Drennan R.D. Statistics for Archaeologists: A Common Sense Approach
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316 CHAPTER 25
artifacts already in the cluster. If we hope for clusters whose basis is shared raw
material source, then we should insist, just as complete linkage clustering does, that
each member of the group be similar to each other member. In fact, complete link-
age clustering often does produce very convincing results in raw material sourcing
studies.
The most important concern really is whether a particular linkage criterion pro-
duces interpretable results. The aim is to produce clusters that make good sense
according to whatever considerations can be applied to their interpretation. If one
linkage criterion produces sensible clusters, then that is the best solution, even if
there were good a priori reasons to think that a different linkage criterion might
work well with that dataset. Fortunately, once similarity scores are calculated sat-
isfactorily, it is extremely easy to get a statpack to produce hierarchical clusterings
based on different criteria and compare the results.
HOW MANY CLUSTERS TO DEFINE
Since hierarchical clustering starts with each case in a separate cluster and finishes
with all cases in a single large cluster, it is necessary to decide how many clusters to
read out of the results. This is, in effect, a process of deciding where to stop the clus-
tering procedure for purposes of interpretation. If we consider the average linkage
clustering in the example above to have run its useful course after Step 5, then two
clusters are produced (1/2/3/5/7 and 4/6). If, on the other hand, we find meaning in
the three clusters 1/3, 2/5/7, and 4/6, we could cut the process off after Step 4. This
is not an option that must be set when running the analysis, but instead a question
of reading the results. As with the choice of the number of dimensions in a multidi-
mensional scaling or the number of components in a principal components analysis,
though, the decision devolves primarily on what is meaningful. For this, there are
really no rules. It depends on the prior knowledge, intuition, and inventiveness of
the analyst.
CLUSTERING BY VARIABLES
Usually hierarchical clustering is done by cases, and that is what all of the above
concerns. Occasionally, however, a hierarchical clustering of variables can be useful
and enlightening. The starting point for a hierarchical clustering of variables is to
measure the similarities between variables. Thinking about the similarities between
variables requires a shift of mental gears after thinking about similarities between
cases. Similarities between variables amount to a consideration of how similarly
the variables behave across cases. Two variables that vary together across the cases
in the dataset are quite similar. If the variables are all measurements, correlation
CLUSTER ANALYSIS 317
Statpacks and Reporting Results
Cluster analysis, like multidimensional scaling, is performed by many but
not all large statpacks. The measurement of similarities, whether between
variables or between cases, sometimes appears as an option to be selected as
part of the clustering routines. It is conceptually a separate step, however, and
more options for different measures of similarity may be available elsewhere
in your statpack or in stand-alone programs. Either way, it is important in
reporting results to be explicit about the exact nature of the variables, the
similarity coefficient used, and the clustering criterion selected. The resulting
dendrogram is the essential result to present to back up your discussion of
what you’ve found out.
As with multidimensional scaling and principal components analysis, it is a
good idea to limit the number of variables in a cluster analysis to no more than
about half the number of cases. If the number of variables is much larger than
this, there is substantial risk of finding spurious patterns that are no more than
the product of random noise in the data.
coefficients (r) are a good measure of similarity between variables. This is, after
all, the starting point for principal components analysis, which is also an analysis
by variables. In many instances it may make most sense to use the absolute values
of correlations, rather than the values with signs. This is because a strong negative
correlation may be just as meaningful a similarity between two variables as a strong
positive correlation. The dissimilar variables, then, would be those that show little
relationship to each other at all. The decision about whether to use correlations or
absolute values of correlations depends on the specific context and content of the
variables in a particular dataset.
If the variables are not all measurements, but include ranks or presence/absence
variables, correlation coefficients may still be a perfectly reasonable choice, for the
same reasons that such variables often turn out to create no real obstacles to prin-
cipal components analysis. If there are variables with multiple unranked categories,
then the use of correlation coefficients is not appropriate. If all the variables are
categorical, we can use V or
φ
or
φ
2
, the measures of strength that accompany the
chi-square test (Chapter
14).
Either way, the starting point for the clustering is a square symmetric matrix of
scores (r or V values) of each variable with each other variable. The same measure
must be chosen for the entire matrix, however, since r and V values are not really
comparable. This matrix is handled in the clustering procedure exactly as it would
be if it measured similarities of each case with each other case. A linkage criterion
must be chosen, and clustering can proceed.
