Level of Measurement and Archaeological Dating

Mathematics Building, University of Colorado, Boulder, Colorado

In 1946 the psychologist Stanley Smith Stevens, founder and director of Harvard’s Psycho-Acoustic Laboratory, published a short article in Science laying out a classification scheme for scales of measurement.  This system, and the four scales it proposed, would go on to become extremely influential in the quantitative sciences, and it is still widely used.  I learned about it in the statistics class I took in graduate school.  It doesn’t seem to get much attention among archaeologists, however, which is unfortunate because the distinctions between different types of data and the scale by which they should be measured are quite important to a proper study of the archaeological record.  This is particularly the case with different types of archaeological dating, which fall on different scales of measurement and are thus not directly comparable to each other.  This post is an attempt to explain Stevens’s system and show where different types of archaeological dating fall within it and why it matters.

Stevens proposed four scales (today often called “levels“) of measurement: nominal, ordinal, interval, and ratio.  They are distinguished by the type of empirical operation required to create them, as well as by their formal mathematical properties.  As a result of these distinctions, different statistical techniques can be applied to scales at different levels.

• Nominal scales are the most basic, and result from essentially arbitrary labels being applied to either individual data points or groups of data points.  Variables at this level of measurement are really just grouped into categories, and are therefore often called categorical variables.  The labels can be numbers, which is how Stevens justifies including nominal scales within his classification system, but they can also be letters, words, or any other set of unique identifiers.  Because they are really qualitative rather than quantitative classifications, nominal scales are not susceptible to many statistical analyses.  The statistics than can be used are number of cases, mode (in the case that each class includes more than one data point), and certain other statistics such as chi-squared tests.
• Ordinal scales result from the rank-ordering of data points.  The numerical labels in this case do have a real mathematical meaning, unlike with nominal variables, but it is a very rudimentary one, limited to the determination of relative order.  The distances between values are not defined.  The statistics available for ordinal variables are more extensive than for nominal ones, but still somewhat limited.  The median can be calculated as a measure of central tendency, and interquartile range can serve as a measure of dispersion.  Other statistics such as mean and standard deviation are often calculated for ordinal data, but as Stevens notes this is not really appropriate since they assume that the intervals between values are equal, which may not be the case.
• Interval scales have values with equal intervals between them, and most quantitative statistics, including mean and standard deviation, are appropriate with them.  The zero point on an interval scale is arbitrary, however, and negative values can be meaningful.
• Ratio scales are similar to interval scales but have non-arbitrary zero points and cannot take negative values.  They are used in contexts involving the counting of actual objects.  In addition to mean and standard variation, statistics such as coefficient of variation that depend on meaningful ratios can be used with.

There are numerous examples of variables that are measured at these different levels.  Before getting into examples specific to archaeological dating, here are some general ones:

• Nominal: Anything that is grouped into categories with no relative order.  Stevens gives the example of the numbers assigned to football players as a situation in which each category (i.e., number) applies only to a single data point, but there are also many systems in which multiple data points fall into a single category.  Gender, race, color, shape, and texture are examples.
• Ordinal: Stevens notes that most of the scales used by psychologists are ordinal.  Subjective scales ranking perceptions numerically but without explicitly defining the differences between values fall into this category.  The Mohs scale of mineral hardness, which is based on whether one mineral can scratch another, is another example of an ordinal scale.
• Interval: Probably the best-known example of a variable measured at the interval level is temperature.  Both the Fahrenheit and Celsius temperature scales have consistent intervals and arbitrary zero points, both of which differ between the two.  Also an interval scale, and important for the discussion of dating below, is calendar time.
• Ratio: Any scale based on counting physical object is at the interval level, as is any scale that has a real zero point and no negative values, such as measurement of length or weight using any consistent system of units.  Returning to the example of temperature, the Kelvin scale is at the ratio level, since zero Kelvin is equivalent to absolute zero.

With that in mind, let’s move on to archaeological dating.  There are a wide variety of dating systems used in archaeology, and I’m just going to focus on a few that are particularly common especially in the Southwest.  They are organized by level of measurement; note that this is based on the way the dates are conventionally presented, which in some cases is different from the way they are initially derived.

