, and methods have been
developed for recognizing these [28]. Archaeologists,
too, have long been cognizant of such assemblages,
referring to them as resulting from ‘blending and smearing’
[2] that produces “coarse-grained assemblages” [6].
Such spatio-temporal palimpsests are the bane of those
seeking fine-scale resolution, but this is not meant to
suggest that spatially or temporally lumped or blended
assemblages are to always be avoided. Palimpsest assemblages
may be useful and appropriate for some questions
[47], and I return to this point later.
“Palimpsest,” “blended,” and “averaged” are reasonable
terms for those spatio-temporally unique materials
that were lumped during formation of the zooarchaeological
record or that have been analytically produced.
The important point in the context of how foraging
theory has been applied to zooarchaeological data is that
such averaging comprises the conversion of ecological
time and space into archaeological time and space.
Choosing to plot an assemblage against the temporal
midpoint of the period during which it was deposited, or
lumping data of various ages into a single temporal or
cultural period to derive a measure of diet—a more or less
continuous variable—per period results in some unknown
degree of time averaging. Similarly, lumping zooarchaeological
data from multiple sites that are variously geographically
distant from one another and that date to the
same period (if not precisely contemporaneous) in order
to derive a measure of diet results in some unknown
degree of spatial averaging. Cannon [19] has recently
addressed the possible effects of the spatial averaging and
suggested ways to analytically contend with such effects.
In the following sections, I use real data sets to explore the
potential effects of temporal averaging, spatial averaging,
and simultaneous averaging of both time and space on
interpretations. In doing so, I am less interested in the
interpretations that one might derive from a data set than
on the effects on those interpretations that result from
temporal and spatial averaging.
3. Materials and methods
For purposes of detecting the influences of spatiotemporal
lumping on analyses and interpretations, the
ideal situation would be to have assemblages meeting
three criteria. First, the assemblages would be from
various geographic locations in one physiographic
area. Such cases are commonplace in western North
America where many archaeological projects have
been undertaken under the auspices of cultural-resource
management. Second, each assemblage would be of
sufficient size in terms of NISP to allow analytical
lumping of more-or-less contemporaneous and geographically
propinquitous assemblages. This is sometimes
found in western North America, but many
assemblages there are also small, given the contingencies
of the particular project. Third, the assemblages would
be of relatively short duration—say, deposited in a
single season in temperate latitudes—in order to allow
analytical lumping dictated by research questions rather
than by the realities of the variously time-averaged
zooarchaeological record. No such zooarchaeological
record in terms of this third criterion is known to me.
Several regional archaeological projects can, however,
be analytically made to come close to meeting these
criteria.
In the 1970s and 1980s, a number of archaeological
sites located adjacent to a stretch of the Columbia River
of north-central Washington state were excavated as
part of two reservoir projects [17,21]. These sites contained
evidence of human occupation spanning the last
7000 years and representing multiple local cultural
periods. Many, but not all, of the sites produced
numerous faunal remains. I compiled data on NISP per
taxon per assemblage, assemblage age, and assemblage
location from reports resulting from the excavations.
The data comprise 31 assemblages (NASM = 31)
recovered from 18 sites (NCTXspace = 18) distributed
along approximately 420 km of the Columbia River
(Table 1). Five individual site assemblages from the
westernmost portion of the area (Wells Reservoir) were
spatio-temporally lumped for purposes of this analysis
to avoid possible effects of sample size rendered as
NISP, so NCTXspace has already been reduced to 12.
Assemblages used in my analyses are of various ages
(NCTXtime = 30) and span the last 5000 radiocarbon
years; some of them are relatively synchronic and represent
brief time intervals (ca. 200 years) whereas others
represent longer periods of duration (e.g. 2200 years).
For analytical and graphing purposes, I follow previous
researchers and use the chronological midpoint (to the
nearest 100 years) of each assemblage. This means that
varying degrees of time averaging have already been
built into the analysis.