318 CHAPTER 25
Figure 25.4. Clustering of the household units from Ixcaquixtla.
CLUSTERING THE IXCAQUIXTLA HOUSEHOLD DATA
The starting point for a clustering by cases of the household units from Ixcaquixtla
is the matrix of similarity coefficients in Table
22.9. The dendrogram in Fig. 25.4
shows a single-linkage clustering. It might be read in several ways. Sometimes a
small grouping can be found that makes some sense to work from. Household Units
15, 9, 2, and 5, for example, form a cluster sharing the large amounts of energy
invested in burials and the high proportions of bowls, decorated ceramics, and faunal
remains previously interpreted in this dataset as associated with wealth. In the other
clearest cluster, Household Units 20, 7, and 1 have high proportions of debitage,
although Household Units 12, 4, and 10 do not share this characteristic even though
they also appear in this cluster. All six, however, seem to be of very similar modest
wealth. Units 6, 11, and 13 are the households with high proportions of obsidian.
Both households 8 and 17 have high proportions of kiln wasters. Households 11 and
13 have high proportions of both kiln wasters and obsidian. Household Units 14 and
18 have mace heads in burials and the houses are on platforms, but they do not join
well, even though they turn up next to each other in the list. Households 3 and 11,
which also have both these things, are widely separated. Some of the patterns seen
before in this dataset are, thus, visible here, although mostly in fragments rather than
in integrated ways.
This is the least satisfactory of all the analyses we have tried so far on this dataset.
The reason is clear. The nature of the patterns we have seen in the Ixcaquixtla house-
hold data is just not at all well expressed by simply grouping the household units
together into several mutually exclusive clusters, but this is the only kind of pattern
hierarchical clustering can delineate. We probably would expect a dataset like this to
CLUSTER ANALYSIS 319
Figure 25.5. Clustering of the variables from the Ixcaquixtla household dataset.
have more complicated patterns that, if thought of as groups of households, would
involve patterns of multiple overlapping and crosscutting groups. If we expected
that we would need to recognize patterns of this sort, we would not choose hier-
archical clustering as an analytical approach. Hierarchical clustering does provide
an effective solution, however, when mutually exclusive groupings accurately char-
acterize the patterns we need to find. This is why it has proven to be especially
useful in questions of raw material sourcing. In raw material sourcing, each artifact
is expected to fit comfortably into one and only one group which represents a distinct
raw material source, and this is precisely the kind of pattern hierarchical clustering
inevitably produces.
A clustering by variables of the Ixcaquixtla household data (Fig.
25.5) is only
slightly more satisfactory than the clustering by cases. The similarities between vari-
ables for this analysis were measured with correlation coefficients (r), so the starting
point was the same as for the principal components analysis. There is a clear cluster
of the four variables that formed a gradient in Dimensions 1 and 2 of the multidimen-
sional scaling and that loaded very strongly on the first component in the principal
components analysis. Proportion of lithic debitage does not join with any other vari-
able until very near the end of the clustering, and this is consistent with what we
have been seeing through several analyses. The pattern of clustering between the
other five variables though is less satisfactory. We have seen relationships between
the presence of mace heads and high proportions of marine shell, so it makes sense
that these two variables cluster together. Platform, however, does not appear in this
cluster but rather in a cluster with proportion of obsidian. That is a relationship we
have seen before, but then obsidian’s connection to kiln wasters pulls that cluster
together with kiln wasters rather than with the shell and mace head cluster. Again
the complexity of crosscutting relationships has confounded the dendrogram struc-
ture created by hierarchical clustering. Just as with cases, when variables really need
to belong to multiple groups, clustering is likely to be predicated on too simple a
structure of relationships to give very satisfactory results.
320 CHAPTER 25
The dataset we have been using as an example, then, is one whose structure just
does not match the only kind of structure that hierarchical clustering can find. In
this sense, hierarchical cluster analysis is the narrowest of the approaches we have
considered. Both multidimensional scaling and principal components analysis offer
a wider view of patterning in a multivariate dataset and are thus applicable in a
broader range of contexts. When a clear and simple structure of mutually exclu-
sive groupings is indicated, though, hierarchical clustering is probably the approach
of choice. It will delineate such clusters more clearly and effectively than either
multidimensional scaling or principal components analysis.