Nominal: Since nominal variables have no order, they are generally not well suited for dating.  However, the traditional use of “phases” consisting of certain combinations of artifact assemblages, architectural styles, and so forth might fit into this category, especially in areas where the phases are not easily placed in a relative order through stratigraphy or absolute dating.  This has generally not been the case in the Southwest, where stratified sites are common, but until the development of other dating techniques it was widely used in areas like the Plains and the Great Basin where stratified sites are uncommon and many sites are surface artifact scatters that can only be dated by the presence of diagnostic artifacts.

Ordinal: Stratigraphy, the determination of the relative ages of components within a site based on their relative order in the ground, is an ordinal-scale dating system.  It’s impossible to tell either the absolute date of a given layer or the length of time during which it was deposited in the absence of other evidence, such as material within the layer that can be dated by other means.  The relative thickness of different layers may give some clue as to differences in how long it took them to be deposited, but there are so many other potential factors that could determine the thickness of layers that this is a highly unreliable method.

Another, less obvious, ordinal-scale dating system is uncalibrated radiocarbon.  This is an important point, and I don’t think it is widely understood.  Radiocarbon dating is based on the ratio of one isotope of carbon to another within organic material, and radiocarbon dates are expressed in years “before present” (BP), with “present” now conventionally defined as AD 1950.  If all else were equal, these BP dates would be interval-scale (not ratio, because the zero point at 1950 is arbitrary).  All else is not equal, however, and it turns out that one of the basic assumptions behind the radiocarbon technique, that the background level of carbon in the atmosphere is constant, is actually false.  The amount of background carbon has varied over time, so radiocarbon dates must be calibrated by calculating radiocarbon ages for materials of known age (for relatively recent periods this mostly means tree rings) and developing a calibration curve to convert straight radiocarbon ages to calendar dates.  There are now several programs freely available to do these calibrations, the best known being CALIB and OxCal, and there is really no excuse these days for only reporting uncalibrated dates.  Unfortunately, many archaeologists continue to do this, in some cases justifying it by the fact that so many dates already in the literature are uncalibrated.  This is true, but it’s still no excuse.  Calibrating those old dates is totally possible.  Furthermore, and this is where the ordinal nature of the dates becomes important, while uncalibrated radiocarbon dates can be compared with each other they cannot be directly compared with dates derived from any other dating method.  This is the crucial difference between ordinal and interval scales when it comes to dating.  In many places virtually the only dating method used is radiocarbon, so this doesn’t seem like such a big deal, but in places where other methods are used its importance becomes clear.   This is especially the case in the Southwest, with its long history of (interval-scale) tree-ring dating.  Without calibration, a radiocarbon date cannot be compared to a tree-ring date.  They aren’t measuring on the same scale.

Interval: Calendar dates (in any calendar), and dates derived from any method that are expressed as calendar dates, are interval-scale data.  In an archaeological context this includes dates from historical records, tree-ring dates, archaeomagnetic dates, and calibrated radiocarbon dates.  These can all be compared to each other, although they vary in precision and some “dates” end up as ranges rather than point estimates when put on this scale.  This is true for archaeomagnetic and calibrated radiocarbon dates.

Ratio: Any dates expressed as “before present” with “present” meaning the actual present rather than an arbitrary date like 1950 are ratio-scale data.  The main example I know of is luminescence dating (although not all luminescence dates are reported this way), though there are probably others.  While this seems like a good way to express dates at first glance, it’s actually somewhat problematic because the “present” keeps on moving as time goes on.  If dates are going to be expressed this way it is absolutely crucial to say what the “present” date is.  Once you do that, though, it’s trivially easy to convert the date to a  calendar date, and this is probably the best way to go.  This would need to be done anyway to compare dates like this to those from the more numerous interval-scale methods.  What is particularly annoying is the tendency for some calibrated radiocarbon dates to be cited as “BP” values without explicitly indicating if the “present” refers to 1950, the actual present, or something else.  In general I think people should report dates as calendar years whenever possible.  “BP” is just too confusing and tricky a concept.