In light of a recent study of the early historic faunal
situation in eastern Washington that suggested that
prehistoric human foragers there had depressed large
game populations [44,45], I designed three indices to
monitor changes in prey abundances over time. One
index is based on the fact that as large mammalian prey
decrease in abundance, one response of human foragers
might be to broaden the diet to include more smallmammalian
prey and more individuals of smallmammal
taxa. Because few small-mammal taxa seem to
have served as human prey throughout the Holocene
(last 10,000 years) of eastern Washington [41], and
because NISP sample sizes for these taxa tend to be
598 R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610
small, I lumped all small-mammal prey taxa into a single
category—small mammals. Similarly, because virtually
all large-mammal prey are artiodactyls, and because
deer (Odocoileus hemionus (mule deer) and O. virginianus
(white-tailed deer)) tend to dominate most collections, I
lumped the five taxa of artiodactyls (Odocoileus spp.,
Ovis canadensis (bighorn sheep), Antilocapra americana
(pronghorn antelope), Cervus elaphus (wapiti, or elk),
Bison bison (bison)) into one category—artiodactyls.
The first AI is termed the small-mammal index and is
calculated as

artiodactyl NISP/(
artiodactyl NISP

small mammal NISP)
To calculate this index, I included only taxa the
remains of which displayed evidence of human exploitation.
That evidence typically was in the form of
butchering marks, burning, bone breakage, and skeletal
disarticulation [42]. Small-mammal taxa include marmot
(Marmota flaviventris), beaver (Castor canadensis),
muskrat (Ondatra zibethicus), red fox (Vulpes vulpes),
canids (Canis latrans, C. lupus), mustelids (Mustelidae),
and bobcat (Lynx rufus). Bears (Ursus americanus, U.
arctos) and cougars (Felis concolor) were omitted from
the analysis because they are very rarely represented.
Note that the greater the abundance of artiodactyl
remains relative to small-mammal remains, the closer
the small-mammal AI will be to 1.0.
The specific response of human foragers in eastern
Washington to depressed large-game populations hypothesized
by Martin and Szuter [44,45] involved the
intensification of fishing. To determine if this in fact
happened, a measure of the abundance of artiodactyls
relative to that of fish is necessary. The second AI, then,
is termed the fish index and is calculated as

artiodactyl NISP/(
artiodactyl NISP
fish NISP)
Fish exploited by prehistoric foragers included
several species of salmonids and steelhead trout
Table 1
Zooarchaeological data from north-central Washington state by site and age
Site/location Age Age
midpoint
Artiodactyl NISP Small mammal
NISP
Fish
NISP
Small mammal
AI
Fish
AI
Nonartiodactyl
AI
Western
Wells Reservoira 4350–3800 4100 247 151 303 0.62 0.45 0.35
Wells Reservoira 3300–2200 2700 88 82 4812 0.52 0.02 0.02
Wells Reservoira 900–100 500 2 13 7235 0.13 0.00 0.00
45DO-326 300–100 200 81 21 8 0.79 0.91 0.74
45DO-326 1500–800 1100 64 52 3 0.55 0.96 0.54
45DO-326 4500–3000 3700 12 51 10 0.19 0.55 0.16
45DO-326 5000–4500 4700 45 166 67 0.21 0.40 0.16
Eastern
45OK287/288 1500–850 1200 167 1 5 0.99 0.97 0.96
45OK287/288 4400–1500 2900 426 11 4 0.97 0.99 0.97
450K287/288 4800–4400 4600 103 22 0 0.82 1.00 0.82
45OK-2 300–100 200 581 22 61 0.96 0.90 0.88
45OK-2 1300–500 900 662 22 84 0.97 0.89 0.86
450K-2 3000 3000 346 6 95 0.98 0.78 0.77
45OK-2 4000–3000 3500 506 8 117 0.98 0.81 0.80
45OK-250 2800–1000 1900 158 4 16 0.98 0.91 0.89
45OK-250 3800–2800 3300 492 37 340 0.93 0.59 0.57
45OK-250 5000–3800 4400 136 5 20 0.96 0.87 0.84
45OK-4 2000–200 1100 91 6 48 0.94 0.65 0.63
45OK-4 3200–2000 2600 744 16 1108 0.98 0.40 0.40
45OK-4 5000–3200 4100 72 2 60 0.97 0.55 0.54
45OK-258 800–100 400 1872 41 27 0.98 0.99 0.96
45OK-258 3600–2400 3000 2030 170 88 0.92 0.96 0.89
45DO-242 4000–2000 3000 537 46 42 0.92 0.93 0.86
45OK-11 4000–2000 3000 283 94 201 0.75 0.58 0.49
45DO-214 1200–1000 1100 228 27 264 0.89 0.46 0.44
45DO-211 3500–2000 2700 41 25 1026 0.62 0.04 0.04
45OK-197 800–100 400 201 52 110 0.79 0.65 0.55
45OK-197 1100–800 900 157 20 47 0.89 0.77 0.70
45OK-197 1500–1200 1300 456 19 30 0.96 0.94 0.90
45OK-197 1950–1600 1700 354 3 1 0.99 1.00 0.99
45DO-189 2950 2900 273 8 609 0.97 0.31 0.31
aData from five sites and 13 assemblages.