In the end, the approaches to multivariate analysis described in these final
chapters are tools for exploring multivariate datasets, looking for patterning both
suspected and unsuspected. It often pays to try out different approaches and different
options within each approach (including different similarity coefficients for multi-
dimensional scaling and clustering). It is often not possible to predict in advance
which approach will work well with a particular dataset, and experimenting with
different ones can produce insights that would not be obtained in any other way.
Suggested Reading
Bibliographic citations have been avoided in this book in order to streamline the
presentation and because careful tracing of the intellectual pedigree of many of the
ideas and techniques discussed here is a scholarly endeavor in itself, and one not
comfortably combined with an introduction to their application in archaeology. The
books and articles listed below, however, are places to go for further information
on statistics in archaeology. The literature on statistics in archaeology has become
very large, and the list below is very short. Consequently, a large number of perfectly
relevant references have not been included – the selection is idiosyncratic rather than
comprehensive. Some of the items included are relatively new; some are not so new.
Some are included because they share the general outlook of this book (and indeed
in some cases are the specific inspiration for it); some, because they complement it
(which is to say they take a different perspective).
GENERAL STATISTICS BOOKS
Exploratory Data Analysis, by John W. Tukey (Reading, MA: Addison-Wesley,
1977), is one of the classic presentations of an approach to statistics from which
much in this book is derived, and its author is the father of the approach. Not sur-
prisingly, there is a great deal more to exploratory data analysis (EDA) than has
been presented in this volume, and readers who would like to go directly to the
source to find out about it should read Tukey’s book, which is a full-scale intro-
ductory text in EDA. Although EDA is now more than 40 years old, only parts of
the prescription Tukey laid out for EDA have been much applied in archaeology
(and even those parts that have been do not yet constitute the “standard” statistical
approach in the archaeological literature). Many of the techniques Tukey discusses
in his book were intended to be easily accomplished with pencil and paper, or, at
most, with a calculator, but more widespread availability in the most commonly used
computer statpacks would undoubtedly encourage greater use of EDA techniques in
archaeology and in other fields.
Exploratory Data Analysis,byFrederickHartwigandBrianE.Dearing(Beverly
Hills, CA: Sage, 1979), is a brief presentation of the basic techniques of EDA. It
nevertheless includes a number of EDA topics not covered in this volume.
321
322 Suggested Reading
Applications, Basics, and Computing of Exploratory Data Analysis,byPaul
F. Velleman and David C. Hoaglin (Boston, MA: Duxbury Press, 1981), is yet
another introduction to EDA techniques, less formidable than Tukey’s and more
comprehensive than Hartwig and Dearing’s.
Understanding Data, by Bonnie H. Erikson and T. A. Nosanchuk (Toronto:
McGraw-Hill Ryerson, 1977), is an introductory statistics text that combines EDA
with more traditional statistical approaches. It advocates two different and comple-
mentary kinds of work with numbers (exploratory and confirmatory), keeping the
two strongly separated and emphasizing the differences between their goals. The
presentation is especially accessible and free of jargon and abstract mathematics.
Introduction to Contemporary Statistical Methods, by Lambert H. Koopmans
(Boston, MA: Duxbury Press, 1987), also combines EDA with more traditional
statistics. A very wide range of methods is covered, and the logic behind the meth-
ods is presented in more abstract mathematical terms than in most of the other
books listed here. Instead of focusing on the difference between exploration and
confirmation throughout the book, Koopmans considers statistical exploration at
the beginning, and then complements the discussion of the usual significance test-
ing techniques with a wide array of robust techniques suitable for use on data that
present problems for the usual techniques.
Nonparametric Statistics for the Behavioral Sciences, by Sidney Siegel (New
York: McGraw-Hill, 1956), is a classic presentation of a full array of robust tech-
niques for evaluating significance – that is, ones that are not much affected by things
like very asymmetrically shaped batches for which means and standard deviations
are not useful. Many of these techniques require special tables in which to look up
the results, and Siegel provides them.
Sampling Techniques, by William G. Cochran (New York: Wiley, 1977), descri-
bes itself (quite accurately) as “a comprehensive account of sampling theory.” It
is, perhaps, the ultimate source on this subject. Estimating means and proportions,
sample selection, stratified sampling, cluster sampling, sampling with and without
replacement, determining necessary sample size, and many other topics are covered
in detail. The full logic behind the techniques presented is given in mathematical
terms.