The key thing I would like people to take away from this discussion is that data can only be compared directly if they are at the same level of measurement and using the same scale.  When it comes to dates specifically, we already have this nifty interval-level scale called the calendar, and consistently using it to express dates, no matter how they were derived, is the best way to ensure that data from different researchers working in different places can be compared.

Stevens, S. (1946). On the Theory of Scales of Measurement Science, 103 (2684), 677-680 DOI: 10.1126/science.103.2684.677

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Steve Lekson Hearts Jared Diamond

Chaco Wash and Escavada Wash Near Their Confluence

I’ve never read any of Jared Diamond‘s books, so I’ve been reluctant to say much about him and his ideas.  Chaco is one of his main case studies in Collapse, however, so I really should read it at some point and try to figure out what I think of it.  I’ve heard conflicting things about how accurately it presents and interprets the evidence he gathers from archaeologists.  A lot of people seem to really like it, but most archaeologists seem to hate it and think that it’s riddled with errors.  I browsed through it a little once in the Chaco bookstore (which, yes, carries it, or at least did at the time), and I didn’t see any obvious errors of fact in the parts of the Chaco chapter I looked at, but the caption for one of the pictures, an overview of the canyon as it appears now, seemed to imply that the current desolate look of the area was the result of the overexploitation of the local environment by the Chacoans, which presumably led to their collapse.  My understanding of Diamond’s message, based mainly on the subtitle of the book (“How Societies Choose to Fail or Succeed”), is that the main driver of collapse he sees is environmental degradation, and the book’s popularity in environmentalist circles certainly makes sense in this light.

In any case, I’m skeptical about the whole idea that Chaco “collapsed” in the way that Diamond seems to think.  I’ve put forth my case in detail elsewhere and won’t repeat it now, but the basic idea is that what happened at Chaco is more complicated than a simple catchword like “collapse” (however it’s defined) implies.  On the narrow point of whether whatever happened at Chaco was the result of “choices” the Chacoans made about whether to “succeed or fail,” I guess it depends on what choices you mean by that.  David Stuart argues that the rigid, hierarchical social structure that allowed Chaco to become so impressive in the first place made the system too brittle to withstand severe climatic fluctuations, with the result that it was replaced by the more egalitarian and resilient social structures of the modern Pueblos.  He sees some clear lessons for our own society from this, primarily about the problems with economic inequality (a timely topic these days).  That’s one way of looking at “collapse.”

Southeast Corner of Pueblo Bonito

I’m not sure if it’s what Diamond is talking about, though.  I’ve seen him described as an “environmentalist” in the old sense, i.e., an environmental determinist who sees major aspects of human societies as inevitable results of their environmental situations, with the twist that he obviously doesn’t have a completely deterministic view of human reactions to the environment but rather, more in line with the modern meaning of “environmentalism,” he recognizes that the interaction between humans and their environments goes both ways.  Under this view, presumably the most enlightening examples of past “collapses” to look at for insights into how we should address our own environmental problems are those where collapse was the result of ecological “overshoot,” or human use of natural resources outstripping the ability of the environment to provide them.  Joseph Tainter, who knows a lot about “collapse” from an archaeological perspective, has vigorously criticized Diamond’s (and others’) use of this approach, and I think choosing Chaco as an example of this type of collapse is particularly questionable.

It’s not that the Chacoans didn’t have major effects on their local environment.  The permanent resident population of the canyon may not have been very large, but it’s not an area that’s exactly abounding in resources, and the fact that the Chacoans imported all kinds of stuff from outside the canyon strongly implies that there wasn’t enough of all sorts of things locally to support the community.  I believe Diamond makes a big deal specifically out of the evidence for importing wood from the distant mountains, which I presume he sees as evidence that the Chacoans had deforested their local area more or less completely, with the attendant implications for overshoot and collapse.  Hence the caption on the picture I noticed when leafing through the book: the implied sequence of events is rise of Chaco leading to deforestation leading to collapse leading to a treeless desert wasteland even 1000 years later.