R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610 599
(Salmonidae), several species of cyprinid (Cyprinidae),
and several species of catastomids (Catastomidae). Most
fish bones were identified only to family. Given that the
genus and species of fish is not central to the issue of
concern here, and that Martin and Szuter’s [44,45]
hypothesis did not include expectations with respect to
which fish species would have been more intensively
exploited, finer taxonomic distinctions are unnecessary.
The greater the abundance of fish relative to
artiodactyls, the lower the fish AI.
Because the response to depressed abundances of
artiodactyls could have involved an intensification of the
exploitation of both small mammals and fish, I also
determined what I call the nonartiodactyl AI, calculated
as

artiodactyl NISP/(
artiodactyl NISP

small mammal NISP
fish NISP)
If artiodactyl abundances were progressively more
depressed over time—irrespective of why they were
depressed—then small mammals, fish, or both should
increase in abundance as human foragers attempted to
maintain their food intake levels. Again, low values of
the nonartiodactyl AI suggest that artiodactyls were of
low availability whereas high AI values indicate artiodactyls
were of high availability. Technological change,
environmental change, change in the human-population
level, or some combination of these factors might be the
causal variable driving any perceived changes in the AIs,
and although I call on these variables in the following,
they are of less concern to the discussion than the effects
of time averaging and of space averaging.
It is possible in the geographic area and temporal
period of concern that intensification of the exploitation
of river mussels (Pelecypoda) might have compensated
in part for depressed ungulate populations (Refs. [39,40]
and references therein), but shellfish remains were not
consistently identified among the 31 assemblages.
Further, so few bird remains were recovered that it is
apparent that exploitation of this taxon would not have
made any significant difference in measures of human
subsistence. I follow analytical tradition and use the
temporal midpoint of an AI’s age for graphing and
statistical purposes, and plot a simple best-fit regression
line to identify trends in prey abundances graphed in
scatterplots. I do not reproduce all scatterplots here, but
instead show only representative ones. Similarly, I do
not always discuss all possible indices.
4. Results
To examine the influence of spatio-temporal lumping
of assemblages, I first calculated the three AIs for all 31
assemblages. Here, NASM = 31, NCTXspace = 1, and
NCTXtime = 30; Fig. 1 shows the scatterplot for the
nonartiodactyl index. All three indices tend to increase
over time—the slopes of the best-fit regression lines are
positive—suggesting that the overall trend was for the
abundance of artiodactyls to increase relative to small
mammals (r0.15), fish (r0.17), or small mammals
+ fish (r0.21). The slopes are not steep, suggesting that
change in the structure of the exploited resource pool
was gradual and not very marked. The correlation
coefficients are, however, low and insignificant (P>0.25
in all), indicating that the conclusion that there was in
fact change in the exploited resource pool is statistically
unwarranted. What happens when assemblages are
lumped into a limited number of temporal periods?
4.1. Time averaging
To address the preceding question, the temporal
midpoint of each assemblage was used as its ‘age’ to
assign it to one of five time periods of 1000 radiocarbon
years duration each beginning with the period 0–1000
BP. (All included assemblages are in fact R100-year
old.) This reduces NCTXtime from 30 to 5. Values of
the nonartiodactyl AI were plotted against their time
period. The resulting scatterplot (Fig. 2) is a bit less
noisy than that produced when the midpoint age of each
assemblage is used (Fig. 1) because the ages have been
‘averaged’ to fall within a particular 1000-year period.
