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the group with which I am most familiar. However, virtually every thing that is said
about quantifying vertebrate remains and their attributes holds with equal force for
invertebrates (e.g., Claassen 1998:106“107).
In many discussions of how paleofaunal remains are tallied, and even in some dis-
cussions of how modern animal bones should be counted, the reader may encounter
the term “skeletal element.” Or, one might encounter the term “bone,” or “tooth,”
or “shell,” or any of many other similar, more or less synonymous general terms
for skeletal remains. But if one collection comprises ten “bones” of a skeleton and
another consists of eleven “bones” of another skeleton of the same species as the ¬rst,
is the latter more anatomically complete than the former? Is the taxon less abundant
in the ¬rst collection than in the second? If you think the answer is “Yes” to either
question, you might be correct. But you could be wrong if when the analyst tallied
specimens no distinction was made between anatomically complete bones and frag-
ments of bones. The lesson is simple. If we are going to tally up skeletal parts and want
to compare our tally with that of another analyst working with another collection,
we had best be sure that we counted skeletal parts the same way that the other person
did. What, then, exactly is a skeletal element?
Paleontologist Michael Voorhies (1969:18) distinguished between “fragments” and
“elements or bones,” but we need something more explicit and inclusive because
not all skeletal elements are, technically, bones. Some are teeth, some are horns, and
some are antlers, and so on. Following Arnold Shotwell (1955, 1958), Donald Grayson
(1984) and Catherine Badgley (1986) provide useful terminology and de¬nitions. A
tallying and counting: fundamentals 5


skeletal element is a complete discrete anatomical unit such as a bone, tooth, or shell.
The critical phrase is complete discrete anatomical unit. Each such item is a discrete
“anatomical organ” (Francillon-Vieillot et al. 1990:480) that does not lose its integrity
or completeness when it is removed from an organism. A humerus, a tibia, a carpal,
a ¬rst lower molar “ each is a skeletal element. One might correctly note that “dis-
creteness” depends on the age or ontogenetic stage of development of the organism,
but many paleozoologists would not tally the proximal epiphysis of a humerus and
the diaphysis of that humerus as two separate specimens if it was clear that the two
specimens went together (an issue we return to in Chapter 2). Those same paleozo-
ologists usually don™t tally up each individual tooth ¬rmly set in a mandible, along
with the dentary or mandible bone. These are potentially signi¬cant concerns but
may ultimately be of minimal analytical import once we get into tallying specimens.
Not all faunal remains recovered from paleozoological deposits are anatomically
complete; some are represented by only a part of the original skeletal element because
of fragmentation. Thus, another term is necessary. A specimen is a bone, tooth, or
shell, or fragment thereof. All skeletal elements are specimens, but not all specimens
are skeletal elements. A distal humerus, a proximal tibia, and a fragment of a premolar
are all specimens that derive from skeletal elements; phenomenologically they are
not, technically, anatomically complete skeletal elements. Specimen is an excellent
term for many counting operations because it is value-free in the sense that it does
not reveal whether specimen A is anatomically more complete, or less complete,
than specimen B. We can record whether specimen A is anatomically complete, and
if it isn™t, we can record the portion of a complete element that is represented by a
fragment, if our research questions demand such. Specimen is also a better generic
term than skeletal element for the individual skeletal remains we study because skeletal
element implies that a complete anatomical unit is represented. The problem with
the terms “bone” and “tooth” and the like are that sometimes when analysts use
them they mean both anatomically complete skeletal elements as de¬ned here and
incomplete skeletal elements. Failure to distinguish the two kinds of units “ skeletal
element and specimen “ can render separate tallies incomparable and make the
signi¬cance of various analyses obscure. Throughout this volume, I use the term
skeletal part as a synonym for specimen, but whereas the latter is a general category
that can include many and varied anatomical portions, skeletal part is restricted to
a particular category of anatomical portion, say, distal humerus. Skeletal portion is
sometimes used in the same category-speci¬c way that skeletal part is but will usually
mean a multiple skeletal element segment of a skeleton, such as a forelimb.
Henceforth, in this volume, specimen will be used to signify any individual skeletal
remain, whether anatomically complete or not. Unfortunately, the terms “skeletal
quantitative paleozoology
6