Intact Roof at Pueblo Bonito

But of course the evidence for importing timber from 50 miles away also implies that the Chacoans had the ability to organize some seriously impressive procurement for those resources they were lacking locally (whether because they had outstripped them or because they were never there to start with).  It’s not that they didn’t deforest their local area; they totally did, and fast!  But if that had been enough to make the system collapse, it never could have gotten going in the first place.  The abiding mystery of Chaco, after all, is not that a major center of its scale arose in the Southwest but that it arose where it did, in one of the least inviting environments in the whole region.  Somehow, the people at Chaco were able to marshal the resources of a much bigger area with many more resources, until suddenly they couldn’t.  The thing that needs to be explained by any “collapse” narrative is why that social power stopped so abruptly, which presumably also requires an answer to the question of how it developed in the first place.  We don’t know the answers to any of these questions, which is why Chaco remains such a fascinating and mysterious place even after over a century of intensive study.

“Overshoot” is not a very helpful explanation in this context.  Stripping the canyon of all its productive potential clearly didn’t lead to the collapse of Chaco, as the Chacoans were able to draw on the much greater potential of the whole region, at least for a while.  Overshoot doesn’t really explain why that control ended, either, since the overall resources of the region that the Chacoans apparently had access to were much too abundant for them to deplete.  They easily deforested the mesas above the canyon, but they never came close to deforesting the Chuskas or Mt. Taylor.  Those are big mountains, covered in trees!  And the same goes for all the other imported goods.  You could perhaps make a case for overshoot in some particular area perhaps contributing to the collapse of Chacoan power in some roundabout way, but it would definitely not be as simple as a straightforward story of overshoot leading to collapse implies.  That picture doesn’t show the enduring effects of Chacoan deforestation on the canyon; it shows what the canyon probably looked like when the Chacoans first encountered it.  Indeed, the canyon ecosystem we see today is the result of over fifty years of protection from grazing, and over a hundred years of protection from most other impacts.

Juniper Trees on the South Mesa Trail

So those are my thoughts on Diamond, and I really should read the book at some point to get a better sense of what he actually argues and whether this is a fair interpretation.  What I find interesting, though, is that noted archaeological iconoclast Steve Lekson has recently written an impassioned post in support of Diamond.  He points out that most archaeologists seem to hate Diamond’s books and spend a lot of time pointing out the flaws in them, but he argues that doing this is missing the more important point:

I’m sure there are errors – real errors.  Any work of this scope will have errors.  But much of the carping seems to concern not facts, but interpretations.  Diamond necessarily works from other archaeologists’ interpretations and I suspect the authors upon whom he relies would have something to say about all this.  The interpretations he accepts are not necessarily wrong; they are simply inconsistent with those of his critics.

I’m not saying that Diamond gets it “right.”  It’s hard to get things completely “right,” especially in science when many very reasonable hypotheses are probably wrong.  But the vehemence of academic reaction to Diamond is, I think, far disproportionate to his sins – sins of omission, commission or (worst of all) failure to cite the critic.  It is my opinion that much of the heat comes from Diamond’s success as a popular writer.  It’s not jealousy — well, maybe a little: after all, the guy won the Pulitzer with our data.  We don’t want anyone else to tell our story, even though we almost never tell it ourselves – accessibly.  And, it must be said, there is antipathy, even hostility from academics towards popular writers, even when that popular writer is an academic.     We all should re-read Article 4 of the SAA’s Principles of Archaeological Ethics, especially the bit about “Archaeologists who are unable to undertake public education and outreach directly should encourage and support the efforts of others in these activities.”