Elementary Survey Sampling, by Richard L. Scheaffer, William Mendenhall, and
Lyman Ott (Boston, MA: Duxbury Press, 1986), covers much of the same ground
that Cochran does. The presentation is largely in terms of abstract mathematics, but
it is considerably less detailed and formidable than Cochran’s.
INTRODUCTIONS TO STATISTICS FOR (AND OFTEN BY)
ARCHAEOLOGISTS
Sampling in Archaeology, by Clive Orton (Cambridge: Cambridge University Press,
2000), explores sampling in archaeology at length. The emphasis is on sensible use
of sampling theory in the varied array of circumstances archaeological data col-
lection presents. There is especially extended treatment of sampling in the field,
Suggested Reading 323
which usually means spatially based sampling. Examples are drawn from real
archeological datasets.
Quantifying Archaeology, by Stephen Shennan (Edinburgh: Edinburgh Univer-
sity Press, 1997), is an introductory statistics text (and more) specifically for
archaeologists. Mostly traditional statistical methods are covered, but some EDA
techniques are also included. Shennan goes beyond basic statistical principles to
deal with multivariate analysis (with emphasis on multiple regression, clustering,
and principal components and factor analysis). Methods for estimating population
means and proportions are presented (not usual in introductory statistics books) and
the special issues that sampling raises in archaeology are discussed.
Statistics in Archaeology, by Michael Baxter (London: Arnold, 2003), reviews
basic statistical techniques (including the fundamental ones from EDA) very com-
pactly, and considers a number of special topics from an archaeological perspective.
These include several multivariate approaches, spatial analysis, radiocarbon dating,
seriation, and assemblage diversity. Baxter’s earlier book, Exploratory Multivariate
Analysis in Archaeology (Edinburgh: Edinburgh University Press, 1994), consid-
ers multivariate analysis in greater depth. Extended treatment is given to principal
components analysis, correspondence analysis, cluster analysis, and discriminant
analysis, and numerous examples of multivariate analyses of real archaeological
data are woven into the explanations.
Digging Numbers: Elementary Statistics for Archaeologists, by Mike Fletcher
and Gary R. Lock (Oxford: Oxford University Committee for Archaeology, 2005),
applies basic statistical techniques (both traditional and EDA) specifically to archae-
ology. The presentation is informal, avoids jargon, and is designed to be very
accessible, especially to those suffering math anxiety.
Refiguring Anthropology: First Principles of Probability and Statistics,byDavid
Hurst Thomas (Prospect Heights, IL: Waveland Press, 1986), is an introduc-
tory statistics text specifically for anthropologists (including archaeologists). The
approach is purely traditional (that is, it does not incorporate an EDA perspective
or techniques), and some rules are laid down that this volume has argued against,
but numerous robust methods are discussed. There are abundant examples of the
application of all the techniques presented to real data from archaeology, cultural
anthropology, and biological anthropology.
ARCHAEOLOGISTS CONSIDER STATISTICS
IN OUR DISCIPLINE
“The Trouble with Significance Tests and What We Can Do About It,” by George
L. Cowgill (American Antiquity 42:350–368, 1977), makes the case for an attitude
about significance testing that has inspired much in the perspective taken on this
subject in this volume. It is a view distinctly different from that often adopted in
introductory statistics texts – indeed it is branded as heresy by the rules often found
in introductory statistics texts. This article is fundamental for those interested in
324 Suggested Reading
a fuller presentation of the arguments that archaeologists will often find it useful
to use samples directly to make estimates about populations and that it is usually
a mistake for archaeologists to force significance tests into the mold of a yes-or-
no decision. Cowgill’s suggestions about the most useful ways to approach these
issues in archaeology go well beyond what is presented in this volume, which
has stopped at the point where the information commonly provided by computer
statpacks imposes a limitation.
“A Selection of Samplers: Comments on Archaeo-statistics,” by George L.
Cowgill [In Sampling in Archaeology, edited by James W. Mueller (Tucson: Uni-
versity of Arizona Press, 1975)], prefigures some of the issues Cowgill argues more
fully in his later paper (above), and focuses especially on sampling, criticizing many
of what he sees as erroneous notions that appear in other papers in the same volume.
“On the Structure of Archaeological Data,” by Mark S. Aldenderfer [In Quantita-
tive Research in Archaeology: Progress and Prospects,editedbyMarkS.