Not surprisingly, given the reduced variation in the
independent variable, the correlation coefficient
increases slightly, although it is still insignificant
(r0.23, P>0.1). The midpoint age was then used to
assign each assemblage to one of three cultural phases
recognized in the area [17]. These phases and their
temporal durations are Coyote Creek (0–2000 BP),
Hudnut (2000–4000 BP), and Kartar (4000–7000 BP).
This reduces NCTXtime from 30, or 5, to 3. Once
assigned to a phase, the nonartiodactyl AI for each
assemblage was plotted against the temporal midpoint
of that phase. Because all Kartar samples date between
4000 and 5000 BP, the age midpoint of 4500 BP was
retained. The resulting scatterplot (Fig. 3) is again just
what might be expected. Because temporal variation has
been removed by ‘time averaging’, the scatterplot is
cleaner visually and thus perhaps more easily interpreted
subjectively. As well, the correlation coefficient increases
a bit more, although it remains insignificant (r0.28,
P>0.1).
Although clarity remains imperfect, successively
greater time averaging affected by reducing NTCXtime
such as that represented by the progression of Figs. 1–3
seems to produce a clearer result. And, the statistical
correlation also, not unexpectedly, improves. Of course
the price paid is a loss of temporal resolution; that is,
there is a loss of the ability to detect variation across
time and also within a particular temporal span. This
600 R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610
begs the question of why the analyst might time average
in the first place. The answer is simply that time averaging
such as that represented in Fig. 3 relative to Fig. 1
might be said to be warranted by the fact that cultural
phases are often construed as periods of cultural stability
(e.g. Ref. [56]; see also Ref. [43] and references
therein), despite the fact that it has long been known
that this is seldom the case in reality [49]. Perhaps the
most important thing to note with respect to Figs. 1–3 is
that all of them involve plotting an AI value against a
single age assigned by determination of a temporal
midpoint, whether of the assemblage (Fig. 1), of an
arbitrary temporal period (Fig. 2), or of a cultural
period (Fig. 3). The differences apparent in those three
figures should give us pause when we use the temporal
midpoint of an assemblage as the plotting algorithm.
When we use temporal midpoints, we are using an
estimate of the ‘average’ age of the assemblage. Two
problems can be identified: the estimate may be to some
unknowable degree off the mark, and we have converted
a duration or period of time to a single year. The former
is an archaeological problem; the latter is a statistical
problem. Do similar problems attend the averaging of
variation in geographic space?
4.2. Space averaging
As noted earlier, the area from which the 31 assemblages
listed in Table 1 were recovered comprises a
stretch of the Columbia River valley. All assemblages
were collected from sites adjacent to the river. This
stretch of river cannot be subdivided into multiple
physiographically distinct areas on the basis of significant
vegetation differences from one portion of this
stretch of river to the next [25,27]. However, one might
divide the stretch into two subareas on the basis of the
fact that the western half of this stretch of river valley
today receives less than 25 cm mean annual precipitation
whereas the eastern half receives over 25 cm mean
annual precipitation [26]. In addition and corresponding
to this climatic difference is the fact that the western half
includes the mouth of the Okanogan River; the eastern
half includes only the mouth of the much smaller
Nespelem River. Salmon could and did make spawning
Fig. 1. Nonartiodactyl-index values for 31 assemblages plotted against ages (NASM = 31, NCTXspace = 1, NCTXtime = 30). Pearson’s r0.21 for
best-fit regression line.
R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610 601
runs up the Okanogan as well as the Columbia, but
could not proceed more than 1 km up the Nespelem due
to an impassable falls. Finally, topographic maps indicate
that the western third of the stretch of Columbia
River valley comprises broad expanses of level terrain
only a few meters above the river whereas the eastern
two-thirds of the valley are more deeply incised and level
terrain is significantly farther above the active floodplain.