Fair enough.  I do obviously agree with the value of outreach and it’s true that Diamond has been a wildly successful popularizer of archaeology.  Lekson goes on to give a very interesting account of what he sees as the important “collapses” in Southwestern prehistory.  I note that Chaco, specifically, doesn’t appear on the list, although the depopulation of the Four Corners around AD 1300 does.  I have my doubts about that one too, but it really depends a lot on how you define “collapse.”  It’s not clear if Lekson has actually read Diamond’s book(s) (although obviously I’m hardly one to judge on that score), and he doesn’t directly address any of Diamond’s claims or interpretations about Chaco specifically, even though he is of course much more of an expert on Chaco than either Diamond or me.  Still, his general points about the reaction to Diamond are fair.  It would probably be more helpful for archaeologists who object to interpretations of their data put forth in popular accounts like Diamond’s to explain their objections in similarly popular fora, rather than just whining amongst themselves.  Diamond’s work may have a lot of problems, but at least he’s trying to draw conclusions from archaeological data and apply them to modern issues in accessible way, which is much more than you can say for most archaeologists, with a few notable exceptions like Stuart and, to a lesser extent, Lekson himself.  In any case, I think it’s clear that this conversation is really just getting started, so anyone who is really upset by the direction it’s taken so far has plenty of opportunity to jump in and contribute a different perspective.

View from Doorway at Pueblo del Arroyo

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What Are Museums For?

Anchorage Museum, Anchorage, Alaska

In the same post about photography I discussed earlier, Matthew Yglesias also has some thoughts about museums:

It’s extremely difficult for me to avoid the conclusion that these super-gigantic collections represent an inefficient allocation of global resources. If 15 percent of the stuff on display at the Louvre vanished at random, the impact on the experience of visiting the museum in particular or Paris in general would be minimal. But in the majority of the cities of the world, that 15 percent would be the basis for an excellent new museum.

This touches on a longstanding debate about the purpose of museums in general.  One school of thought holds that amassing “super-gigantic collections” is precisely the point of having “universal museums” like the Louvre, the British Museum, the Smithsonian, etc.  The idea is that these institutions are the ones that have the resources to care for their collections properly, which can be quite a challenge with certain types of specimens, and that their locations in major cities and general cultural clout make the portions of their collections they display (necessarily a tiny fraction of the total) more accessible to more people than would be the case if they were scattered among numerous smaller museums.  The result of this approach is that while most people go to the Louvre to see the Mona Lisa, whoever also wants to see the antiquities collections and so forth can do so all in one trip to Paris rather than having to go all over the world.  This view was also traditionally associated with the less savory imperialist attitudes of the elites who founded these museums in the nineteenth century, of course, which is one of the major reasons that it’s not the only view out there these days.

Entrance to Carnegie Museums, Pittsburgh, Pennsylvania

The main opposing viewpoint is basically the one Yglesias takes: cultural treasures should be distributed more equitably, rather than amassed in a handful of huge museums.  A more specific variant holds that these treasures should really be kept and displayed as close as possible to their places of origin, which is a viewpoint particularly held by the governments of countries like Greece, Italy, Egypt, and Peru, i.e., the places of origin of all that stuff being collected by the museums during their more overtly imperialistic eras.  There have been some recent high-profile cases of these governments demanding (and in some cases getting) their stuff back from major museums.  You also see this sort of thing within the US, especially with regard to archaeological collections.  One of the main points of criticism of the Hyde Expedition at Chaco, for example, was that the artifacts were being shipped off to New York, to be kept by the American Museum of Natural History, rather than staying in New Mexico where they belonged.   Tellingly, however, many of the most strident (and effective) critics of the Hyde excavations were closely associated with the Museum of New Mexico in Santa Fe, preeminently the founder of the museum, Edgar Lee Hewett.  Although the dispute was often portrayed as between “looters” shipping the stuff off with no regard for its scholarly value and professional archaeologists who wanted to preserve it, it was in reality primarily a dispute between two sets of professional archaeologists with different institutional sponsors.  Both groups wanted the sites to be excavated, and neither had excavation techniques that were up to modern standards.  Both also wanted the artifacts to leave Chaco, too.  It was really a matter of whether they went to New York or Santa Fe.  Nobody wanted to leave the stuff at Chaco (except the Indians, of course, but in those days nobody with any power cared what they wanted).  Hewett and the Santa Fe side ended up winning, and a few years after the AMNH gave up on Chaco for good Hewett began his own excavations there, only to be immediately pushed aside by Neil Judd and his institutional sponsor, the Smithsonian.  Hewett did get his chance once Judd was done, and the institutions with which he was affiliated, including the University of New Mexico and the School of American Research in addition to the MNM, have dominated research at Chaco ever since.