Table 1.1. An example of the Linnaean taxonomy

Taxonomic level Taxonomic name Common name
Kingdom Animalia Animals
Phylum Vertebrata Vertebrate
Class Mammalia Mammals
Infraclass Eutheria Placental mammal
Order Carnivora Carnivores
Family Canidae Canids
Genus Canis Dogs, coyotes, wolves, and allies
Species— latrans coyote


Technically, the species name is Canis latrans; latrans is the speci¬c epithet.


element” and “element” are still often used to denote anatomically incomplete items.
An effort is made throughout this book to make clear what exactly is being tallied
and how it is being tallied. In this respect, what are usually tallied are what are
termed “identi¬ed” or “identi¬able” specimens. Typically, this means identi¬ed as
to biological taxon, usually genus or species, represented by a bone, tooth, or shell
(Driver 1992; Lyman 2005a). To identify skeletal remains, one must know the structure
of the Linnaean taxonomy, an example of which is given in Table 1.1 . One must also
know the basics of skeletal anatomy, by which is meant that one must know the
difference between a scapula and a radius, a femur and a cervical vertebra, a clavicle
and a rib, and so on. Finally, the person doing the identi¬cations must be able to
distinguish intertaxonomic variation from intrataxonomic variation. Intrataxonomic
variation is also sometimes termed “individual variation” within the species level of
the taxonomy. I presume that readers of the book know these things, along with
anatomical location and direction terms used in later chapters.
The importance of the requirements for identi¬cation should be apparent when
one realizes that “identi¬cation” involves questions such as: Is one dealing with a
mammal or a bird? If it is a mammal, is it a rodent or a carnivore? If it is a car-
nivore, is it a canid, a felid, a mustelid, or any of several other taxa of carnivores?
The importance of the other knowledge requirement “ basic skeletal anatomy “ will
assist in answering the questions just posed. The importance of distinguishing inter-
taxonomic from intrataxonomic variation is usually (and best) met by consultation
of a comparative collection of skeletons of known taxonomic identity. The proce-
dure is simple. Compare the taxonomically unknown paleozoological specimen with
comparative specimens of known taxonomy until the best match is found. Often the
closest match will be obvious, and the unknown specimen is “identi¬ed” as belonging
to the same taxon as the known comparative specimen. Sometimes this means that
tallying and counting: fundamentals 7


one may be able to determine the species represented by the paleozoological speci-
men, but other times only the genus or perhaps only the taxonomic family or order
will be distinguishable.
Taxonomic identi¬cation is a complex matter that is discussed at length in other
contexts (e.g., Driver 1992; Lyman 2005a; and references therein). Blind tests of iden-
ti¬cation results (e.g., Gobalet 2001) highlight the practical and technical dif¬culties.
For one thing, what is “identi¬able” to one analyst may not be to another (e.g.,
Grayson 1979). Gobalet (2001) provides empirical evidence for such interanalyst
variation. It is precisely because of such interobserver differences and the interpre-
tive signi¬cance of whether, say, a bone is from a bobcat (Lynx rufus) or a North
American lynx (Lynx canadensis) that paleontologists developed a standardized for-
mat for reporting their results. Specimens (not necessarily anatomically incomplete
skeletal elements) are illustrated and are verbally described with taxonomically dis-
tinctive criteria highlighted so that other paleontologists can independently evaluate
the anatomical criteria used to make the taxonomic identi¬cation. Zooarchaeolo-
gists have been slow to understand the importance of this reporting form (see Driver
[1992] for a noteworthy exception). This is not the place to delve further into the
nuances of taxonomic identi¬cation and how to report and describe identi¬ed speci-
mens. What is important here is to note that skeletal remains “ faunal specimens “ are
usually tallied by taxon. “There are X remains of bobcats and Y remains of lynx.” So,
identi¬cation must precede tallying. To make taxonomic identi¬cations, one must
¬rst determine which skeletal element is represented by a specimen is order to know
whether the paleozoological unknown should be compared to femora, humeri, tib-
iae, and so on. And sometimes the frequencies of each skeletal element or each part
thereof are analytically important.
The ¬nal paleozoological concept that requires de¬nition is taphonomy. The term
was originally coined by Russian paleontologist I. A. Efremov (1940:85) who de¬ned
it as “the study of the transition (in all details) of animal remains from the bio-
sphere into the lithosphere.” Although not without precedent, Efremov™s term is the
one paleozoologists (and an increasing number of paleobotanists) use to refer to
the processes that in¬‚uence the creation and preservation (or lack thereof) of the
paleobiological record. We will have reason to return again and again to this basic
concept; here it suf¬ces to note that a taphonomic history concerns the formation
of an assemblage of faunal remains. Such a history begins with the accumulation
and deposition of the ¬rst specimen, continues through the deposition of the last
specimen, through the preservation, alteration, and destruction of remains, and up
to collection of a sample of the remains by the paleozoologist (see Lyman [1994c]
for more complete discussion). Along the way, faunal remains are modi¬ed, broken,
and even destroyed. The modi¬cation, fracture, and destruction processes create
quantitative paleozoology
8