Thus, topographic differences may have contributed
to geographic variation in human predation
behaviors (and perhaps prey—particularly salmonid—
availability) between the western and eastern portions of
the stretch of Columbia River that produced the 31
assemblages under consideration here.
Seven assemblages were collected from sites in the
western half of the valley and the remaining 24 assemblages
from the eastern half (Table 1). Bivariate scatterplots
for each areally designated set of assemblages
(NCTXspace = 2) were generated. The tendency is for
the western assemblages to have more small mammals
and more fish relative to artiodactyls than the eastern
assemblages (Fig. 4). Such is not noticeable if NCTXspace
= 1, as in Figs. 1–3. With spatial variation
included, the correlation coefficients are a bit stronger
and the slopes of the simple best-fit regression lines are a
bit steeper for the western assemblages (r0.09 for the
small-mammal AI, r0.11 for the fish AI) than for the
eastern assemblages (r0.30 for the small-mammal AI,
r0.20 for the fish AI), suggesting artiodactyls became
more important (relative to small mammals and fish)
over time more rapidly in the west than in the east,
although none of these differences are statistically significant
(P>0.05). The important point here is that by
considering all 31 assemblages simultaneously without
regard for geographic variation (NCTXspace = 1), as in
Figs. 1–3, the differences between the eastern and western
assemblages are undetectable because in effect the
geographic signal has been muted by spatial averaging.
4.3. Simultaneous time averaging and space averaging
Thus far, index values for individual assemblages
have been lumped by plotting them as groups according
to their particular temporal or spatial coordinates. That
Fig. 2. Nonartiodactyl-index values for 31 assemblages plotted against 1000-year period (NASM = 31, NCTXspace = 1, NCTXtime = 5). Pearson’s
r0.23 for best-fit regression line.
602 R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610
is, NCTX has been reduced but not NASM. What
happens if the NISP values for various assemblages are
combined on the basis of spatio-temporal propinquity,
and the indices recalculated, such that NASM is
reduced? Answer this question requires that the NISP
values per assemblage for each taxonomic category be
summed if they are not only similar in geographic
location, but also similar in age. First, assemblages that
fell in the same 1000-year period were lumped together
by western or eastern provenience as indicated in
Table 1 (NCTXspace = 2, NCTXtime = 5 for each
subarea, NASM = 5 for each subarea). The fish index
and the nonartiodactyl index were then recalculated for
each of the two sets of spatio-temporally lumped data
(Table 2) and the index values were plotted against the
temporal midpoint of each 1000-year period. The resulting
scatterplots (Figs. 5 and 6) are based on 5 points and
are cleaner and not as noisy as when all 31 data points
are plotted.
The scatterplots suggest that at about 2500 BP, things
were different in the two areas. In particular, whether
one considers the fish index or the nonartiodactyl index,
and whether one considers the west area (Fig. 5) or the
east area (Fig. 6), artiodactyls were exploited the least
intensively between about 3000 and 2000 BP, and this
trend seems to have begun about 4500 BP or earlier. One
might be tempted to argue that this decrease was in
response to a relatively cool–moist climatic interval
between about 4000 and 2200 BP [22,23]. Computer
simulations of anadromous fish runs in light of models
of climate and river runoff [24,46] suggest fish would
have been more abundant in the Columbia River during
this cool–moist interval than they were 4500 or 1500
years ago. This, however, leaves unexplained the marked
difference between the two areas between 1000 and 0 BP.
At that time, there seems to have been decreased exploitation
of ungulates in the west area (Fig. 5), but not the
east area (Fig. 6). There is no obvious technological
Fig. 3. Nonartiodactyl-index values for 31 assemblages plotted against cultural period (NASM = 31, NCTXspace = 1, NCTXtime = 3). Pearson’s
r0.28 for best-fit regression line.
R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610 603
difference between the two areas in terms of artifact
inventories, so I am tempted to suggest that the geographic
difference is the result of fish being more available
(or perhaps more technologically accessible) in the
western area where the Okanogan River is found.
Whatever the explanation might be, the influence of
spatio-temporal lumping can be shown by comparing
the results in Figs. 5 and 6 with the results of lumping all
assemblages irrespective of geographic provenience.