Maxwell Museum of Anthropology, University of New Mexico

This local approach generally sounds better when viewing these issues from a distance, I think, although it’s important to note that the upshot of the various institutions competing over Chaco was that the collections from excavations there are scattered all over the country with very few of them on public display.  From the perspective of the residents of a given place, whichever approach results in more of the stuff being near them sounds good, however, and the impetus for the founding of local museums in places like New Mexico has to be understood in the context of the general spirit of boosterism and economic development that pervaded small western cities in the late nineteenth and early twentieth centuries, just as the universal museum concept resulted from the rapid growth of major industrial cities slightly earlier.  From the perspective of the visitor, concentration has the virtue of reducing the number of trips necessary but with the potential for increasing the distance traveled, while dispersion has the opposite effects (all depending, of course, on where a given visitor lives).  Basically, as with so many disputes, this one ultimately comes down to a clash of fundamental values.  Thus, there isn’t really a “solution” to it, just the necessity in each specific context of deciding which values to prioritize.

Panhandle Plains Historical Museum, Canyon, Texas

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The Plight of the Amateur Photographer

Mendenhall Glacier, Juneau, Alaska

Matthew Yglesias, on vacation in Paris, says:

In the age of the Internet, I think it’s often hard to know what to take photos of. I got a lovely shot of the gardens at Versailles, to be sure, but Flickr and Wikipedia and all the rest are already loaded with pictures of everything obvious. Pictures taken by more skilled photographers. At the same time, I like taking pictures of things. It’s fun. So I often end up taking pictures of tourists taking pictures of things.

This is an issue I’ve encountered myself as well.  I’ve traveled a lot in the past few months, and taken a lot of pictures.  It does often seem silly, though, to take a picture of something I can find numerous pictures of with the click of a mouse.  I do still generally take those obvious pictures, mostly because I figure I might want to write blog posts about them at some point and my policy on this blog of only using my own pictures, while particularly silly in this sort of situation, does give me a pretext for taking pictures of many things.

The Hammer Museum, Haines, Alaska

There’s only so many pictures like that you can take, though, and once I’ve got one shot of each famous place I might want to write about I face the issue that I want to keep taking pictures, because it’s fun, but I no longer have any real idea what purpose those pictures might serve.  Now that huge memory cards are cheap, this doesn’t matter as much as it used to, but it’s still an issue.  One approach is to try to actually become a good photographer and take pictures that have intrinsic artistic merit.  I’ve considered this, and certainly tried to improve the quality of my pictures over the years, but really doing a good job of taking pictures would require a substantial investment of time that I’m not sure I want to make.

Passengers on Alaska State Ferry Taking Pictures of Mendenhall Glacier

Another option is to take pictures of unusual things.  Yglesias notes above that he likes to take pictures of other tourists taking pictures of famous things, and I’ve done the same thing at times.  What I’ve done more often, though, is focus more on documenting the mundane: city streets, houses, office buildings, and especially signs.  I take a lot of pictures of signs these days.  I like signs because people rarely take pictures of them, they’re often surprisingly photogenic (and make great generic illustrations for all sorts of topics I might want to discuss here), and they sometimes contain a remarkable amount of information beyond the literal message of the sign itself.  It’s also generally easier to take good pictures, with compelling composition of so forth, of simple things like signs than of more complicated scenes.

Sign for Petroglyph Beach, Wrangell, Alaska

I’m not sure that there’s really a “right” answer to this question, or even that it’s an important enough problem to worry about, but I thought it was interesting to see Yglesias’s post about it because it’s something I’ve been thinking about recently.  The post also contains some interesting thinking about museums, but that’s a topic for another post.

"No Parking" Sign at Ferry Terminal, Juneau, Alaska

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