Aldenderfer (Newbury Park, CA: Sage, 1987)], is a discussion of the fundamen-
tal nature of data in archaeology, the position that numbers occupy in such data,
and the implications that this has for how we think about and analyze data. Four
other articles in this same volume are also of special interest. “Quantitative Meth-
ods Designed for Archaeological Problems,” by Keith W. Kintigh, discusses the
issue of the extent to which standard statistical techniques and those borrowed
directly from other disciplines are suited to the particular needs of archaeology.
“Simple Statistics,” by Robert Whallon, stresses the importance of exploring the
patterns in numbers in batches before proceeding to more complicated analyses.
And “Archaeological Theory and Statistical Methods: Discordance, Resolution, and
New Directions,” by Dwight W. Read, and “Removing Discordance from Quantita-
tive Analysis,” by Christopher Carr, try to place archaeological data analysis firmly
in a broader context. Both authors are concerned that data analysis is too often con-
ceived and carried out in isolation from the theoretical questions that analysis aims
to help answer. As a consequence “discordance” between data, analysis, and theory
arises and seriously impedes the archaeological endeavor.
“Statistics for Archaeology,” also by Aldenderfer [In Handbook of Archaeo-
logical Methods, edited by Herbert G.D. Machsner and Christopher Chippindale
(Lanham, MD: Altamira Press, 2005)], reviews the history of statistical analysis in
archaeology and considers a series of topics of special importance in archaeology,
including spatial analysis, EDA techniques, and Bayesian analysis.
MULTIVARIATE ANALYSIS
Since the chapters here on multivariate analysis are only brief explanations, further
reading on these techniques is likely to be especially important. Several of the vol-
umes listed above on statistics in archaeology also provide introductory treatments,
and some are more detailed than those in the final chapters of this book. In addition,
the sources below may be helpful.
Suggested Reading 325
In Multidimensional Scaling (Quantitative Applications in the Social Sciences,
Paper 11, Beverly Hills, CA: Sage, 1972), Joseph B. Kruskal and Myron Wish pro-
vide a clear introduction to this multivariate approach. Kruskal also wrote an article
directed specifically at archaeologists [“Multi-Dimensional Scaling in Archaeology:
Time Is Not the Only Dimension.” In Mathematics in the Archaeological and Histor-
ical Sciences, edited by F.R. Hodson, D.G. Kendall, and P. Tautu, eds., Edinburgh:
Edinburgh University Press, 1971]. Multidimensional Scaling: Theory and Applica-
tions in the Behavioral Sciences, edited by Roger N. Shepard, A. Kimball Romney,
and Sara Beth Nerlove (2 volumes, New York: Seminar Press, 1972), is a compi-
lation of studies making use of multidimensional scaling in all parts of the social
sciences. Ingwer Borg and Patrick J.F. Groenen bring the subject more up to date in
Modern Multidimensional Scaling: Theory and Applications (New York: Springer,
1997).
Principal Components Analysis, by George H. Dunteman (Quantitative Appli-
cations in the Social Sciences, Paper 69. Newbury Park, CA: Sage, 1989), is
an accessible but comprehensive account, with numerous examples. Two other
volumes in the same series of publications, both by Jae-On Kim and Charles
W. Mueller, discuss factor analysis, as opposed to principal components analy-
sis, in fairly intuitive terms. They are Introduction to Factor Analysis: What It
IsandHowtoDoIt(Quantitative Applications in the Social Sciences, Paper 13.
Beverly Hills, CA: Sage, 1978) and Factor Analysis: Statistical Methods and Prac-
tical Issues (Quantitative Applications in the Social Sciences, Paper 14. Beverly
Hills, CA: Sage, 1978).
A very accessible discussion of measures of similarity and basic principles of
cluster analysis can be found in Cluster Analysis by Mark S. Aldenderfer and
Roger K. Blashfield (Quantitative Applications in the Social Sciences, Paper 44.
Newbury Park, CA: Sage, 1984). Numerical Taxonomy, by Peter H.A. Sneath
and Robert R. Sokal (San Francisco, CA: Freeman, 1973), is one of the classic
sources on hierarchical clustering. Cluster Analysis for Applications, by Michael R.
Anderberg (NewYork: Academic, 1973), is another. Both these books also address
the measurement of similarities between cases at length.