Now, NCTXspace = 1, NCTXtime = 5, and NASM = 5.
Fig. 4. Nonartiodactyl-index values for 24 assemblages (filled symbols) from the eastern area plotted against their ages (NASM = 24, NCTXspace
=1, NCTXtime = 23). Pearson’s r0.15 for upper best-fit regression line. Nonartiodactyl-index values for seven assemblages (open symbols)
from the western area plotted against their ages (NASM = 7, NCTXspace = 1, NCTXtime = 7). Pearson’s r0.38 for lower best-fit regression
line.
Table 2
Zooarchaeological data from north-central Washington state by area and 1000-year temporal period (derived from Table 1)
Location Age Age midpoint Artiodactyl NISP Small mammal NISP Fish NISP Fish AI Nonartiodactyl AI
Western 5000–4001 4500 292 317 370 0.44 0.30
4000–3001 3500 12 51 10 0.55 0.16
3000–2001 2500 88 82 4812 0.02 0.02
2000–1001 1500 64 52 3 0.96 0.54
1000–0 500 83 34 7243 0.01 0.01
Eastern 5000–001 4500 311 29 80 0.80 0.74
4000–3001 3500 998 45 457 0.69 0.67
3000–2001 2500 4639 351 2147 0.68 0.65
2000–1001 1500 1454 60 364 0.80 0.77
1000–0 500 3473 157 329 0.91 0.88
604 R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610
The small-mammal index (Fig. 7) suggests artiodactyls
were not very important relative to small mammals at
the end of the mid-Holocene climatic interval generally
referred to as the Altithermal. The Altithermal period
ended about 5000 BP and is locally characterized as
comprising warmer, wetter winters than at present;
stream flow was reduced as were salmon runs, and
grasslands were replaced by shrub–steppe habitats [23].
These environmental differences suggest a possible cause
of apparently low artiodactyl abundances at about 4500
BP. The nonartiodactyl index (Fig. 7), which is very
similar to the fish index, suggests something quite different.
Similar to Figs. 5 and 6, the lumped data suggest
much fluctuation in the abundance of artiodactyls over
the last 5000 years (Fig. 7). As indicated in the preceding
paragraph, explanations might comprise elements of
environmental fluctuation. They might also include evidence
of fluctuation in the size of the human population
[22] and changes in fishing technology such as increased
use of nets [1]. Whatever the case, the explanation would
be complex.
Explanatory complexity largely disappears if the
index values are calculated on the basis of cultural
period and irrespective of geographic position. Here,
NCTXspace = 1, NCTXtime = 3, and NASM = 3. The
plot of the small-mammal index per cultural period
indicates that artiodactyls were not frequently exploited
relative to small mammals at the end of the Altithermal,
but subsequent to that point small mammals were
significantly overshadowed by artiodactyls (Fig. 8). This
trend aligns nicely with the standard conception of the
Altithermal and subsequent climates and habitats and
human adaptations [5,38]; small mammals were more
economically important during the Altithermal than
subsequently. Similarly, the fish index also aligns with
standard explanations [13,62] that the exploitation of
fish intensified after the Altithermal as a result of
increased run sizes and improved technology (Fig. 8).
These results align with inferences produced by
archaeologists who were working 35 years ago and were
interested in region-wide economic trends, a point I will
return to subsequently. Of more concern here is the
Fig. 5. Fish-index (open squares) and nonartiodactyl-index values (x) for seven summed assemblages from the western area plotted against 1000-year
period (NASM = 5, NCTXspace = 1, NCTXtime = 5). Upper best-fit regression line is for fish index (r0.18); lower best-fit regression line is for
nonartiodactyl index (r0.14).
R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610 605
following simple fact: interpreting a bivariate scatterplot
of many points representing relatively precise
spatio-temporal data (Fig. 1) is a rather different matter
than interpreting a bivariate scatterplot of few points
representing the same data in spatio-temporally lumped
form (Fig. 8).