and destroy different kinds of phenomena the observation of which can generate
quantitative data.
A ¬nal note about how paleozoological data are presented in the book. Capital
letters are used to denote upper teeth, lower case letters to denote lower teeth, and
a lowercase d to denote deciduous premolars. Thus, a permanent upper second
premolar is P2, a deciduous lower third premolar is dp3, and a lower ¬rst molar is
m1. The capital letter L is used to signify the left element of bilaterally paired bones,
and the capital letter R is used to signify the right element. In general, D stands for
distal, and P stands for proximal. The critical thing to remember is the difference
between a specimen and a skeletal element; both terms will reappear often in what
follows, and both kinds of units can be identi¬ed and tallied.



M A T H E M A T I C A L A N D S T A T I S T I C A L CO N CE P T S


This book is about quanti¬cation, but the topics covered include different sorts of
quanti¬cation, particularly counting or tallying units, methods of counting, and
analyzing counts. A term that might have been used in the title of the book, were
it not for its generality, is measurement. Typically this term is de¬ned as assigning a
numerical value to an observation based on a rule governing the assignment. The
rule might be that length is measured in linear units of uniform size, such that we
can say something like “Pencil A is 5 cm long and pencil B, at 10 cm of length,
is twice as long as pencil A.” Measurement more generally de¬ned concerns writing
descriptions of phenomena according to rules. An estimate is a measurement assigned
to a phenomenon (making a measurement) based on incomplete data. The process
of estimation can involve judging how tall someone is in centimeters without the
bene¬t of a tape measure, or studying a ¬‚ock of birds and suggesting how many
individuals there are without systematically tallying each one. Making estimates,
like taking measurements, is a way to describe phenomena. Descriptions involve
attributes of phenomena that may or may not have numerical symbols or values
associated with them. Whether they do or not concerns what is often referred to as
the scale of measurement of the attribute that is under scrutiny.



Scales of Measurement

Stock™s census of Rancho La Brea mammals (Figure 1.1 ) illustrates that quanti-
tative data describing taxonomic abundances are more revealing than taxonomic
tallying and counting: fundamentals 9