5. Discussion
The general significance of the example discussed
above for applications of models derived from foraging
theory (or any explanatory theory) to zooarchaeological
data should be clear. On the one hand, the
manner in which zooarchaeological samples are
lumped by space or by time—whether taphonomically
or analytically—may reveal regionwide trends precisely
because that lumping masks spatio-temporal variation
within the region. Lumping data without consideration
of the included spatio-temporal variation, on the
other hand, could mask what may otherwise comprise
significant differences in resource exploitation
patterns.
Notice that I said “may comprise significant differences.”
Does spatio-temporal lumping in fact mask
significant differences? That depends on the spatiotemporal
scale of the question being asked, and this is
the most critical point of this discussion. Paleobiologists
who have examined time averaging argue that it only
occurs when the process or property one seeks to
measure occurs or was created, respectively, over a
shorter time span than is represented by the fossil record
under study [28,36,47,57]. Thus, time averaging is a
problem when two or more analytically interesting
events of distinct ages appear to be contemporaneous. A
similar argument is made for spatial averaging. If the
property of analytical interest is a biological community’s
structure in terms of taxonomic richness,
evenness, or diversity, and if the fossil record under
study comprises organisms from multiple communities,
Fig. 6. Fish-index (open squares) and nonartiodactyl-index values (x) for 24 summed assemblages from the eastern area plotted against 1000-year
period (NASM = 5, NCTXspace = 1, NCTXtime = 5). Upper best-fit regression line is for fish index (r0.55); lower best-fit regression line is for
nonartiodactyl index (r0.66).
606 R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610
then that record is spatially averaged. The critical issue,
as paleobiologists note, is one of ascertaining the spatiotemporal
scale of the fossil record and asking questions
of the same scale.
In terms of applying explanatory models derived
from foraging theory to the zooarchaeological record,
the critical issue is the same as that in paleobiology.
Asking if prehistoric hunters depressed prey populations
by exploiting them is so general as to have no spatiotemporal
coordinates. Adding the phrase “in area X
during cultural periods A, B, and C” to the question
provides coordinates. The zooarchaeologist must,
however, determine if the assemblages included in the
analysis fall within those coordinates and thus not be
spatio-temporally averaged. To make this determination
demands that explicit spatio-temporal parameters be
built into the research questions we ask. If they are
not, then zooarchaeologists will be open to criticisms
that their analyses mask potentially significant
spatio-temporal variation at finer scales.
6. Conclusions
The observation of Grayson and Delpech [31] that
the application of models derived from foraging theory
requires the translation of ecological time into archaeological
time must be expanded to include the notion that
those applications must also translate ecological space
into archaeological space. Such translations have, to
date, not been accomplished by literally converting or
rewriting the ecological model in terms of archaeological
variables. Rather, the models have simply been applied
to archaeological data, and the latter interpreted in
terms of the former. The influence on interpretations of
the resultant muting of temporal and/or spatial variation
should be clear and is the central point of the example
Fig. 7. Small-mammal-index values (open squares) and nonartiodactyl-index values (x) for 31 summed assemblages plotted against 1000-year period
(NASM = 5, NCTXspace = 1, NCTXtime = 5). Upper best-fit regression line is for small-mammal index (r0.78); lower best-fit regression line is
for nonartiodactyl index (r0.10).
R.L. Lyman / Journal of Archaeological Science 30 (2003) 595–610 607
discussed here. However, lumping might be affected—
decreasing NASM by summing assemblages, ignoring
temporal variation by decreasing NCTXtime, ignoring
spatial variation by decreasing NCTXtime—it mutes,
smooths, and averages what might otherwise be significant
differences. This demands a decision regarding
what comprise significant differences. Such a decision
can be made at the initiation of analysis, but may have
to be modified once the degree of spatio-temporal averaging
of the assemblages at hand has been determined.
The lessons here, then, are two. We must first be
explicit about the values of the spatio-temporal variables
within the questions we ask of the zooarchaeological
record. What should be abundantly clear is that spatiotemporal
palimpsests are averaged, blended, or coarsegrained
relative to some preconceived and generally
implicit size of spatio-temporal unit [7]. And second, we
must be cognizant of the influences of spatial and
temporal averaging on our analytical results in lieu of
rewriting the ecological models we apply to the zooarchaeological
record in terms of the spatio-temporal
scale of that record.