presence“absence data. Quantitative data often are subjected to a variety of math-
ematical manipulations and statistical analyses. Those manipulations and analyses
are only valid if the data are of a certain kind. Four distinct scales of measurement
are often distinguished (Blalock 1960; Shennan 1988; Stevens 1946; Zar 1996), and it
is important that these be explicitly de¬ned at the start because they will be referred
to throughout the book.
Nominal scales of measurement are those that measure differences in kind. Of
the several scales they contain the least amount of information. Numbers may be
assigned to label nominal scale phenomena, such as 0 = male, 1 = female; or 11 =
quarterback, 32 = fullback, and 88 = wide receiver on a football team. Or, numbers
need not be assigned, but rather labels used such as Italian citizen, French citizen,
and German citizen; or coyote (Canis latrans), wolf (Canis lupus), and domestic
dog (Canis familiaris). Nominal scales of measurement do not include an indication
of magnitude, ordering, or distance between categories, and are sometimes labeled
qualitative attributes or discontinuous variables. They are qualitative because they
record a phenomenon in terms of a quality, not a magnitude or an amount. They are
discontinuous (or discrete) because it is possible to ¬nd two values between which
no other intermediate value exists; there is (normally) no organism that is halfway
between a male and a female within a bisexual species. Other scales of measurement
tend to be quantitative because they specify variation more continuously. Continuous
variables are those that can take any value in a series, and there is always yet another
value intermediate between any two values. A tally of skeletal specimens of coyote
in an archaeological collection may be 127 or 128, but there won™t be a collection
in which there are 127.5 or 127.3 or 127.924 specimens of coyote. But the lengths of
coyote humeri are continuous; think about the numbers just noted as millimeters of
length.
Ordinal scales of measurement are those that record greater than, less than relation-
ships, but not the magnitude of difference in phenomena. They allow phenomena
to be arranged in an order, say, from lesser to greater. “I am older than my children”
is a statement of ordinal scale difference, as is “The stratum on the bottom of the
stratigraphic column was deposited before the stratum on the top of the column”
and “A year is longer than a month.” There is no indication of the magnitude of
difference in my age and the ages of my children, or in the length of time between
the deposition of the bottom and top strata, nor in the duration of a year relative
to the duration of a month. Instead, we only specify which phenomenon is older
(or younger), or which was deposited ¬rst (or last), or which is longer in duration
(or shorter). Sometimes when one uses an ordinal scale, measurements are said to
be relative measurements because a measure of phenomenon A is made relative to
quantitative paleozoology
10


phenomenon B; A is older/shorter/heavier than B. Ordinal scale measurements may
be (and often are said to be) rank ordered from greatest to least, or least to greatest,
but the magnitude of distance between any two measurements in the ordering is
unknown. Ordinal scale measurements are discrete insofar as there is no rank of
“¬rst and a half” between the rank of ¬rst and second (ignoring tied ranks).
Interval scales of measurement are those that record greater than or less than
relationships and the magnitude of difference between phenomena. Both the order
of measurements and the distance between them are known. My children are 23 and
25 years old; I am 56 years old, so I am 33 and 31 years older than my two children,
respectively. The stratum on the bottom of the stratigraphic column has an associated
radiocarbon date of 3000 BP and the stratum on top has an associated date of 500 BP,
so the stratum on the bottom was deposited about 2,500 (14 C) years before the stratum
on top (assuming the dated materials in each stratum were formed and deposited at
the same time as the strata were deposited). On average, a year is 365.25 days long
whereas an average month is about 30.4 days in duration; the difference in duration
of an average year and an average month is thus 334.85 days. The distance between 10
and 20 units (days, years, centimeters) is the same as the distance between 244 and
254 of those units, the same as the distance between 5337 and 5347 of those units, and
so on. Interval scales are typically used to measure what are referred to as quantitative
variables. Interval scale measurements, like ordinal scale ones, can be rank ordered
from greatest to least, or least to greatest, but unlike with ordinal scale measures, the
distance between any two interval scale measurements is known. Indeed it must be
known else the variable is not interval scale. Interval scale measurements are generally
continuous but may be discrete. If age is recorded only in whole years, then age is
continuous but it is also discrete (ignoring for the sake of discussion that one might
be 53.7 years old). Importantly, interval scale measures have an arbitrary zero point.
It can be 0—¦ Celsius outside, but there is still heat (if seemingly only a little) caused
by the movement of molecules. The zero point on the Celsius scale is placed at a
different location along the continuum of amount of molecular movement than is
the zero point of the Fahrenheit temperature scale. Both zero points are arbitrary
with respect to the amount of heat (molecular movement), thus both measures of
temperature are interval scale.
Ratio scales of measurement are identical to interval scales but have a natural zero.
Thus, the theoretical natural zero of temperature is “273 —¦ Celsius (or 0 Kelvin, or “
459—¦ Fahrenheit). There is no molecular movement at that temperature. Similarly,
a mammal in a cage comprises 1 individual consisting of more than 100 bones and
teeth, but if the cage is empty there are 0 (zero) individuals, 0 bones, and 0 teeth in
the cage. Thus, if a taxon is represented by 0 skeletal specimens in an assemblage, it is
tallying and counting: fundamentals 11


absent; that is a natural zero. Essentially all quantitative measures in paleozoology “
taxonomic abundances, frequency of gnawing damage, and so on “ are potentially
ratio scale. Whether they are in fact ratio scale or not is another matter.
Measurements of different scales allow (or demand, depending on your perspec-
tive) different statistical tests of different scales or power. Thus, ordinal scale mea-
surements require ordinal scale statistical tests; interval/ratio scale measurements
can be analyzed with either interval/ratio scale statistics or ordinal scale ones, but the
reverse “ applying interval/ratio scale statistics (or parametric statistics) to ordinal
scale data “ will likely violate various statistical assumptions.
Most paleozoologists working in the twentieth century sought ratio scale measures
of the attributes of the ancient faunal remains they studied, just as Stock and Howard
did. Although the optimism that such measures would eventually be designed has
waned somewhat, there are still many who hope for such, whether working with
human remains (e.g., Adams and Konigsberg 2004), paleontological materials (e.g.,
Vermeij and Herbert 2004), or zooarchaeological collections (e.g., Marean et al.
2001; Rogers 2000a). We now know a lot more about taphonomy than we did even
20 years ago when Grayson (1984), Klein and Cruz-Uribe (1984), and Hesse and
Wapnish (1985) noted that many problems with quantitative zooarchaeology orig-
inated in taphonomic histories. And we also know that many taphonomic analyses
and interpretations of taphonomic histories require quantitative data and analy-
ses of various sorts. Where taphonomy can in¬‚uence quantitative paleozoology is
noted throughout this volume, and it is occasionally suggested what we might do
about those in¬‚uences. The point here is that ratio scale measurements of faunal
remains and many of their attributes may be precluded because of taphonomic
history.



Measured and Target Variables: Reliability and Validity

Other important statistical concepts concern the difference between a measured vari-
able and a target variable. A measured variable is what we actually measure, say, how
many gray hairs I have on my head. A target variable is the variable that we are
interested in, say, my age. The critical question is this: Are the measured variable and
the target variable the same variable, or are they different? If the latter, the question
becomes: Are the two variables suf¬ciently strongly correlated that measuring one
reveals something about the other? It is likely that the number of gray hairs on my
head will be correlated with my age, assuming I do not arti¬cially color my hair
(either not gray, or gray). But although the color of the shirt I am wearing today
quantitative paleozoology
12


can be measured rather precisely, it is unlikely to indicate or correlate with my age
(although the style of my shirt might).
The concepts of measured variable and target variable can be stated another way.
When we measure something, are we measuring what we think we are measuring?
Does the attribute we are measuring re¬‚ect the concept (e.g., length, age, color) we
wish to describe (Carmines and Zeller 1979)? These questions serve to de¬ne the
concept of validity. Is a radiocarbon age on a piece of burned wood a valid measure
of the age of deposition of a fossil bone with which the wood is stratigraphically
associated? Assuming no contamination of the sample of wood, and that the wood
was deposited more or less simultaneously with the bone, it will be a valid measure if
it derives from a plant that was alive at about the same time as the animal represented
by the bone. Validity is a different property of a measurement than reliability, which
simply de¬ned means replicability, or, if we measure something twice, do we get the
same answer? If, on the one hand, we measure the length of a femur today and get
12.5 cm, tomorrow we measure it and get 12.4 cm, and the next day we measure it and
get 12.5 cm, then we are producing rather consistent and thus reliable measures of
that femur™s length. On the other hand, femur length is unlikely to be a valid measure
of the time period when the represented animal was alive, regardless of the reliability
of our measurements of length.
Another set of measurements will help underscore the signi¬cance of the preceding
paragraph, and help highlight the differences between a target variable and a mea-
sured variable. A fundamental measurement (sometimes referred to as primary data
[Clason 1972; Reitz and Wing 1999]) is one that describes an easily observed property
of a phenomenon. Length of a bone, stage of tooth eruption in a mandible, and
taxon represented by a shell are all fundamental measurements. A derived measure-
ment (sometimes referred to as secondary data) is more complex than a fundamental
one because it is based on multiple fundamental measurements. Derived measure-
ments are de¬ned by a speci¬ed mathematical (or other) relation between two or
more fundamental measurements. A ratio of length to width exempli¬es a derived
measure. Derived measurements require analytical decisions above and beyond a
choice of scale; do we calculate the ratio of length to width, or width to length,
or width to thickness? As a result, derived measurements are sometimes dif¬cult
to relate clearly to theoretical or interpretive concepts. Derived measurements may
nevertheless reveal otherwise obscure patterns in data even though relating those
patterns to a target variable may be dif¬cult.
The MNI measure mentioned above is the most widely known derived measure-
ment in paleozoology. It depends on (i) tallies of (ii) each kind of skeletal element
of (iii) each taxon in a collection, and often (iv) (but not always) other information,
such as size of bones of a taxon. Each of the lower case Roman numerals denotes
tallying and counting: fundamentals 13


a distinct fundamental measurement; each plays a role in deriving an MNI, as can
several other fundamental measurements (considered in more detail in Chapter 2).
A ¬at or proxy measurement will likely be more complex than either a fundamental
or derived measurement because a ¬at measurement is more conceptual or abstract
and less easily observed. The distinction of fundamental, derived, and proxy mea-
surements is relevant to a measurement™s accuracy. “Accuracy” refers to “the nearness
of a measurement to the actual value of the variable being measured” (Zar 1996:5).
Throughout this volume, major concerns are the accuracy and validity of derived
measures or secondary data, and fundamental measures or primary data with respect
to a target variable of interest. Does a particular derived measure, such as MNI, accu-
rately re¬‚ect the abundance of individual organisms in a collection of bones and
teeth (or shells)? Of organisms in a deposit? Of organisms on the landscape?
Stock™s census of Rancho La Brea mammals was, he hoped, an accurate proxy
measure of the structure and composition of the mammalian fauna on the landscape
at the time of the deposition of the remains. That long-dead fauna is not directly
visible or measurable, so how well the remains from the tar pits actually re¬‚ect or
measure that fauna in terms of which taxon was most abundant and which was
least abundant and a host of other properties (how accurately MNI measures the
landscape fauna) cannot be determined. The validity of a ¬at or proxy measurement,
or a measured variable, for re¬‚ecting a target variable of some sort is the key issue
underpinning much of the discussion in this volume. This is so for the simple reason
that many target variables in paleozoology cannot be directly measured reliably or
validly with broken bones, isolated teeth, and fragments of mollusk shell. What this
book is in part about is how well the measured variables and proxy measurements
commonly used by paleozoologists measure or estimate the target variable(s) of
interest. Two key questions to keep in mind throughout this book are: What is the
target variable? How is the measured variable related to the target variable of interest?
As a prelude to how important these questions are, think about this. Was Stock wise
to use MNI (the derived and measured variable) to estimate the abundances of
mammals on the landscape (the target variable) given that he only had animals that
became mired in the pits of sticky tar at Rancho La Brea? Would he have been
better off using, say, the tally of skulls (a different measured variable) to estimate the
abundances of mammals trapped in the tar pits (a different target variable)?



Absolute and Relative Frequencies and Closed Arrays

An absolute frequency is a raw tally of some set of entities, usually all of a particular
kind. To note that there are ten rabbit bones and ¬ve turkey bones in a collection is
quantitative paleozoology
14

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